Fiber Optic Acoustic Transduction

—

7

Fiber Optic Acoustic

Transduction

J. A . B U C A R O , N . L A G A K O S , J. H . C O L E , a n d T . G . G I A L L O R E N Z I Naval Research

Laboratory,

Washington,

1. Introduction 2. Optical Fiber Types 3. M a c h - Z e h n d e r Fiber Interferometer 3.1 Interferometer 3.2 Threshold Pressure Detectability 3.3 G e o m e t r i c Versatility 3.4 Acoustic T r a n s d u c t i o n 4. Single-Fiber Interferometer 5. Polarization Sensors 6. Optical Intensity Fiber Sensors 6.1 Threshold Pressure Detectability 6.2 M i c r o b e n d Sensor 6.3 M a c r o b e n d Sensor 7. Evanescent Field Fiber C o u p l e r Sensors 8. Hybrid Fiber Sensors 8.1 Moving Mirror . . . , 8.2 Laser D i o d e 8.3 Fiber-to-Fiber T r a n s m i s s i o n 8.4 Fresnel Reflection Sensors 9. Practical Sensor I m p l e m e n t a t i o n 9.1 Diode Laser Technology 9.2 Sensor C o m p o n e n t s 9.3 Fiber Sensor Design References

DC. 385 386 389 389 390 391 392 415 420 424 425 426 435 436 439 439 441 441 443 445 446 452 453 455

1. Introduction Optical fiber technology was born a decade ago with the first demonstrations of low-loss (<1000 dB/km) optical fiber waveguides. Steady progress in material processing and fiber fabrication techniques now allows repeaterless light transmission over great distances (tens of kilometers) with almost lim385 PHYSICAL ACOUSTICS, VOL. XVI

Copyright © 1982 by Academic Press, Inc. All rights of reproduction in any form reserved. ISBN 0-12-477916-6

/. A. Bucaro et al.

386

itless bandwidth. Such optical fibers, together with associated component technology—diode lasers, couplers, etc.—have found rapidly growing use in complex communication systems and multichannel data links. More recently, noncommunication applications for fiber optic technology have emerged. Most notably, optical fiber sensor technology has begun to mature to the point that the impact of this new technology is now evident. Fiber sensors—including acoustic, magnetic, thermal, rotation, strain, radiation, and pressure sensors—offer a number of advantages over their conventional counterparts. These include: geometric versatility allowing arbitrary configurations; dielectric construction permitting usage in highvoltage, electrically noisy, high-temperature, corrosive, or other hostile environments; ultrahigh sensitivity; and inherent compatibility with optical fiber data link technology. Fiber acoustic sensors were among the first such devices demonstrated (Bucaro et al, 1977; Cole et al, 1977) and at this writing a significant degree of progress has been achieved, both in the understanding of the relevant acousto-optic transduction mechanisms as well as in component development and sensor packaging. In this chapter, we attempt to review in a comprehensive manner the acousto-optic transduction mechanisms upon which these fiber optic acoustic sensors are based. Although we will discuss many different sensor types, transduction can be derived from acoustic modulation of any one of only three fundamental optical field parameters: phase, intensity, and polarization. In Section 2 a brief review of the fundamentals of optical fiber propagation is presented. Fiber interferometric sensors based on acoustically induced phase modulation are discussed next, two-fiber interferometers in Section 3 and single-fiber types in Section 4. Polarization-based sensors are treated in Section 5. Intensity-based sensors are considered next, loss sensors in Section 6 and coupler sensors in Section 7. In Section 8 hybrid sensors, in which transduction takes place external to the optical fiber, are reviewed. Finally, in Section 9, practical implementation of these sensors is discussed. 2. Optical Fiber Types Since we are mainly concerned in this article with acoustic transduction effects within optical fibers, it would be useful to review briefly the basic principles involved in the propagation of optical modes within an optical fiber waveguide. The simplest way to consider transmission over optical waveguides is to think in terms of total reflection in a medium of refractive index nco (the fiber core) at the boundary with a medium of refraction index nc\ (the fiber

7. Fiber Optic Acoustic Transduction

387

cladding) where nco is greater than nc\. This is the situation in a typical fiber, such as the one shown in Fig. 1. Light is guided in an optical fiber either along the fiber axis in the lowest order mode or at an angle in a higher order mode. When this angle exceeds a critical value, the light radiates away from the core into the cladding or into a radiated mode. Optical fibers carrying more than one mode are called multimode fibers. The simplest multimode fibers are step index, so named because of their constant refractive index core profile. For such a fiber, the total number of modes Μ propagating in the fiber can be calculated from the expression (Gloge, 1971) Μ = ViV = V2ka(n co - n d) 2

2

2

l/2

= ka- ΝΑ,

(1)

where V is the normalized frequency, k is the wave number of light in vacuum, a is the core radius, and NA is the numerical aperture of the fiber. Figure 2 shows the normalized propagation constant β/k as a function of V for a few of the lowest order modes of a step-index fiber. As can be seen from this figure, or Eq. (1), as V is reduced, either by reducing the core radius or the fiber numerical aperture, the number of modes propagating is steadily reduced. When breaches the critical value 2.405, the last of the

RADIATED

MODE \

^

CLAD

MODE

(b) FIG. 1. M o d e s excited in m u l t i m o d e fibers: different order guided or core m o d e s , clad a n d radiated m o d e s . Right: refractive index profile of fibers; (a) step index; (b) graded index.

388

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FIG. 2. N o r m a l i z e d propagation c o n s t a n t as a function of V p a r a m e t e r for a few of the lowest order m o d e s of a step waveguide. After Keck (1976).

higher order modes is cut off and only the dominant HE!, mode propagates. Such a waveguide is called a single-mode fiber. Actually, a single-mode fiber carries two degenerate modes with polarizations perpendicular to each other. This degeneracy is lifted in a nonisotropic fiber or when uniaxial pressure is applied to the fiber. Such birefringent fibers can be utilized as polarizationpreserving waveguides. Step-index multimode fibers have limited bandwidth due to the dif­ ferent propagation times of different-order modes. The difference in prop­ agation times can be significantly decreased by utilizing an appropriate graded fiber index profile. A general form for the refractive index profile of a fiber is (Gloge and Marcatili, 1973) 2 h n\r) = n (0)[\ - 2d(r/a) ], where 2 2 2 <5 = [n (0) - n (a)]/2n (0). (2) Here, n(0), n(r), and n(a) are the refractive indices at distances 0, r, and a from the fiber axis, respectively, and b is a constant. For b = oo we have a step profile, and for b = 2, a parabolic profile. In graded-index fibers maximum bandwidth is obtained with an optimum b which is a function of the composition of the core glass and the light wavelength. Optical fibers are usually composed of high-content silica glasses which exhibit an optical window in the visible and near infrared (IR), and, in particular, at —1.6 μτη wavelength, where loss and bandwidth are significantly optimized. The glass waveguide is usually coated with two elastomers. The first coating is

7. Fiber Optic Acoustic Transduction

389

a soft elastomer, such as a thermoset rubber or a soft ultraviolet (U V) curable elastomer, which is applied directly to the glass waveguide in order to min­ imize microbending losses (see Section 6.2.1). The outer coating is hard, such as a thermoplastic or a hard UV curable elastomer, introduced for preserving glass strength, protecting the fiber from adverse environments, and for facilitating fiber handling. Metals have also been used successfully as hermetic seals to achieve high-strength fibers. 3. Mach-Zehnder Fiber Interferometer 3.1

INTERFEROMETER

Fiber optic interferometric acoustic sensors generally employ the Mach-Zehnder arrangement shown in Fig. 3. A laser beam is a split, with part sent into a reference fiber. In the reference arm, some means is provided for either shifting the optical frequency (e.g., a Bragg modulator) or for phase modulation [e.g., a fiber stretcher (Jackson et al, 1980) or an inte­ grated optic phase shifter]. The other beam is sent into a sensing fiber exposed to the acoustic field which induces in it a phase shift Αφ. After passing through the fibers, the two beams are then recombined and allowed to interfere on the surface of a photodetector. There results in the photodetector current a signal which is related to the acoustically induced phase shift Δ0. For homodyne operation (no frequency shift of the reference beam) LASER

MOD

FIBER SENSOR

DEMODULATOR

F I G . 3. Fiber optic M a c h - Z e h n d e r interferometer.

390

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the photodetector output signal is given by ί al

i s = W0\0- ^

OO

0

0

Σ Λ , + ι(Δ0) X sin([2n + l]Qt) sin 0 O - Σ Λ„(Δ0)

X cos(2ntit) cos 0o I ·

(3)

Here W0 is the input optical power, h is Planck's constant, ν is the optical frequency, e is the electronic charge, and q is the detector quantum effi­ ciency. The quantities a and / are the optical attenuation coefficient and length of the fiber, respectively, Ω is the acoustic frequency, Jn is an integerorder Bessel function, and 0 O is the "static" phase difference between the reference and sensing optical beams at the photodetector surface. It is clear from Eq. (3) that in this mode of operation, path length stability is essential, and 0o should be close to π/2 for maximum signal at frequency Ω. Various techniques have been utilized to obtain stable signals. Jackson et al. (1980) employed a phase-stabilized feedback loop utilizing a fiber stretcher, Cole et al. (1981) developed a synthetic heterodyning demodulator utilizing a rf phase-shifted reference beam, and Bucaro and Hickman (1979) utilized heterodyning with FM discrimination. 3.2

THRESHOLD PRESSURE DETECTABILITY

In order to calculate the threshold detectability of this sensor, we write Eq. (3) for the small-signal case with 0 O = π/2. Then the signal current for a pressure AP is 4 = W0\0-

al

f

al

Δ0 = W0lO~

f

^

AP,

(4)

where άφ/dP is the change of phasejn the sensing fiber per unit pressure. The mean-square shot noise signal zN in a band Δ/ is al

% = 2e\q/hv)W0\0-

Af.

(5)

The signal-to-noise ratio (SNR) is then

SNR=

i ^7 " W^) =

/N

10 a

2nv Af

2

<> 6

\dPJ

For SNR = 1, we obtain the minimum detectable pressure P m i n: The extreme sensitivities possible with such a device are illustrated in

7. Fiber Optic Acoustic Transduction

391

V

60

\ V

50 -

^ H 5 6 V

40

\ X

V— H U M A N \ AUDIBILITY

30 LU

ESSUR Ε (dB

oc

oc OL

O P T I C A L FIBER

20 10

10m

0 -10 -20

100m

1000m

-30 -40 Ι

Ι Ι Ι Ι N IL

0.01

Ι Ι

Ι 11 N IL

0.1 1.0 FREQUENCY IN kHz

1 Ι Ι ΙΙ Ι NL

10

F I G . 4. M i n i m u m detectable pressure versus frequency for: H 5 6 piezoelectric h y d r o ­ p h o n e ; four lengths of plastic-coated I T T fiber for shot-noise-limited operation; a n d the h u m a n ear (pure t o n e at e n t r a n c e t o external ear canal). T h e curves for the fiber a n d H 5 6 are for 1 H z b a n d w i d t h s (Bucaro a n d H i c k m a n , 1979).

Fig. 4 for shot-noise-limited performance. This assumes a value for άφ/dP as measured by Bucaro and Hickman (1979) on a typical plastic-coated fiber. For comparison, the threshold detectability of a state-of-the-art piezoceramic sensor, the H56 hydrophone, as measured by Henriquez (1972) is shown as well as the threshold detectability of the human ear. As can be seen, these threshold detectabilities can be exceeded with relatively short lengths of optical fiber. 3.3

GEOMETRIC VERSATILITY

A principal advantage possessed by fiber acoustic sensors is their geo­ metric versatility (Bucaro, 1978). Some examples are shown in Fig. 5. As shown in the upper left, planar sensor elements can be fabricated which are light and flexible. Linear arrays can be made as shown in the lower left. Single long fiber elements can be made much longer than an acoustic wave­ length, resulting in a highly directional receiver. Alternatively, individual sensors with desired properties can be placed in a line, resulting in a small lightweight linear array. As shown on the right, simple gradient sensors can be made by placing the "reference" fiber (in this case a loop) adjacent to

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392

FIG. 5. Various fiber optic sensor configurations.

the "sensing" fiber. If the pressure sensitivities of the two loops are matched, the pair does not respond to the pressure, but rather to the spatial pressure gradient. Such a gradient sensor can be utilized to detect the direction of an acoustic wave, yet the sensor can be made small and rugged. Most of these fiber sensors can be spatially shaded to achieve predetection signal-tonoise processing advantages by varying the density of fiber windings. Finally, as illustrated in the upper right, sensitive omnidirectional elements can be made by simply keeping the size of the sensing element small compared to an acoustic wavelength. 3.4

ACOUSTIC TRANSDUCTION

The phase φ of light propagating through an optical fiber is defined as φ = βΙ = knl,

(8)

where / is the fiber length and β is the wave propagation constant (for example, see Fig. 2 ) , k is the free-space optical wave number, and η is the

7. Fiber Optic Acoustic Transduction

393

effective index of refraction for the mode. An acoustic wave interacting with the fiber results in a change in phase Δ0 given by Αφ = β Al + l Δβ = β Al + l[k An + (δβ/da) Δα].

(9)

It can be shown (Hocker, 1979a) that the waveguide dispersion (last) term in Eq. (9) is small so that k An > (θβ/da) Δα. Thus, (Δφ/φ)^(Δΐ/1)

+ (Δη/η).

(10)

The first term in Eq. (3) is the axial strain. Except in the special case of an airborne acoustic wave (which we discuss later) the second term can be computed from the equation for the optical indicatrix: 2

A(l/n )ij=

Σ Pijk,ekl9

(11)

where the ek! are Cartesian strain components and the Pijki are the photo­ elastic constants. Typical fibers are fabricated from isotropic glass materials so that there are only two independent photoelastic constants. In order to utilize Eqs. (10) and (11) to calculate the acoustically in­ duced phase shift, it is convenient to define three separate frequency regimes. To define these regions it is useful to consider a typical sensor arrangement, for example, a loop of diameter D formed from fiber having a diameter d (Fig. 6). At low frequencies, where the acoustic wavelength Λ is much larger than Z>, the acoustic wave exerts a hydrostatic pressure on the fiber. At intermediate frequencies, where Λ is comparable to D but still much smaller than d9 pressure gradients along the sound propagation direction become

F I G . 6. V a r i o u s frequency regimes for t h e interaction of a typical fiber configuration with an acoustic wave.

394

/. A. Bucaro et al

important. Finally, at ultrasonic frequencies where Λ is comparable to or smaller than d, the optical mode distribution (polarization and amplitude) must be taken into account as well as the anisotropic character of the elastic strain. 3.4.1 Acoustic Sensitivity at Low Frequencies So long as Λ > d no shear strains are generated in the fiber. In contracted notation (Nye, 1976) there are then only three strain vector components in Eq. (11), namely e z = Al/l and ex = ey = Ar/r = er, where er is the radial strain. Equation (10) can then be written as Αφ/φ = ez-

(n /2)[(pn

+ pl2)er

2

+

pl2ez].

(12)

A typical optical fiber is composed of a core, cladding, and a substrate from glasses having slightly different properties. This glass fiber is usually coated with a soft rubber and then with a hard plastic. To calculate the sensitivity as given in Eq. (12) the strains in the core, ez and en must be related to the parameters of the various layers of the fiber. Budiansky et al (1979) and Hocker (1979b) have calculated the pressure sensitivity of a onelayer fiber, Hughes and Jarzynski (1980) that for two layers, and Lagakos and Bucaro (1981) that for the more typical four-layer case. At low frequencies, the acoustic wave exerts a hydrostatic pressure on the fiber. For sensor sizes of the order of inches, this is the case for fre­ quencies up to several kilohertz. In contracted notation, the polar stresses ar, σθ, and σζ in the fiber are related to the strains er, ee, and ez as follows: λ'

λ'

λ'

(λ' + 2μ·)

λ'

λ'

λ'

"(λ' + 2μ·) σ'β =

(λ' +

4 2μ')_ -4

(13)

where i is the layer index (0 for the core, 1 for the cladding, etc.), and λ' 1 and μ are the Lame parameters, which are related to Young's modulus E' and Poisson's ratio y' as follows: yE i

i

E 2(1+7') l

( 1 + 7 ' ) ( 1 ~2y )

i 9

"

(14)

For a cylinder the strains can be obtained from the Lame solutions as calculated by Timoshenko and Goudier (1970): l

l

e r=

U 0 + U\/r\

e> = UO-

2

(U\/r ),

tz = W0 .

(15)

where UO, U u and W0 are constants to be determined. Since the strains must be finite at the center of the core, U°x = 0. l

7. Fiber Optic Acoustic Transduction

395

For a fiber with m layers the constants £/ 0 , U\, and are determined from the boundary conditions: Gr\r=ri Ui\r=n

Gr \ r

= U r+l\r= n °r

=n

in Eq. (15)

{I

= 0 , 1 , . . . , m - 1),

(16)

U

= 0, 1, . . .

(17)

\r=rm

, m -

1),

(18)

= -Λ

m

Σ i=0

σ'ζΑί

e° = el =

=

~PAm , t

z

(19) ?

(20)

where u\ (=Je^ dr) is the radial displacement in the z'th layer, and η and At are the radius and the cross-sectional area of the /th layer, respectively. Equations (16) and (17) describe the radial stress and displacement conti­ nuity across the boundaries of the layers. Equations (18) and (19) assume that the applied pressure is hydrostatic. Equation (20) is the plane strain approximation which ignores end effects. For long thin cylinders such as fibers the error introduced by this approximation is negligible (Budiansky etal, 1979). Using the boundary conditions described by Eqs. (16)-(20), the con­ l stants U o, U ί, and W0 are determined, and e°r and are calculated from Eq. (15). Then, from Eq. (12) the sensitivity Αφ/φ AP can be found. Figure 7 shows the calculated pressure sensitivity of a typical com­ mercially available (ITT) single-mode fiber as a function of the plastic coat­ ing (trade name Hytrel) thickness, which usually varies among fibers. The fiber is nominally composed of a fused silica core with traces of G e 0 2 , a cladding of 5% B 2 0 3 + 95% Si0 2 , and a fused silica substrate in a w-shaped index profile. The fiber jacket consists of a silicone layer and a Hytrel layer. The acoustic response of this fiber has been studied both experimentally (Bucaro and Hickman, 1979) and analytically (Hughes and Jarzynski, 1980) in some detail. Table I lists all the parameters used to calculate the sensitivity Αφ/φ AP of this fiber (Lagakos et al, 1980). From Fig. 7 it is seen that the largest contribution to Αφ/φ AP is due to the fiber length change [first term in Eq. (12)]. The photoelastic terms [the last two terms in Eq. (12)] give smaller contributions of opposite polarity. As the Hytrel thickness increases (Fig. 7), the magnitude of the pressure sensitivity increases rapidly due primarily to the length change. Commonly used coating materials for optical fibers include rubbers, thermoset plastics, and U V curable elastomers. The coating which is applied directly to the waveguide is typically a soft material, such as a rubber, introduced for minimizing microbend loss. The outer coating is hard and

396

J. A. Bucaro et al.

0.0

100

200

300

400

500

600

HYTREL THICKNESS (μπι) FIG. 7. Calculated pressure sensitivity Αφ/φ AP of the I T T fiber as a function of Hytrel thickness. After Lagakos a n d Bucaro (1981).

TABLE I FIBER COMPOSITION, G E O M E T R Y , A N D ELASTIC A N D O P T I C A L COEFFICIENTS O F A TYPICAL S I N G L E - M O D E FIBER ( N A

Characteristic Composition D i a m e t e r (μπι) Young's modulus 10 2 (10 dyn/cm ) Poisson's ratio Pi l Pl2

Refractive index

Core

Clad

=

0.1)

Substrate

First coating (soft)

Second coating (hard)

S i 0 2 -1- traces of G e 0 2 (0.1%) 4.5

95% S i 0 2 5% B 2 0 3 30

Si02

Silicone

Hytrel

85

220

450

72.45 0.17 0.126 0.27 1.4580

64.14 0.149

72.45 0.17

0.0035 0.49947

0.39 0.483

1.4546

7. Fiber Optic Acoustic Transduction

397

is introduced for preserving glass strength, protecting the fiber from adverse environments, and facilitating fiber handling. The soft inner coating plays little role in determining the acoustic sensitivity. Accordingly, optimization of the fiber acoustic response involves selecting proper outer jacket materials. Figure 8 shows the calculated fiber pressure sensitivity of such a fiber for various outer jacket moduli as a function of coating thickness. Here, Young's modulus of the coating is varied, but the bulk modulus is fixed at 10 2 a value of 4 Χ 10 dyn/cm . As can be seen from this figure, as the coating thickness becomes large, the fiber sensitivity approaches a limit which is independent of the coating Young's modulus. In the case of a thick coating, hydrostatic pressure in the form of a low-frequency acoustic wave results in isotropic strains in the fiber waveguide whose magnitudes depend only upon the coating compressibility (inverse bulk modulus). Thus, for the thick coating case the pressure sensitivity is governed entirely by the coating bulk modulus, independently of the other elastic moduli. An example of a fiber having a thick coating (6 mm) is shown in Fig. 9 where the sensitivity is plotted versus the inverse of the bulk modulus of the fiber coating. As can be seen from this figure, the fiber acoustic sensitivity is approximately pro­ portional to the inverse of the bulk modulus of the fiber coating. For fibers with more typical coating thicknesses (<1 mm), the sensi­ tivity becomes a more complicated function of the elastic moduli. In this case the waveguide experiences anistropic strains, and knowledge of two independent elastic moduli is required to predict the acoustic sensitivity.

ο

2

3

7 4 5 6 FIBER DIAMETER (mm)

8

9

10

F I G . 8. Calculated pressure sensitivity versus fiber d i a m e t e r for different Y o u n g ' s m o d u l i 2 10 2 Ε (in units of d y n / c m ) with c o n s t a n t bulk m o d u l u s (4 X 1 0 d y n / c m ) of the o u t e r coating.

398

/. A. Bucaro et al.

ο

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.1 C O M P R E S S I B I L I T Y O F O U T E R COATING ( 1 0 " I Oc m 2 / d y n )

1.0

FIG. 9. Calculated pressure sensitivity versus compressibility (inverse of bulk m o d u l u s ) of the fiber o u t e r coating. Fiber d i a m e t e r is 6 m m a n d Ε = k/2.

Figure 10 shows the pressure sensitivity of a typical 0.7 mm fiber diameter, whose parameters are given in Table I, as a function of bulk modulus for various Young's moduli of the outer coating. As can be seen from this figure, for high Young's moduli the fiber sensitivity is a strong function of the bulk modulus. This dependence becomes weaker as the Young's mod­ ulus decreases. This can be understood in the following way. For a composite fiber geometry, the axial stress carried by a particular layer is governed by the product of the cross-sectional area and the Young's modulus of that layer. Thus for high-Young's-modulus materials, very little coating thickness is required to reach the "thick" coating case in which the sensitivity is governed essentially by the bulk modulus of that layer. For low-Young'smodulus materials, however, the degree to which the coating contributes

7. Fiber Optic Acoustic Transduction 1

c ·>>

1

1

1

1

1

12

-

II

-

10

-

-

9

-

-

-

E = l x l 0 10

8

T3

I

399

A - E = 0 . 5 x l 0 K)

-

7

b 6

a.

-

<

-β- 5 \ -e-

<

%ε*3χΐο

10

4

Milk.

3 2 1

π

°C )

O . O I x l O 1 1

1 2

1 3

-

.^flf

\ \ = 0 . l x l 0 10 1

-

0

^ — 1 4

1 5

1 6

1 7

8

BULK MODULUS {10 °dyn/cm2) F I G . 10. Calculated pressure sensitivity versus bulk m o d u l u s for various Y o u n g ' s m o d u l i 2 (in units of d y n / c m ) of the o u t e r coating of 0 . 7 - m m fiber diameter. Shaded areas d e n o t e plastics (upper), U V curable coatings (middle), a n d rubbers (lower).

to the axial strain is diminished and we begin moving toward the limit in which the glass waveguide plays the major role in the sensitivity. Accord­ ingly, for typical coating thicknesses, high acoustic sensitivity requires a material with high Young's modulus and low bulk modulus. Figure 11 shows the calculated frequency response of the acoustic sen­ sitivity of fibers coated with various elastomers whose moduli were measured 2 4 by Lagakos et al. (1982a) in the frequency range of 10 H z - 1 0 Hz. As can be seen from this figure, the frequency dependence of the sensitivity of fibers with hard coatings is relatively small. Among these coatings, nylon gives the weakest frequency dependence, and the soft UV curable elastomer the strongest. The highest sensitivity is obtained with Teflon TFE, while the

J. A. Bucaro et al.

400

Q_ < -O-

< 2

0.1

J L_L

10

FREQUENCY (kHz) FIG. 11. Calculated frequency d e p e n d e n c e of pressure sensitivity of fibers with a 0.7 m m diameter coated with various elastomers at 2 7 ° C : ( O ) Teflon T F E type I; ( A ) Teflon T F E type II; ( X ) Polypropylene 6 8 2 3 ; ( + ) Polypropylene 7 8 2 3 ; ( · ) N y l o n ; ( • ) Hytrel 7246; a n d U V curable acrylate based elastomers: hard ( • ) a n d soft ( • ) .

least is achieved with the soft UV coating. With this latter coating, the sensitivity decreases rapidly as the frequency is lowered below 2 kHz. This and similar coatings are not compatible with broadband acoustic perfor­ mance. However, such a coating can be utilized as a low-frequency fiber filter allowing the detection of high-frequency acoustic signals only. 3.4.2 Acoustic Desensitization Minimizing the pressure sensitivity of optical fibers is also important for acoustic fiber sensors. It is generally required that the fiber acoustic sensitivity be localized in the sensing fiber and that the reference and lead fibers be insensitive to the acoustic field. The pressure sensitivity of an optical fiber Αφ/φ AP is due to the effect of the fiber-length change [first term in Eq. (12)] and the effect of the re­ fractive index modulation of the core, which is related to the photoelastic effect [second and third terms in Eq. (12)]. These effects are generally of opposite polarity. The acoustic sensitivity of typical optical fibers coated with plastics can be significantly enhanced over that of the bare fiber due to the large contribution of the fiber length change. On the other hand, as first pointed out by Hocker (1979c), substantially reduced sensitivity can be achieved if the fiber is coated with high-bulk-modulus materials. These materials can be glasses (Lagakos and Bucaro, 1981) or metals (Hocker, 1979c).

7. Fiber Optic Acoustic Transduction

401

Lagakos et al. (1981a) have demonstrated this principle utilizing alu­ minum coated fibers. Their results are shown in Fig. 12 for a 20 μτη jacket of aluminum. The close agreement between measurement and theory is evidence that the bonding of the aluminum to the glass is sufficiently good and that the elastic parameters of the thin jacket are comparable to those measured for the bulk material. Figure 13 shows the calculated pressure sensitivity of glass- and metalcoated fibers as a function of coating thickness. For a nickel jacket of ~ 13 μπι, an aluminum jacket of ~ 9 5 μπι, or a calcium aluminate glass thickness of ^-70 μίτι, the fiber has zero pressure sensitivity. For this case, nickel requires the smallest jacket thickness. However, the sensitivity versus thick­ ness curve is rather steep, requiring critical control of thickness. The cor­ responding slopes for aluminum, and in particular for the glass, are much 1001

1

1

I

1

I

I

I

1400

1600

PLASTIC-COATED FIBER

Όοοο°ο

OO

0 0

οο

ο <

10

> Ε

BARE FIBER

ζ oc D

Δ Δ " | " Δ

if)

Δ

*

Δ ^

^

Δ Δ

*

Δ

Τ

A L U M I N U M - C O A T E D FIBER

200

400

600

800 1000 F R E Q U E N C Y (Hz)

1200

FIG. 12. F r e q u e n c y response of the pressure sensitivity of optical fibers. ( Δ ) E x p e r i m e n t a l d a t a obtained from m e a s u r e m e n t s o n a n a l u m i n u m - c o a t e d fiber (Lagakos et al., 1981a); ( ) calculated sensitivity Αφ/φ AP of the a l u m i n u m - c o a t e d fiber; ( ) calculated sensi­ tivity Αφ/φ AP of the bare fiber; ( O ) e x p e r i m e n t a l d a t a o b t a i n e d from m e a s u r e m e n t s o n a typical plastic (Hytrel)-coated I T T fiber. After Lagakos et al. (1981a).

402

A. Bucaro et al. ι

1

1

I

1

COATING THICKNESS

Γ

( Mm)

FIG. 13. Calculated pressure sensitivity versus coating thickness of a typical glass singlem o d e fiber for three different coatings: nickel, a l u m i n u m , a n d a h i g h - b u l k - m o d u l u s glass (cal­ cium aluminate); ( ) zero pressure sensitivity.

less steep—making them more attractive from a dimensional tolerance point of view. 3.4.3 Acoustic Response at Intermediate Frequencies Consider again the sensing element in Fig. 6 with a characteristic di­ mension Z), made from fiber having a diameter d. At intermediate fre­ quencies where the acoustic wavelength Λ is comparable to D, pressure gradients across the sensor become important. In this case, the acoustic response characteristics such as bandwidth and directivity depend upon the geometry of the sensor element. Some typical fiber elements are shown in Fig. 5. Useful configurations include fiber loops, fibers wound helically on compliant mandrels, flat helically wound elements, a straight fiber embed­ ded in a compliant rod, etc. Two cases have been considered in some detail. Jarzynski et al. (1981) have studied fiber loops while Africk (1980) has treated mandrel-wound sensors.

7. Fiber Optic Acoustic Transduction

403

a. Fiber Loop. The fiber loop is an interesting configuration both from a sensor point of view (i.e., it is easy to fabricate in a compact, rugged element) and from a transduction measurement point of view (i.e., histor­ ically most measurements of the pressure sensitivity of optical phase have been carried out in this geometry). Accordingly, we describe in some detail here the calculations of Jarzynski et al. (1981) for the loop. Consider a small element in a loop of fiber of radius r c located by the position vector r c (see Fig. 14). The pressure field of the incident sound acting on this element can be expressed as Ρ = Pc exp(/K» ΔΓ), where Pc is the incident pressure at r c , Δ Γ is a position vector from the center of the element to a point on the surface of the element, and Κ = 2π/Α. Only forces acting in the plane of the coil contribute to the net optical phase shift, and this contribution AF is obtained by integration over the surface of the coil element, i.e., AF = ds) · n]n where As is the surface area of the element and η is a unit vector parallel to r c . Taking into account the curvature of the element, this integral yields ΔΡ = [-w(d/2) Pc 2

Δξ]η + i[-w(d/2) PcKrc 2

sin β sin ξ Δ£]η,

(21)

where Αξ is the angle subtended by the element at the center of the coil. In addition to the acoustic force there are axial elastic stresses on each coil element, determined by the instantaneous axial strain ez and the acous­ tic pressure P c . The total axial force across a cross section of the coil is the sum of contributions from the axial stresses in the fibers and the force due

FIG. 14. G e o m e t r y for calculating the response of a n optical fiber coil (Jarzynski et al., 1981).

/. A. Bucaro et al.

404

to the hydrostatic pressure in the fluid between the fibers. When the wave­ length of the incident sound is long compared to the cross section of the coil, the instantaneous hydrostatic pressure in the fluid can be assumed to be approximately constant across the cross section of the coil and equal to the acoustic pressure Pc. Then the contribution of the fluid pressure to the 2 axial tension is T w = —(1 — f)T(d/2) Pc, where (1 — / ) is the fraction of the coil cross section occupied by the fluid. The total tension in the fibers is 2 7} = fw(d/2) az, where σζ is the average axial stress in the fiber. The stress σζ can be expressed in terms of the axial strain ez and the instantaneous pressure Ρ on the fiber, where Ρ is taken to be equal to the acoustic pressure Pc. The final expression for the instantaneous tensile force on a cross section of the coil is then Tf =

Md/2)\Ee

2

z

- 2yPc) - (1 - f)ir{d/2) Pc

.

(22)

where Ε and y are the effective fiber's Young's modulus and Poisson ratio. The incident sound wave will excite various modes of vibration of the coil. When the wavelength of the sound is long (A > d) the vibrations of the coil will be similar to the modes of vibration of a thin circular ring. Flexural vibrations will not give a measurable optical phase shift and the dominant contribution is from the breathing mode. The driving force F0 for this mode is obtained by integrating around the coil the force AF and that part of - 7 } Δ£ which is directly determined by the acoustic pressure Pc, 2

Pc dt - i(w(d/2) Krc sin Θ) 0 2

+ *(d/2) (2yf

Pc sin £ d£ ^0

+ 1 - /) JΓ

o

P c dt

(23)

Substituting the expression for Pc and evaluating the integrals gives F0 = -2wAP[f(\

- 2y)J0(v)

- η^η)],

(24)

where A is the cross-sectional area of the coil, Ρ is the pressure amplitude of the incident sound, 7 0(^) is a Bessel function, and η = Krc sin Θ. The Λ (77) term originates from the surface pressure term and the 7} Δ£ force. The equation of motion can be expressed as a balance between the driving force F0 and the elastic force, the inertial term, and the reaction force of acoustic radiation from the vibrating coil. For the breathing mode, the radial displacement u and the axial strain ez are iw

u = u0e~ ,

mt

ez = u0e- /rc.

The elastic force on this element is -fAEez

(25)

Δξ. Damping forces can be

7. Fiber Optic Acoustic Transduction

405

formally included by making the eifective elastic modulus complex, Ε = E' + iE". The E" term includes the effects of internal friction in the fibers and viscous damping in the fluid between the fibers. The inertial term for the coil element is p Ar where u is the acceleration and p is the effective density, which is an average between the density of the fibers and the density of the fluid between them. The vibrating coil will radiate sound and this acoustic radiation will exert a reaction force on the coil element which is approximately given by Δ£, where p is the density of water. The equation of AF = Ap ti u e~ r motion for an element Δξ of coil can now be expressed as: c

2

T

c

c

lQt

w

0

c

w

-tl u p Ar

Δ£ = F Δ£/2ττ - fAEu A£/r + n u p r

2

0

c

2

c

0

0

c

0

w c

Δ£,

(26)

where the e~ factor is omitted. Rearranging terms yields the following solution for the amplitude of vibration: mt

=

°

U

If (I - 2y)M ) - ηΜη)]Ρ [U (p + p„)r -fE'/R]-ifE''/r ' V

2

c

c

1

c

'

The above expression for w can be substituted into Eq. (14) to obtain the axial strain. The radial strain can be written in terms of the axial strain and applied pressure Ρ as e = Ge + HP (28) 0

r

z

where the coefficients G and Η can be determined by application of Eqs. (13)-(20). Finally, the normalized optical phase shift is determined from Eq.(12): Αφ/klP = Re + / Im, (29) where Re =

{Gw fl okc)

cos u - n\p p

u

-

p )HJ (v)}n, 44

0

Im = {Q(\u \/r ) sin w }«, 0

with

c

Q = n (p

-

2

n

p

+ /2n (p l

p44)G

2

u

- 2p ) - 1, 44

and |w |, u are the magnitude and phase, respectively, of the vibration amplitude u . In Figs. 15 and 16 are shown the measured acoustic response of a loop of Hytrel coated fiber compared with theory, as calculated from Eq. (29). The effect of the fiber loop resonance is apparent in Fig. 15. The directional response shown in Fig. 16 clearly indicates the effect due to the addition of pressure gradient forces. In the limit of low frequency, η = Kr sin θ —* 0 so that J (v) —* 1 and 77/1(77) —> 0. The amplitude of vibration and the corresponding axial strain are u — - ( 1 - 2y)r P/E and e = u /r - > - ( ! - 2y)P/E', which is 0

p

0

c

0

f

0

c

z

0

c

J. A. Bucaro et al.

406 10"

n—ι

ι ι ι ι

11

-i

π

1—ι—I I I I I I

1—ι—I I I I J

110-"

CO ω

CO < ζ

Q. Q ID Ν -J < £

Ι Ο " 12

ο

10"

_J 10^

i I I IIII

J

I

I

I» I

t t

I

10*

FREQUENCY (Hz) F I G . 15. F r e q u e n c y response of the fiber optic coil at the θ = 0 ° orientation. T h e m e a s ­ ured ( Δ ) a n d calculated ( ) phase shifts (per u n i t fiber length, 1 c m , a n d unit pressure, 2 1 d y n / c m ) are plotted for the frequency range 100 H z - 5 0 k H z of the incident s o u n d (Jarzynski et al., 1981).

equivalent to the result obtained in Section 3.3.1 for hydrostatic conditions. As the frequency is increased, the pressure gradient term r\J\{ri) becomes important. At some orientations, there is a cancellation between the pressure and pressure gradient terms which leads to complicated beam patterns, one of which is shown in Fig. 16. As the frequency is raised beyond the resonant frequency, the inertial term in Eq. (27) begins to dominate, and the axial strain falls off. At sufficiently high frequencies, the axial strain becomes negligible and the optical phase shift is due only to changes in the refractive index caused by radial strains in the fiber. b. Mandrel. The fiber-wound mandrel sensor is of interest due to its ease of fabrication, its high sensitivity, and its amenability to spatial shading. Consider the configuration shown in Fig. 17. A homogeneous cylindrical mandrel of length L m and radius Rm is radially wrapped with a fiber over a length lm. Assuming that the fiber sensor response is solely driven by the mandrel response, it is clear that a circumferential change in the mandrel couples directly to a length change in the fiber. Since the circumferential

7. Fiber Optic Acoustic Transduction

407

F I G . 16. Beam pattern of t h e fiber optic coil at 15 k H z .

FIG. 17. C o m p l i a n t m a n d r e l configuration for fiber optic interferometric sensing.

408

/. A. Bucaro et al.

change is directly proportional to the radial change, the radial strain induced in the mandrel by the acoustic wave must be calculated. Africk (1980) has determined the fiber sensor response by calculating a pressure response term which neglects the cylinder end response and by separately determining the term associated with that response. The length response of the cylinder consists of a forced response, in which the velocity within the cylinder follows the driving pressure, and a resonant extensional response. The radial strain per unit pressure for the length term has been calculated by Africk (1980) and is: UcosKLJ2\ / s i n KLJ2\ \ il „ ,~ I cos K z + il . I sin K z\ , (30a) IVcos K LJ2J Vsin K LJ2J J E where y is the mandrel Poisson's ratio, E is the mandrel Young's modulus, Κ is the acoustic wave number, and K is the wave number of the exten­ sional waves. The radial strain due to the pressure field on the cylinder ends can be determined by representing the velocities induced in the mandrel as a sum of rightward- and leftward-propagating longitudinal waves and matching the velocities of these waves with the velocities induced on the ends by the incident pressure field: -2y

2

trr = - Τ Γ —

T

m

E

e

e

e

m

Q

7 J/cosAX /2\

^

m

E

m

T

IVcos K LJ2j e

/sinAX /2\ .

)

Vsin K L /2J

J

m

e

m

;

The normalized shift can now be calculated by integrating the phase shift induced in a single fiber loop by the radial strain over the fiber wrap length / : Αφ J_ f /sin KIJ2\ Β /sin ^ / /2\j φ E \ \ KIJ2 ) cos K LJ2 \ KJJ2 )) where A = -'/2(1 - C sin Θ/Cl) m

=

e

m

9

M

2

(3

)

e

2

dil

1-ΎΓ/

TT^

MRU/CA

Lv ~~T ^ 7 J

"T"

.

KL l

sin θs i n

—J

po (QRMCQ\

KLM c o s

.

\T^)

+ 2 7

(l-y)Em

m

Cdil = V£dil/p , e

and C is the sound speed in water, θ is the incidence angle of the acoustic wave, Ω is the acoustic angular frequency, p is the density of water, p is the density of the cylinder material, and R is equal to 0 . 6 1 3 / ^ . The results of this model are shown in Figs. 18 and 19. Figure 18 0

0

M

E

7. Fiber Optic Acoustic Transduction 3 X 1 0 ~ 11

I

I

I

409 I

ι

ι

I

I

I

2

/EXPERIMENT /THEORY

/

/

~/\

0

I

"

200

400

I 600

I 800

/

I I I I 1000 1200 1400 1600

I

1800 2000

F R E Q U E N C Y (Hz) FIG.

18. I n t e r f e r o m e t e r a c o u s t i c s e n s i t i v i t y for a Teflon

mandrel

configuration

( L m = 5.1 c m ; Rm = 1.9 c m ; / M = 1.62 c m ) (Africk et al., 1981).

demonstrates very good agreement between theory and experiment for the case of a relatively hard mandrel like Teflon. Figure 19 illustrates that for soft materials like rubbers, the experimentally observed sensitivity is sig­ nificantly lower than that calculated. This is attributed to restriction of the mandrel's radial displacement by the fiber. 3.4.4 Ultrasonic Response Consider an optical fiber of radius a, with an isotropic strain distri­ bution. The fiber is infinite in length, aligned along the ζ axis, and interacts with an ultrasonic wave incident along the χ direction. At ultrasonic fre­ quencies, the strains induced in the fiber are in general no longer uniform across the fiber cross section and we must take into account the state of polarization of the optical beam. If £^0 and Ey0 represent the electric field components of the input optical beam (and thus its polarization state), the output fields after inter­ acting with the ultrasonic wave can be described by the Jones matrix Exl Ey\

ia

=

l(rJt)

[cos T(r, Θ) - s i n T(r, 0)Te- * Lsin T(r, Θ) cos T(r, Θ) J|_

0 0

-' β

Ί θ) > ]

Αφ2{Γ

cos T(r, Θ) sin T(r, 0)ΊΓ£*ο" - s i n T(r, Θ) cos T(r, θ)]ΐΕγ0_ Here the A„(r, Θ) are the induced phase shifts along each of the acoustically induced principal strain axes, Γ is the angle between the χ axis and the

/. A. Bucaro et al.

410

0

0.2

0.4

0.6

0.8

I

1.2

1.4

1.6

1.8

2.0

FREQUENCY (kHz) F I G . 1 9 . Interferometer acoustic sensitivity for a iron oxide-filled silicone m a n d r e l (Lm - 7 . 7 5 c m , Rm = 1 . 2 7 c m , lm = 4 . 6 c m ) : ( — ) experimental; ( ) theory.

principal strain axes, and r and θ are polar coordinates locating the point of interest in the fiber. The phase shifts per unit length Δβη are related to the principal strains ci and e 2 in the following manner (Nye, 1976): Αβι = -k0n\pn€y

+ p 1 2c 2 ) / 2 ,

(33a)

Δ02 = -kon\px2ex

+ P n € 2) / 2 .

(33b)

In general, the principal strains as well as the principal axis angle will vary with position across the fiber cross section. Usually the acoustically induced strains are sufficiently small so as to act only as a perturbation to the fiber

7. Fiber Optic Acoustic Transduction

411

waveguide. This condition can be stated: Αβη <ζ k An, where An is the refractive index difference between the fiber core and cladding. The ultrasonically induced strains can be calculated by solving the displacement wave equation (Rax et al, 1981; Sittig and Coquin, 1970) u = - ν Φ + V Χ Ψ,

(34)

where u is the displacement vector, Φ is the scalar and Ψ the vector potential. can be calculated from Eq. (34) by expanding The strains err, eee, and Φ and Ψ in terms of cylindrical basis functions and applying the four bound­ ary conditions which hold at the surface of the fiber: the displacement and normal stress must be continuous and the tangential stress must be zero at the surface boundary. Diagonalization of the strain matrix then yields the principle strain components: € i = (*n + eee)/2

+ Vi[(err - €θθ)

2

+ 4e e] ,

2 l/2

(35a)

6 2 = (err + eee)/2

- V2[(err - eee)

2

+ 4e e] .

2 1/2

(35b)

Using the standard method of axis transformation, one can find the angle Γ defining the principle strain axes (Sittig and Coquin, 1970): tan[2(r - Θ)] = 2e /(e re

rr

- eee).

(36)

For the special case of a single-mode fiber, the condition usually holds that a/d < 1, where d and a are the fiber diameter and fiber core radius, respectively. In this case e{, e 2, and Γ are very nearly constant across the fiber core area for ultrasonic wavelengths Λ such that Λ > a. Thus, the calculation of the ultrasonically induced phase shift described above can be simplified by assuming the core strains to be everywhere equal to those at the core center r = 0. This approximation is valid for ultrasonic frequencies as high as several hundred megahertz for typical single-mode fibers. For the case of normal incidence, at the core center, the principle strain directions are in fact parallel and perpendicular to the incident sound direction. The results of such a calculation are shown in Fig. 20 as a function of 4 Kd/2 for an acoustic pressure of 10 Pa. In general, we see significant dif­ ferences in the magnitudes of the phase shifts for each polarization and a significant frequency dependence to each. The major peak at Kd/2 ~ 6 results from excitation of the quadrupole radial resonance of the fiber. At low frequencies Kd/2 ~ 0 the acoustically induced phase shift is indepen­ dent of polarization direction. In this limit the value obtained here is exactly that calculated by a constrained radial model, for which the optical fiber experiences a uniform radial pressure but is constrained not to expand in the axial direction. For a homogeneous glass fiber, it can be shown that the

/. A. Bucaro et al.

412 ULTRASONIC

WAVE

J

I

I

I

I

I

I

I

1.0

2.0

3.0

4.0 Κ (d/2)

5.0

6.0

7.0

L_

8.0

F I G . 20. T h e acoustically i n d u c e d phase shifts are s h o w n as a function of Kd/2 for a single-mode fiber where the induced principal axes are p e r p e n d i c u l a r a n d parallel to the acoustic 4 wave. T h e phase shift has been calculated for a incident acoustic pressure of ΙΟ Pa a n d an interaction length of 2.54 c m ( D e P a u l a et al.9 1981).

phase shift for this case is given by Αφ/φ = (n /2){Pn 2

+ P 1 2)(l - 7 " 2y )(P/E).

(37)

2

This can be compared to the corresponding expression for low frequencies (hydrostatic case), namely, Αφ/φ = (P/3B) - {n /6)(Pu

+ 2Pn){P/B),

2

(38)

and to that for the unconstrained radial pressure model, viz. ^

2

φ

|

(

Λ

,

+

™

,

^

-

^

)

.

(39,

Relatively little experimental work has been carried out at megahertz ultrasonic frequencies. Cole et al (1977) measured the angular dependence of the ultrasonic sensitivity at 1 MHz. At normal incidence they observed phase shifts whose magnitude agreed with those calculated from a con­ strained radial model. However, at nonnormal incidence they observed excess phase shifts which they attributed to the generation of axial propa­ gating modes, but which may in fact be due to tank reverberation. Kingsley (1978) measured ultrasonically induced phase shifts in silica and borosilicate single-mode fibers employing a novel experimental ar-

7. Fiber Optic Acoustic Transduction

413

rangement shown in Fig. 21 to obtain calibrated ultrasonic pressure levels. In this arrangement, one beam is guided via an optical fiber through the acoustic interaction region. A second unguided optical beam propagating through the fluid is used to calibrate the ultrasonic pressure level. The unguided beam displays diffracted orders whose amplitudes are given by Bessel functions whose arguments are proportional to the ultrasonic pres­ sure. Kingsley's (1978) measurements at 7.8 MHz yielded values again pre­ dicted by the constrained radial model. At this frequency however (corre­ sponding to a Kd/2 value of about 1.2) there is a significant acoustically induced birefringence (see Fig. 20). We thus conclude that Kingsley's mea­ surements were made with the perpendicular polarization state, although the author makes no such statement. DePaula et al. (1981) have reported measurements of the ultrasonically induced birefringence in single mode fibers. This birefringence is essentially given by Αβι - Δβ 2 , and their measurements agree with those calculated by the above theory, which can simply be obtained by taking the difference between the two curves shown in Fig. 20. Transduction measurements have also been made by bonding piezoactive elements directly to the optical fiber. Howard and Hall (1978) excited multimode fibers by bonding the fiber to a piezoelectric plate. Here strains at frequencies as high as 50 MHz were generated. Large signals were observed at frequencies corresponding to the first radial resonance position (see Fig. 20). However, no detailed quantitative analysis could be carried out due to the presence of many optical modes, and to the complicated

F I G . 2 1 . R a m a n - N a t h diffraction t e c h n i q u e for d e t e r m i n i n g the high-frequency pressure sensitivity of optical fibers (After Kingsley, 1978).

/. A. Bucaro et al.

414

strain condition. Davies and Kingsley (1974) studied a fiber-wound piezo­ electric shell. Measured values for phase shift were in excellent agreement with those calculated by a model which assumed the fiber was pulled at its ends axially. Kingsley (1975) studied a fiber embedded in the center of a radially stressed cylinder. His measurements agree well with those calculated from a constrained radial model [see Eq. (40)]. 3.4.5 Acoustic Response in Air Since acoustic waves are adiabatic pressure fluctuations, there exists a periodic temperature change associated with the propagation of the wave through a medium. This temperature change is given simply by Δ θ = [ ( 7 s - 1)/7.1(Δ/>/*0ν),

(40)

where ys is the ratio of specific heats, Β is the isothermal bulk modulus, and j8v is the volume expansion coefficient. Fibers respond not only to pressure but also to temperature. The modulation in optical phase due to the temperature change Δ θ is (Lagakos et al, 1981c)

where ef and ef are the temperature-induced strains. Using Eqs. (40) and (41), and Eq. (12), and noting that ef = e? = (β ν /3) Δ θ and ez = er = l (B~ /3) ΔΡ, we can compute in the static limit the ratio, Rep, between the pressure- and temperature-induced phase shifts due to the adiabatic pressure wave:

where Bg and Β are the glass fiber and acoustic medium bulk moduli, respectively. For water, the ratio of specific heats is very near one, and ys — I ~ 6 6 4 Χ 10" giving a ratio Rep ~ 10" . This small value for Rep justifies our neglect of temperature-induced transduction in the previous sections. For 7 3 air, however, ys ~ /s and Rep ^ 2.5 Χ 10 and this effect cannot be ignored. Maurer et al. (1981) have measured the phase response of a fiber optic loop in air. These results are shown in Fig. 22. As can be seen, the trans­ duction coefficient in air is substantially higher than that in water, partic­ ularly at low frequencies. We can understand this in the following quan­ titative manner. In the low-frequency limit as Ω —• 0, the temperature effect in the bare fiber can be estimated from Eq. (42). In this limit, the phase 3 shift due to the temperature effect is calculated to be 2.5 Χ 10 greater than

7. Fiber Optic Acoustic Transduction

415 (b)

AIR

WATER

I

ι

ι

ι

20

40

60

ι

ι

80 100 20 40 60 ACOUSTIC FREQUENCY (Hz)

ι

ι

I I

ι

ι

80

ι 100

I

F I G . 22. F r e q u e n c y response of optical fibers in air a n d water. T h e phase shifts per unit pressure a n d per u n i t length m e a s u r e d in air are c o m p a r e d t o those calculated for fibers in water; (a) sensitivity for a polypropylene-coated fiber of 0.6 m m diameter; (b) sensitivity for a bare fiber ( M a u r e r et al, 1981).

that due to pressure. The latter is given by the curve for water, in which only the pressure term is operative. Thus in the limit of low frequencies, 9 2 -1 ΑφIφ ΔΡ for air should be —10~ (dyn/cm ) . Inspection of Fig. 22 shows that Maurer's results begin to approach this value as Ω —• 0. At higher frequencies nonzero thermal transfer times diminish the temperature effect (see Fig. 22) and the values begin to approach that for water. At sufficiently high frequencies, one would expect the temperature ef­ fect to have vanished, leading to identical results for air or water. Indeed, this has been observed by Maurer et al (1981) for plastic-coated fibers. As can be seen in Fig. 22, this occurs at relatively low frequencies for these fibers because of the low plastic thermal conductivity. Ultimately, in the high-frequency limit where Ω > 2wC/d (C is the sound speed in the fiber) the pressure sensitivity will decrease substantially due to the impedance mismatch between the glass fiber and air. This latter effect has not yet been observed. 4. Single-Fiber Interferometer The Mach-Zehnder interferometer requires two single-mode optical fibers. This necessitates the use of suitable fiber-to-fiber 3 dB couplers in order to form the fiber interferometer. In an attempt to eliminate this complexity,

416

/. A. Bucaro et al.

Layton and Bucaro (1979) and Bucaro and Carome (1978) proposed es­ tablishing the interferometer within a single fiber. Bucaro and Carome (1978) utilized a single-mode fiber whose total length was less than half the coherence length of the optical source. At the output face of the fiber, due to Fresnel reflection at the fiber ends, there exists a series of coherent beams, the first of amplitude TE0 and second of 2 amplitude TR E0, where Τ and R are the amplitude transmission and re­ flection coefficients, respectively, and E0 is the optical amplitude input to the fiber. The first beam, having traveled once through the fiber has an acoustically induced phase shift of Δ0ι, which is given by Eq. (12). The second beam, having traveled three times through the same fiber, has a phase shift Αφ2 = 3 Αφχ. The interference of these two beams on a photodetector surface would generate signals of the general form of Eq. (3), but 2 for which the nXh harmonic would be proportional to R Jn(2 Αφ). For a glass silica fiber output face, the signal at the fundamental would be about 6% of that achievable with a Mach-Zehnder interferometer, which for most applications yields sufficient sensitivity. As in the two-fiber interferometer, the interfering beams should be maintained in quadrature to assure opti­ mum, stable operation, although the associated fading problem is much less severe than in the two-fiber interferometer. However, no technique has yet been reported which eliminates the fading problem completely. A second single-fiber approach has been reported by Layton and Bucaro (1979) utilizing a multimode fiber. The intensity distribution in the exit plane of a multimode fiber represents a complicated interference pattern which is determined by the phased addition of the amplitude of the various modes propagating through the fiber. Acoustic pressures which change the phase shifts differentially for these modes thus lead to intensity fluctuations in the output pattern which can be used for acoustic detection. Consider the zth mode propagating in a step-index fiber of length / with core/cladding indices nCOi nch respectively. Upon exiting the fiber, the phase of this mode is given by 0/ = ft/ + Ψΐ, (43) where ft is the ζ component of the propagation constant for the waveguided mode, and φι is its phase upon entering the fiber. From Eq. (9) we can write the acoustically induced phase shift for this mode as: Αφϊ = ft Al + / Aft

(44)

The transverse components of the electric and magnetic fields in the fiber core are of the form Am(r, Θ, t) = Am(r, Θ) exp[/(j8m/ - ωί + φ„ + Αφ„)].

(45)

7. Fiber Optic Acoustic Transduction

417

where r, 0, ζ are the appropriate cylindrical coordinates. For simplicity, consider the interference of only two modes and note that in principle it is straightforward to generalize to any number of modes. The intensity / resulting from the combination of the two modes at the fiber output end face is then the time average of the real part of the ζ component of the complex Poynting vector. The resulting expression is 7(r, 0, t) = A(r, 0) + 72(r, 0) + Vi{[Ex(r9 0) X 77f(r, 0)] βχρ[/(Δ0 1 2/ + Δ* ι 2)] + [£ 2(r, 0) X 77ftr, 0)] βχρ[-ι(Δ0 Ι 2 / + Δ* Ι 2)]} (cos(A^, 2 + / Δ0 1 2)[/ο(Δ0ι 2 ) + 2 Σ Λ^(Δ0 1 2) cos(2fc«/)] sin(A^ 12 + / Δ0 1 2){2 £ ^ + ι ( Δ φ 1 2 ) sin[(2/c + 1)0ί]|),

(46)

where 7, = « Λ ^ Χ 77ft,

7 2 = V2(£ 2 Χ 772*),

Δ 0 1 2 = Δ0, - Δ 0 2 ,

Αφi2 = Φι ~ Φι ,

Δ/? 12 = 0, - β2 ,

and £ and 7/ are the electric and magnetic fields, respectively. For small differences Δ 0 1 2 <^ 1, the expression reduces considerably to 7(r, 0, t) = 7,(r, 0) + 72(r, 0) + ^ { [ ^ ( r , 0) X 772*(r, 0)] exp(/A0 1 2/) + [E2(r9 0) X 77T(r, 0)] exp(-z Δ/9 1 2/)} Χ [ c o s ^ 1 2 + / Αβχ2) - Δ 0 1 2 sin(Ai//12 + / Δ0 1 2) Δ φ 1 2 sin Ω*]. (47) Note that the interference term in Eq. (47) is a product of a spatial term and a term depending upon A 0 J 2 and /. As with all interferometers, the detected signal fades as Αφχ2 + / Δβι2 follows slow environmental perturbations. For the quadrature condition (Αφχ2 + / Δβι 2 = π/2) the sensitivity depends only on Αφχ2 and the field structures of the respective modes. The pressure-induced phase shift between the modes Δψι 2 is the sum of two terms:

^-δ4^) AP

μι2

\ΘΡ/

+

/ ^ . dP

(48)

;

The first term due to the pressure-induced length change is a factor of 3 ~ 1 0 " smaller than the corresponding term for the Mach-Zehnder inter-

V

/. A. Bucaro et al.

418

ferometer [see Eq. (9)]. The second term can be expressed as

Pressure-induced variations in the index η and the core radius a cause changes in Δ β 1 2 which are essentially given by the differences in slope of the curves shown in Fig. (2). These terms are typically of the same order of magnitude as the length change terms [Eq. (48)]. Thus, the transduction 3 coefficient for mode-mode beating is typically 10" times less than that for the two-fiber Mach-Zehnder interferometer, its exact value being dependent upon the specific modes in question. In order to achieve even this magnitude for the transduction coefficient a proper choice of interfering modes must be made. The dependence of β on the mode type is shown in Fig. (2). Two modes are required which exhibit a large difference in β [first term in Eq. (48)] and in the slope of β [second term in Eq. (48)], and which possess transverse field components for which the magnitude of the interference term [see Eq. (47)] is maximized. In addition, it is desirable that the interference term should depend in a simple way upon r and 0, and that the phase of the interference term should remain invariant with 0. The H E ! m modes satisfy the above conditions for a number of reasons. The H E i m modes are all linearly polarized (for nco ~ nc\\ and, for two modes polarized in the same direction, intensity modulation will result at all points on the fiber output end face where the modes have finite field amplitudes. This can be seen by inspection of Fig. 23, where the transverse electric field is sketched for four of the modes. From this figure it is also obvious that no beat signal would result for a combination of T E 0i and T M 0 i modes. Also, if the H e M mode is combined with the T M 0 i mode, beating occurs over specific regions of the fiber end face. In this case, the phase of the interference term will differ by π in opposite semicircular regions defined by a diameter perpendicular to the HEj ι polarization direction. Other possibilities exist involving EHnm modes and HEnm (n > 1) modes, but these have somewhat complicated transverse field patterns (Snitzer, 1961; Snitzer and Osterberg, 1961). The spatial dependence of the interference term at the fiber end face

HE — 1 1

HE-12

TM-01

TE-01

FIG. 2 3 . Schematic drawing showing t h e direction of the transverse electric field for four of the lowest waveguide m o d e s for a step-index fiber.

7. Fiber Optic Acoustic Transduction

419

resulting from a combination of the H E M and T M i modes is shown in Fig. 24. The intensity in the figure corresponds to the magnitude of the inter­ ference term and, as expected, drops to zero where the polarizations of the two modes are orthogonal. A phase reversal exists between the interference term for each lobe. If all the light from the fiber is detected (i.e., no optical mask is used), a phase shift of π would need to be introduced into the light exiting from one of the two regions. In addition to exhibiting a simple interference pattern, the H E and HE i 2 mode combination as well as the H E and T M i combination pos0

n

n

0

FiG. 2 4 . Spatial variation of the m a g n i t u d e of t h e interference t e r m o n t h e fiber e n d face for a c o m b i n a t i o n of H E n a n d T M 0 i m o d e s in a V = 4 fiber. T h e intensity of t h e image is proportional t o t h e m a g n i t u d e , a n d t h e dashed line indicates t h e core circumference (Layton a n d Bucaro, 1979).

420

/. A. Bucaro et al.

sesses propagation constants which differ by a substantial amount for a lowV fiber, as can be seen in Fig. 2. Furthermore, in any working device, the modes chosen will need to be selectively excited, and as Kapany and Burke (1972) have demonstrated, this can be done without resorting to elaborate techniques for these modes. Layton and Bucaro (1979) were able to isolate the interference of the HEn and TM 0 i modes in a fiber which propagated only a few modes. Their measurements of acoustically induced mode-mode beating agreed reasonably well with that predicted from the analysis above. Carome and Satyshur (1979) utilized other modes and found similar results. 5. Polarization Sensors Single-mode fibers are in fact bimodal in that they can propagate two de­ generate HEj ι modes having identical propagation constants but orthogonal polarizations. Polarization fiber sensors which exploit this fact depend upon detecting phase differences acoustically induced between these polarization states. For appropriate polarization input, the acoustically induced phase difference between the modes can also be considered as a polarization ro­ tation of the beam propagating through the fiber. In this case, a polarization analyzer can provide an intensity-modulated output proportional to the induced birefringence. Figure 25 shows a typical fiber polarization sensor as described by Rashleigh (1980). Light from a laser is linearly polarized and launched into a fiber with birefringence β\2 at 45° to the eigenmode direction. This insures that both eigenmodes are equally exited. Upon interaction with an acoustic

COMPLIAN T CYLINDE R

FIG. 2 5 . Schematic a r r a n g e m e n t for a low-frequency birefringent acoustic sensor; (LP) linear polarizer, (SBC) Soliel-Babinet c o m p e n s a t o r , ( W P ) a Wallaston prism (Rashleigh, 1980). 1980).

7. Fiber Optic Acoustic Transduction

421

wave of pressure AP the electric field amplitudes exiting the fiber of length / are of the form: Eu2

= (E/y/2) exp(*/){±/[0 1 2//2 + ά(βι21/2)/άΡ

AP]},

(50)

where Ε is the incident amplitude, k is the average propagation constant, and d(fil2l/2)/dPis the transduction coefficient. The beams exiting the fiber are passed through a Wollanston prism oriented to transmit beams linearly polarized at 45° relative to the fiber eigenmodes and a Soleil-Babinet com­ pensator is used to provide quadrature. The intensities of these beams are of the form: 2 2 /, = Ε cos [0 1 2//2 + ά{βχ21/2)/άΡ AP], 2

2

I2 = Ε sin [0 1 2//2 + ά(βχ2Ιβ)ΙάΡ

AP].

These can each be photodetected and electronically processed to give the signal I: I - y^Y Ji -t-

12

= cos[0 1 2/ + ά(βχ21)/άΡ AP].

(52)

Such a signal is analogous to the signal generated in the two-fiber interfer­ ometer case and can be expanded to give the same signal expression de­ scribed by Eq. (3) with Αφ —> Α(βΙ) and φ0 —• βΐ. Thus as for the interfer­ ometer, the signal at the acoustic frequency fades as ambient conditions change βί. However, unlike the two-fiber interferometer, the polarization sensor utilizes only one optical fiber and thus substantially decreases the magnitude of the environmentally induced fading. Polarization sensors can utilize statically isotropic fibers (Rax et al, 1981) or statically birefringent fibers (Rashleigh, 1980). Consider the general case of a birefringence βχ2. This may be a fiber fabricated with an elliptical core (Dyott et al, 1979) or with an anisotropic cladding stress (Stolen et al, 1978), or it can be a fiber originally isotropic, mounted in such a way as to induce anisotropic stress (e.g., wound tightly around a rigid mandrel). The static birefringence βΧ2 is 0i2 = (0i " 0 2 )// = k0(nx - n2),

(53)

where φ„ is the optical phase shift of each polarization state and nn is the corresponding effective index of refraction. An acoustically induced change in βχ2 is related to variations in both the length and the fiber birefringence:

Which term dominates in Eq. (54) is dependent upon the birefringent prop­ erties of the fiber as well as the geometrical details of the acoustically induced

422

/. A. Bucaro et al.

strains. The latter, in turn, depend upon the mechanical arrangement and mounting of the fiber. Flax et al. (1981) and DePaula et al. (1981) have studied ά{βηΙ)/άΡ for the case of a fiber standing alone. In this case for typical fibers, the second term in Eq. (54) which depends on P44 is several orders of magnitude larger than the first term which depends on the static fiber birefringence; 3 the latter rarely exceeds one part in 10 . For this case

ίι^ρ

(βιι1)

^ίϊ> ~ ~ ^ " ^ {ηχ

ni) =

n]p

ex

e

(55)

where ex and e2 are the acoustically induced strains as given in Eqs. (35) and (38). DePaula et al. (1981), utilizing the polarization sensor shown in Fig. 26, measured the transduction coefficient [Eq. (55)] for bare silica fibers. Their results are shown in Fig. 27 along with the values calculated utilizing Eqs. (55) and (35) for the case of an ITT single-mode fiber with 80 μηι o.d. and 2.5 μπι core. As can be seen in Fig. 27, the induced modal birefringence is very small for Kd/2, less than 0.1, since in this region e\ « e 2. At first, d Δβχ2/άΡ rises

H e - N e LASER

0

LP.

λ/4 PLATE

20X OBJ.

[/

OPTICAL FIBER ULTRASONIC TRANSDUCER

I

WATER TANK

PHOTOMULTIPLIER

Τ

ANALYZER

λ/4 PLATE

20X OBJ.

SPECTRUM ANALYZER F I G . 26. S c h e m a t i c a r r a n g e m e n t for a n u l t r a s o n i c birefringent s e n s o r ( D e P a u l a et al., 1981).

7. Fiber Optic Acoustic Transduction

423

F R E Q U E N C Y ( M H z - F O R d / 2 = 40 μηι) 0

3

6

I

I

0

0.5

1.0 Κ (d/2)

9

.

12

I 1.5

ι

ι

.

2.0

FIG. 2 7 . A plot of the i n d u c e d m o d u l a t i o n index for birefringence as a function of 4 Kd/2 for an incident acoustic pressure of 10 Pa a n d an interaction length of 2 . 5 4 c m ( D e P a u l a et al., 1981).

linearly with Kd/2, and then becomes roughly constant up to the quadrupole resonance of the fiber, where the acoustic wavelength Λ is half the fiber diameter. For a typical single-mode fiber with 80 μπι o.d., the resonance occurs at a frequency of 36.8 MHz. For such a fiber, d Δβη/dP is approx­ imately constant between 10 and 30 MHz. Rashleigh (1980) has demonstrated a birefringent sensor fabricated by wrapping a single-mode fiber on a compliant cylindrical mandrel with the fiber under tension (Fig. 25). In this geometry, unlike the unsupported fiber case, a low-frequency hydrostatic acoustic pressure induces significant bi­ refringence in the fiber. This induced birefringence is derived, in general, from the radial, axial, and azimuthal mandrel strains in a complicated way (Rashleigh, 1980). This technique, which also introduces a large static bi­ refringence in a conventional single-mode fiber, provides independent prop­ agation of the two polarization states. The polarization eigenmodes are aligned perpendicular and parallel to the cylinder axis. The Soleil-Babinet compensator (SBC) is utilized to provide the quadrature condition, i.e.,

J. A. Bucaro et al.

424

βι21 = π/2 in Eq. (52). For this case, the fading problem is eliminated and the demodulated output is simply /= -ύη[ά{βηΙ)/άΡ

AP].

(56)

Finally, Rashleigh and Taylor (1981) have demonstrated a beam-form­ ing polarization sensor. As shown in Fig. 28, a single-mode birefringent fiber is twisted uniformly along its length at a rate τ (rad/m). The fiber is then embedded in the wall of a hollow cylinder with its axis parallel to that of the cylinder. The acoustically induced birefringence [Eq. (50)] is now spatially modulated by the term cos(2rz), where ζ is along the cylinder axis. This leads to a transduction coefficient of the form: d

ί* η sin(
(57)

with q = Kcosd.

(58)

Here Κ is the acoustic wavenumber and θ is the angle of incidence with respect to the cylinder axis. Thus the response is highly directional (see Fig. 28) and is steered in a direction determined by r. 6. Optical Intensity Fiber Sensors Any fiber optic mechanism which in response to an acoustic field produces a change in the total optical power carried by the fiber or a redistribution of the power carried in its various modes can be utilized for acoustic sensing. Figure 29 shows schematically such an intensity sensor. Light from a light

FIG. 2 8 . E x p e r i m e n t a l a r r a n g e m e n t for investigating sensor response as a function of acoustic frequency a n d angle of incidence: (---) theory, (—) e x p e r i m e n t (Rashleigh a n d Taylor, 1981).

7. Fiber Optic Acoustic Transduction

425

LIGHT SOURCE

PHOTODETECTOR

4 LOSS MODULATOR

MODE SELECTOR

FIG. 2 9 . Schematic of i n t e n s i t y - m o d u l a t e d fiber optic sensor.

source such as a laser or light-emitting diode is coupled into an optical fiber and propagates through an acoustically sensitive section of the fiber (sensing element). When an acoustic wave is incident on the sensing element, it causes a modulation of the intensity of light propagating in the various fiber modes. In the general case, this modulation can be detected by utilization of a mode filter and subsequent photodetection. This mode power modu­ lation can be a redistribution of power in the propagating core modes or a transition from core modes to radiation (loss) modes. In the latter case, some of the radiated light can be recaptured by the fiber cladding and subsequently guided down the fiber as cladding modes (Fig. la). 6.1

T H R E S H O L D PRESSURE DETECTABILITY

Following Fields et al (1979), let W0Tbe the light power incident on the detector, where Τ is the optical power transmission coefficient for the specified modes through the acoustic sensor and W0 is the input light power to the fiber. When an acoustic pressure ΔΡ is applied to the sensor, the transmission coefficient will change by Δ Γ and the detector signal current / s is given by the following equation: (59) where q is the quantum efficiency of the detector, h is Planck's constant, ν is the light frequency, and ΔΤ/ΔΡ is the transduction coefficient of the sensor.

/. A. Bucaro et al.

426

The mean-square shot noise is given by (Yariv, 1971) i N = 2e(qe\V0T/hv)Af, 2

(60)

where Δ/ is the detection bandwidth. The SNR ratio can be found from Eqs. (59) and (60): SNR = The minimum detectable pressure can be found from the above equation by equating the signal to the noise:

When the power in the core modes is monitored (Fig. 30, insert) the optical power transmission coefficient Τ is typically in the range 0.1 to 1. When the power in the cladding modes is monitored, however, Τ can be several orders of magnitude smaller, since these modes can be easily stripped (dark field) just before the sensing element (Lagakos et al, 1981b). Thus, as can be seen from Eq. (62), dark field detection allows in principle the detection of significantly smaller pressure fields. Two distinct loss mechanisms have been utilized to achieve fiber in­ tensity sensors. Fields et al (1980) and Lagakos et al (1981b) employed periodic microbending with a spatial periodicity of the same order as that of the inverse wave number difference of neighboring guided modes. J. H. Cole, N. Lagakos, and J. A Bucaro, private communication (1981) utilized a long, continuous bend obtained by looping a fiber around a compliant mandrel. 6.2

MICROBEND SENSOR

In a microbend sensor, an optical fiber is periodically deformed (see Fig. 30) by mechanical means. This deformer is usually a pair of ridged plates, wire grids, etc. The fiber deformation has two critical parameters, namely, the spatial period of the bending A f and the amplitude of the de­ formation x. The transduction coefficient of the sensor ΑΤ/ΔΡ used in Eq. (59) can be written in terms of the maximum displacement of the fiber deformation χ as follows: AT/AP = (AT/Ax)(Ax/AP). (63) In this equation, AT/Ax depends on the sensitivity of the optical fiber to microbending losses, and Ax/AP depends on the acoustical and mechanical design of the sensor.

7. Fiber Optic Acoustic Transduction

All

t Ρ

2X

FIG. 3 0 . M i c r o b e n d sensor. (A F) M e c h a n i c a l periodicity, (x) fiber displacement. Insert: coupling of core to clad m o d e s .

6.2.1 Microbend Loss It is well known that any fiber bending introduces excess transmission loss, which is particularly enhanced with periodic deformation along the fiber axis (Fig. 30). Such a deformation causes strong mode coupling, which redistributes the light among core modes and couples some light from core to radiation and clad modes. Thus, by monitoring the light power in certain modes an applied pressure can be detected. These modes could be core modes (Fields et al, 1979) or cladding modes (Lagakos et al, 1981b). Since the early stages of fiber optics, microbending losses have deliberately been minimized for communication applications by making strongly guiding fi­ bers (high δ) with thick cladding and soft concentric primary coating. Optical fibers having opposite characteristics, particularly very weakly guiding fibers (small δ) exhibit high microbending loss. Indeed, Miller (1976) has exper­ imentally demonstrated that microbending losses increase with decreasing δ, and with decreasing cladding-to-core thickness ratio. Increased micro­ bending sensitivity is also expected when high-order guided modes are ex­ cited, as has been shown experimentally by Fields et al (1981). These higher

/. A. Bucaro et al.

428

order modes are easily coupled to radiation modes (loss) and are very sus­ ceptible to any fiber deformation. From the theory of mode coupling (Gloge, 1972; Marcuse, 1974) it is well known that when a microbend of periodicity A f is imposed along the fiber axis, light power is coupled between modes with propagation constants kt and kj such that h -kj

= ± ^ .

(64)

Applying the WKB approximation (Merzbacker, 1961) to the solution of the dielectric waveguide problem (Gloge and Marcatili, 1973; Kurtz and Streifer, 1969) and expressing the fiber profile as given by Eq. (1), it can be shown (Gloge and Marcatili, 1973) that the difference in adjacent mode propagation constants is

^-^¸ )

w

m

’

(65)

where m is the mode label and Μ is the total number of modes. For step-index fibers (b = oo), Eq. (64) becomes: km+i

-km

]/2

= (2d /a)(m/M).

(66)

This means that the separation of modes in k space depends on the order of the mode m. From Eqs. (63) and (65) we see that high-order modes (large m) can be coupled with small periodicity Af, whereas low-order modes (low m) are coupled with large Af. For parabolic index fibers (b = 2), Eq. (64) gives Ak = km+x -km

,/2

= (2δ) Μ

(67)

This means that in a parabolic index fiber Ak is independent of m, that is, all modes are equally spaced in k space (to within the WKB approximation). In this case, the same bending periodicity couples all adjacent modes effi­ ciently. This critical mechanical wavelength A c is given by: ι/2

A c = 2πα/(2δ) .

(68)

In Fig. 31 the value of A f required to couple adjacent modes is shown versus mode number for step- and parabolic index fibers. For scale, we note that for typical sensing fiber parameters (<5 = 0.15, a = 50 μπι) A c for the parabolic fiber is ~~1 mm [Eq. (66)]. As can be seen from Eq. (66) for a fiber sensor which operates by switching low-order modes, much longer deformer periodicities are required for the step-index fibers. For sensors which operate by switching high-order modes to radiation modes, however, somewhat shorter deformer periods are required for the step-index case. In

7. Fiber Optic Acoustic Transduction

429

Μ F I G . 3 1 . Periodicity for m o d e coupling for step index a n d parabolic index (m) M o d e n u m b e r , {M) total n u m b e r of guided m o d e s .

fibers.

any case, the parabolic fiber with its single value of A c has ideal mode properties for microbend sensors. Indeed, large enhancement of the bending loss can be achieved in a parabolic fiber when the bending periodicity is A c as given by Eq. (68). In this case efficient coupling takes place from one propagating mode to the next, and from higher order modes to radiation modes. Indeed, this effect has been demonstrated experimentally by Fields et al. (1981) who studied the loss of a Corning graded fiber as a function of deformer periodicity. As can be seen in Fig. 32, they observed a strong resonant peak indicating the existence of a critical periodicity at 1.6 mm, in agreement with the prediction of Eq. (68) (δ = 0.008, 2a = 65 μπι) for this fiber. 6.2.2 Mechanical Design Having discussed the optical loss term (AT/Ax) in Eq. (63), we now consider the mechanical factor (Ax/AP). In most microbend sensors, pres­ sure is applied to the fiber indirectly through acoustic couplers (pistons, diaphragms, etc.) which are used as pressure multipliers. If A is the area of the acoustic coupler against which the acoustic pressure is applied, Eq. (63) can be written as: AT/AP = (AT/Ax)(ACm), (69) where C m is the mechanical compliance of the sensor. The mechanical term, ACm in Eq. (69) depends on the sensor design. The mechanical compliance of the sensor C m can be found from the equivalent acoustic circuit of the sensor (Lagakos et al., 198 Id). When the compliance of the surrounding

/. A. Bucaro et al.

430 Ο.Ο61-

0.05

ε

o.04

0.02

0.01

I 2

A 3

,

4

5

6

7

Af ( m m ) FIG. 3 2 . Experimentally obtained m i c r o b e n d i n g sensitivity {AT/Ax) versus mechanical periodicity (A f) for a C o r n i n g 1150 graded index fiber; A c = 1.6 m m . After Fields et al. (1979).

fluid is substantially larger than the mechanical compliance of the sensing fiber element Q the compliance of the sensor C m is proportional to Q: C m oc C f =

G(A /Ed )(\/Ndl 3

4

(70) where Nd is the number of the deformer intervals, and G is a constant which depends on how the fiber is loaded and suspended. For example, for a fiber clamped at its ends and deformed by a load at the center, G = 1/3π (Singer, 1962). From Eqs. (68) and (69) we see that high sensitivity is obtained with highly compliant fibers. This can be achieved with a large deformer peri­ odicity Af. Also, for large C m , Nd, and d, the Young's modulus Ε should be small. However, the Young's modulus of typical glasses used for optical fibers does not vary very much, certainly not by an order of magnitude. For example, the presence of B 2 0 3 in silicate glasses tends to lower Ε by 20-40%. Also, very small values of d, the fiber diameter, are prevented due to light coupling and fiber mechanical strength considerations. And finally, even though utilizing a small number of deformer intervals Nd increases C f proportionally, it decreases Δ Τ/χ by the same amount. Thus the sensor sensitivity is approximately independent of Nd. In practice, the optimum value of Nd is determined from sensor mechanical design considerations, for example the position of the mechanical resonance and thus the required sensor bandwidth.

7. Fiber Optic Acoustic Transduction

431

The characteristics of the microbend sensor, such as sensitivity, minimum detectable pressure, and resonant frequency, depend strongly upon the acoustical and mechanical design of the sensor and, in particular, on the acoustic coupler and the deformer. The acoustic coupler is usually a piston or a diaphragm which multiplies and transfers pressure of an applied acoustic wave to the sensing element. The design of the deformer which bends the sensing element can vary widely, being periodic or nonperiodic, spatially short or extended, in accordance with the desired sensor performance. For compactness and versatility the deformer can even be an integrated part of the fiber coating. A spatially extended deformer can be designed, one example of which is a fiber wrapped around a cylindrical tube in a machined thread. The fiber is periodically exposed to the sound pressure in axial slots that cut below the root of the threads. In this case, pressure can be applied directly to the fiber using a rubber boot which encloses the tube and the fiber. Such a sensor has been tested as a hydrophone (Lagakos et al. 1972b), demonstrating very promising performance. 6.2.3 Acoustic Measurements The first microbend fiber optic acoustic sensor measurements were made by Fields and Cole (1980) utilizing a Corning multimode step-index fiber sandwiched between a pair of ridged plates containing ten periods of wavelength A f = 2 mm. One of the plates was driven by a Mylar diaphragm exposed to the acoustic field, while the other plate was attached to the sensor housing through a spider support assembly. The experimental setup used in this experiment was similar to that shown in Fig. 29 and the light power modulation in the core modes was monitored. Figure 33 (circles) shows the results for the sensitivity of the hydrophone versus frequency (J. N. Fields and J. H. Cole, unpublished results, 1980). As can be seen from this figure, the sensitivity exhibits large peaks, indicating a primitive mechanical design. Figure 34 shows the minimum detectable pressure of the hydrophone from 200 to 1500 Hz. As can be seen from this figure, the minimum detectable pressure is ~ 1 1 0 dB re 1 MPa, much higher than theoretically possible. Lagakos (1981) constructed and tested a microbend fiber optic hydrophone, which is shown schematically in Fig. 35. In this hydrophone, an acoustically stiff cylindrical container was filled with liquid. The fiber was sandwiched between the two corrugated plates of the deformer which were attached to two independent pistons where pressure was applied. A small port was utilized as a low-pass filter to equalize hydrostatic pressure. The whole container was embedded in a rubber boot filled with the same liquid as the container. The fiber used in these experiments was a Hughes Research Laboratory step-index aluminum-coated multimode fiber with a 0.133 numerical aperture. A small section of the fiber was stripped from its aluminum

/. A. Bucaro et al.

432

a.

> CO

500

1000

1500

F R E Q U E N C Y (Hz) F I G . 3 3 . E x p e r i m e n t a l l y o b t a i n e d sensitivity versus frequency for t h e two-piston h y d r o p h o n e ( Δ ) a n d the h y d r o p h o n e of J. N . Fields a n d J. H. Cole (unpublished results, 1980) ( O ) .

coating and etched down to 116 μτη o.d. in order to increase the fiber compliance [Eq. (70)] and the sensor sensitivity [Eq. (69)]. The deformer periodicity was 2.5 mm and the total deformed fiber length was 1.2 cm. The compliance of the sensor C m needed to calculate the sensor character­ istics, such as sensitivity, minimum detectable pressure, and resonant fre­ quency, can be found from the equivalent circuit of the hydrophone which is shown in Fig. 35. C Ai and CA2 are the acoustic compliances of the liquidfilled chamber and the bending fiber, respectively, Z A is the acoustic imped­ ance of the port, and mA is the acoustic mass of the piston and the deformer plus the radiation reactance.

7. Fiber Optic Acoustic Transduction

433

120

F R E Q U E N C Y (Hz) FIG. 34. Experimentally o b t a i n e d m i n i m u m detectable pressure versus frequency for the two-piston h y d r o p h o n e ( Δ ) a n d t h e h y d r o p h o n e of Fields a n d Cole (1980) ( O ) ; ( ) sea state zero.

Figure 33 shows the frequency response of the hydrophone sensitivity from 200 to 1500 Hz (triangles). As can be seen from this figure, the sen­ sitivity is approximately - 2 0 5 dB re 1 ν/μΡ& and is flat to within ±0.5 dB. These results are in agreement with analytically predicted values (-210 dB re 1 Υ/μΡζ) (Lagakos et al, 1980c). Significant improvement in bandwidth has been achieved over the previously reported microbend sensor (J. N. Fields and J. H. Cole, unpublished results, 1980). Figure 34 shows the measured minimum detectable pressure from 200 to 1500 Hz. For fre­ quencies higher than 500 Hz, the minimum detectable pressure is at 60 dB re 1 ^Pa, much lower than that of the previous microbend sensor. For

/. A. Bucaro et al.

434

u

I

m

A

1

m

A

2

L/YYYYWJ

SENSOR

F I G . 35. (a) Two-piston acceleration-insensitive h y d r o p h o n e , (b) Equivalent circuit of the two-piston h y d r o p h o n e . After Lagakos et al. (1981b).

acoustic

frequencies lower than 500 Hz, the electronic noise of the laser significantly deteriorates the sensor performance. Finally, Lagakos et al (1981b) have utilized the microbend sensor as a displacement sensor and have monitored the dark field, i.e., the light power in the cladding modes. Any light in these modes was stripped before entering the sensing element. One piece of the deformer was displaced relative to the other by a piezoelectric transducer. This microbend displacement sensor was found to have a fraction of an angstrom displacement detectability, a dynamic range of more than 110 dB, and excellent linearity (within 1%).

7. Fiber Optic Acoustic Transduction 6.3

435

MACROBEND SENSOR

A different type of loss sensor can be obtained by modulating a long, continuous optical fiber bend. This is readily accomplished, for example, by wrapping a weakly guiding fiber tightly around a compliant mandrel. It is well known that such a uniformly bent fiber experiences added losses due to radiation of the propagating core modes out of the fiber waveguide. This loss is a strong function of the radius of curvature, so that acoustically induced changes in the mandrel radius modulate the light transmitted through the fiber. If a is the loss due to bending, the transmission coefficient Τ in Eqs. (59)-(62) is simply α/

T= 1(Γ .

(71)

The transduction coefficient is then ^=-2.3/10-^ = - 2 . 3 / 1 0 - ^ ^ , dP dP 3Km dRm

(72)

where Rm and Km are the mandrel radius and bulk modulus, respectively. Figure 36 shows experimentally measured bend loss α as a function of bend radius Rm for some typical fibers. This strong dependence on Rm is generally predicted theoretically (Marcuse, 1976; Sakai and Kimura, 1978) and is of the form B R

=A'e- ' ™lfRm,

a

(73)

where A' and Β' are functions which depend on the core and clad refractive indices, the core radius, and the light wavelength. Figure 37 shows a computed from Eq. (73) for a single-mode fiber for various values of the numerical aperture. As can be seen, a is higher for lower NA although (for a given bend radius), higher values for da/dRm are obtained for the larger NA fibers. We can calculate the minimum detectable pressure by inserting Eq. (72) into Eq. (62):

Inspection of Eq. (74) shows that there exists an optimum length / o p which minimizes P m i n. Maximizing Eq. (74) with respect to / gives / o p = 0.87/a.

(75)

Thus, a trade-off exists between total fiber length /, bend loss a, and bend

436

/ . A. Bucaro et al.

0

J 0.2 0.4 0.6 0.8 1.0 1.2

L 1.4 1.6

BEND RADIUS IN cm FIG. 36. T h e excess power loss d u e to a bent fiber is plotted as a function of the b e n d radius. ( · ) Single-mode fiber ( N A = 0 . 1 0 - 0 . 1 2 ) [after Salisbury (1980)]; ( Ο , Δ ) single-mode (NA = 0.11) a n d m u l t i m o d e fiber, respectively ( L a g a k o s a n d Bucaro, 1981). T h e solid line is a least-squares fit of the data.

radius Rm (which determines a). For small mandrel radii, the bend loss is high and / o p is small. Alternatively, if long fiber lengths are desired for extended elements or for shaded windings, a must be low, which requires a large Rm. 7. Evanescent Field Fiber Coupler Sensors The field amplitude of a guided mode in an optical fiber exists primarily in the core of the waveguide. There is, however, an evanescent field which extends a substantial distance into the cladding. This phenomena has been exploited in the development of fiber-to-fiber couplers (Sheem and Giallorenzi, 1979). In such devices, as shown in Fig. (38), the cores of two fibers are brought close together by, for example, cladding etching or thermal fusion. If the evanescent field from one fiber extends into the core region of the second over an appreciable interaction length L some power is cou-

7. Fiber Optic Acoustic Transduction HE,,

BEND

BEND

RADIUS

437 LOSS

(mm)

F I G . 37. T h e theoretical excess power loss for single-mode fibers is plotted as in Fig. 36 as a function of b e n d radius with fiber N A as a p a r a m e t e r : V = 2.405, ncX = 1.458, λ = 0.84 n m (Bucaro, 1979).

pled from one fiber to the other. If the interaction length is equal to, or some multiple of, the beat length L B , all the power is coupled and the splitting ratio SR is unity, as can be seen from the following expression for a lossless coupler: SR = s i n V L / 2 L B ) . (76) The beat length depends upon a number of parameters—including the coreto-core separation d, the core and clad refractive indices nco and nc\9 and the core radius a. Sheem and Cole (1979) and Carome and Koo (1980) have observed that SR can be quite sensitive to acoustic fields and can thus form the basis

438

J. A. Bucaro et al.

\

INTERACTION LENGTH

L FIBER CORES

LIGHT OUTPUT FIG. 3 8 . S c h e m a t i c of evanescent wave t r a n s d u c t i o n ( B u c a r o a n d Cole, 1979).

for an acoustic sensor. In such sensors, an optical beam of power W is input to one of the fibers. The light level coupled to the second fiber is monitored with a photodetector. An acoustic pressure AP incident on the coupling region changes the splitting ratio by an amount (dS /dP) AP. The photodetector signal current is then: 0

R

and the mean-square shot noise i = 2e(qe/hv)(W S )Af.

(78)

2

N

0

The minimum detectable pressure P ratio of one as:

m i n

R

, is then given for a signal-to-noise

The transduction coefficient dS /dP can be expressed in terms of the relevant parameters by differentiation of Eq. (76) with respect to those vari­ ables. Using the expression for L given by Sheem and Cole (1979) and differentiating gives: R

B

(dSA

.

UL\\^L\V V C x

2

d

+ (3 + Κ,Α,)(2™/λ)(κ - « ,r («,C„ 2

2 c

L/2

Α))(2τΓ/λ)(« 0 - AJc, )

l / 2

0

+ (3 +

2

2

CO

nCJ cl

n

c ] + 2TT/LBC L} .

(80)

7. Fiber Optic Acoustic Transduction

439

Here Vx and V2 are parameters which depend on the modal properties of the fiber (Sheem and Cole, 1 9 7 9 ) , D0 + 2 is the ratio of core separation d to core radius a, and C d , C„ co, C„cl, C f l, and CL are the rate of change with pressure of the core separation, core index, cladding index, core radius, and interaction length, respectively. Thus, several physical effects can take place to yield the resulting sensitivity. Sheem and Cole ( 1 9 7 9 ) measured the transduction coefficient for a single-mode fiber coupler. The dominant terms for their case are those associated with Cd and Cn where the cladding medium was actually water. Acoustically induced accordian-like motion of their slender, unclad coupler led to an enhanced contribution from the Cd terms and a relatively high sensitivity. Carome and Koo ( 1 9 8 0 ) have described the optical coupling mechanisms associated with a multimode evanescent sensor. Beasley ( 1 9 8 0 ) fabricated a sensor which incorporated mechanical amplification utilizing a diaphragm configuration, and observed very low detection thresholds ( 5 0 dB re 1 MPa). Carome and Koo ( 1 9 8 0 ) have also investigated a multimode device, but without mechanical amplification. 8. Hybrid Fiber Sensors The sensors discussed thus far involve transduction mechanisms which take place within an optical fiber. There has been considerable work on optical fiber sensors in which the transduction takes place external to the optical fiber. We call these hybrid fiber sensors.

8.1 M O V I N G M I R R O R

Figure 3 9 illustrates the moving mirror hydrophone studied by Shajenko ( 1 9 7 6 ) . As shown, an optical beam is permitted to reflect back and forth between two optical mirrors as it travels from left to right. One mirror is rigidly mounted, and the other is attached to a diaphragm. Upon acoustic excitation, the movable mirror is displaced at the acoustic frequency, thus modulating the total optical path length through the device. The output beam, thus modulated in phase, is allowed to interfere with a reference optical beam, thereby converting this phase modulation to intensity modulation. The mixed beams are then carried back to a photodetector where the optical intensity fluctuations are converted to electrical signals. The minimum detectable pressure can be shown to be: (81)

440

/. A. Bucaro et al.

FIG. 3 9 . M o v i n g m i r r o r h y d r o p h o n e . After Shajenko ( 1 9 7 6 ) .

Here / / e is the homodyne/heterodyne efficiency, Nr is the number of re­ flections, dm is the mirror separation, AC is the medium's compressibility, and a is the optical angle of incidence. With air between the mirrors the sensitivity predicted can be quite high (and the P M low) for practical choices of these parameters. For example, 7 2 5 taking Hc = 1, Nr = 14, dm = 0.5 cm, κ = 7 X 10~ cm /dyn (air), k = 10 1 c m , a = 22°, q = 0.5, and W0 = 1 mW yields PM = - 4 8 dB re 1 MPa. For operation at moderate ocean depths, however, air can no longer be utilized between the mirrors since the static pressure would result in collapse. To prevent this, a low-compressibility material must be employed; however, this results in a loss of sensitivity. For example, if a silicone rubber 10 2 were used (κ = 1.09 X 10" dyn/cm ), a reduction of performance of 76 dB would result. Furthermore, the optical quality of such transparent rub­ bers must be very good in order to avoid substantial optical loss.

441

7. Fiber Optic Acoustic Transduction 8.2

LASER D I O D E

Dandridge et al. (1980) have described a laser diode acoustic sensor. The principle of operation of this sensor is depicted in Fig. 40. Here, a translatable external reflector feeds light back into the laser cavity. The phase of the reflected light is determined by the separation between the reflector and the laser. When the reflected light is fed back in phase, the effective laser facet reflectance is increased, and when it is fed back out of phase, the facet reflectance is lowered. Since the laser gain is proportional to the facet reflectance, small movements in the reflector position result in relatively large output intensity variations. A sensitive acoustic device is obtained by mounting a mirror, which acts as the external reflector, on a diaphragm that is displaced by the acoustic field. 8.3

FlBER-TO-FlBER

TRANSMISSION

A number of conceptually simple intensity sensors in which the amount of light coupled into a fiber is modulated by the acoustic wave have been studied. Spillman and Gavel (1980) employed two single-mode fibers ACOUSTIC WAVES BIAS SUPPLY

EXTERNAL CAVITY

LASER C A V I T Y

FIG. 4 0 . D i o d e laser acoustic sensor ( D a n d r i d g e et al., 1980).

442

/. A. Bucaro et al. ACOUSTIC WAVE

FERRULE

MOVING FIBER

FERRULE

FIG. 4 1 . M o v i n g fiber h y d r o p h o n e (Spillman a n d Gavel, 1980).

mounted such that their end surfaces were parallel, coaxial, and separated by several microns (See Fig. 41). Acoustically induced relative motion between the fixed and movable fibers modulates the transmitted light. Spillman and McMahon (1980a) and Tietjen (1981) developed a schlieren or grating sensor. This sensor modulates light coupled from one multimode fiber to another by means of a pair of opposed gratings which move relative to one another. Figure 42 shows the arrangement used by Spillman

FIG. 4 2 . Schlieren m u l t i m o d e fiber h y d r o p h o n e (Spillman a n d M c M a h o n , 1980a).

7. Fiber Optic Acoustic Transduction

443

and McMahon (1980a). Here light from one multimode optical fiber is collimated by a graded-index rod microlens and passed through an opposed 5 μτη stripe grating pair. The light is then focused into an output fiber by means of a second microlens. Figure 43 shows minimum detectable pressure levels measured with this device. At frequencies below 1 kHz, sea state zero pressures (or lower) can be detected. At higher frequencies, the response falls off sharply, essentially due to the small displacements at higher fre­ quencies. 8.4

FRESNEL REFLECTION SENSORS

An acoustic wave can modulate the optical intensity reflected or trans­ mitted by an optical interface, Fig. 44. The Fresnel intensity reflection coef­ ficients R±, R\\ for perpendicular and parallel polarizations are: (82) (83) where Ν = n2/nco, nco is the fiber core index, n2 is the index of the medium, and θ is the incidence angle. Following the work of Phillips (1980), the modulation index on which the acoustic sensitivity and minimum detect­ able pressure depend is found by differentiating Eqs. (82) and (83) with respect to pressure and noting that acoustic transduction takes place due to acoustically induced changes in TV. For the usual case in which the re­ flection takes place at the interface between the fiber end and a fluid, pres100 ω 3.

r.

75

.Ω Ό

c

50

Έ

Q.

SEA STATE ZERO

25

0 100

200

500

1000

2000

5000

FREQUENCY (Hz)

(

F I G . 4 3 . M i n i m u m detectable pressure as a function of frequency: ( O ) ) shot noise limit a s s u m e d (Spillman a n d M c M a h o n , 1980a).

measured;

J. A. Bucaro et al.

444

SIGNAL PROCESSING

DETECTOR

SENSOR y 2 ^

FIBER

LASER

FIBER CORE n

LIGHT INPUT, O U T P U T -

co

3 ^

CLADDING

FIG. 44. Critical angle acoustic sensor. After Phillips (1980).

sure-induced changes in n2 are much larger than those in nco yielding:

2

,i

VN

m"co

'

Jy

2-

sin Θ(Ν cos θ + VjV - sin 0) ' 2

2n 2h cos θ (2 sin θ - N )(N 4

2

nloKjN

2

2

2

2

cos θ - ^N - sin Θ)

- sin Θ(Ν cos θ + 2

2

2

2

2

VTV

2

3

(84)

2

- sin 6>) 2

3

(85)

Here h and A^m are the photoelastic constant and bulk modulus of the medium, respectively. Acoustic sensing can be achieved by either moni­ toring the light reflected back into the original fiber (Phillips, 1980), or by monitoring the light transmitted to a second fiber (Hull, 1981). As can be seen from Eqs. (84) and (85), the pressure sensitivity is a strong function of θ and in fact becomes very large near the critical angle 0C defined by 1 flc = snT n. (86) Accordingly, Phillips (1980) has proposed a fiber acoustic probe in which a single-mode fiber end is cut to an angle just below 0C (to within a few minutes of arc. In principle, the difference between this angle cut and 6C can be controlled as suggested by Phillips (1980) using dispersion tuning

7. Fiber Optic Acoustic Transduction

445

of 0C. As can be seen from Eq. (86), this requires that the dispersions of n2 and nco differ by a sufficient range consistent with the total optical wavelength range available. Finally, Spillman and McMahon (1980b) have demonstrated a frustrated-total-internal-reflection sensor which operates just above the critical angle (Fig. 45). In this configuration, the optical beam in the first fiber is confined by total internal reflection at its end until a second fiber is allowed to move close to the first. The proximity of the second fiber allows evanescent field coupling and destroys the condition for total internal reflection. Light is transmitted from the first fiber to the second when an acoustic field modulates the gap between the two fibers. 9. Practical Sensor Implementation The earliest fiber sensors were principally laboratory devices constructed using large lasers (HeNe), bulk optical components, and short lengths of

/. A. Bucaro et al.

446

readily available fiber. It was quickly realized practical implementation re­ quired substantial packaging and ruggedization. Fiber sensors must with­ stand substantial environmental stresses while maintaining performance. Additionally, size and cost considerations led sensor developmental efforts towards utilizing all-fiber components whenever possible, and replacing large components such as gas lasers with the much smaller solid state laser diodes. With these improvements, practical, reliable sensors could be re­ alized. In this section the issues attendant upon replacing gas lasers with diode lasers will be considered. All-fiber couplers (beam splitters) and a packaged interferometric sensor will be briefly examined. 9.1

DIODE LASER TECHNOLOGY

Solid state GaAlAs diode lasers are ideal sources for fiber optic sensor systems because of their small size, excellent efficiency, and high output power. However, the replacement of conventional gas lasers, which have excellent coherence and spectral properties, with semiconductor diode lasers with nonideal characteristics leads to several problems. The output of diode lasers exhibits increased amplitude noise, short coherence length, and phase noise when compared with gas lasers. If not compensated for, such degraded laser output manifests itself as a loss of sensor detectivity. Although there are several different diode laser structures, they all show similar behavior with regard to their optical output characteristics. While the output char­ acteristics of diode lasers impose certain restrictions on the design of fiber sensors, it is still possible to build very compact sensors with high sensitivity. In amplitude or intensity sensors, where coherence properties are un­ important, light emitting diodes (LEDs) may also be used as the light source. LEDs can usually provide several hundred microwatts of power to these sensors and are ideal when simplicity of design is required. However, diode lasers, compared with LEDs, usually offer an order of magnitude improve­ ment in the optical power, which can be coupled into a multimode fiber. Diode lasers must be used with single-mode fibers, since LED/single-mode fiber coupling efficiencies are unacceptably low. 9.1.1 Diode Laser Amplitude Noise Amplitude noise degrades the performance of all sensor types (both amplitude and interferometric) since it is an additive noise. For example, the presence of amplitude noise in a conventional homodyne fiber inter­ ferometer produces a signal indistinguishable from the signal of interest. A phase shift Δ0 at an angular frequency Ω in one arm of the interferometer appears as a signal on the output of the interferometer as Λ(Δ0) (where Jx is the first-order Bessel function). If the phase shift is below 0.1 rad, it is

7. Fiber Optic Acoustic Transduction

447

directly proportional to the intensity output of the interferometer. The am­ plitude of J\(A^) reaches a maximum at a phase shift of approximately 1 rad (all values in radians are rms), and this corresponds to nearly complete modulation of the interferometer output. Consequently, to measure phase 6 shifts of approximately 10" rad, it is necessary to measure intensity fluc­ 6 tuations at the signal frequency which are approximately 10" of the dc 6 output. Thus to detect 10" rad, the relative amplitude noise of the source 6 must be 120 dB (10~ ) below the dc output. If the interferometer fringe visibility υ is less than one, the minimum detectable phase shift φηι is ap­ proximated by 0 m^ ( A S ' m/ S ' ) ( l / i O ,

(87)

when AS'm represents the amplitude noise at the frequency of interest and S' is the dc level. Shown in Fig. 46 is the amplitude noise plotted as a function of fre­ quency for a number of single-mode diode lasers. All of these lasers show a characteristic 1/frequency noise dependence. From this figure, it can be seen that for frequencies below 1 kHz laser amplitude noise may be a potential problem. The problem may be reduced by using amplitude sub­ traction schemes which have been shown to reduce the noise by more than a factor of ten at low frequencies. The effectiveness of amplitude subtraction techniques is shown in Fig. 47. Here, the minimum detectable phase shift 0 m (obtained from a bulk interferometer system powered by a Hitachi HLP 1400 diode laser) is plotted as a function of frequency. The solid line is that theoretically predicted from measurements of the laser's amplitude noise, the squares are experimentally determined values of m, and the circles represent experimental values of 0 m after applying amplitude subtraction techniques. At 50 Hz the subtrac­ tion technique has increased the detectability of the interferometer by a factor of 10, thus also increasing the dynamic range by a factor of 10. For 6 frequencies above 10 Hz, 10" rad phase shifts may be achieved by using amplitude subtraction and an interferometer with good fringe visibility. From Fig. 46 it can be seen that some form of noise reduction will always 6 be necessary for good (10~ rad) sensor performance below 1 kHz. A second effect which must be carefully controlled is feedback into the laser. It has been shown that relatively small amounts of light [approximately 3 10" %) fed back into the laser cavity can cause the laser to operate at lon­ gitudinal modes other than the previously lasing mode (Kanada and Nawata, 1979; Miles et al, 1981)]. The phase and amplitude of the feedback may be adjusted such that the laser oscillates between longitudinal modes. This has been shown to produce 20-60 dB of excess noise which is clearly unacceptable.

/. A. Bucaro et al.

448 ι

J

1

1

Γ

1

I

I

I

I

10

100

1000

1Q000

F R E Q U E N C Y (Hz) FIG. 46. T h e lasers shown in this figure are ( • ) Hitachi H L P (1400 c h a n n e l substrate p l a n n a r ) ; ( Δ ) Hitachi H L P 2400 (buried heterostructure); ( O ) Hitachi 3400 (buried heterostructure); ( · ) Mitsubishi M L 4 3 0 7 (transverse j u n c t i o n stripe); ( • ) Laser D i o d e L a b CS234 (channel substrate p l a n n a r ) ; a n d ( • ) G e n e r a l O p t r o n i c s T B 4 T ( p r o t o n - b o m b a r d e d buried het­ erostructure with a tellurium mask to reduce s p o n t a n e o u s emission).

t

5 ^

10

100

1000

1QQ00

F R E Q U E N C Y (Hz) FIG. 4 7 . A m p l i t u d e noise reduction in fiber optical interferometeric sensors. ( ) Theo­ retical noise prediction, ( • ) experimentally m e a s u r e d noise, a n d ( · ) interferometer noise level after c o m m o n - m o d e noise subtraction ( D a n d r i d g e a n d T v e t e n , 1981b).

7. Fiber Optic Acoustic Transduction

449

Therefore, the suppression of feedback induced noise is required. This noise can be minimized by ensuring that the fiber ends are either cut at an angle or index-matched to eliminate reflections. Fiber splices and couplers 2 3 in sensor systems must also be of high quality to avoid the 10" -10~ % reflections which are sufficient to produce feedback-induced noise. 9.1.2 Coherence Length The linewidth of cw diode lasers determines the laser coherence. Recent measurements by Miles et al. (1980) and Okoshi et al. (1980) have indicated that typical linewidths in the free running GaAlAs are between 5 and 10 MHz. Linewidth measurements of both channel substrate planar and buried heterostructure lasers have indicated that linewidths for similar lasers may vary from 5 to 100 MHz, presumably dependent on sample variations and age. Recently, it has been shown that controlled feedback into the laser via a grating, etalon, or a simple reflector can lead to line narrowing to linewidths below 100 kHz. While this requires external bias controls, it provides a method for obtaining a substantial increase in the diode laser coherence length. The lowest values of coherence length still exceed 3 m, therefore path matching of the interferometer is required only to approximately 1 m for good fringe visibility. However, in the presence of optical feedback due to coupling the laser to an optical circuit, line broadening, line narrowing, satellite-mode generation and multimode generation have all been observed (Figueroa et al., 1980). Figure 48 shows the diode laser linewidths and mode patterns for various feedback conditions. When no feedback exists, an excellent single-mode output (Fig. 48a) can be achieved. Different modal structures observed under different values of feedback (<0.04%) (Miles et al., 1980) are required to induce the relaxation oscillations shown in Fig. 48b and c. Figure 48d and e shows the laser running simultaneously on different longitudinal modes of the laser cavity. It should also be noted that at these latter levels of feedback the laser is operating on many longitudinal modes of the external cavity which, in conjunction with broadening of the external cavity modes, appears as severe line broadening of the laser's longitudinal cavity modes. In Fig. 48 these convoluted values of linewidth Av are given. Broadening and satellite mode generation require path lengths to be matched to approximately 1 cm, whereas multilongitudinal mode operation of the laser cavity requires path length matching to 0.1 mm to ensure good fringe visibility (Dandridge and Miles, 1981). Fortunately relatively large amounts of feedback (>0.1%) are required to induce the deleterious effects associated with multimode operation. Backscattering from long lengths of fiber may however be a problem, and some form of isolation may be re-

/. A. Bucaro et al.

450

(α)

(b)

(c)

(d)

(e)

FIG. 4 8 . Effects of reflection o n the m o d a l c o n t e n t of single-frequency d i o d e laser (Miles et al., 1980). T h e a m o u n t of reflected light increases m o n o t o n i c a l l y from (a) t o (e).

quired. Line narrowing occurs with small amounts of feedback (approxi­ 3 mately 10" %), but line narrowing has been shown not to induce noise in interferometer systems (Goldberg et al, 1981). It is clear that diode laser coherence problems are not a serious problem in interferometer systems provided that excessive values of optical feedback are not encountered. As mentioned above, care in design is necessary to 2 ensure that reflections are kept below approximately 10~ %. 9.1.3 Phase Noise Phase noise is a problem mostly confined to interferometer systems and is due to a change in wavelength of the light source. This noise is strongly dependent on the optical path difference between arms of the in­ terferometer. If the interferometer is powered by a source with good fre­ quency stability such as a HeNe laser, and the path length difference is changed between 0 and 1 m, almost no increase in output noise is noted. If, however, a diode laser is substituted as the source, the noise is observed to increase linearly (Dandridge and Tveten, 1981a) and Yamamoto et al., 1981) with path difference and at 1 m path difference the noise (at quad­ rature) may be approximately 3000 times greater than that observed at zero path difference. Results for a typical diode laser (a channel substrate planar

7. Fiber Optic Acoustic Transduction

451

Hitachi HLP 1400) are shown in Fig. 49 where the bandwidth is 1 Hz. The phase noise has a frequency dependence similar to that observed for the amplitude noise (Dandridge and Tveten, 1981a). From Fig. 49 it can be seen that at 50 Hz the minimum detectable phase shift for a 40 cm path difference is approximately 30 dB above the desired - 1 2 0 dB. To achieve sensitivities below 1 (-120 dB) it is necessary to match the paths of the interferometer to better than 1 mm, which can prove difficult for sensors whose arms are long. The phase noise of other laser structures have also been investigated (Dandridge and Tveten, 1981a) and these results are very similar to those shown in Fig. 49. To reduce the effect of the phase noise contribution to the interfer­ ometer, it is possible to set up two interferometers powered by the same source where only one arm of one interferometer is coupled to the external field to be measured. This combination enables approximately 1 Mrad sen­ sitivity to be achieved for path length differences below 1 m. However, this method is unattractive and in practice may be unworkable. If small amounts 5 3 of light are fed back (10" % to 10" %) into the laser cavity, the linewidth of some lasers is observed to narrow (Yamamoto et al, 1981; Saito and Yamamato, 1981). Typically, with lasers that exhibit this effect, the line

I

0.1

I

1

I

10

1

100

1

1000

P A T H - L E N G T H D I F F E R E N C E (mm) FIG. 4 9 . Noise induced in a interferometer as a function of p a t h length difference between the two a r m s of the interferometer. T h i s noise arises because of frequency (phase) fluctuations in the o u t p u t of diode lasers: ( Δ ) 50 H z , ( O ) 500 Hz, ( T ) 2 k H z .

/. A. Bucaro et al.

452

may be narrowed by up to a factor of 100. Under these conditions, it has been noted that the diode laser phase noise has been reduced by up to a factor of ten. However, the line narrowing and phase noise reduction effects are strongly dependent on the phase and amplitude of the light fed back into the laser cavity, as well as on the external cavity length. Consequently, some form of active phase compensation may be necessary to use these narrowed line lasers. At present to avoid excess phase noise, it is necessary to balance the Mach-Zehnder fiber interferometers path difference to approximately 1 mm, or to use single-fiber sensors such as the polarimetric or mode-beating devices which intrinsically have a small path difference. Various properties of diode lasers thus impose limitations and restric­ tions on fiber sensors, particularly on the design of fiber optic interfer­ _3 ometers. However, when optical feedback levels are below 10 %, when path length differences are approximately 1 mm, and when amplitude sub­ traction schemes are used, 1 μτ&ά performance down to 10 Hz should be possible. 9.2

SENSOR COMPONENTS

Regarding the fabrication of hybrid sensors, the necessary components usually either existed or could be modified. This was particularly true for the fiber-to-fiber transmission, Fresnel reflection, and moving mirror sen­ sors. Diode laser sensors required the placement of an external reflector in proximity to a diode laser. Fiber loss sensors involve the optimization of fiber parameters to enchance mode coupling, as discussed in Section 6. However, in the case of the evanescent and interferometric sensors, new components were required; in particular, the development of optical fiber couplers was mandatory before these devices could be realized in practical configurations. Evanescent couplers and their characteristics are therefore important and will be discussed in order to outline their characteristics. In many sensor embodiments, it is necessary to couple light from one single-mode fiber to another. This coupling is effected via evanescent field interactions in which guided light propagating in the core of one fiber in­ teracts with a second fiber in close proximity. In order to get the two cores of the fibers close enough to effect evanescent coupling, most of the cladding between the two cores has to be eliminated. The first successful realization of single-mode evanescent couplers involved etching the claddings and bringing the fragile cores together (Sheem and Giallorenzi, 1979). A second technique to fabricate evanescent directional couplers (Bergh et al, 1980) involved embedding the fibers in substrates and polishing the fiber cladding until the cores are nearly exposed. The embedded, polished

7. Fiber Optic Acoustic Transduction

453

fibers are then brought in close contact to effect the coupling desired above. This technique may prove to be particularly useful to maintaining the separation of the polarization states. These couplers are however structurally more complex (when polarization-preserving fibers are employed for couplers) than the etched type and do require precise alignment. In order to package and stabilize evanescent couplers fabricated by the above two techniques, etched fiber couplers have to be encapsulated in a stable protective jacket (Koo et al, 1981). This is accomplished by depositing layers of epoxy, RTV, or gel glass around the etched (polished) parts of the coupler. Using these techniques, temperature stability can be greatly enhanced. Coupling efficiencies typically can be made constant (to within 1%) over tens of degrees of temperature excursions, which is required for sensor applications. Recently, fused fiber evanescent couplers have been fabricated using single-mode fibers. In this case, the two fibers to be coupled are twisted, heated to their softening point, and elongated. This action effectively brings the cores closer together enabling evanescent coupling to take place. When cooled, fused couplers with good stability and low loss result. These all-glass fiber couplers have characteristics equal in quality to the characteristics of the two previously mentioned coupler types, and possess additionally the inherent advantage of all-glass construction. Typically, 3 dB power division with 0.5 dB insertion loss can be realized with these couplers. 9.3

FIBER SENSOR DESIGN

Many of the sensors fabricated and tested in laboratory environments have to be substantially modified for use in uncontrolled environments. In the cases of the moving mirror, diode laser, fiber-to-fiber transmission, and Fresnel reflection sensors, drifts caused by ambient conditions can cause serious degradation in performance. For maximum sensitivity, these sensors usually require the transduction elements be accurately aligned. This alignment must be maintained even as the pressure and temperature drift. Mechanical designs possessing the desired characteristics are evolving; however, to date only preliminary verifications of these techniques exist. Feedback techniques can obviously be used to maintain alignment. However, the attractive feature of these sensors is their passive nature, which is destroyed by the addition of feedback. The single-fiber, birefringent fiber, and microbend sensors on the other hand do not necessarily lose sensitivity as the ambient conditions change. In this case, the sensor elements (fibers) may be presented to the environment without substantial modification or feedback stabilization, thus maintaining their simplicity of design.

454

/. A. Bucaro et al.

Interferometric sensors to date require active feedback in order to maintain sensitivity in the presence of pressure or temperature drifts. Initially these interferometers were fabricated using bulk components. These bulk interferometers were not suitable for field use and remained a laboratory tool. However, the use of diode lasers and fiber optical beam splitters made practical packaging of these interferometers possible. In Fig. 50, a packaged interferometric fiber acoustic sensor is shown. The sensor coil is mounted on a massive pressure cell penetrator to permit the sensing coil to be placed in environmental test chambers. The sensing coil can be routinely cycled under high pressure and temperature in this configuration. To the right of the pressure cell penetrator body is the reference arm of the interferometer and associated electronics. This rugged sensor construction has been successfully tested in environments simulating the most stringent conditions encountered in practice. Recently, interest has developed in existing techniques which would permit the interferometric sensors to be operated without feedback. These passive techniques appear capable of eliminating signal fading and do not require servodriven piezoelectrically stretched fiber coils. The principle em-

FIG. 50. Optical fiber acoustic interferometeric sensor packaged for environmental testing.

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ployed relies on the fact that if the interferometer output is composed of two terms which are proportional to sin £ and cos £, respectively, where ξ is the deferential optical phase shift induced by the signal, then complete signal fading does not occur even though the magnitude of the signal may vary with time. The first embodiment of these passive techniques utilized two wavelengths to obtain the sin ξ and cos ξ terms. Alternative approaches utilized either two polarizations or the properties of the ( 3 X 3 ) directional coupler (Sheem, 1981) as the second interferometer beam splitter to obtain sine and cosine outputs. If passive homodyne techniques prove to be successful, then the con­ struction of the interferometer sensor will be greatly simplified. In any case, it is now becoming possible to package interferometric as well as amplitudemode fiber sensors in practical configurations and to achieve performances competitive with conventional sensors using these packaged devices. REFERENCES

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