## Complex, 3D modeling of the acousto-optical interaction and experimental verification |

Optics Express, Vol. 22, Issue 9, pp. 10165-10180 (2014)

http://dx.doi.org/10.1364/OE.22.010165

Acrobat PDF (1810 KB)

### 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

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]

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]

^{3}, 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, 9] – 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 ~10

^{10}.

*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 cm

^{3}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

*c*is the elastic stiffness tensor,

_{KL}*ρ*is the density of the medium and

*u*

_{i}is the displacement vector. For simplicity, we use an abbreviated subscript notation according to [1, 10]. Introducing the complex amplitude

*U*

_{i}for a given angular frequency Ω: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**:Expanding and grouping the terms of the left side and returning to the classical notation yields:The

*A*

_{k}matrices are yielded from the

*c*elastic stiffness tensor. The wave equation may be written further:

_{KL}*K*

_{x},

*K*

_{y}) pair, the only unknown variable is

*K*

_{z}. The mentioned determinant thus makes a sixth degree (polynomial) equation. For small values of (

*K*

_{x},

*K*

_{y}), the polynomial equation describes three forward and three backward propagating eigenpolarizations. In case of high values of (

*K*

_{x},

*K*

_{y}), the equation may have non-realistic roots also e.g. Re{

*K*

_{z}}>0 and Im{

*K*

_{z}}<0 which implies that some boundary conditions cannot arise. For each

*K*

_{z}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]

### 2.2 The complex AO model

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]

### 2.3 The issues of determining physical coefficients of TeO_{2}

**310**(1), 31–34 (2014). [CrossRef]

*c*): there are more accurate values in the literature, yet their accuracy is questionable [1, 2, 10]. 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].

_{KL}*p*

_{44},

*p*

_{66}) which affect the measured slow shear wave mode, however the effect of the second one is smaller by orders of magnitude, hence even

*p*

_{66}is negligible.

## 3. Experimental setup

_{2}. 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

**310**(1), 31–34 (2014). [CrossRef]

**310**(1), 31–34 (2014). [CrossRef]

*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

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]

*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.

^{2}, 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

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

*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.

#### The investigation of the initial acoustic wavefront (IAWF)

*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)

*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

*is in linear relationship with the IAWF (in the linear domain). So resolving the IAWF into basis elements (segments), we calculated the electric field*

_{d,spot}*(τ,*

_{d,spot}*X*,

*Y, k*

_{x}

*,k*

_{y}) 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

*(τ,*

_{d,spot}*X*,

*Y, k*

_{x}

*,k*

_{y}) 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. Sincethe 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

*Y*(equidistantly partitioned in the whole range) on the filter, for

*X*

_{1}= 6 mm and

*X*

_{2}= 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 X*

_{1}

*or X*

_{2}, separately, but could not find one, which would satisfy all the measured values at the same time (

*X*

_{1}

*and X*

_{2}) with the same, good correspondence. In case of the Bragg angle the IAWF we found fitted the efficiencies η(τ

_{Bragg},

*X*,

*Y*) relatively well for both

*X*

_{i}, for other angles the difference |η

_{mes}– η

_{sim}| was significantly higher than the error of the measurement. That raises the question, what causes the deviations?

*X*

_{1}or

*X*

_{2}, affirms the correctness, since there are significantly more measured values of η (τ,

*Y*) than the independent values of the IAWF.

*X*coordinates is that the inhomogeneity of the medium quality filter is too high. The question of inhomogeneity is particularly discussed in [5

**310**(1), 31–34 (2014). [CrossRef]

**310**(1), 31–34 (2014). [CrossRef]

_{off}) is certainly smaller than the error of the IAWF fitting: η

_{off}< |η

_{sim}– η

_{mes}|.

*X*

_{i}value than for both

*X*

_{i}.

### 4.4 Detailed 3D investigation of the diffraction in AO deflector

**310**(1), 31–34 (2014). [CrossRef]

*X*

_{1}= 3 mm,

*X*

_{2}= 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. As mentioned earlier, we experienced a small diffraction even when the cell was switched off (η

_{off}). Since small efficiencies (η < 1.2%) are measured, η

_{off}is subtracted from η

_{mes}.

#### 4.4.1 The polynomial (P) IAWF

*p*

_{4}describes the relative position). 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: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

*p*

_{1}, the IAWF is constant (in the hexagon).

_{mes}) and simulated (η

_{sim}) efficiencies are illustrated in the following (Section 4.4.3, Figs. 7-11).

*p*

_{2}and

*p*

_{3}correspond to the size of the basis element (the discrete resolution) used for the IAOM (

*p*

_{1}is determined by the 2D model).

#### 4.4.2 The high resolution (HR) IAWF

*Y*(

*p*

_{4}) is determined only by fitting, but the fit has an uncertainty of 0.05 mm.

#### 4.4.3 Discussion

#### The η(τ_{Bragg}, *X*, *Y*) curves

*Y*) in case of τ = τ

_{Bragg}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. (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 = τ

_{Bragg}, 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).

*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).

*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

*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. 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. 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

#### The analysis of the correspondence

*p*parameters agree well with measurement (Table 1), the

_{i}*q*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,

_{i}*X*,

*Y*), see Figs. 7-10. That is why we find indisputably that the very good correspondence verifies the correctness.

*X*

_{0}) has, however, a greater error of 0.1 mm, since the location of the incident beam spot and the micrometer gauge has to be aligned (the radius of the spot size is 0.38 mm, and

*X*

_{0}cannot be estimated by fitting with the IAOM). Toward the edges in the

*Y*direction, the diffraction efficiency changes orders of magnitude; here the maximal 0.03 mm discrepancy in

*Y*may cause greater relative error. The acoustical inhomogeneity of the cell is much less than in case of the investigated AO filter (Section 4.3), but it is still not completely negligible. Next to the side surfaces (top and bot-tom), the effect of the reflected acoustic waves may emerge, however it is surely smaller than the effect of the non-reflected waves, which rapidly vanish toward these surfaces (Fig. 7). We deem, that here the reflected waves have no practical significance.

## 5. Conclusion

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]

**310**(1), 31–34 (2014). [CrossRef]

## Acknowledgments

## References and links

1. | J. Xu and R. Stroud, “Acousto-Optic Interaction,” in |

2. | A. Korpel, |

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. |

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. |

5. | G. Mihajlik, A. Barocsi, and P. Maák, “Measurement and general modeling of optical rotation in anisotropic crystal,” Opt. Commun. |

6. | Y. Hao and R. Mittra, |

7. | A. Deinega, S. Belousov, and I. Valuev, “Hybrid transfer-matrix FDTD method for layered periodic structures,” Opt. Lett. |

8. | G. Dhatt, E. Lefrançois, and G. Touzot, |

9. | R. J. LeVeque, |

10. | B. A. Auld, |

11. | B. E. A. Saleh and M. C. Teich, “Polarization and Crystal Optics,” in |

**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

- J. Xu and R. Stroud, “Acousto-Optic Interaction,” in Acousto-Optic Devices (Wiley-Interscience, 1992). pp. 61–94.
- A. Korpel, Acousto-Optics (Marcel Dekker Inc., 1997).
- 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]
- 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]
- G. Mihajlik, A. Barocsi, P. Maák, “Measurement and general modeling of optical rotation in anisotropic crystal,” Opt. Commun. 310(1), 31–34 (2014). [CrossRef]
- Y. Hao and R. Mittra, FDTD Modeling of Metamaterials, Theory and Applications (Artech House, Inc., 2009).
- A. Deinega, S. Belousov, I. Valuev, “Hybrid transfer-matrix FDTD method for layered periodic structures,” Opt. Lett. 34(6), 860–862 (2009). [CrossRef] [PubMed]
- G. Dhatt, E. Lefrançois, and G. Touzot, Finite Element Method (Wiley, 2012).
- R. J. LeVeque, Finite Difference Methods for Ordinary and Partial Differential Equations: Steady-State and Time-Dependent Problems (SIAM, 2007).
- B. A. Auld, Acoustic Fields and Waves in Solids (Krieger Pub Co. 1990).
- 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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