
- Chinese Optics Letters
- Vol. 20, Issue 2, 021101 (2022)
Abstract
1. Introduction
Standard coherent optical image processing employs a Fourier plane in a 4−f system[
To have a self-contained review, in Section 2, we discuss some of the fundamentals of acousto-optics, introducing some important parameters of the AOM, and, in Section 3, we summarize the Korpel–Poon multiple plane-wave theory. The presentation in these two sections closely follows the book by Poon and Kim[
2. Fundamentals of Acousto-Optics
In acousto-optics, we deal with the interaction between sound and light. An AOM consists of a transparent acoustic medium, such as dense glass. A piezoelectric transducer is bonded to the acoustic medium to provide propagating sound waves into it. When a sound wave of wavelength
Sign up for Chinese Optics Letters TOC. Get the latest issue of Chinese Optics Letters delivered right to you!Sign up now
Figure 1.AOM illustrating diffraction of light by sound.
There are a variety ways to explain the interaction between sound and light. When we consider the interaction of plane waves of light and sound, we assume that the length of the transducer
The corresponding law of conservation of energy gives us (after division by
Figure 2.Upshifted Bragg diffraction: (a) wavevector diagram and (b) experimental configuration. Adapted from Ref. [
The two conservation laws can be employed again to give two equations similar to Eqs. (2) and (3) if we exchange the directions of incident and diffracted light. With the so-called downshifted interaction in acousto-optics and corresponding to Eqs. (2) and (3), we have
Figure 3.Downshifted Bragg diffraction: (a) wavevector diagram and (b) experimental configuration. Adapted from Ref. [
From Figs. 2(a) and 3(a), we note that the wavevector diagrams are closed for both cases of the interaction. As a result, there can only be one critical incident angle, i.e., the Bragg angle, such that plane waves of sound and light can interact. By inspecting either Fig. 2(a) or 3(a), we find the Bragg angle
Figure 4.Multiple diffraction. Adapted from Ref. [
In addition to the Bragg angle of the AOM, there is another important parameter called the Klein–Cook parameter
If
3. Korpel–Poon Multiple Plane-Wave Scattering Theory
In the previous section, we used the simple particle approach to describe the necessary conditions for Bragg diffraction to occur. Often, we are interested in knowing how the acousto-optic interaction process affects the amplitude distribution among the different diffracted beams. We shall adopt the Korpel–Poon multiple plane-wave theory to understand this aspect, which is summarized as follows[
Figure 5.AOM modeled by a column of sound of width L. Adapted from Ref. [
For a given value of
4. Transfer Functions and Acousto-Optic Spatial Filtering
For many decades, the use of acousto-optics has been extensively confined to signal processing. The reason is that AOMs are one-dimensional (1D) devices, and the interaction between light and sound is confined on a plane defined by the wavevectors of sound and light. The use of AOMs operating in the Bragg regime for 2D image processing was pioneered by Xia et al.[
We consider upshifted Bragg diffraction with off-Bragg angle incidence and limit to two diffracted orders. Hence, we let
These solutions represent the plane-wave solutions that are due to oblique incidence and have been used for thick hologram gratings[
Equation (11) motivated Poon and Chatterjee[
These transfer functions show angular selectivity, and they depend on the angle of incidence of the light incident on the AOM. The transfer functions can be written as a function of spatial frequency if we inspect the interaction geometry shown in Fig. 6.
Figure 6.Diffraction geometry for upshifted Bragg operation. Adapted from Ref. [
A similar expression exists for the first-order diffracted beam:
Figure 7.Characteristics of |H0(kx′)| and |H1(kx′′)| as a function of Q and α. (a) and (b) Transfer function for the zeroth-order beam and the first-order beam at Λ = 0.01 mm with Q = 14, respectively; (c) and (d) transfer function for the zeroth-order beam and the first-order beam at Λ = 0.01 mm with Q = 28, respectively.
Acousto-optic spatial filtering to the incident beam
Figure 8.Flaptop beams obtained by the fine tuning of Q or α (alpha) through H1(kx′′Λ/π). Input laser beam is of the profile e−x2/2σ2. Reprinted with permission from Ref. [
The transfer function of the zeroth-order beam has been used for the investigation of image processing. We place an AOM near the object in an optical imaging setup, as shown in Fig. 9. The object is placed on the input plane, and the output plane is the image plane. The AOM is rotated by the Bragg angle, i.e.,
Figure 9.Diffraction by AOM and image formation by lens.
Figure 10 displays the first experimental results using an AOM for image processing.
Figure 10.Experimental results on the output plane: (a) image of the object on the output plane when the AOM is turned off; (b) images of the zeroth-order (left) and the first-order (right) beams. Reprinted from Ref. [
5. Illustrative Examples
In this section, we illustrate that AOMs can perform some of the optical computing operations such as the important differentiation operations.
5.1. First–order partial derivative
Let us assume that
Assuming the incident beam is of two transverse dimensions, i.e.,
Note that we can only process the image in one dimension. If we choose the correct value for
Figure 11.(a) Input square object, (b) magnitude spectrum of (a), and (c) intensity of the zeroth-order light.
5.2 Higher–order partial derivative
To obtain higher derivative operations, we can, for example, have two AOMs cascaded. The situation is shown in Fig. 12(a). We use the zeroth-order light output of the first modulator as the input to the second modulator. We then track the zeroth-order light of the second modulator as a final output. Mathematically, the output after the first AOM, from Eq. (20), is
Figure 12.(a) Cascaded AOM system and (b) intensity of the zeroth-order light |ψ0(2)(x′, y′)|2 at the exit of the cascade AOM system illustrating second-order differentiation operation.
Similarly, the zeroth-order light after the second AOM is
Note that if
5.3 Mixed partial derivative
The acousto-optic interaction is confined to two dimensions, i.e., in the
Figure 13.Intensity of the zeroth-order light |ψ0(2)(x′, y′)|2 at the exit of the cascade AOM system.
Depending on applications, the first derivative gives a maximum at the edge location in image processing, and the second derivative gives a zero at the edge location. The mixed derivative provides corner detection commonly used in computer vision to extract certain kinds of features and infer the contents of an image. In addition, corner detection is often used in image registration and image recognition.
6. State-of-the-Art Considerations
In the previous section, we found that the use of AOMs effectively perform a variety of partial derivatives. In this section, we discuss a couple of the latest considerations that would enhance the capability of using AOM(s) for image processing applications.
6.1 AOMs within a Mach–Zehnder interferometer
We consider two AOMs to be used within a Mach–Zehnder interferometer, as shown in Fig. 14. In principle, the two AOMs can be rotated arbitrarily along the xy plane. Irises 1 and 2 are used to select the different diffracted orders for display. The upper arm and the lower arm of the interferometry can perform different processing, depending on the orientation of each of the AOMs in the arm. The shutter has control if we have processing operations by a single arm or by both arms of the interferometer. Beamsplitter BS2 would then sum the contributions from each arm. For example, by aligning one AOM along the
Figure 14.Dual AOMs in a Mach–Zehnder interferometer.
Figure 15(a) shows the original input image. With the shutter being on and AOM1 oriented at the angle of 135° in the second quadrant in the xy plane, we see that the first-order differentiation operation is performed along the 135° angle, as shown in Fig. 15(b). At the angle of 45° in the first quadrant, processing is missed. The physical reason is that sound waves propagate along the 135° angle, and hence the 2D image is only processed along this direction. Now, with the operation realized by Eq. (28), where one AOM is along the
Figure 15.(a) Input, (b) image processing by a single AOM, and (c) image processing by dual AOMs in a Mach–Zehnder interferometer realizing the computing operation given by Eq. (
6.2 Off-Bragg angle incidence
We consider the angular misalignment of the AOM by letting
The highpass characteristic of the zeroth-order transfer function shown in Figs. 7(a) and 7(c) has become a single-sided notch filter with the center frequency given by Eq. (30). The amount of shift depends on the tilt angle. Figure 16(a) shows the image of a 1D chirp grating
Figure 16.(a) Image of a 1D chirp grating and (b) line trace across the red line in (a).
For
Figure 17.(a) Spectrum of the chirp grating and shifted zeroth-order transfer function for Δδ = 0.15, (b) processed chirp grating, and (c) line trace across (b).
Figures 18(a) and 18(c) show the processed chirp grating for
Figure 18.Processed images for (a), (b) Δδ = 0.2 and (c), (d) Δδ = 0.25.
While single-sided notch filtering has been previously investigated[
7. Concluding Remarks
We have reviewed Bragg processing using AOMs for real-time programmable spatial filtering. In the review, we have discussed the fundamentals of acousto-optics, which is followed by the summary of the multiple plane-wave theory. From the theory, we have discussed the concept of the acousto-optic transfer function, leading to the applications of spatial filtering. We have then given some illustrative examples on how to implement some of the optical computing operations. Finally, we have mentioned a couple of state-of-the-art considerations that would enhance the processing capabilities of Bragg processing. The first consideration is the use of AOMs within a Mach–Zehnder interferometer to perform the summation of two partial differentiation operations. Conceptually, the Mach–Zehnder interferometer system is elegant. However, practical implementation of the idea is quite challenging, as we need to carefully align the two images for summation. In the second consideration, we have looked at the situation when the incident angle is not exactly at the Bragg angle, thereby introducing the tilt angle. The tilt angle gives rise to single-sided notch filtering or half-plane filtering[
References
[1] T.-C. Poon, J.-P. Liu. Introduction to Modern Digital Holography with MATLAB(2014).
[2] D. Peri, A. A. Friesem. Image restoration using volume diffraction gratings. Opt. Lett., 3, 124(1978).
[3] S. Case. Fourier processing in the object plane. Opt. Lett., 4, 286(1979).
[4] Y. Yang, X. Liu, Y. Wu, T. Li, F. Fan, H. Huang, S. Wen. Optical edge detection with adjustable resolution based on liquid crystal polarization gratings. Chin. Opt. Lett., 18, 093501(2020).
[5] L. Zhou, X. Huang, Q. Fu, X. Zou, Y. Bai, X. Fu. Fine edge detection in single-pixel imaging. Chin. Opt. Lett., 19, 121101(2021).
[6] V. I. Balakshy. Acousto-optic cell as a filter of spatial frequencies. Radiotekh. Elektron., 29, 1610(1984).
[7] V. I. Balakshy. Acousto-optic visualization of optical wavefronts [Invited]. Appl. Opt., 57, C56(2018).
[8] J. Xia, D. Dunn, T.-C. Poon, P. P. Banerjee. Image edge enhancement by Bragg diffraction. Opt. Commun., 128, 1(1996).
[9] P. P. Banerjee, D. Cao, T.-C. Poon. Basic image-processing operations by use of acousto-optics. Appl. Opt., 36, 3086(1997).
[10] D. Cao, P. P. Banerjee, T.-C. Poon. Image edge enhancement with two cascaded acousto-optic cells with contrapropagating sound. Appl. Opt., 37, 3007(1998).
[11] V. M. Kotov, S. V. Averin, E. V. Kotov. High-frequency acousto-optic light modulation by double propagation of the beam through two Bragg cells. J. Opt. Technol., 86, 129(2019).
[12] V. M. Kotov. Broadband acousto-optic modulation of optical radiation. Acoust. Phys., 65, 369(2019).
[13] P. P. Banerjee, D. Cao, T.-C. Poon. Notch spatial filtering with an acousto-optic modulator. Appl. Opt., 37, 7532(1998).
[14] J. A. Davis, M. D. Norwak. Selective edge enhancement of images with an acousto-optic light modulator. Appl. Opt., 41, 4835(2002).
[15] M. R. Chatterjee, T.-C. Poon, D. N. Sitter. Transfer function formalism for strong acousto-optic Bragg diffraction of light beams with arbitrary profiles. Acustica, 71, 81(1990).
[16] M. McNeill, T.-C. Poon. Gaussian-beam profile shaping by acousto-optic Bragg diffraction. Appl. Opt., 33, 4508(1994).
[17] Y. A. Abdelaziez, P. P. Banerjee, D. R. Evans. Beam shaping by use of hybrid acousto-optics with feedback. Appl. Opt., 44, 3473(2005).
[18] T. Wang, C. Zhang, A. Aleksov, I. Salama, A. Kar. Gaussian beam diffraction by two-dimensional refractive index modulation for high diffraction efficiency and large deflection angle. Opt. Express, 25, 16002(2017).
[19] K. Yushkov, V. A. Molchanov, Y. I. Balakshy, S. N. Mantsevich. Acousto-optic transfer functions as applied to S laser beam shaping. Proc. SPIE, 10744, 107440Q(2018).
[20] V. B. Voloshinov, T. M. Babkina, V. Y. Molchanov. Two-dimensional selection of optical spatial frequencies by acousto-optic methods. Opt. Eng., 41, 1273(2002).
[21] V. I. Balakshy, D. E. Kostyuk. Acousto-optic image processing. Appl. Opt., 48, C24(2009).
[22] K. B. Yushkov, V. Y. Molchanov, P. V. Belousov, A. Y. Abrosimov. Contrast enhancement in microscopy of human thyroid tumors by means of acousto-optic adaptive spatial filtering. J. Biomed. Opt., 21, 016003(2016).
[23] K. B. Yushkov, V. Y. Molchanov. Hyperspectral imaging acousto-optic system with spatial filtering for optical phase visualization. J. Biomed. Opt., 22, 066017(2017).
[24] V. M. Kotov, S. V. Averin, E. K. Kotov, G. N. Shkerdin. Acousto-optic filters based on the superposition of diffraction fields [Invited]. Appl. Opt., 57, C83(2018).
[25] V. M. Kotov, S. V. Averin. Two-dimensional image edge enhancement using two order of Bragg diffraction. Quantum Electron., 50, 305(2020).
[26] V. M. Kotov. Processing 2D images using the Bragg diffraction. J. Commun. Technol. Electron., 65, 1331(2020).
[27] V. M. Kotov. Two-dimensional image edge enhancement using a spatial frequency filter of two-color radiation. Quantum Electron., 51, 348(2021).
[28] T.-C. Poon, T. Kim. Engineering Optics with MATLAB(2018).
[29] W. R. Klein, B. D. Cook. A unified approach to ultrasonic light diffraction. IEEE Trans. Sonics Ultrason., 14, 123(1967).
[30] R. Appel, M. G. Somekh. Series solution for two-frequency Bragg interaction using the Korpel–Poon multiple-scattering model. J. Opt. Soc. Am. A, 10, 466(1993).
[31] T.-C. Poon, M. R. Chatterjee. Transfer function approach to acousto-optic Bragg diffraction of finite optical beams using Fourier integral. International IEEE/Antennas and Propagation Society Symposium and URSI Radio Science Meeting, 1(1988).
[32] A. V. Gorevoy, A. S. Machikhin, G. N. Martynov, V. E. Pozhar. Spatiospectral transformation of noncollimated light beams diffracted by ultrasound in birefringent crystals. Photonics Res., 9, 687(2020).
[33] P. Phariseau. On the diffraction of light by progressive supersonic waves. Proc. Indian Acad. Sci., 44, 165(1956).
[34] H. Kogelnik. Coupled wave theory for thick hologram gratings. Bell Sys. Tech. J., 48, 2909(1969).
[35] S. Lowenthal, Y. Belvaux. Observation of phase objects by optically processed Hilbert transforms. Appl. Phys. Lett., 11, 49(1967).
[36] X. Yang, W. Jia, D. Wu, T.-C. Poon. On the difference between single-and double-sided bandpass filtering of spatial frequencies. Opt. Commun., 384, 71(2017).
[37] Y. Zhang, T.-C. Poon, P. W. M. Tsang, R. Wang, L. Wang. Review on feature extraction for 3-D incoherent image processing using optical scanning holography. IEEE Trans. Industr. Inform., 15, 6146(2019).
[38] T.-C. Poon, A. Korpel. High efficiency acousto-optic diffraction into the second Bragg order. IEEE Ultrasonics Symposium, 751(1983).

Set citation alerts for the article
Please enter your email address