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Agile interferometry: a non-traditional approach
Optics Communications 241 (2004) 419–428 www.elsevier.com/locate/optcom
Agile interferometry: a non-traditional approach Nabeel A. Riza, Zahid Yaqoob
*
Photonic Information Processing Systems Laboratory (PIPS), The School of Optics/Center for Research and Education in Optics and Lasers (CREOL), University of Central Florida, 4000 Central Florida Buliding, Orlando, FL 32816-2700, USA Received 17 February 2004; received in revised form 19 July 2004; accepted 20 July 2004
Abstract A new approach called agile interferometry is introduced to attain interferometric information with high sensitivity and scenario-based intelligence. Compared to traditional interferometric techniques, the proposed method thrives on dynamic control of the reference signal strength and detector integration time for efficient interferometric detection with high signal-to-noise ratio and significantly improved detected signal dynamic range capabilities. Theoretical analysis is presented with the operational methodology of the new approach. A high-speed optical attenuator is required in the interferometer reference arm to implement the proposed agile interferometer. 2004 Elsevier B.V. All rights reserved. PACS: 07.60.L; 42.79.Y; 42.81.Q Keywords: Interferometers; Optical attenuators; Fiber-optic
1. Introduction Optical heterodyne [1] and phase-shifting [2] interferometry techniques provide highly accurate non-invasive optical measurements [3–5]. In recent years, the availability of high-quality low-loss optical fibers that allow miniaturization has made it possible to realize fiber-optic interferometric sen-
* Corresponding author. Tel.: +1 407 823 6829/6973; fax: +1 407 823 6880/3347. E-mail addresses: [email protected] (N.A. Riza), [email protected] (Z. Yaqoob).
sors [6] for non-destructive testing such as laser vibrometry [7] and minimally invasive clinical procedures such as laser Doppler velocimetry [8]. In interferometric or direct detection optical systems using low-noise lasers, the noise in the optical detection process is generated in the photodetector. There are several sources of this noise, but the most important ones are Johnson noise (i.e., thermal noise) and shot noise. Since photodetectors are only sensitive to the energy (or photon flux), the phase information in the optical signal is gathered by mixing the optical signal with a coherent reference signal with known stable phase. One approach is to mix 2-weak Doppler shifted
0030-4018/$ - see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2004.07.043
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information signals and getting a Doppler difference signal [9,10]. This technique is also known as balanced mixing and is generally recognized as effective means for noise suppression. Amplitude and angle modulation schemes have also been proposed for signal-to-noise improvements but they are specific to dual beam velocimeters only [11]. In another method which is more traditional, the strength of the reference signal is kept very high so that after mixing it with even a weak information signal, the resulting interferometric information signal gets heterodyne gain [8] and stays above the noise floor. Because photodetectors are sensitive to the photon flux, a very strong reference optical signal means lots of photons (without any information) reaching the photodetector. Note that for a weaker information carrying optical signal (which is usually the case), the photons from the much stronger reference optical signal quickly eat up most of the dynamic range of the photodetector leaving very little dynamic range for the useful interferometric information signal. This can be clearly understood by the analogy of a bucket and its volume representing the photodetector and its available dynamic range, respectively. Let blue and red balls represent the information and reference optical signal photons, respectively. The time required to fill the whole bucket can be regarded as the optimum integration time of the photodetector to maximize the full dynamic range of the detection process. A strong reference optical signal means lots of red balls and fewer blue balls filling the bucket (with a finite volume) within the given detector integration time. It is clear that the bucket will be filled up quickly and will have plenty of red balls but only a few blue balls. It would seem appropriate to reduce the rate at which red balls are entering the bucket so as to let the bucket fill up with relatively more blue balls (that correspond to the information carrying photons). In other words, it would be appropriate to reduce the reference signal strength and increase the photodetectorÕs integration time in order to efficiently utilize the detectorÕs available dynamic range for collecting signal photons. The forthcoming analysis is performed to favorably show how agile control of the reference signal strength and integration time (or bandwidth) of a photodetector positively
affect the signal-to-noise ratio (SNR) and dynamic range of the detected interferometric electrical signal.
2. Principles of agile interferometry Fig. 1 shows the schematics of classic Michelson and Mach–Zehnder interferometers for reflective and transmissive topologies, respectively. In the Michelson interferometer, shown in Fig. 1(a), light from a laser source with intensity I0 is split into the reference and signal arms of the interferometer via a beam splitter (BS) with optical power split factor g, where 0 < g < 1 After the BS, the intensity of light in the reference and signal arms can be given by I0g and I0(1 g), respectively. Let a and b be the reflection coefficients of the reference and signal beams, respectively. Thus, the light intensity of the reference and signal beams at the output
Fig. 1. Schematics of classic interferometric setups such as: (a) the Michelson and (b) the Mach–Zehnder Interferometers. BS: beam splitter; PD: photodetector.
N.A. Riza, Z. Yaqoob / Optics Communications 241 (2004) 419–428
of the Michelson interferometer will become I0g(1 g)a and I0g(1 g)b, respectively. In the case of Mach–Zehnder interferometer, shown in Fig. 1(b), light from a laser source with intensity I0 is split into the reference and signal arms of the interferometer via the first beam splitter (marked as BS1) with optical power split factor g1, where 0 < g1 < 1. After the BS1, the intensity of light in the reference and signal arms of the Mach–Zehnder interferometer can be given as I0g1 and I0(1 g1), respectively. Let a and b be the transmission coefficients of the reference and signal beams, respectively, indicating that the intensity of light in the reference and signal arms at the second beam splitter (marked as BS2), which works as a beam combiner, can be given by I0g1a and I0(1 g1)b, respectively. After BS2 with optical power split factor g2, where 0 < g2 < 1 the light intensity of the reference and signal beams at the output of the Mach–Zehnder interferometer will become I0g1(1 g2)a and I0g2(1 g1)b, respectively. As a practical case, g1 can be made equal to g2, indicating that the light intensity in the reference and signal beams at the output of Mach– Zehnder interferometer can be given by I0g(1 g)a and I0g(1 g)b, respectively, which are the same as that for the Michelson interferometer. This indicates that the forthcoming analysis is valid for either of the interferometric setups shown in Fig. 1. Do note that for the Mach–Zehnder case, choosing different beam splitters can lead to optimization of output SNR, although at the expense of dynamic range. Note that a frequency shifter/ modulator may also be introduced in the reference or signal arms of the interferometric setups (shown in Fig. 1) to generate an intermediate frequency (IF) at the photodetector (PD). Therefore, in general, the reference and signal optical fields, denoted by ER and ES, respectively, at the photodetector can be written as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ER ¼ I 0 gð1 gÞa expfið2pfR t þ uR Þg; ð1Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ES ¼ I 0 gð1 gÞb expfið2pfS t þ uS Þg: Here fR and fS are the optical frequencies of the reference and signal beams, respectively. In addition, /R and /S are the optical phase values of the reference and signal beams, respectively.
421
Therefore, the total intensity at the photodetector will become 2
I d ðtÞ ¼ jES þ ER j ; 2
2
¼ jES j þ jER j þ 2RefES ER g; pffiffiffiffiffiffi ¼ I 0 gð1 gÞfa þ b þ 2 ab cosð2pfRF t þ uÞg: ð2Þ In Eq. (2), fRF is the IF generated at the photodetector such that fRF = fR fS and / = /R /S is the phase of the interferometric electrical signal. The optical power at the photodetector is given by P d ðtÞ ¼ I d ðtÞAd ;
ð3Þ
where Ad is the area of the photodetector under illumination. From Eqs. (2) and (3), we can write pffiffiffiffiffiffiffiffiffiffiffi P d ðtÞ ¼ P S þ P R þ 2 P S P R cosð2pfRF t þ uÞ; ð4Þ where P S ¼ P 0 gð1 gÞb; P R ¼ P 0 gð1 gÞa; P 0 ¼ I 0 Ad :
ð5Þ
The electrical current signal i(t) generated by the photodetector will be directly proportional to Pd(t) such that iðtÞ / P d ðtÞ ¼ S P d ðtÞ;
ð6Þ
where S is the spectral sensitivity of the photodetector and is given by [12] S¼
g0 e ; hm
ð7Þ
such that g 0 is the quantum efficiency, e is the electronic charge, and m is the photon frequency. Using Eqs. (4) and (6) simultaneously gives pffiffiffiffiffiffiffiffiffiffiffi iðtÞ ¼ SðP S þ P R Þ þ 2S P S P R cosð2pfRF t þ uÞ; ¼ iDC þ iAC cosð2pfRF t þ uÞ; ¼ iDC þ iac ðtÞ:
ð8Þ
In Eq. (8) iDC ¼ SðP S þ P R Þ ¼ SP 0 gð1 gÞða þ bÞ; iAC ¼ 2S
pffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi P S P R ¼ 2SP 0 gð1 gÞ ab:
ð9aÞ ð9bÞ
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3. SNR analysis interferometry
for
classic
versus
agile
If PN is the total detection noise power such that PN = Pshot + Ptherm, where Pshot and Ptherm denote shot noise and thermal noise powers, respectively, the SNR can be written as SNR ¼
P AC P AC ¼ ; PN P shot þ P therm
ð10Þ
where PN can also be written as P N ¼ i2N RL such that RL is the load resistor, iN is the noise current, and total signal electrical power PAC is given by P AC ¼ ¼
hi2ac ðtÞiRL ; i2AC hcosð2pfRF t
þ uÞiRL ;
1 ¼ i2AC RL : 2 Combining Eqs. (9b) and (11) gives 2
P AC ¼ 2S 2 P 20 g2 ð1 gÞ abRL :
ð11Þ
ð12Þ
P shot ¼ hi2shot iRL ¼ 2eB½iDC þ id RL ; ð13Þ
and P therm ¼ 4kTB;
ð14Þ
where k is the Boltzmann constant, T is the temperature in (K), and B is the bandwidth of the photodetector. In Eq. (13), id is the dark current. For the photocurrent iDC to dominate the dark current, Eq. (13) gives the following condition: ða þ bÞ >
id : SP 0 gð1 gÞ
ð15Þ
Therefore, in the limit b ! 0, i.e., infinitesimally small information carrying signal, Eq. (15) reduces to id : SP 0 gð1 gÞ
P min shot ¼ 2eB½SP 0 gð1 gÞða þ bÞRL :
ð17Þ
To make sure of being limited by shot noise only, the reference signal power has to be larger than the minimum value of reference power, i.e., PR > PR,min. With reference to Eq. (10), this lower limit for reference signal power can be determined by using the following condition: P min shot ¼ P therm :
ð18Þ
Substituting Eqs. (14) and (17) in Eq. (18) and rearranging terms, we get ða þ bÞ ¼
Shot noise and thermal noise power can be written as [12]
¼ 2eB½SP 0 gð1 gÞða þ bÞ þ id RL ;
noise. When the condition given by expression (16) is satisfied, Eq. (13) simplifies to
2kT : eSP 0 gð1 gÞRL
ð19Þ
In the limit b ! 0, i.e., an infinitesimally small information signal, Eq. (19) reduces to amin;therm ¼
2kT : eSP 0 gð1 gÞRL
ð20Þ
The corresponding value of reference signal power can therefore be written as P R;min ¼ P 0 gð1 gÞamin;therm ¼
2kT : eSRL
ð21Þ
In conclusion, for PR > PR,min [given by Eq. (21)], the interferometric information signal at the output dominates the thermal noise and the SNR is limited only by the shot noise. However, in a general case, the SNR (dB) [under condition given by expression (16)] can be written as P AC SNRðdBÞ ¼ 10log10 min : ð22Þ P shot þ P therm Substituting Eqs. (12), (14) and (17) in Eq. (22) gives SNRðdBÞ ¼ 10log10
"
# 2 2S 2 P 20 g2 ð1 gÞ abRL : 2eBfSP 0 gð1 gÞða þ bÞgRL þ 4kTB
ð16Þ
ð23Þ
Expression (16) gives the minimum value of a so as to generate an interferometric signal (at the output) strong enough to dominate the dark current
For a b (which is usually the case with classic indicating ab/ interferometry), P min shot P therm (a + b) @ b. Under these conditions, Eq. (23) simplifies to
amin;dark >
N.A. Riza, Z. Yaqoob / Optics Communications 241 (2004) 419–428
SNRðdBÞ ¼ 10log10
g0 P 0 gð1 gÞb : hmB
ð24Þ
It is evident from Eq. (23) that for a b, the SNR is proportional to b and is maximized for g = 0.5 (since g(1g) attains its maximum value of 0.25). On the other hand, for a = b (i.e., reference signal power equal to the information signal power), the SNR given in Eq. (23) will become " # S 2 P 20 g2 ð1 gÞ2 b2 RL SNRðdBÞ ¼ 10log10 ; 2B½efSP 0 gð1 gÞbgRL þ kT
423
approach (a b). Fig. 2 also indicates that the SNR (for both the classic as well as the agile approach) increases with a decrease in photodetector bandwidth B (or an increase in photodetector integration time). The difference in SNR, DSNR, available from the two approaches is however not a function of photodetector bandwidth B and is shown versus b in Fig. 3. The plot shows that the difference in SNR for the two approaches is not significant and it decreases with increasing b Specifically, the difference in SNR decreases from 4.5 dB and tends to reach 0 dB as b approaches 1.
ð25Þ where S is given by Eq. (7). However, if the condition P min shot P therm is still valid, Eq. (25) simplifies to 0 g b P 0 gð1 gÞ : SNRðdBÞ ¼ 10log10 ð26Þ 2 hmB This indicates that as a is reduced (such that a is not b), the SNR drops. Specifically, under the condition P min shot P therm , the SNR drops by a factor of [a/(a + b)] and at a = b, it drops by 3 dB [see Eq. (26)]. Nonetheless, if the output signal is not limited by shot noise only (but through the thermal noise as well), the SNR (dB) for the a = b case will be given by Eq. (25). Subtracting Eq. (25) from Eq. (24), we get the difference in SNR available from the two photo-detection approaches giving 2½efSP 0 gð1 gÞbgRL þ kT DSNR ¼ 10log10 : eSP 0 gð1 gÞbRL ð27Þ Eq. (27) indicates that the defined DSNR is not a function of photodetector bandwidth B (or the integration time). A comparison of SNR available from the classic approach (a b) as well as the agile approach (a = b) for interferometric detection is provided in Fig. 2. Specifically, the SNR versus b for both the classic and the agile approaches, given by Eqs. (24) and (25), respectively, is shown for two different values of photodetector bandwidth B. It is clear from Figs. 2(a) and (b) that regardless of the photodetector bandwidth, the SNR available from the agile approach is slightly less than that is obtainable using the classic
4. Dynamic range comparison of classic versus agile interferometry To determine the dynamic range available from the classic and the agile interferometry approach, Eq. (4) is rewritten as P d ðtÞ ¼ P 0 gð1 gÞða þ bÞ pffiffiffiffiffiffi 2 ab 1þ cosð2pfRF t þ uÞ : aþb
ð28Þ
From Eq. (27), the visibility parameter V can be written as pffiffiffiffiffiffi 2 ab V ¼ : ð29Þ aþb At a b,a + b @ a. Therefore, Eq. (29) simplifies to rffiffiffi b : ð30Þ V ¼2 a Eq. (28) thus becomes "
# rffiffiffi b P d ðtÞ ¼ P 0 gð1 gÞa 1 þ 2 cosð2pfRF t þ uÞ ; a
which indicates that the electrical current signal at the output becomes " # rffiffiffi b iðtÞ ¼ A 1 þ 2 cosð2pfRF t þ uÞ ; ð31Þ a where A ¼ SP 0 gð1 gÞa:
ð32Þ
424
N.A. Riza, Z. Yaqoob / Optics Communications 241 (2004) 419–428 100 98
Single to Noise Ratio (dB)
96 94 92
Classic Method ( α >> β ) Agile Method ( α = β )
90
B = 10 MHz
88 86 84 82
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
β
(a) 152 150
Single to Noise Ratio (dB)
148 146 144
Classic Method ( α >> β ) Agile Method ( α = β )
142 140
B = 60 Hz
138 136 134
0
0.1
0.2
0.3
0.4
(b)
0.5
0.6
0.7
0.8
0.9
1
β
Fig. 2. Signal-to-noise ratio (SNR) in dB versus b for photodetector bandwidth of (a) B = 10 MHz and (b) B = 60 Hz. P0 = 100 mW, T = 300 K, RL = 50 X, k = 1550 nm, g = 0.5, g 0 = 0.7, a = 1.
Hence, the dynamic range (dB) of the output current signal, defined as the ratio of maximum to minimum current signal amplitudes, will be given by pffiffiffiffiffiffiffiffi! 1 þ 2 b=a pffiffiffiffiffiffiffiffi Dab ¼ 10log10 ð33Þ 1 2 b=a
comes equal to b, the visibility parameter V given by Eq. (29) simplifies to
that will reduce as b drops (since a is constant) till it approaches zero. On the other hand, as a be-
which indicates that the electrical current signal at the output becomes
V ¼ 1:
ð34Þ
Therefore, Eq. (28) becomes P d ðtÞ ¼ 2P 0 gð1 gÞb½1 þ cosð2pfRF t þ uÞ;
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425
4.5 4 For classic case, α = 1 For agile case, ( α = β ); however, it remains > α
3.5 Difference in SNR (dB)
min,therm
3 2.5 2 1.5
α min,therm
1 0.5 0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
β
Fig. 3. Difference in SNR, DSNR, for the traditional and proposed agile approaches to interferometric detection. P0 = 100 mW, T = 300 K, RL = 50 X, k = 1550 nm, g = 0.5, g 0 = 0.7, a = 1.
iðtÞ ¼ C½1 þ cosð2pfRF t þ uÞ;
ð35Þ
where C ¼ 2SP 0 gð1 gÞb:
ð36Þ
Thus in the proposed agile case of a = b, the dynamic range D (in dB) of the output current signal is given by 0 1 B 2C C Da¼b ¼ 10log10 @qffiffiffiffiffiffiffiffiA hi2N i ¼ 10log10
! 2C pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; P N =RL
ð37Þ
where under condition (16), P N ¼ P min shot þ P therm . As a result, Eq. (37) becomes ! C Da¼b ¼ 10log10 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : BfeSP 0 gð1 gÞb þ kT =RL g ð38Þ P min shot
P therm , Eq.
! C pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : eBSP 0 gð1 gÞb
ð39Þ
Note that under the condition (38) simplifies to Da¼b ¼ 10log10
A comparison of dynamic range available from the traditional approach (a b) as well as the new agile approach (a = b) for interferometric detection is provided in Fig. 4. In particular, plots for the dynamic range versus b for both the classic and the new approaches, given by Eqs. (33) and (38), respectively, are shown for two different values of photodetector bandwidth B. It is clear from Figs. 4(a) and (b) that regardless of the photodetector bandwidth, the dynamic range available from the agile interferometry approach is significantly more than that obtainable using the traditional approach. Moreover, the dynamic range response for the agile approach remains relatively flat over a wide range of b. The difference in available dynamic range (in dB) from the two approaches goes to zero at b = 1. However, as the value of b drops, the difference in available dynamic range from the two approaches starts increasing drastically, thus indicating that the newly proposed agile method is superior to the classical approach for detection of an interferometric information signal. A comparison of Figs. 4(a) and (b) also shows that the dynamic range increases further by increasing the photodetector integration time.
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N.A. Riza, Z. Yaqoob / Optics Communications 241 (2004) 419–428 60
Dynamic Range (dB)
50
40 Classic Method ( α >>β ) Agile Method ( α = β ) 30
10MHz MHz BB=B=10 =10 Hz 20
10
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.6
0.7
0.8
0.9
1
β
(a) 80 70
Dynamic Range (dB)
60
Classic Method (α >>β) C A Agile Method (α =β)
50
B = 60 Hz
40 30 20 10 0
0
0.1
0.2
0.3
0.4
(b)
0.5
β
Fig. 4. Output interferometric signal dynamic range versus b for photodetector bandwidth of (a) B = 10 MHz and (b) B = 60 Hz. P0 = 100 mW, T = 300 K, RL = 50 X, k = 1550 nm, g = 0.5, g 0 = 0.7, a = 1.
5. Agile interferometry implementation issues The earlier sections have theoretically shown that agile interferometry (i.e., when a = b) compared to the classical approach (a b) for interferometric information signal detection can provide a comparable SNR with a much improved detection dynamic range. However, when the sig-
nal strength, i.e., b is unknown, which is usually the case, one does not know where to set the value of a so that the a = b condition is satisfied for the best dynamic range possible via agile interferometry. This means that one needs to sweep a across the entire band and get many readings for the detected interferometric signal dynamic range, one of which is the best. For shot noise limited detection,
N.A. Riza, Z. Yaqoob / Optics Communications 241 (2004) 419–428
427
Fig. 5. Hardware implementation scenarios for agile interferometry. For test objects with non-uniform reflectivities, shown are pointby-point scan (a) free-space and (b) fiber-optic interferometers with full test data collected via test object x–y–z motion. For the special case where reflectivity over object is constant, (c) shows a free-space agile interferometer using a two-dimensional (2D) sensor device such as a charge coupled device (CCD).
the range over which a can be swept will be [amin,therm, 1], where amin,therm is given by Eq. (20). Fig. 5 shows schematics of free-space and fiber-optic versions of agile interferometry setups that deploy high-speed variable optical attenuators (VOAs) for agile control of reference signal strength. Options for ultrafast reference beam power control modules include microsecond speed VOAs such as the ones using acousto-optics [13,14]. It is also clear from Figs. 5(a) and (b) where b is non-uniform on the test object, agile interferometry works on a point-by-point information scan basis, where for each information point in the signal sensing zone, the reference beam
power is swept to match the given signal scan point location power. Therefore, agile interferometry requires high-speed optical scanners [15], VOAs, point optical detectors, and electronic processing, all of which are a reality today. In the special Fig. 5(c) case when the object reflectivity is uniform across the object, the object can be illuminated by a plane wave (instead of a point) and the point detector can be replaced by a two-dimensional photodetector array such as a CCD. In addition, test object x–y motion is not required; only the object can be translated in z (along optical beam) to acquire three-dimensional object information coded in the phase of the detected signal.
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6. Conclusion
References
In conclusion, a new approach called agile interferometry is proposed for attaining interferometric information with high sensitivity. Specifically, compared to classical interferometry with a strong fixed reference optical beam, the proposed method utilizes dynamic control of the reference beam strength and the detector integration time for efficient interferometric detection leading to a high SNR and significantly improved detection signal dynamic range. A theoretical analysis of the agile approach indicates that equalizing the reference signal strength with the information signal strength drops the detected SNR by <4.5 dB. However, agile interferometry compared to fixed reference interferometry leads to a significant advantage in the final output detection dynamic range. In addition, an appropriate increase in the photodetector integration time also improves both SNR as well as output signal dynamic range. In effect, agile interferometry proposes appropriate reduction in the reference signal strength and increase in the photodetectorÕs integration time to utilize the photodetectorÕs available dynamic range. From a practical implementation point of view, agile interferometry requires the use of a high-speed variable optical attenuator to sweep the reference beam power strength through a series of settings to obtain a set of detected interferometric data. This stored data is then compared to extract the detected signal with the highest dynamic range. Thus, agile interferometry relies on the storage of many signals that are then processed by a computer to deliver the optimized output data.
[1] G.E. Sommergren, Optical heterodyne interferometry, Appl. Opt. 20 (1981) 610. [2] K. Creath, Progress in Optics, Elsevier, New York, 1988, p. 3874. [3] P. Hariharan, Interferometers, in: M. Bass (Ed.), Handbook of Optics, vol. II, McGraw-Hill, New York, 1995 (Chapter 21). [4] W.H. Steel, Interferometry, Cambridge Press, Cambridge, MA, 1967. [5] M. Francon, Optical Interferometry, Academic Press, New York, 1966. [6] E. Udd, Fiber Optic Sensors: An Introduction for Engineers and Scientist, Wiley, New York, 1991. [7] T. Miles, M. Lucas, S. Rothberg, Bending vibration measurement on rotors by laser vibrometry, Opt. Lett. 21 (4) (1996) 296. [8] N. Nishihara, K. Matsumoto, J. Koyama, Use of a laser diode and an optical fiber for a compact laser-Doppler velocimeter, Opt. Lett. 9 (2) (1984) 62. [9] G.L. Abbas, V.W.S. Chan, T.K. Yee, A dual-detector optical heterodyne receiver for local oscillator noise suppression, J. Lightwave Technol. LT-3 (5) (1985). [10] R. Ba¨ttig, R. Stierlin, P.-D. Henchoz, H.P. Weber, New monostatic balanced Doppler velocimeter using a high birefringence fiber, J. Lightwave Technol. 6 (1) (1988). [11] P. Le-Cong, R.H. Lovberg, Signal-to-noise improvement in laser Doppler velocimetry, Appl. Opt. 19 (24) (1980). [12] E.L. Dereniak, G.D. Boreman, Infrared Detectors and Systems, Wiley, New York, 1996. [13] N.A. Riza, Z. Yaqoob, Sub-microsecond speed variable optical attenuator using acousto-optics, IEEE Photon. Technol. Lett. 13 (7) (2001) 693. [14] M.J. Mughal, N.A. Riza, Compact acousto-optic highspeed variable attenuator for high-power applications, IEEE Photon. Technol. Lett. 14 (4) (2002) 510. [15] N.A. Riza, S.A. Khan, Programmable high-speed polarization multiplexed optical scanner, Opt. Lett. 28 (7) (2003) 561.
Agile interferometry: a non-traditional approach Nabeel A. Riza, Zahid Yaqoob
*
Photonic Information Processing Systems Laboratory (PIPS), The School of Optics/Center for Research and Education in Optics and Lasers (CREOL), University of Central Florida, 4000 Central Florida Buliding, Orlando, FL 32816-2700, USA Received 17 February 2004; received in revised form 19 July 2004; accepted 20 July 2004
Abstract A new approach called agile interferometry is introduced to attain interferometric information with high sensitivity and scenario-based intelligence. Compared to traditional interferometric techniques, the proposed method thrives on dynamic control of the reference signal strength and detector integration time for efficient interferometric detection with high signal-to-noise ratio and significantly improved detected signal dynamic range capabilities. Theoretical analysis is presented with the operational methodology of the new approach. A high-speed optical attenuator is required in the interferometer reference arm to implement the proposed agile interferometer. 2004 Elsevier B.V. All rights reserved. PACS: 07.60.L; 42.79.Y; 42.81.Q Keywords: Interferometers; Optical attenuators; Fiber-optic
1. Introduction Optical heterodyne [1] and phase-shifting [2] interferometry techniques provide highly accurate non-invasive optical measurements [3–5]. In recent years, the availability of high-quality low-loss optical fibers that allow miniaturization has made it possible to realize fiber-optic interferometric sen-
* Corresponding author. Tel.: +1 407 823 6829/6973; fax: +1 407 823 6880/3347. E-mail addresses: [email protected] (N.A. Riza), [email protected] (Z. Yaqoob).
sors [6] for non-destructive testing such as laser vibrometry [7] and minimally invasive clinical procedures such as laser Doppler velocimetry [8]. In interferometric or direct detection optical systems using low-noise lasers, the noise in the optical detection process is generated in the photodetector. There are several sources of this noise, but the most important ones are Johnson noise (i.e., thermal noise) and shot noise. Since photodetectors are only sensitive to the energy (or photon flux), the phase information in the optical signal is gathered by mixing the optical signal with a coherent reference signal with known stable phase. One approach is to mix 2-weak Doppler shifted
0030-4018/$ - see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2004.07.043
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information signals and getting a Doppler difference signal [9,10]. This technique is also known as balanced mixing and is generally recognized as effective means for noise suppression. Amplitude and angle modulation schemes have also been proposed for signal-to-noise improvements but they are specific to dual beam velocimeters only [11]. In another method which is more traditional, the strength of the reference signal is kept very high so that after mixing it with even a weak information signal, the resulting interferometric information signal gets heterodyne gain [8] and stays above the noise floor. Because photodetectors are sensitive to the photon flux, a very strong reference optical signal means lots of photons (without any information) reaching the photodetector. Note that for a weaker information carrying optical signal (which is usually the case), the photons from the much stronger reference optical signal quickly eat up most of the dynamic range of the photodetector leaving very little dynamic range for the useful interferometric information signal. This can be clearly understood by the analogy of a bucket and its volume representing the photodetector and its available dynamic range, respectively. Let blue and red balls represent the information and reference optical signal photons, respectively. The time required to fill the whole bucket can be regarded as the optimum integration time of the photodetector to maximize the full dynamic range of the detection process. A strong reference optical signal means lots of red balls and fewer blue balls filling the bucket (with a finite volume) within the given detector integration time. It is clear that the bucket will be filled up quickly and will have plenty of red balls but only a few blue balls. It would seem appropriate to reduce the rate at which red balls are entering the bucket so as to let the bucket fill up with relatively more blue balls (that correspond to the information carrying photons). In other words, it would be appropriate to reduce the reference signal strength and increase the photodetectorÕs integration time in order to efficiently utilize the detectorÕs available dynamic range for collecting signal photons. The forthcoming analysis is performed to favorably show how agile control of the reference signal strength and integration time (or bandwidth) of a photodetector positively
affect the signal-to-noise ratio (SNR) and dynamic range of the detected interferometric electrical signal.
2. Principles of agile interferometry Fig. 1 shows the schematics of classic Michelson and Mach–Zehnder interferometers for reflective and transmissive topologies, respectively. In the Michelson interferometer, shown in Fig. 1(a), light from a laser source with intensity I0 is split into the reference and signal arms of the interferometer via a beam splitter (BS) with optical power split factor g, where 0 < g < 1 After the BS, the intensity of light in the reference and signal arms can be given by I0g and I0(1 g), respectively. Let a and b be the reflection coefficients of the reference and signal beams, respectively. Thus, the light intensity of the reference and signal beams at the output
Fig. 1. Schematics of classic interferometric setups such as: (a) the Michelson and (b) the Mach–Zehnder Interferometers. BS: beam splitter; PD: photodetector.
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of the Michelson interferometer will become I0g(1 g)a and I0g(1 g)b, respectively. In the case of Mach–Zehnder interferometer, shown in Fig. 1(b), light from a laser source with intensity I0 is split into the reference and signal arms of the interferometer via the first beam splitter (marked as BS1) with optical power split factor g1, where 0 < g1 < 1. After the BS1, the intensity of light in the reference and signal arms of the Mach–Zehnder interferometer can be given as I0g1 and I0(1 g1), respectively. Let a and b be the transmission coefficients of the reference and signal beams, respectively, indicating that the intensity of light in the reference and signal arms at the second beam splitter (marked as BS2), which works as a beam combiner, can be given by I0g1a and I0(1 g1)b, respectively. After BS2 with optical power split factor g2, where 0 < g2 < 1 the light intensity of the reference and signal beams at the output of the Mach–Zehnder interferometer will become I0g1(1 g2)a and I0g2(1 g1)b, respectively. As a practical case, g1 can be made equal to g2, indicating that the light intensity in the reference and signal beams at the output of Mach– Zehnder interferometer can be given by I0g(1 g)a and I0g(1 g)b, respectively, which are the same as that for the Michelson interferometer. This indicates that the forthcoming analysis is valid for either of the interferometric setups shown in Fig. 1. Do note that for the Mach–Zehnder case, choosing different beam splitters can lead to optimization of output SNR, although at the expense of dynamic range. Note that a frequency shifter/ modulator may also be introduced in the reference or signal arms of the interferometric setups (shown in Fig. 1) to generate an intermediate frequency (IF) at the photodetector (PD). Therefore, in general, the reference and signal optical fields, denoted by ER and ES, respectively, at the photodetector can be written as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ER ¼ I 0 gð1 gÞa expfið2pfR t þ uR Þg; ð1Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ES ¼ I 0 gð1 gÞb expfið2pfS t þ uS Þg: Here fR and fS are the optical frequencies of the reference and signal beams, respectively. In addition, /R and /S are the optical phase values of the reference and signal beams, respectively.
421
Therefore, the total intensity at the photodetector will become 2
I d ðtÞ ¼ jES þ ER j ; 2
2
¼ jES j þ jER j þ 2RefES ER g; pffiffiffiffiffiffi ¼ I 0 gð1 gÞfa þ b þ 2 ab cosð2pfRF t þ uÞg: ð2Þ In Eq. (2), fRF is the IF generated at the photodetector such that fRF = fR fS and / = /R /S is the phase of the interferometric electrical signal. The optical power at the photodetector is given by P d ðtÞ ¼ I d ðtÞAd ;
ð3Þ
where Ad is the area of the photodetector under illumination. From Eqs. (2) and (3), we can write pffiffiffiffiffiffiffiffiffiffiffi P d ðtÞ ¼ P S þ P R þ 2 P S P R cosð2pfRF t þ uÞ; ð4Þ where P S ¼ P 0 gð1 gÞb; P R ¼ P 0 gð1 gÞa; P 0 ¼ I 0 Ad :
ð5Þ
The electrical current signal i(t) generated by the photodetector will be directly proportional to Pd(t) such that iðtÞ / P d ðtÞ ¼ S P d ðtÞ;
ð6Þ
where S is the spectral sensitivity of the photodetector and is given by [12] S¼
g0 e ; hm
ð7Þ
such that g 0 is the quantum efficiency, e is the electronic charge, and m is the photon frequency. Using Eqs. (4) and (6) simultaneously gives pffiffiffiffiffiffiffiffiffiffiffi iðtÞ ¼ SðP S þ P R Þ þ 2S P S P R cosð2pfRF t þ uÞ; ¼ iDC þ iAC cosð2pfRF t þ uÞ; ¼ iDC þ iac ðtÞ:
ð8Þ
In Eq. (8) iDC ¼ SðP S þ P R Þ ¼ SP 0 gð1 gÞða þ bÞ; iAC ¼ 2S
pffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi P S P R ¼ 2SP 0 gð1 gÞ ab:
ð9aÞ ð9bÞ
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3. SNR analysis interferometry
for
classic
versus
agile
If PN is the total detection noise power such that PN = Pshot + Ptherm, where Pshot and Ptherm denote shot noise and thermal noise powers, respectively, the SNR can be written as SNR ¼
P AC P AC ¼ ; PN P shot þ P therm
ð10Þ
where PN can also be written as P N ¼ i2N RL such that RL is the load resistor, iN is the noise current, and total signal electrical power PAC is given by P AC ¼ ¼
hi2ac ðtÞiRL ; i2AC hcosð2pfRF t
þ uÞiRL ;
1 ¼ i2AC RL : 2 Combining Eqs. (9b) and (11) gives 2
P AC ¼ 2S 2 P 20 g2 ð1 gÞ abRL :
ð11Þ
ð12Þ
P shot ¼ hi2shot iRL ¼ 2eB½iDC þ id RL ; ð13Þ
and P therm ¼ 4kTB;
ð14Þ
where k is the Boltzmann constant, T is the temperature in (K), and B is the bandwidth of the photodetector. In Eq. (13), id is the dark current. For the photocurrent iDC to dominate the dark current, Eq. (13) gives the following condition: ða þ bÞ >
id : SP 0 gð1 gÞ
ð15Þ
Therefore, in the limit b ! 0, i.e., infinitesimally small information carrying signal, Eq. (15) reduces to id : SP 0 gð1 gÞ
P min shot ¼ 2eB½SP 0 gð1 gÞða þ bÞRL :
ð17Þ
To make sure of being limited by shot noise only, the reference signal power has to be larger than the minimum value of reference power, i.e., PR > PR,min. With reference to Eq. (10), this lower limit for reference signal power can be determined by using the following condition: P min shot ¼ P therm :
ð18Þ
Substituting Eqs. (14) and (17) in Eq. (18) and rearranging terms, we get ða þ bÞ ¼
Shot noise and thermal noise power can be written as [12]
¼ 2eB½SP 0 gð1 gÞða þ bÞ þ id RL ;
noise. When the condition given by expression (16) is satisfied, Eq. (13) simplifies to
2kT : eSP 0 gð1 gÞRL
ð19Þ
In the limit b ! 0, i.e., an infinitesimally small information signal, Eq. (19) reduces to amin;therm ¼
2kT : eSP 0 gð1 gÞRL
ð20Þ
The corresponding value of reference signal power can therefore be written as P R;min ¼ P 0 gð1 gÞamin;therm ¼
2kT : eSRL
ð21Þ
In conclusion, for PR > PR,min [given by Eq. (21)], the interferometric information signal at the output dominates the thermal noise and the SNR is limited only by the shot noise. However, in a general case, the SNR (dB) [under condition given by expression (16)] can be written as P AC SNRðdBÞ ¼ 10log10 min : ð22Þ P shot þ P therm Substituting Eqs. (12), (14) and (17) in Eq. (22) gives SNRðdBÞ ¼ 10log10
"
# 2 2S 2 P 20 g2 ð1 gÞ abRL : 2eBfSP 0 gð1 gÞða þ bÞgRL þ 4kTB
ð16Þ
ð23Þ
Expression (16) gives the minimum value of a so as to generate an interferometric signal (at the output) strong enough to dominate the dark current
For a b (which is usually the case with classic indicating ab/ interferometry), P min shot P therm (a + b) @ b. Under these conditions, Eq. (23) simplifies to
amin;dark >
N.A. Riza, Z. Yaqoob / Optics Communications 241 (2004) 419–428
SNRðdBÞ ¼ 10log10
g0 P 0 gð1 gÞb : hmB
ð24Þ
It is evident from Eq. (23) that for a b, the SNR is proportional to b and is maximized for g = 0.5 (since g(1g) attains its maximum value of 0.25). On the other hand, for a = b (i.e., reference signal power equal to the information signal power), the SNR given in Eq. (23) will become " # S 2 P 20 g2 ð1 gÞ2 b2 RL SNRðdBÞ ¼ 10log10 ; 2B½efSP 0 gð1 gÞbgRL þ kT
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approach (a b). Fig. 2 also indicates that the SNR (for both the classic as well as the agile approach) increases with a decrease in photodetector bandwidth B (or an increase in photodetector integration time). The difference in SNR, DSNR, available from the two approaches is however not a function of photodetector bandwidth B and is shown versus b in Fig. 3. The plot shows that the difference in SNR for the two approaches is not significant and it decreases with increasing b Specifically, the difference in SNR decreases from 4.5 dB and tends to reach 0 dB as b approaches 1.
ð25Þ where S is given by Eq. (7). However, if the condition P min shot P therm is still valid, Eq. (25) simplifies to 0 g b P 0 gð1 gÞ : SNRðdBÞ ¼ 10log10 ð26Þ 2 hmB This indicates that as a is reduced (such that a is not b), the SNR drops. Specifically, under the condition P min shot P therm , the SNR drops by a factor of [a/(a + b)] and at a = b, it drops by 3 dB [see Eq. (26)]. Nonetheless, if the output signal is not limited by shot noise only (but through the thermal noise as well), the SNR (dB) for the a = b case will be given by Eq. (25). Subtracting Eq. (25) from Eq. (24), we get the difference in SNR available from the two photo-detection approaches giving 2½efSP 0 gð1 gÞbgRL þ kT DSNR ¼ 10log10 : eSP 0 gð1 gÞbRL ð27Þ Eq. (27) indicates that the defined DSNR is not a function of photodetector bandwidth B (or the integration time). A comparison of SNR available from the classic approach (a b) as well as the agile approach (a = b) for interferometric detection is provided in Fig. 2. Specifically, the SNR versus b for both the classic and the agile approaches, given by Eqs. (24) and (25), respectively, is shown for two different values of photodetector bandwidth B. It is clear from Figs. 2(a) and (b) that regardless of the photodetector bandwidth, the SNR available from the agile approach is slightly less than that is obtainable using the classic
4. Dynamic range comparison of classic versus agile interferometry To determine the dynamic range available from the classic and the agile interferometry approach, Eq. (4) is rewritten as P d ðtÞ ¼ P 0 gð1 gÞða þ bÞ pffiffiffiffiffiffi 2 ab 1þ cosð2pfRF t þ uÞ : aþb
ð28Þ
From Eq. (27), the visibility parameter V can be written as pffiffiffiffiffiffi 2 ab V ¼ : ð29Þ aþb At a b,a + b @ a. Therefore, Eq. (29) simplifies to rffiffiffi b : ð30Þ V ¼2 a Eq. (28) thus becomes "
# rffiffiffi b P d ðtÞ ¼ P 0 gð1 gÞa 1 þ 2 cosð2pfRF t þ uÞ ; a
which indicates that the electrical current signal at the output becomes " # rffiffiffi b iðtÞ ¼ A 1 þ 2 cosð2pfRF t þ uÞ ; ð31Þ a where A ¼ SP 0 gð1 gÞa:
ð32Þ
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N.A. Riza, Z. Yaqoob / Optics Communications 241 (2004) 419–428 100 98
Single to Noise Ratio (dB)
96 94 92
Classic Method ( α >> β ) Agile Method ( α = β )
90
B = 10 MHz
88 86 84 82
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
β
(a) 152 150
Single to Noise Ratio (dB)
148 146 144
Classic Method ( α >> β ) Agile Method ( α = β )
142 140
B = 60 Hz
138 136 134
0
0.1
0.2
0.3
0.4
(b)
0.5
0.6
0.7
0.8
0.9
1
β
Fig. 2. Signal-to-noise ratio (SNR) in dB versus b for photodetector bandwidth of (a) B = 10 MHz and (b) B = 60 Hz. P0 = 100 mW, T = 300 K, RL = 50 X, k = 1550 nm, g = 0.5, g 0 = 0.7, a = 1.
Hence, the dynamic range (dB) of the output current signal, defined as the ratio of maximum to minimum current signal amplitudes, will be given by pffiffiffiffiffiffiffiffi! 1 þ 2 b=a pffiffiffiffiffiffiffiffi Dab ¼ 10log10 ð33Þ 1 2 b=a
comes equal to b, the visibility parameter V given by Eq. (29) simplifies to
that will reduce as b drops (since a is constant) till it approaches zero. On the other hand, as a be-
which indicates that the electrical current signal at the output becomes
V ¼ 1:
ð34Þ
Therefore, Eq. (28) becomes P d ðtÞ ¼ 2P 0 gð1 gÞb½1 þ cosð2pfRF t þ uÞ;
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425
4.5 4 For classic case, α = 1 For agile case, ( α = β ); however, it remains > α
3.5 Difference in SNR (dB)
min,therm
3 2.5 2 1.5
α min,therm
1 0.5 0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
β
Fig. 3. Difference in SNR, DSNR, for the traditional and proposed agile approaches to interferometric detection. P0 = 100 mW, T = 300 K, RL = 50 X, k = 1550 nm, g = 0.5, g 0 = 0.7, a = 1.
iðtÞ ¼ C½1 þ cosð2pfRF t þ uÞ;
ð35Þ
where C ¼ 2SP 0 gð1 gÞb:
ð36Þ
Thus in the proposed agile case of a = b, the dynamic range D (in dB) of the output current signal is given by 0 1 B 2C C Da¼b ¼ 10log10 @qffiffiffiffiffiffiffiffiA hi2N i ¼ 10log10
! 2C pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; P N =RL
ð37Þ
where under condition (16), P N ¼ P min shot þ P therm . As a result, Eq. (37) becomes ! C Da¼b ¼ 10log10 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : BfeSP 0 gð1 gÞb þ kT =RL g ð38Þ P min shot
P therm , Eq.
! C pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : eBSP 0 gð1 gÞb
ð39Þ
Note that under the condition (38) simplifies to Da¼b ¼ 10log10
A comparison of dynamic range available from the traditional approach (a b) as well as the new agile approach (a = b) for interferometric detection is provided in Fig. 4. In particular, plots for the dynamic range versus b for both the classic and the new approaches, given by Eqs. (33) and (38), respectively, are shown for two different values of photodetector bandwidth B. It is clear from Figs. 4(a) and (b) that regardless of the photodetector bandwidth, the dynamic range available from the agile interferometry approach is significantly more than that obtainable using the traditional approach. Moreover, the dynamic range response for the agile approach remains relatively flat over a wide range of b. The difference in available dynamic range (in dB) from the two approaches goes to zero at b = 1. However, as the value of b drops, the difference in available dynamic range from the two approaches starts increasing drastically, thus indicating that the newly proposed agile method is superior to the classical approach for detection of an interferometric information signal. A comparison of Figs. 4(a) and (b) also shows that the dynamic range increases further by increasing the photodetector integration time.
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Dynamic Range (dB)
50
40 Classic Method ( α >>β ) Agile Method ( α = β ) 30
10MHz MHz BB=B=10 =10 Hz 20
10
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.6
0.7
0.8
0.9
1
β
(a) 80 70
Dynamic Range (dB)
60
Classic Method (α >>β) C A Agile Method (α =β)
50
B = 60 Hz
40 30 20 10 0
0
0.1
0.2
0.3
0.4
(b)
0.5
β
Fig. 4. Output interferometric signal dynamic range versus b for photodetector bandwidth of (a) B = 10 MHz and (b) B = 60 Hz. P0 = 100 mW, T = 300 K, RL = 50 X, k = 1550 nm, g = 0.5, g 0 = 0.7, a = 1.
5. Agile interferometry implementation issues The earlier sections have theoretically shown that agile interferometry (i.e., when a = b) compared to the classical approach (a b) for interferometric information signal detection can provide a comparable SNR with a much improved detection dynamic range. However, when the sig-
nal strength, i.e., b is unknown, which is usually the case, one does not know where to set the value of a so that the a = b condition is satisfied for the best dynamic range possible via agile interferometry. This means that one needs to sweep a across the entire band and get many readings for the detected interferometric signal dynamic range, one of which is the best. For shot noise limited detection,
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427
Fig. 5. Hardware implementation scenarios for agile interferometry. For test objects with non-uniform reflectivities, shown are pointby-point scan (a) free-space and (b) fiber-optic interferometers with full test data collected via test object x–y–z motion. For the special case where reflectivity over object is constant, (c) shows a free-space agile interferometer using a two-dimensional (2D) sensor device such as a charge coupled device (CCD).
the range over which a can be swept will be [amin,therm, 1], where amin,therm is given by Eq. (20). Fig. 5 shows schematics of free-space and fiber-optic versions of agile interferometry setups that deploy high-speed variable optical attenuators (VOAs) for agile control of reference signal strength. Options for ultrafast reference beam power control modules include microsecond speed VOAs such as the ones using acousto-optics [13,14]. It is also clear from Figs. 5(a) and (b) where b is non-uniform on the test object, agile interferometry works on a point-by-point information scan basis, where for each information point in the signal sensing zone, the reference beam
power is swept to match the given signal scan point location power. Therefore, agile interferometry requires high-speed optical scanners [15], VOAs, point optical detectors, and electronic processing, all of which are a reality today. In the special Fig. 5(c) case when the object reflectivity is uniform across the object, the object can be illuminated by a plane wave (instead of a point) and the point detector can be replaced by a two-dimensional photodetector array such as a CCD. In addition, test object x–y motion is not required; only the object can be translated in z (along optical beam) to acquire three-dimensional object information coded in the phase of the detected signal.
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6. Conclusion
References
In conclusion, a new approach called agile interferometry is proposed for attaining interferometric information with high sensitivity. Specifically, compared to classical interferometry with a strong fixed reference optical beam, the proposed method utilizes dynamic control of the reference beam strength and the detector integration time for efficient interferometric detection leading to a high SNR and significantly improved detection signal dynamic range. A theoretical analysis of the agile approach indicates that equalizing the reference signal strength with the information signal strength drops the detected SNR by <4.5 dB. However, agile interferometry compared to fixed reference interferometry leads to a significant advantage in the final output detection dynamic range. In addition, an appropriate increase in the photodetector integration time also improves both SNR as well as output signal dynamic range. In effect, agile interferometry proposes appropriate reduction in the reference signal strength and increase in the photodetectorÕs integration time to utilize the photodetectorÕs available dynamic range. From a practical implementation point of view, agile interferometry requires the use of a high-speed variable optical attenuator to sweep the reference beam power strength through a series of settings to obtain a set of detected interferometric data. This stored data is then compared to extract the detected signal with the highest dynamic range. Thus, agile interferometry relies on the storage of many signals that are then processed by a computer to deliver the optimized output data.
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