Experimental study of light diffraction by standing ultrasonic wave with cylindrical symmetry

Optics Communications 267 (2006) 14–19 www.elsevier.com/locate/optcom

Experimental study of light diffraction by standing ultrasonic wave with cylindrical symmetry Ireneusz Grulkowski *, Piotr Kwiek Institute of Experimental Physics, University of Gdansk, ul. Wita Stwosza 57, 80-952 Gdansk, Poland Received 2 February 2006; received in revised form 17 May 2006; accepted 7 June 2006

Abstract We report a new acousto-optic arrangement based on ultrasonic wave with cylindrical symmetry. The theory of light interaction with standing cylindrical ultrasonic wave is experimentally verified in the Fraunhofer region. A very good agreement of experimental results with numerical calculations based on the proposed theory is found. The diffraction pattern consists of ring-shaped diffraction orders which posses a fine structure. The time average light intensity of the whole zeroth diffraction order as a function of the Raman–Nath parameter is investigated. The modulation properties of presented system are examined by means of single photon counting technique. Finally, some potentially useful applications in the laser and fibre technology are suggested. Ó 2006 Elsevier B.V. All rights reserved. PACS: 78.20.Hp; 42.79.Jq; 43.35.+d Keywords: Acousto-optical effects; Acousto-optical devices; Ultrasound

1. Introduction Since the first experimental demonstration of light interaction with ultrasound by Lucas, Bicquard [1], Debye and Sears [2], this phenomenon has been receiving considerable attention in pure and applied physics. Most of the theoretical and experimental investigations focused on the diffraction of light (plane wave or Gaussian beam) by a plane ultrasonic wave or more complex acoustic fields such as: superposed, adjacent or spatially separated ultrasonic beams progressing either in the same or opposite directions [3]. All those studies were performed in the Bragg regime, the Raman–Nath regime or within the transition region. Invention of coherent light sources [4] and the first application of acousto-optic device in laser technology [5] gave rise to the continuous development of acousto-optics. Advances in laser technology provided many powerful *

Corresponding author. Tel.: +48 58 5529264; fax: +48 58 3413175. E-mail address: [email protected] (I. Grulkowski).

0030-4018/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2006.06.006

tools such as: modulators [6], deflectors [7], filters [8], frequency shifters [9], signal processing devices [10], imagers [11] or Q-switchers [12]. Acousto-optic phenomenon has been also used in light diffraction tomography for ultrasonic field mapping [13]. What is more, interaction of light with surface acoustic waves was successfully applied in the integrated and guided-wave optics [10,14,15]. The development of the laser systems based on optical fibres stimulates implementation of acousto-optic devices into fibre technology for laser beam controlling [14–19]. On the other hand, in most cases the proposed arrangements use only conventional acousto-optic modulators, which take advantage of generation of plane acoustic fields in the piezoelectric crystals [16–18]. However, due to the fibre geometry, generation of ultrasonic fields having cylindrical symmetry seems to be more suitable. In this context, understanding the phenomenon of acousto-optic interaction in the case of cylindrical ultrasound serves as a crucial step in optimizing the conditions of laser beam manipulation (e.g. Q-switching) in all-fibre lasers.

I. Grulkowski, P. Kwiek / Optics Communications 267 (2006) 14–19

2. Theory Theoretical model requires considering the ultrasonic wave generated by harmonic vibrations of a cylindrical transducer (shell) and a plane light wave propagating along the shell’s axis of symmetry, as shown in Fig. 1. Generally, piezoelectric shell can be a source of radial, axial or circumferential acoustic waves as well as more complex modes like flexural or torsional ones [27]. One can assume that in the first (linear) approximation acoustic pressure Dp causes proportional changes Dl of refractive index, so that Dl  Dp. What is more, the results of holographic and interferometric investigations revealed that our cylindrical

Ω, K

2R ω, k

P ∼w

r θ

ϕ

z=0

diaphragm

L z=L transducer

screen

Fig. 1. Geometry of acousto-optic interaction.

transducer generates the simplest (zeroth order) radial mode. Consequently, uniform radial vibrations of the hollow cylinder produce the variations of the refractive index of the medium inside the shell which are given in terms of the Bessel function of the first kind of zero order [28]: lðr; tÞ ¼ l0 þ Dl ¼ l0 þ l1 J 0 ðKrÞ cosðXtÞ;

ð1Þ

1

where l0 denotes the refractive index of the undisturbed medium, l1 is the amplitude of variations of the refractive index, and K and X refer to the wave number and the circular frequency of ultrasound, respectively. By reference to Eq. (1) the normalised refractive index distribution inside vibrating shell for experimental conditions was shown in Fig. 2. The plane electromagnetic wave propagating in the z-direction illuminates the standing cylindrical ultrasonic wave (Fig. 1). We assume after Raman and Nagendra Nath that ultrasound can act as a pure phase grating [29]. As the incident light field is limited by a circular diaphragm of the radius R, we take advantage of the Fresnel–Kirchhoff diffraction theory in the Fraunhofer approximation [30]. Thus, the problem is reduced to the calculation of diffraction integral with phase modulated light wave over the circular aperture. Let us define a point P at the screen in the far field by two coordinates: w – the sine of the angle

Normalised refractive index Δ μ /μ

Despite the fact that the experiments on light diffraction have led to an understanding of many aspects of acousto-optics, the phenomenon of light diffraction by ultrasound having more complicated shape of wave fronts has not been extensively studied. Huang, Nissen and Bodegom presented a theory of light interaction with focused ultrasonic wave that has been used in the determination of pressure amplitude in the focal point of a spherical transducer [20]. Korpel reported numerical results of Bragg diffraction by curved wave front of ultrasound [21]. It was Hargrove, who for the first time dealt with the interaction of Gaussian laser beam (of the waist equal to the first nodal diameter) by standing cylindrical ultrasonic wave [22]. Unfortunately, Hargrove’s theoretical approach did not let him to show the diffraction orders in the far field. The theory of Windels and Leroy, developed in 2001 [23] and experimentally verified two years later [24], predicts that the phenomena of focusing, defocusing and raybending are observed when the width of light beam is comparable to the sound wavelength. However, these predominant effects were not taken into account by Hargrove, who concentrated on on-axis light intensity. It should be mentioned that the authors observed and investigated the focusing and defocusing while Gaussian light beam of the width comparable to the first nodal diameter was passing through the cylindrical ultrasound [25]. Recently, some preliminary studies of light diffraction by ultrasound having cylindrical symmetry were presented by Grulkowski and Kwiek [26]. Although the light intensity measurements were limited only to the centre of the zeroth diffraction order, the experimental results revealed a very good agreement with numerical predictions. In this paper, we propose a new acousto-optic arrangement which is regarded as an efficient laser light modulator. We present the results of comprehensive studies on light diffraction by cylindrical ultrasound in the Fraunhofer zone with regard to their applications. The profiles of a few diffraction orders (up to the third one) are demonstrated. Then, we focus on the zeroth diffraction order. The modulation properties of proposed system are reported and possible applications are suggested.

15

1.0 t=0 t = 0.5T

0.5

0.0

-0.5

-1.0

0

2

4

6

8

10

12

14

16

Radial distance r [mm] Fig. 2. Normalised variations of refractive index Dl/l1 inside vibrating cylindrical shell (F = 1.323 MHz) for two different instants of time.

16

I. Grulkowski, P. Kwiek / Optics Communications 267 (2006) 14–19

between the direction of observation (defined by P) and the axis of transducer (w is proportional to the radial coordinate at the screen, thereby not being dependent on the location of the screen), and u – the angle indicated in Fig. 1 [30]. The light amplitude at the point P(w, u) on the screen in the far field can be expressed by the integral Z R Z 2p Eðw; u; tÞ ¼ C expfi½krw cosðh  uÞ 0

0

þ kl1 LJ 0 ðKrÞ cosðXtÞgr dr dh;

PN LASER λ = 658 nm

PMT D TR

TLN

LN

LG

Fig. 3. Experimental setup.

3. Experimental arrangement

ð2Þ

where the coordinates (r, h) define a point in the aperture (Fig. 1), C is a constant including among others the amplitude of the electric field and factor exp(iXt), R is the radius of the circular diaphragm, x and k stand for the circular frequency and the wave number of light wave, respectively. Using the identity: Z 2p 1 J 0 ðxÞ ¼ expðix cos hÞdh; ð3Þ 2p 0 Eq. (2) can be simplified and given by the expression [26] Z R J 0 ðkrwÞ exp½ivmax J 0 ðKrÞ cosðXtÞrdr: Eðw; tÞ ¼ 2pC 0

ð4Þ A thorough description of all the algebraic steps leading to Eq. (4) is given in our previous paper [26]. We introduced here the maximum value of the Raman–Nath parameter vmax = kl1L, where L is the width of the sound field (length of the transducer). In general, the Raman– Nath parameter describes the phase change that light wave accumulates during the passage through an acousto-optic cell, and depends proportionally on the amplitude l1 of refractive index variations. In case of standing ultrasound the Raman–Nath parameter always reaches maximum values at the antinodes. Contrary to the plane standing ultrasound, the antinodes of the cylindrical standing wave are not equally distant and there is no one particular amplitude of refractive index variations at the antinodes (Fig. 2). Due to the focusing of ultrasound, the highest variations of the refractive index take place at the axis of the transducer (r = 0), so that vmax in Eq. (4) can be interpreted as a maximum phase change of the light wave propagating in the ultrasonic field with cylindrical symmetry. While the wave number k of light and the length L of the ultrasonic field are kept constant during the experiments, the value of vmax can be easily changed by changing the amplitude of acoustic pressure variations, which depends on the amplitude of refractive index variations, and – in turn – on the voltage applied to the transducer. Since the integral in Eq. (4) can not be solved analytically, we used numerical integration to calculate the normalised light intensity I(w, t) = EE*/I0(w), where the normalisation factor I0 describes the light intensity in the absence of ultrasound (vmax = 0). All the numerical calculations were carried out by means of MathematicaÒ 4.

In order to verify the theoretical predictions, we designed the experimental setup, illustrated in Fig. 3. The ultrasonic wave was generated by a piezoelectric circular shell TR (96.0 mm in length, 51.0 mm in external diameter, 36.1 in internal diameter) made of PZT and closed at both ends by glass windows. The cell designed in this way was filled with water, and all presented experiments were performed for the third harmonic (F = X/2p = 1.323 MHz). Light was provided by a semiconductor laser (LDCU 12/ 6284; Power Technology) emitting 45 mW at k = 658 nm. Having left the acousto-optic cylindrical cell, the diffracted light beam was transformed into the far field by a transforming lens TLN. The intensity of diffracted light in the far field was examined in the focal plane of a lens LN. In order to scan the profiles of the diffraction orders, the photomultiplier PMT (H5783P; Hamamatsu) with a pinhole PN (12 lm in diameter) was attached to the system of two perpendicular linear guides LG (Isel Automation) that provided the positioning of the photomultiplier with an accuracy of 0.5 lm. As the incident light was to propagate exactly along the axis of the cylindrical transducer TR, before each experiment we controlled the symmetry of Airy pattern, produced by the circular diaphragm D limiting the dimensions of the light field. The experimental conditions provided a constant value of the Klein–Cook parameter Q  K2L/(l0k) = 0.24 < 1, thus lying within the Raman– Nath regime (phase diffraction grating limit). The values of rf voltage applied to the transducer were correlated with the corresponding values of the Raman–Nath parameter vmax by means of a standard calibration procedure described elsewhere [26]. The measurements were made automatically by GPIB interface. 4. Results Due to the cylindrical symmetry of ultrasound generated by the transducer, the diffraction pattern consists of a set of concentric rings (diffraction orders). The images of the diffracted light in the Fraunhofer zone for two different values of vmax are shown in Fig. 4a and b. It should be noted that even for small values of the voltage driving ultrasonic transducer it was possible to obtain relatively high value of the Raman–Nath parameter at the axis of the shell. As a consequence, quite large number of diffraction orders appeared on the screen. Additionally, the light intensity does not depend on the angular coordinate / at the screen, as indicated in Eq. (4). Fig. 4c presents a magnified

I. Grulkowski, P. Kwiek / Optics Communications 267 (2006) 14–19

vibrating cylinder. The standing cylindrical wave, described by the Bessel function of the first kind of zero order J0, has no constant distance between adjacent nodes (or antinodes). In fact, the function J0 can be approximated by cosine function only for large values of its argument. Additionally, our studies have shown that there is no rule governing the symmetry of the profiles. Although the profile of the first diffraction order seems to be symmetric, there is a small asymmetry predicted by the theory and revealed by experimental results. From the practical point of view, the zeroth diffraction order serves as the most important. In our previous paper, we examined the light intensity only at the central point of this diffraction order (w = 0). In the present study, we addressed the question how the light intensity from the whole spot of the zeroth order depends on the maximum value of the Raman–Nath parameter vmax. In order to do this, the pinhole PN (Fig. 3) was replaced by a circular aperture so that photomultiplier captured only the zeroth diffraction order whereas higher orders were cut off. The results of both numerical and experimental studies, presented in Fig. 6, made it possible to correlate the Raman–Nath parameter value with a corresponding rf driving voltage of the transducer. As a consequence, obtained curve can be used in calibration procedure. Due to the fact that the ultrasonic wave was a standing one, time average light intensity never falls to zero. It is also evident that the diffraction efficiency of considered system is raising with increasing value of vmax, and the highest increase can be observed up to vmax = 25. Because the experimental results are in excellent agreement with theo-

Fig. 4. Diffraction patterns (F = 1.323 MHz): (a) vmax = 20, (b) vmax = 60, (c) the fine structure of the first, the second and the third diffraction order; vmax = 50.

fragment of the first, second and third diffraction order which exhibit a fine structure. In order to confirm these observations, we performed a scan across the profiles of the orders 0, 1, 2 and 3. The results of experimental studies for vmax = 50 together with the theoretical predictions are illustrated in Fig. 5. The obtained data agree very well with the numerical calculations based on the presented theory. Generally, it is easy to recognise that each particular diffraction order is composed of one or two main rings and a few side rings. Such a diffraction pattern is the result of the limited aperture of the incident light (Airy diffraction) as well as the specific spatial distribution of the refractive index inside

0.20

(a)

0.0012

0.010

(b)

0.0009

0.15

Time average normalised light intensity

17

0.005

0.0006

0.10 0.000 0

0.05

10

20

0.0003 0.0000

0.00 0

5

10

15

20

80

85

kwR 0.0004

0.00020

(c)

0.0003

0.00015

0.0002

0.00010

0.0001

0.00005

0.0000 165

170

175

kwR

180

90

95

100

275

280

kwR

185

190

(d)

0.00000 260

265

270

kwR

Fig. 5. Profiles of diffraction orders (F = 1.323 MHz; vmax = 50): 0th order (a), 1st order (b), 2nd order (c), 3rd order (d). Solid line – theory, dots – experiment.

18

I. Grulkowski, P. Kwiek / Optics Communications 267 (2006) 14–19

parameter and reaches its maximum value of ca. 96.3% for vmax P 25.

Time average normalised light intensity

1.0 theory experiment

0.8

5. Discussion and conclusions

0.6 0.4 0.2 0.0 0

20

40

60

80

100

Raman-Nath parameter vmax Fig. 6. Time average normalised light intensity of the zeroth diffraction order as a function of the Raman–Nath parameter (F = 1.323 MHz).

retical predictions, we are able to precisely determine the value of vmax and related diffraction efficiency. Since the transducer generates the standing ultrasonic wave, the light in the far field is temporarily modulated, so in the final stage of our studies we checked the modulation properties of our system. Temporal changes of the diffracted light intensity were investigated by a single photon counting technique with the time resolution of 5 ns. Fig. 7 illustrates the modulation of light from the whole zeroth diffraction order over a period T of ultrasound. Those experiments were carried out for three values of the Raman–Nath parameter. The results were compared with the numerical data and there is no doubt that also in this case our theory was confirmed by the experiment. What is more, the depth of temporal modulation of the non-diffracted light intensity increases with the Raman–Nath

Normalised light intensity

1.0 0.8 0.6 0.4 0.2 0.0 0

0.2T

0.4T

0.6T

0.8T

T

Time t vmax = 10 (theory) vmax = 10 (experiment) vmax = 20 (theory) vmax = 20 (experiment) vmax = 35 (theory) vmax = 35 (experiment)

Fig. 7. Modulation of light intensity of the zeroth order (F = 1.323 MHz) for three values of the Raman–Nath parameter.

In conclusion, since the distribution of the refractive index regarded as a standing cylindrical ultrasonic wave is not periodic in space, it is not possible to derive a set of Raman–Nath equations. This fact makes the problem of light interaction with cylindrical ultrasound more complex. However, the theory was verified by the results of experimental investigations in the zeroth and higher diffraction orders. We have also proven the validity of phase grating approximation even for large values of the Raman– Nath parameter. It must be emphasized that the presented results revealed more quantitative facts associated with the light diffraction by cylindrical ultrasound than the Hargrove’s work did. In particular, since the light field is limited and due to the fact that the refractive index distribution inside vibrating shell is represented by the Bessel function J0, which is not periodic in space, each diffraction order exhibits the fine structure. The intensity of the diffracted light is modulated with the double angular frequency of ultrasonic wave. As a consequence, vibrating cylinder can serve as an efficient modulator placed either inside or outside a laser cavity. Such a modulator, based on cylindrical standing wave, operates at the zeroth diffraction order. The modulation properties of such a system can be easily controlled by rf voltage applied to the piezoelectric shell. It should be emphasized that it is easier to obtain an ideal standing cylindrical wave than a plane one. The ultrasound is focused inside vibrating cylindrical shell and we do not have to use any reflecting element. The advantage of considered acousto-optic cell over the plane ultrasound-based systems arises from its compactness and low power consumption. As the conditions of the standing wave inside our acousto-optic cell are stable and well defined, the performance of this device might be precisely determined. The diffraction phenomenon presented in this paper offers also a new opportunity for laser beam modulation in optical fibres. If an optical fibre is placed exactly at the axis of cylindrical shell or the fibre’s core is coated with a piezoelectric material using e.g. sol–gel technology, one can generate cylindrical ultrasonic wave in order to modulate the light propagating in such a fibre. The latter approach has been already presented [31–33]. However, the related theoretical considerations (given by Jarzynski in [34]) do not take into account the specific distribution of refractive index, which is generated inside fiber’s core at characteristic frequencies, thereby being not able to explain diffraction phenomenon. As a consequence, that theory refers rather to the acoustic waves of the wavelengths, which are much longer than the diameter of the fiber. If adapted to the case of optical fibres, our approach would provide additional details of the performance of allfibre modulators.

I. Grulkowski, P. Kwiek / Optics Communications 267 (2006) 14–19

Acknowledgements The research grants from the University of Gdansk (# BW/5200-5-0164-5, # BW/5200-5-0212-6) are gratefully acknowledged. This research was supported by the European Social Fund (project # Z/2.22/II/2.6/002/05 Scholarship programme for Ph.D. students of the University of Gdansk engaged in the research on the development of Pomerania’s innovative economy) as well as by the governmental resources. The authors are deeply indebted to the reviewers for their useful comments. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]

R. Lucas, P. Biquard, J. Phys. Radium 3 (1932) 464. P. Debye, F.W. Sears, Proc. Natl. Acad. Sci. Wash. 18 (1932) 409. O. Leroy, R.A. Mertens, Opt. Eng. 31 (1992) 2049. T.H. Maiman, Nature 187 (1960) 493. L.E. Hargrove, R.L. Fork, M.A. Pollack, Appl. Phys. Lett. 5 (1964) 4. A. Korpel, Acousto-Optics, Marcel Dekker, New York, 1988. P. Maa´k, L. Jakab, A. Baro´csi, P. Richter, Opt. Commun. 172 (1999) 297. H.C. Park, B.Y. Kim, H.S. Park, Opt. Lett. 30 (2005) 3126. I.K. Hwang, S. Yun, B.Y. Kim, Opt. Lett. 22 (1997) 507. P.K. Das, C.M. DeCusatis, Acousto-Optic Signal Processing: Fundamentals and Applications, Artech House, Boston, 1991. N. Gupta, V. Voloshinov, Opt. Lett. 30 (2005) 985. Q. Li, Y. Zheng, Z. Wang, T. Zuo, Opt. Las. Technol. 37 (2005) 357. R. Reibold, P. Kwiek, in: A. S´liwin´ski, B.B.J. Linde, P. Kwiek (Eds.), Acousto-Optics and Applications III, Proc. SPIE 3581 (1998) 2.

19

[14] C.S. Tsai (Ed.), Guided-Wave Acousto-Optics: Interactions, Devices and Applications, Springer-Verlag, Berlin, 1990. [15] C.S. Tsai, W. Chen, P. Le, S.C. Tsai, J. Opt. A: Pure Appl. Opt. 3 (2001) S46. [16] Y. Wang, A. Martinez-Rios, H. Po, Opt. Commun. 224 (2003) 113. [17] Y.H. Tsang, F. Qamar, T.A. King, D.-K. Ko, J. Lee, J. Phys. D: Appl. Phys. 38 (2005) 1365. [18] K. Yang, S. Zhao, G. Li, H. Zhao, Opt. Laser Technol. 37 (2005) 381. ´ ez, [19] D. Zalvidea, N.A. Russo, R. Duchowicz, M. Delgado-Pinar, A. Dı J.L. Cruz, M.V. Andre´s, Opt. Commun. 224 (2005) 315. [20] J. Huang, J.A. Nissen, E. Bodegom, J. Appl. Phys. 71 (1992) 70. [21] A. Korpel, Opt. Eng. 31 (1992) 2083. [22] L.E. Hargrove, J. Acoust. Soc. Am. 51 (1972) 888. [23] F.W. Windels, O. Leroy, J. Opt. A: Pure Appl. Opt. 3 (2001) S1. [24] F. Windels, P. Kwiek, G. Gondek, K. Van Den Abeele, Opt. Lett. 28 (2003) 40. [25] K. Ferria, I. Grulkowski, P. Kwiek, J. Phys. IV-Proc., submitted for publication. [26] I. Grulkowski, P. Kwiek, Arch. Acoust. 30 (4) (2005) 107. [27] S. Markusˇ, The Mechanics of Vibrations of Cylindrical Shells, Elsevier, Amsterdam, 1988. [28] E. Skudrzyk, The Foundations of Acoustics. Basic Mathematics and Basic Acoustics, Springer-Verlag, Wien, 1971. [29] C.V. Raman, N.S. Nagendra Nath, Proc. Ind. Acad. Sci. A 3 (1936) 75. [30] M. Born, E. Wolf, Principles of Optics, Pergamon Press, London, 1959. [31] N.H. Ky, H.G. Limberger, R.P. Salathe´, G.R. Fox, IEEE J. Lightwave Technol. 14 (1996) 23. [32] C.R. Wuethrich, C.A.P. Muller, G.R. Fox, H.G. Limberger, Sens. Actuators A 66 (1998) 114. [33] Q. Li, A.A. Au, C. Lin, E.R. Lyons, H.P. Lee, IEEE Photon. Technol. Lett. 14 (2002) 1563. [34] J. Jarzynski, J. Appl. Phys. 55 (1984) 3243.