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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 22, Iss. 9 — May. 5, 2014
  • pp: 10165–10180
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Complex, 3D modeling of the acousto-optical interaction and experimental verification

Gábor Mihajlik, Attila Barócsi, and Pál Maák  »View Author Affiliations


Optics Express, Vol. 22, Issue 9, pp. 10165-10180 (2014)
http://dx.doi.org/10.1364/OE.22.010165


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Abstract

The acousto-optical crystals are frequently used, indispensable elements of high technology and modern science, and yet their precise numerical description has not been available. In this paper an accurate, rapid and quite general model of the AO interaction in a Bragg-cell is presented. The suitability of the simulation is intended to be verified experimentally, for which we wanted to apply the most convincing measurement methods. The difficulty of the verification is that the measurement contains unknown parameters. Therefore we performed an elaborated series of measurements and developed a method for the estimation of the unknown parameters.

© 2014 Optical Society of America

1. Introduction

The acousto-optical (AO) devices are frequently used, indispensable elements of high technology and modern science, since they have extraordinary features. Even today, the interaction – with widespread applications and scientific importance – has not yet a precise numerical description. The most comparable results with the measurements are given by the so-called coupled wave equations [1

1. J. Xu and R. Stroud, “Acousto-Optic Interaction,” in Acousto-Optic Devices (Wiley-Interscience, 1992). pp. 61–94.

, 2

2. A. Korpel, Acousto-Optics (Marcel Dekker Inc., 1997).

]. Though they are based on strong approximations they have the advantage of yielding analytic results. Other numerical models cannot be applied to describe the anisotropic AO cells.

We had set ourselves the aim to create an accurate and general description of the phenomena. With the model we intend to help the further optimization of the AO devices to improve, for instance, the apodization of the transducer, or design an ideal configuration for pulse shaping of ultrashort lasers. It is known that the apodization is a cardinal point of the AO devices, yet it seems that there are still significant possibilities in its optimization. For that, an accurate 3D AO simulation is essential. The importance of the apodization motivated also the thorough investigation of the angle dependence of the AO diffraction.

Earlier we introduced our method for describing the light propagation in [3

3. G. Mihajlik, P. Maák, A. Barocsi, and P. Richter, “Novel accurate computer algorithm for modeling light propagation and diffraction in inhomogeneous, anisotropic medium – Applied to the acousto-optic interaction,” Opt. Commun. 282(10), 1961–1968 (2009). [CrossRef]

, 4

4. G. Mihajlik, P. Maák, and A. Barocsi, “Simulation of light propagation in anisotropic, optically active and slightly inhomogeneous medium, concerning the acousto-optic interaction,” Opt. Commun. 285(9), 2255–2265 (2012). [CrossRef]

], which calculates accurately, generally and relatively rapidly the AO interaction, where both the incident light beam and the refractive index change are input quantities. In [5

5. G. Mihajlik, A. Barocsi, and P. Maák, “Measurement and general modeling of optical rotation in anisotropic crystal,” Opt. Commun. 310(1), 31–34 (2014). [CrossRef]

] we presented the calculations of the refraction with the same accuracy and advantages. For the complete description, we need the alteration of the refractive index tensor generated by the anisotropic acoustic wave. In the modeling of the sound, the most important objective was the accuracy. Our goal is to take into account every effect which is not negligible; the only exception is that the inhomogeneities caused by the thermal distribution are not considered. The simulation solves the anisotropic, acoustic vector wave equation in 3D.

The most obvious way to model the acoustic wave propagation precisely would be a finite difference time domain (FDTD) method e.g [6

6. Y. Hao and R. Mittra, FDTD Modeling of Metamaterials, Theory and Applications (Artech House, Inc., 2009).

, 7

7. A. Deinega, S. Belousov, and I. Valuev, “Hybrid transfer-matrix FDTD method for layered periodic structures,” Opt. Lett. 34(6), 860–862 (2009). [CrossRef] [PubMed]

]. The FDTD methods are primarily utilized for problems of electrodynamics, but these problems have very similar mathematical structure. However, here it is not applicable in the whole volume for typical cell sizes and frequencies (0.1 – 10 cm3, 10 – 100 MHz). Nevertheless for a given driver frequency, the wave equation can be made time-independent, i.e. the steady-state wave equation. In the presence of more driver frequencies, the calculated refractive index distribution can be superposed because of the linearity. The solution of the steady-state equation is still problematic on account of the large number of grid points. The solution by the well known finite element or finite difference methods e.g [8

8. G. Dhatt, E. Lefrançois, and G. Touzot, Finite Element Method (Wiley, 2012).

, 9

9. R. J. LeVeque, Finite Difference Methods for Ordinary and Partial Differential Equations: Steady-State and Time-Dependent Problems (SIAM, 2007).

] – where the connection of the grid points and elements are represented by a sparse matrix – cannot be carried out, since the number of the grid points is in the order of ~1010.

The solution of the problem is based on the assumption that the boundary conditions allow that the calculation does not need to be performed simultaneously in the whole volume but there may be chosen an optimal calculation direction, in which the partial differential equations may be integrated. Thus the calculation can be carried out in limited duration time. The drawback of such methods is that they do not take into account the waves reflected from the side surfaces, or at least, the estimation of the reflections demands complicated iterations. However, in many cases – even in case of the measured AO filter and deflector – the effect of these reflected waves is irrelevant, according to our examinations, their neglect is reasonable.

The proposed calculation method can be realized in different ways. Out of these, we find that the most accurate and most practical way is the one that is based on the Fourier transform. In case of the Fourier transform, if the sampling frequency exceeds the band limit, the sought function and its derivatives can be exactly determined. The wave equation is a second order partial differential equation (PDE), which can be thus calculated accurately. A further advantage is that the physical quantities may be accurately interpolated between the grid points. This is necessary because in the calculation plane Δε and ∂/∂z Δε have to be computed precisely, and also because the coordinate systems of the light and sound propagations are not identical, their angle is typically a little less than 90°. The boundary condition is periodic, but that does not mean a strong restriction, the expansion of the periodic distances makes it easy to handle this condition. The calculation of the ultrasound in a deflector with ~10 cm3 volume and 100 MHz frequency, takes about one day on a 4 GHz, quad-core desktop computer. The memory space, containing the information of the refractive index tensor alteration, has the order of magnitude of one terabyte. For smaller frequency or AO filter configuration, smaller resolutions are sufficient to satisfy the sampling condition in 3D, and thus the calculation is faster.

2. Numerical method

2.1 Calculation of the acoustic wave propagation

The wave equation to solve is:
iKcKLLjuj=ρ2ui/t2,
(1)
where cKL is the elastic stiffness tensor, ρ is the density of the medium and ui is the displacement vector. For simplicity, we use an abbreviated subscript notation according to [1

1. J. Xu and R. Stroud, “Acousto-Optic Interaction,” in Acousto-Optic Devices (Wiley-Interscience, 1992). pp. 61–94.

, 10

10. B. A. Auld, Acoustic Fields and Waves in Solids (Krieger Pub Co. 1990).

]. Introducing the complex amplitude Ui for a given angular frequency Ω:
ui(r,t)=Re{Ui(r)exp(iΩt)}.
(2)
We utilize that the medium is acoustically homogeneous, the wave field is written as the sum of plane waves. For a given plane wave direction characterized by the wave vector K:
KiKcKLKLjUj=ρΩ2Ui.
(3)
Expanding and grouping the terms of the left side and returning to the classical notation yields:
KiKcKLKLjUj=(A¯¯1Kx2+A¯¯2Ky2+...+A¯¯6KyKz)U.
(4)
The Ak matrices are yielded from the cKL elastic stiffness tensor. The wave equation may be written further:

(A¯¯1Kx2+A¯¯2Ky2+...+A¯¯6KyKzI¯¯ρΩ2)U=0.
(5)

Apparently, in the case of homogeneous, anisotropic medium, the form of the acoustic wave equation is similar to its optical analogue, see e.g [11

11. B. E. A. Saleh and M. C. Teich, “Polarization and Crystal Optics,” in Fundamentals of Photonics (Wiley & Sons Inc. 1991). pp. 216, 6.310–6.311.

]. Equation (5) has nonzero solution if the determinant of the parenthetic part is zero. For the propagation of a plane wave characterized by a (Kx, Ky) pair, the only unknown variable is Kz. The mentioned determinant thus makes a sixth degree (polynomial) equation. For small values of (Kx, Ky), the polynomial equation describes three forward and three backward propagating eigenpolarizations. In case of high values of (Kx, Ky), the equation may have non-realistic roots also e.g. Re{Kz}>0 and Im{Kz}<0 which implies that some boundary conditions cannot arise. For each Kz making the determinant zero an eigenpolarization direction is defined in Eq. (5). The solution of Eq. (5) is analogous with the Eq. (3) in reference [5

5. G. Mihajlik, A. Barocsi, and P. Maák, “Measurement and general modeling of optical rotation in anisotropic crystal,” Opt. Commun. 310(1), 31–34 (2014). [CrossRef]

], the only essential difference is that there are three independent eigenpolarizations here.

Since the ultrasound propagation is based on the solution of the vector wave equation, consequently e.g. the extreme anisotropic behavior of the sound (when the angle between the wave vector and the direction of the energy propagation is large) is accurately fulfilled with-out any further correction.

2.2 The complex AO model

The light propagation and diffraction is calculated as presented in papers [3

3. G. Mihajlik, P. Maák, A. Barocsi, and P. Richter, “Novel accurate computer algorithm for modeling light propagation and diffraction in inhomogeneous, anisotropic medium – Applied to the acousto-optic interaction,” Opt. Commun. 282(10), 1961–1968 (2009). [CrossRef]

, 4

4. G. Mihajlik, P. Maák, and A. Barocsi, “Simulation of light propagation in anisotropic, optically active and slightly inhomogeneous medium, concerning the acousto-optic interaction,” Opt. Commun. 285(9), 2255–2265 (2012). [CrossRef]

]. The simulation of light refraction is summarized in [5

5. G. Mihajlik, A. Barocsi, and P. Maák, “Measurement and general modeling of optical rotation in anisotropic crystal,” Opt. Commun. 310(1), 31–34 (2014). [CrossRef]

] for the case of complex, e.g. Gaussian beam and anisotropic, optically active medium. Thus, by establishing the simulation of the acoustical wave propagation it became possible to model the complex AO interaction. Furthermore, the experimental verification of all three models became also possible.

2.3 The issues of determining physical coefficients of TeO2

In [5

5. G. Mihajlik, A. Barocsi, and P. Maák, “Measurement and general modeling of optical rotation in anisotropic crystal,” Opt. Commun. 310(1), 31–34 (2014). [CrossRef]

] the measurement method enabled us the precise determination of the optical rotation. Here we also intend to get values as accurate as possible for different parameters and coefficients from the measurement. The difficulty of our intention is that the direction of the transducer cannot be varied. Despite of the limited extent of the acoustic wave, it has a strong plane wave character, hence we cannot have much information about the anisotropic coefficients which are responsible for the direction dependent behavior. (The solution of the problem could be the measurement of several, differently oriented AO cells.)

That is why we could not take into account the anisotropy of the attenuation: neither the measurement nor the literature has reliable information about it [10

10. B. A. Auld, Acoustic Fields and Waves in Solids (Krieger Pub Co. 1990).

]. The same reason induces that, for a given cell orientation, the isotropic acoustic attenuation is quite accurate. Its coefficient was determined from the measurement.

Very similar can be said about the elastic stiffness tensor (cKL): there are more accurate values in the literature, yet their accuracy is questionable [1

1. J. Xu and R. Stroud, “Acousto-Optic Interaction,” in Acousto-Optic Devices (Wiley-Interscience, 1992). pp. 61–94.

, 2

2. A. Korpel, Acousto-Optics (Marcel Dekker Inc., 1997).

, 10

10. B. A. Auld, Acoustic Fields and Waves in Solids (Krieger Pub Co. 1990).

]. From the measurement we only got the value of the sound velocity for the given propagation direction, which means only one dependence between the seven stiffness coefficients – the rest were obtained from the literature [1

1. J. Xu and R. Stroud, “Acousto-Optic Interaction,” in Acousto-Optic Devices (Wiley-Interscience, 1992). pp. 61–94.

].

The elasto-optic coefficient tensor has only two independent elements (p44, p66) which affect the measured slow shear wave mode, however the effect of the second one is smaller by orders of magnitude, hence even p66 is negligible.

3. Experimental setup

The rightness of the simulation is intended to be verified experimentally, for which we wanted to apply the most persuasive measurement methods. However, even upon calculating several quantities (the polarization, phase, diffracted orders, or spatial dependence of the electric field in the whole volume), we cannot use them for verification if their measurement is not possible. Well and (relatively) easily measurable quantities are the intensity and polarization of the first diffracted order as the functions of the following: angle of incidence, acoustic frequency, acoustic power, polarization and position of the incident beam (on the entrance window).

We examined the acoustic efficiency as the function of acoustic power. In case of small sound amplitude, the efficiency is proportional to the sound power, and only the case of high power modifies the linear behavior. Vainly we have tried to analyze the efficiency dependence on the acoustic power and at the same time on the angle of incidence, but we got hardly more information (though the nonlinear behavior near the Bragg angle is well described by the simulation). Therefore in the following, we investigated specifically the case of small acoustic power, the linear behavior. Hereby we could also avoid the warming of the crystal.

The objects of our investigations are a medium quality tunable AO filter and a high quality deflector. The material of both devices is TeO2. First, we investigated the filter, and when the quality of the cell prevented the more accurate examination, we switched to the deflector made of a better crystal. In case of the filter, the angle between the optic axis and the transducer is 3.51°, the same for the deflector is 9.52°. The deflector is the same what has been measured in detail in [5

5. G. Mihajlik, A. Barocsi, and P. Maák, “Measurement and general modeling of optical rotation in anisotropic crystal,” Opt. Commun. 310(1), 31–34 (2014). [CrossRef]

]. The measurement system is also the same as in [5

5. G. Mihajlik, A. Barocsi, and P. Maák, “Measurement and general modeling of optical rotation in anisotropic crystal,” Opt. Commun. 310(1), 31–34 (2014). [CrossRef]

], except that here the first diffracted order is measured, see Fig. 1.
Fig. 1 Arrangement of the measurement.
The rotation axis of the board is within the entrance window of the device so that the angle of incidence (τ) and the beam positions (X, Y) can be independently adjusted at the plane of incidence. The rotation of the cell is driven by a digital stepper motor which is controlled by a computer. The detector signal is read also by the computer, which ensures precise and relatively fast measurement. The motor step is synchronized to the detector readout and the speed is optimized for accuracy and noise. All the measurements were performed twice, with reverse rotation at different ranges of the detector. The translation accuracy is about 30 μm, the angle accuracy is 0.01°. The applied laser is He-Ne (633 nm), the beam waist radius is 0.38 mm.

4. Comparison of simulated and measured diffraction efficiency functions

4.1 2D investigation of the angular dependence of the diffraction in AO filter

First we investigated the filter, and modeled both the optical and acoustic propagation in two dimensions (2D, Fig. 2).
Fig. 2 Measured and simulated diffraction efficiency values as the function of the angle of in-cidence with vertical (↕) or horizontal (↔) polarizations of the incident beam (P1), in case of the filter. Here the P2 polarizer was not applied.
Both the vertical and horizontal polarizations were measured. The acoustic frequency was 40 MHz.

In [4

4. G. Mihajlik, P. Maák, and A. Barocsi, “Simulation of light propagation in anisotropic, optically active and slightly inhomogeneous medium, concerning the acousto-optic interaction,” Opt. Commun. 285(9), 2255–2265 (2012). [CrossRef]

], it is shown that in case of optical activity, the a and b eigenpolarizations can diffract to their respective polarization states also, though in this case the diffracted intensity is certainly smaller. There are maximum eight possible diffraction transitions (in first order): both the incident and diffracted beams may have vertical or horizontal polarizations, and we can consider the plus or minus first orders. Instead of a, b eigenpolarizations, which are slightly elliptical, we measure the linear polarizations.

We used the same transducer width in the simulation, as the measured value: 20 mm. In the literature [1

1. J. Xu and R. Stroud, “Acousto-Optic Interaction,” in Acousto-Optic Devices (Wiley-Interscience, 1992). pp. 61–94.

, 2

2. A. Korpel, Acousto-Optics (Marcel Dekker Inc., 1997).

], the analytic result for the angular dependence of the efficiency is sinc2, a function with periodic zeros, which is not exact as it can be seen. On the contrary, the model gives the locations of the main peaks accurately and the locations of the minima quite well. The values of the main peaks correspond to all of the eight transitions relatively well, thus here the polarizations are relatively well described. Other models do not calculate the diffractions in such a general way. The correspondence of the values of the lateral maxima is worse, only their orders of magnitude agree. It can be concluded, that the 2D model describes some features significantly better than the earlier analytic results, nevertheless it is not sufficient.

4.2 2D investigation of the Bragg angles as functions of the frequency

Next, the measurement was repeated at different frequencies with quite similar results. In Fig. 3, the measured and modeled locations of the eight main peaks are represented for the different driver frequencies in case of the AO filter.
Fig. 3 Measured and modeled Bragg angles of the eight different first order diffractions as a function of the acoustic frequency, in case of the filter.

It can be seen, that the modeled Bragg angles correspond to the measured values well. 4.1 and 4.2 investigations were repeated with the deflector. The correspondence was again similar to 4.1, however, here the shape and location of the main peak was not evident, see Fig. 11. (Moreover, in case of the deflector, under ~100 MHz, there are only six different diffraction transitions in the ± first orders.)

Discussion of the deviations of the 2D and 3D model and the measurement

Investigations 4.1 and 4.2 show that the 2D simulation describes some quantities more accurately than other models, however, there are still well measurable deviations. Therefore, we simulated the 3D acoustic and 3D optical wave propagation. Although the first expectation was that the 3D complex model would be accurate, the obtained results only partly fulfilled this. It is true that the values of the main peaks depend on the values of X, Y. It is also true, that the calculated and measured dependences agree well, but their correspondence was not within the error limit. The angular dependence of the efficiency did not match significantly better, either even though that the rectangular shape of the transducer (of the AO filter) was included in the 3D model.

From our examination it follows clearly that the principal reason of the deviation is that the initial acoustic wavefront is far from uniform. The result is not surprising: according to Fig. 7, the perfect conductivity and zero impedance of the electrodes are only idealistic assumptions.

The investigation of the initial acoustic wavefront (IAWF)

Our primary goal was to reveal the correctness of the method, we deem it right by the theory, so it might be validated by measurement. The experience is that the efficiency depends on τ, X and Y. On the other hand, the distribution of the IAWF is not uniform. Hence, the idea is if we could estimate somehow the IAWF experimentally, which is a 2D quantity, that could be an experimental validation of the simulation, since the measurable quantity, η(τ, X, Y) is 3D. This is valid, because the compared and measured η(τ, X, Y) contains significantly more information than that used for boundary condition. At the same time, the investigation of the IAWF could be more important than our primary aim, since its ideal setting (by the construction of the transducer) may contribute to the optimization of the AO devices.

The inverse acousto-optic method (IAOM)

Because of the reasons above, we afforded more years of intensive research to establish a method which tries to estimate the IAWF from the dependence of the measured diffraction efficiency η(τ, X, Y). For the estimation, we chose a diffraction with optimal polarization transition. An essential element of the IAOM is that it shortens the required computation time from some days-weeks to some seconds-minutes for calculating all the desired efficiencies η(τ, X, Y) from an assumed IAWF. The fast calculation enables to change the IAWF by an iteration technique so that the measured and simulated efficiencies may correspond. The fast computation is made possible by the fact that the electric field distribution of the diffracted spot E˜d,spot is in linear relationship with the IAWF (in the linear domain). So resolving the IAWF into basis elements (segments), we calculated the electric field E˜d,spot(τ, X, Y, kx ,ky) for each element by the original complex AO method. Thus, finally, a large database was created, and the acceleration of the calculation became possible. The price of the acceleration is that the calculation of all E˜d,spot(τ, X, Y, kx ,ky) functions requires high initial computational resources. In case of the deflector it took some months by parallel running on four desktop computers. The execution of the iteration part took some weeks. Since
η(τ,X,Y)kx,kyspot|E˜d,spot|2,
(6)
the linear system becomes strongly nonlinear (η is not in linear relationship with the IAWF). As a consequence, (much) more IAWFs may induce the same η(τ, X, Y) database (surjective mapping). Even if diffractions of other polarization transitions were involved, that would not help (to make a bijective mapping). Unfortunately, this makes it more complicated to figure out exactly the behavior of the transducer.

4.3 Detailed 3D investigation of the diffraction in AO filter

This time, the efficiency as the function of the angle of incidence was measured for several values of Y (equidistantly partitioned in the whole range) on the filter, for X1 = 6 mm and X2 = 11 mm. We looked for such an IAWF, which satisfied the experimental data. For the values of X we could find one, which approximated well the measured efficiencies for either X1 or X2, separately, but could not find one, which would satisfy all the measured values at the same time (X1 and X2) with the same, good correspondence. In case of the Bragg angle the IAWF we found fitted the efficiencies η(τBragg, X, Y) relatively well for both Xi, for other angles the difference |ηmes– ηsim| was significantly higher than the error of the measurement. That raises the question, what causes the deviations?

According to the theory and all precedent examinations, the model is correct. The experience, that there are such IAWFs, for which the simulated and measured efficiencies are in good agreement either for X1 or X2, affirms the correctness, since there are significantly more measured values of η (τ, Y) than the independent values of the IAWF.

We judge that the reason of the model-measurement deviations at different X coordinates is that the inhomogeneity of the medium quality filter is too high. The question of inhomogeneity is particularly discussed in [5

5. G. Mihajlik, A. Barocsi, and P. Maák, “Measurement and general modeling of optical rotation in anisotropic crystal,” Opt. Commun. 310(1), 31–34 (2014). [CrossRef]

], where more techniques are mentioned to estimate the inhomogeneity. Generally, it can be said that the inhomogeneity of the investigated filter is significantly greater than that of the high quality AO deflector, studied in [5

5. G. Mihajlik, A. Barocsi, and P. Maák, “Measurement and general modeling of optical rotation in anisotropic crystal,” Opt. Commun. 310(1), 31–34 (2014). [CrossRef]

]. However, the irregular behavior here is not caused by the inhomogeneity of the refractive index tensor of the switched off crystal. This is implied by the result that the inhomogeneity induced diffraction efficiency of the switched off crystal (ηoff) is certainly smaller than the error of the IAWF fitting: ηoff < |ηsim – ηmes|.

We deem, that the problem is caused by the inhomogeneity of the elastic stiffness tensor, which determines the acoustic wave propagation. The effect of this inhomogeneity is less drastic on the AO diffraction, it only causes a small phase shift depending on the location. That is why the measurements could be matched much better for a single Xi value than for both Xi.

4.4 Detailed 3D investigation of the diffraction in AO deflector

The existence of the inhomogeneity and the reconnaissance of its effect motivated the repeated investigation on a high quality AO cell studied in [5

5. G. Mihajlik, A. Barocsi, and P. Maák, “Measurement and general modeling of optical rotation in anisotropic crystal,” Opt. Commun. 310(1), 31–34 (2014). [CrossRef]

].

We chose for the experiment the driver frequency to 100 MHz, X1 = 3 mm, X2 = 8 mm. The measurements were performed for the values of Y between 0.8 mm and 19 mm with a uniform resolution of 0.2 mm (92 steps). The resolution of the angle of incidence (τ) is 1300 steps on a 12.5° interval. The photograph of transducer can be seen in Fig. 4.
Fig. 4 The transducer of the measured deflector. Only the left hexagon is factory connected.
As mentioned earlier, we experienced a small diffraction even when the cell was switched off (ηoff). Since small efficiencies (η < 1.2%) are measured, ηoff is subtracted from ηmes.

4.4.1 The polynomial (P) IAWF

As the transducer has also high quality, first we fitted a simple IAWF function to the measured data; the shape of the area is a hexagon with four parameters, see Fig. 4 and 5(a) (p4 describes the relative position).
Fig. 5 The parameters of the fitted P IAWF. (a) The parameters of the hexagonal transducer. (b) The absolute value and phase of the fitted polynomial along the symmetric axis. As can be seen the size of the hexagon is not sharp but dithered so as to allow numerical differentiation. The full width at half maximum, FWHM = p2 + 2⋅p3.
The complex amplitude of the IAWF can be written as the product of a third degree polynomial and a linear phase shift along the y direction as:
S(x,y)=(q0+q1y+q2y2+q3y3+iq4y2)exp(2πiq5y).
(7)
The linear phase shift is required because the orientation [1* 1 0] of the crystal axis is not precisely parallel to the plane of the transducer, but makes a small angle, χ ~0.1°. Along p1, the IAWF is constant (in the hexagon).

The p parameters determined by fitting are listed in Table 1.

Table 1. The fitted and measured parameters of the hexagonal transducer for the P IAWF.

table-icon
View This Table

The absolute value of the third degree polynomial can be seen on Fig. 5(b).

In the presented way, the IAWF is fully characterized by a few parameters. The correspondences of the measured (ηmes) and simulated (ηsim) efficiencies are illustrated in the following (Section 4.4.3, Figs. 7-11).

We emphasize that the fitted parameters of the transducer are practically the same as the ones measured by microscope, the errors of p2 and p3 correspond to the size of the basis element (the discrete resolution) used for the IAOM (p1 is determined by the 2D model).

4.4.2 The high resolution (HR) IAWF

Since the simulations performed with the P IAWF were not satisfactorily accurate (see Section 4.4.3 for details), we deem that the real IAWF is not as slowly varying function as a third degree polynomial. Therefore, in the following, we searched a more complicated, continuous IAWF, which could satisfy all measured data the most accurately. Here the number of (complex) parameters is 435 × 27.

The absolute value and phase of the fitted HR IAWF is shown in Fig. 6.
Fig. 6 (a) The absolute value (in arbitrary unit) and (b) phase of the fitted HR IAWF. Here it is important to emphasize that the fitted IAWF is still not the same as the real one, since more different IAWFs can generate the same ηsim(τ, Xi, Y) quantities.

As can be seen, here the phase differs a little from the phase of the P IAWF. The difference means 0.02° uncertainty of the angle of the transducer plane: χ = 0.10° ± 0.01°. The difference follows from the fact that the location of the hexagon along Y (p4) is determined only by fitting, but the fit has an uncertainty of 0.05 mm.

4.4.3 Discussion

The η(τBragg, X, Y) curves

The diffraction efficiency as a function of the vertical position (Y) in case of τ = τBragg gives an important characterization of the 3D interaction, thereby it helps the investigation. Usually its measurement is relatively simple to be performed. Unfortunately in case of the AO deflector, a drastically varying signal is added to the measured efficiency near to the Bragg angle, see Fig. 11. The presence of this additive signal is the effect of the acoustic wave generated by the ground electrode. This can be evidently separated in the simulation, its elimination from the measurement, however, is not so obvious thus complicating the fitting. For the fitting, the efficiency without the effect of the ground electrode had to be estimated, see Fig. 7.
Fig. 7 The diffraction efficiencies at Bragg angle as a function of the vertical position (Y) of the beam at X = 3 mm and X = 8 mm, where the effect of the ground electrode is eliminated by estimation from the measurement. The green line denotes the simulation assuming the P IAWF.
(The estimation cannot be included in the fitting, because presently we do not intend to generalize the IAOM so that to fit even the effect of the ground electrode.) If τ = τBragg, the fitting of the HR IAWF to the estimation is so excellent that practically it has the same value as the estimation, therefore it is not illustrated in Fig. 7. We admit that the error of the estimation may even be somewhat greater than the fitting error of the P IAWF (see Fig. 11).

It is visible that towards the middle region (vertically, 7 mm < Y < 13 mm) the slowly varying P IAWF generates also slowly varying efficiency curves. Despite of the error of the estimation, we find that it differs slightly but certainly from the measurement. That is one of the reasons why we do not think that in the reality the IAWF would be so slowly varying and could be described by such a few parameters. The more complex behavior may be caused by the filaments on the electrode surface which are improving the conductivity (Fig. 4).

Despite of the slight differences, the simulation of the P IAWF describes the measured values quite well. The amplitude of the fitted P IAWF increases with Y, what is physically realistic; it implies the limited conductivity of the electrode. The locations of the descending parts of the efficiency curves are not symmetric because the χ angle (introduced in Section 4.4.1) is not zero but approximately 0.1°

The angular dependence of the efficiency

The accurate illustration of the efficiency as a function of the angle of incidence is not easy, since it changes four orders of magnitude. Even so, we find that it is the most expressive way of the presentation of the correspondence. The most apparent illustration would be a high definition video (where Y changes with the elapsed time). Instead of this, we picked some Y values, for which the efficiency – angle of incidence function is shown in Figs. 8, 9, and 10.
Fig. 8 The measured and modeled diffraction efficiencies are shown as functions of the angle of incidence for several X and Y values. The green line represents the simulation assuming P IAWF without the ground electrode, the red line shows calculations assuming the HR IAWF with the ground electrode. The black line represents the measurements. For each value of (XY) there are two subfigures to illustrate the diffraction curves at different scales with 1× and 10× zoom to show the agreement.
Fig. 9 The measured and modeled diffraction efficiencies are shown as functions of the angle of incidence for several X and Y values. The green line represents the simulation assuming P IAWF without the ground electrode, the red line shows calculations assuming the HR IAWF with the ground electrode. The black line represents the measurements. For each value of (XY) there are four subfigures to illustrate the diffraction curves at different scales with 1×, 10×, 20× and 100× zoom to show the agreement.
Fig. 10 The measured and modeled diffraction efficiencies are shown as functions of the angle of incidence for several X and Y values. The green line represents the simulation assuming P IAWF without the ground electrode, the red line shows calculations assuming the HR IAWF with the ground electrode. The black line represents the measurements. For each value of (XY) there are four subfigures to illustrate the diffraction curves at different scales with 1×, 10×, 20× and 100× zoom to show the agreement.
For each value of (X, Y) there are two or four subfigures to illustrate the diffraction curves at different scales with 1× , 10× , 20× and 100× zoom to show the agreement. At the same time, it is important to emphasize that the goodnesses of fit of the selected functions are completely representative for all other Y values, and there is no other Y value whose goodness of fit would be notably worse, regarding the whole range of the measurement, along with 2 × 92 values of X and Y. For Figs. 8-11, in case of the P IAWF, the ground electrode is not considered, in case of the HR IAWF, it is.
Fig. 11 The measured and modeled diffraction efficiencies are shown as functions of the angle of incidence in the proximity of the Bragg angle. X = 3 mm for all subfigures. The black line represents the measurement, the blue line shows the estimation of the elimination of the ground electrode effect from the measurement. The red line represents the simulation assuming the HR IAWF with the ground electrode, the green line shows calculations assuming the P IAWF without the ground electrode. The same colors denote the same quantities in Figs. 7-11.
If it is calculated for both the polynomial and the HR IAWFs, their behavior is practically the same near the Bragg angle (i.e. the additive, strongly varying signal).

The illustration of the close proximity of the Bragg angle

In Figs. 8-10, the close proximity of the Bragg angle is not visible so it is shown in Fig. 11 for some typical cases. Here again, the correspondence is representative for all (X, Y) values.

The analysis of the correspondence

As mentioned earlier, by the IAOM, our present intention is the estimation of the effect of the transducer and not the ground electrode. Evidently, if practical importance arises for the latter one, that may also be examined accurately. That is why in the model the ground electrode is considered in the simplest way: a uniform 2D IAWF is calculated (the optical calculations remain 3D). The horizontal length of the ground electrode is determined by simple measurement, thus one scalar complex amplitude is the only free parameter.

Analyzing the proximity of the Bragg angle (Fig. 11) we find that the efficiency curves of the simulation and the measurement are quite similar. However they do not correspond within the error limit, because the calculated case does not agree accurately with the reality. For the better accuracy a precise fitting would be necessary again. Thus, the inaccurate fitting complicated our original plan to fit parameters to the measurement without the effect of the ground electrode. Without that effect – according to our experiences – we could fit quite accurately and directly to the measurement, i.e. as in Sections 4.1 and 4.3, and not to an estimation derived from the measurement (i.e. in Fig. 7).

Analyzing all the results we conclude that the polynomial IAWF describes quite well the measured quantities. The correspondence is extraordinarily good if a more complicated, HR IAWF is allowed.

Here a question arises: is the number of the measurement points greater than the number of the fitting parameters? The answer is yes, it is certainly greater, since the number of the measurement points is 1300 × 2 × 92, what is about ten times more than the number of the real fitting parameters (435 × 27 × 2). However, we find that it is not evident to determine the exact rate of the effective measurement points and the independent fitting parameters. In case of the polynomial IAWF, the situation is clear, as there are very few fitting parameters, so there this question is obsolete.

To be very precise, searching the verification in a rigorous sense, two more questions arise. First, whether the goodness of fit of the P IAWF implies the correctness of the AO model? As mentioned earlier, the IAWF is far from uniform, and the minimal parameters used for the P IAWF are necessary for an approximate description. The pi parameters agree well with measurement (Table 1), the qi parameters set a 1D function (Eq. (7), Fig. 5), while the measured and simulated efficiency values agree quite well for all the three independent variables (τ, X, Y), see Figs. 7-10. That is why we find indisputably that the very good correspondence verifies the correctness.

Summarizing the analysis for the second question, again we find it certainly that the small difference may be caused by the error of the measurement.

In general, we conclude that the simulation describes the phenomenon within the error limit of the measurement. As all the measured effects and tendencies are well and reasonable simulated, thus it is verified experimentally. The complex model, which is the result of more than six years of intensive research, is finally completed. Hopefully it will contribute to the further optimization of the AO devices.

5. Conclusion

In this paper, a complex model of the AO interaction is presented. The optical parts of the simulation have been already demonstrated in Refs [3

3. G. Mihajlik, P. Maák, A. Barocsi, and P. Richter, “Novel accurate computer algorithm for modeling light propagation and diffraction in inhomogeneous, anisotropic medium – Applied to the acousto-optic interaction,” Opt. Commun. 282(10), 1961–1968 (2009). [CrossRef]

5

5. G. Mihajlik, A. Barocsi, and P. Maák, “Measurement and general modeling of optical rotation in anisotropic crystal,” Opt. Commun. 310(1), 31–34 (2014). [CrossRef]

], it has now been completed in this paper by the vectorial, 3D, anisotropic model of the acoustic wave propagation. We deem that the established complex model is rapid, quite general and uniquely accurate. We had the intention to verify it experimentally. However the verification is not obvious, since in reality the generated initial acoustic wavefront is not uniform, but contains numerous unknown input parameters which are necessary for the accurate description. To handle the problem, we established an inverse AO method (IAOM) which estimates a possible set of the parameters, i.e. it fits the parameters to the experimental data. The basis of the verification is that the number of measurement points is essentially larger than number of these parameters. The measurement and the model correspond well within the error limit for all measurement values, thus the complex AO simulation is verified experimentally. Both the model and the IAOM may contribute notably to the optimization of the AO devices.

Acknowledgments

References and links

1.

J. Xu and R. Stroud, “Acousto-Optic Interaction,” in Acousto-Optic Devices (Wiley-Interscience, 1992). pp. 61–94.

2.

A. Korpel, Acousto-Optics (Marcel Dekker Inc., 1997).

3.

G. Mihajlik, P. Maák, A. Barocsi, and P. Richter, “Novel accurate computer algorithm for modeling light propagation and diffraction in inhomogeneous, anisotropic medium – Applied to the acousto-optic interaction,” Opt. Commun. 282(10), 1961–1968 (2009). [CrossRef]

4.

G. Mihajlik, P. Maák, and A. Barocsi, “Simulation of light propagation in anisotropic, optically active and slightly inhomogeneous medium, concerning the acousto-optic interaction,” Opt. Commun. 285(9), 2255–2265 (2012). [CrossRef]

5.

G. Mihajlik, A. Barocsi, and P. Maák, “Measurement and general modeling of optical rotation in anisotropic crystal,” Opt. Commun. 310(1), 31–34 (2014). [CrossRef]

6.

Y. Hao and R. Mittra, FDTD Modeling of Metamaterials, Theory and Applications (Artech House, Inc., 2009).

7.

A. Deinega, S. Belousov, and I. Valuev, “Hybrid transfer-matrix FDTD method for layered periodic structures,” Opt. Lett. 34(6), 860–862 (2009). [CrossRef] [PubMed]

8.

G. Dhatt, E. Lefrançois, and G. Touzot, Finite Element Method (Wiley, 2012).

9.

R. J. LeVeque, Finite Difference Methods for Ordinary and Partial Differential Equations: Steady-State and Time-Dependent Problems (SIAM, 2007).

10.

B. A. Auld, Acoustic Fields and Waves in Solids (Krieger Pub Co. 1990).

11.

B. E. A. Saleh and M. C. Teich, “Polarization and Crystal Optics,” in Fundamentals of Photonics (Wiley & Sons Inc. 1991). pp. 216, 6.310–6.311.

OCIS Codes
(070.1060) Fourier optics and signal processing : Acousto-optical signal processing
(160.1050) Materials : Acousto-optical materials
(230.1040) Optical devices : Acousto-optical devices
(070.7345) Fourier optics and signal processing : Wave propagation
(250.4110) Optoelectronics : Modulators

ToC Category:
Optical Devices

History
Original Manuscript: January 15, 2014
Revised Manuscript: March 14, 2014
Manuscript Accepted: March 14, 2014
Published: April 21, 2014

Citation
Gábor Mihajlik, Attila Barócsi, and Pál Maák, "Complex, 3D modeling of the acousto-optical interaction and experimental verification," Opt. Express 22, 10165-10180 (2014)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-9-10165


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References

  1. J. Xu and R. Stroud, “Acousto-Optic Interaction,” in Acousto-Optic Devices (Wiley-Interscience, 1992). pp. 61–94.
  2. A. Korpel, Acousto-Optics (Marcel Dekker Inc., 1997).
  3. G. Mihajlik, P. Maák, A. Barocsi, P. Richter, “Novel accurate computer algorithm for modeling light propagation and diffraction in inhomogeneous, anisotropic medium – Applied to the acousto-optic interaction,” Opt. Commun. 282(10), 1961–1968 (2009). [CrossRef]
  4. G. Mihajlik, P. Maák, A. Barocsi, “Simulation of light propagation in anisotropic, optically active and slightly inhomogeneous medium, concerning the acousto-optic interaction,” Opt. Commun. 285(9), 2255–2265 (2012). [CrossRef]
  5. G. Mihajlik, A. Barocsi, P. Maák, “Measurement and general modeling of optical rotation in anisotropic crystal,” Opt. Commun. 310(1), 31–34 (2014). [CrossRef]
  6. Y. Hao and R. Mittra, FDTD Modeling of Metamaterials, Theory and Applications (Artech House, Inc., 2009).
  7. A. Deinega, S. Belousov, I. Valuev, “Hybrid transfer-matrix FDTD method for layered periodic structures,” Opt. Lett. 34(6), 860–862 (2009). [CrossRef] [PubMed]
  8. G. Dhatt, E. Lefrançois, and G. Touzot, Finite Element Method (Wiley, 2012).
  9. R. J. LeVeque, Finite Difference Methods for Ordinary and Partial Differential Equations: Steady-State and Time-Dependent Problems (SIAM, 2007).
  10. B. A. Auld, Acoustic Fields and Waves in Solids (Krieger Pub Co. 1990).
  11. B. E. A. Saleh and M. C. Teich, “Polarization and Crystal Optics,” in Fundamentals of Photonics (Wiley & Sons Inc. 1991). pp. 216, 6.310–6.311.

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