## Abstract

The selection of the incident polar angle is very important in the entire design of the noncollinear acousto-optic tunable filter (AOTF). The authors discussed how the factors, including tuning range of wavelength, the acoustic frequency, the acousto-optic figure of merit, the spectral bandwidth, the spread of filtered beam and the wavelength dependence, influence the selection of the optimum incident polar angle. By an accurate theoretical analysis, a method of selecting the optimum incident angle was presented. The analysis was significant for improving the performance of the imaging AOTF from the visible to the infrared.

©2007 Optical Society of America

## 1. Introduction

The noncollinear acousto-optic tunable filter (AOTF) is a solid state device which has large angular aperture, fast and electronically tuning property with wide spectral range. It is widely used in the tuning of dye lasers, optical computing, spectral analysis and hyperspectral imaging [1–5]. TeO_{2} is the most commonly used birefringent crystal to fabricate a noncollinear AOTF. The operation of an AOTF is based on the phenomenon of light diffraction by acoustic wave propagating in the crystal. Light can be diffracted in a narrow spectral band centered on a chosen wavelength. The filtered wavelength is changed by tuning the frequency of the acoustic wave. The acoustic wave is generated by applying a radio frequency signal (rf) to a piezoelectric transducer bonded on the birefringent material. A change in the applied rf produces a variation in the acoustic wave frequency. For an imaging AOTF, the diffracted light will be received by a CCD when it exits from the AOTF. In the actual design of AOTF, the selection of the optimum incident polar angle is a key technology, which can influence the entire performance of AOTF. Thus, a discussion on the selection of the incident polar angle is significant in the design. In this topic, researchers have done some valuable work [6, 7]. For example, Binxue had made a discussion to the equivalent point of the incident polar angle realized by two polarized beams [6]. At the equivalent point, if the incident signal was a non-polarized beam, the same wave length portion of both the horizontal and the vertical polarizing part could be filtered by a single acoustic frequency. This arrangement was welcomed to the circumstance that a weak incident optical signal was involved in the spectrum analyzing, since the output signal could be doubled by summing the diffracted signals generated from the horizontal and the vertical polarizing part of incident beams respectively. But the conclusion of the equivalent point was only presented to fulfill the applications in the spectral analysis. As we know, usually in the imaging application of AOTF, only one of two eigen modes for the diffracted optical signal is received for imaging in common AO imaging device [2–5]. Obviously, it is necessary to give an overall investigation on the selecting of the optimum incident optical polar angle for the imaging AOTF. AO interaction equation is the basis of the selection of the optimum incident polar angle. Previous AO interaction equations have certain error degrading the performance of AOTF [8, 9]. In this letter, we gave an accurate AO interaction equation. Parameters such as tuning range of wavelength, acoustic frequency, the acousto-optic figure of merit, bandwidth, the spread of filtered beam with the bandpass and the wavelength dependence of the incident polar angle, are closely relevant with the value of incident polar angle. For the spectral imaging application, the function of the parameters is analyzed systematically and a method to the selecting of the optimum incident optical angle is presented.

## 2. Principle of acoustic-optic interaction

A design of noncollinear AOTF using TeO_{2} is based on AO interaction in [11̄0] plane. There are two eigen wave modes propagating in TeO_{2} when the rotatory property is considered: one is right-handed elliptical polarized mode (the direction of the ellipse’ long axis is parallel with main plane and is the same as that of extraordinary polarized light; we can call the light on this mode as right-handed e light), the other is left-handed elliptical polarized mode (the direction of ellipse’ long axis is perpendicular to main plane and is the same as that of ordinary polarized light; we can call the light on this mode as left-handed o light). If the incident light is right-handed e light, the diffracted one will be left-handed o light. Accordingly, the diffracted light will be left-handed o light when the incident one is right-handed e light. In common spectral imaging applications only one mode light is detected for imaging. Thus we assume that the incident light is right-handed e light and the diffracted one is left-handed o light in the following analysis.

A wave vector diagram of AO interaction is shown in Fig. 1. **k**
* _{i}*,

**k**

*and*

_{d}**K**

*are the incident optical wave vector, the diffracted optical wave vector and the acoustic wave vector, respectively. They are related by*

_{a}**k**

*+*

_{i}**K**

*=*

_{a}**k**

*. The direction of the acoustic wave propagation satisfies the parallel tangents momentum-matching condition. So the refractive indices of the incident beam (*

_{d}*n*

*) and the diffracted beam (*

_{i}*n*

*) are*

_{d}where *θ*
* _{i}* and

*θ*

*are the polar angle for incident and diffracted beams.*

_{d}*n*

*and*

_{o}*n*

*are the ordinary and extraordinary refractive indices in the direction perpendicular to the optical axis, respectively. They are the function of the optical wavelength (e.g.*

_{e}*n*

*=2.2597,*

_{o}*n*

*=2.4119 at 0.6328 µm;*

_{e}*n*

*=2.208,*

_{o}*n*

*=2.352 at 1.0 µm).*

_{e}*σ*is a physical constant relevant with specific rotation (

*ρ*) and satisfies

*σ*=

*λρ*/2

*πn*

*. σ has wavelength dependence (e.g.*

_{o}*ρ*=86.9 deg/mm,

*σ*=6.760677×10

^{-5}at 0.6328 µm;

*ρ*=29.5 deg/mm,

*σ*=3.432966×10

^{-5}at 1.0 µm) [10].

The parallel-tangents momentum-matching condition implies that

The acoustic angle *θ*
* _{a}* fulfilling Eq. (1) and Eq. (2) is given by

While the requirement on the acoustic frequency *f*
* _{a}* is

where *V*
* _{a}* is the acoustic velocity and

*λ*

_{0}is the optical wavelength in free space.

*V*

*is relevant with*

_{a}*θ*

*. From Eqs. (1)–(4), the accurate relationship between*

_{a}*θ*

*and*

_{a}*θ*

*is*

_{i}Comparing with Eq. (6), previous published designs had ignored the rotatory property of TeO2 and had introduced errors to some extent in calculating the optimum incident polar angle for an AOTF [8, 9]. Figure 2(a) gives the relationships between *θ*
* _{a}* and

*θ*

*from Eq. (6) and previous design equations. In Fig. 2(a), the filtered optical wavelength is 0.6328 µm for curve (1) [Eq. (6)] and curve (2) (Gass’ equation). The difference between our relationship and previous designs is obvious when*

_{i}*θ*

*is smaller than about 60°. Corresponding to a fixed acoustic angle of 80°,*

_{i}*θ*

*are 23.804°, 23.305° and 20.698° on curves (1), (2) and (3) respectively and*

_{i}*θ*

*from Eq. (6) is about 0.5° larger than that from Gass’ equation. This 0.5° shift caused by rotatory property can’t be missed, since high accuracy of the incident polar angle design is the basis of AOTF with good performance and modern technique can meet this accurate manufacture [11].*

_{i}From Eq. (1)–(3) and Eq. (5), an exact frequency tuning relationship, considering the rotatory property, is got. In order to indicate the obvious difference between our exact tuning relationship and other published, we have drawn the tuning curves of the acoustic frequency and the optical wavelength in Fig. 2(b).

## 3. Analysis of the optimum incident polar angle

Above analysis indicates that the rotatory property of TeO_{2} crystal is an indispensable factor to be taken into account in the design of AOTF. Considering the rotatory property is necessary in the design, since accurate design is more and more desired in applications such as spectral analysis, hyperspectral imaging, etc. On one hand, accurate relationship of the incident polar angle and the acoustic angle, which is presented in this letter, has made a good basis for the selection of the optimum incident polar angle. On the other hand, the selection of incident polar angle must synthetically consider the influence of many parameters in actual design of AOTF, such as tuning range of wavelength, the acoustic frequency, the acousto-optic figure of merit, bandwidth, the spread of diffracted beam, etc.

## 3.1 The influence of the acoustic frequency and the acousto-optic figure of merit

A desirable acousto-optic property of TeO_{2} is the big acousto-optic figure of merit M_{2} for the transverse acoustic wave propagating along [110] direction. The acousto-optic figure of merit in a certain direction satisfies the following expression [12]:

where *M*
_{2} is 1200×10^{-18} s^{3}/g at 0.6328 µm. The acoustic velocity *V*
* _{a}* can be calculated by

*V*

*(*

_{a}*θ*

*)=(*

_{a}*V*

^{2}

*sin*

_{ss}^{2}

*θ*

*+*

_{a}*V*

^{2}

*cos*

_{fs}^{2}

*θ*

*)*

_{a}^{1/2}, where

*V*

*is 616 m/s and*

_{ss}*V*

*is 2104 m/s (here, it should be mentioned that Eq. (7) is an approximate equation for it takes into account only the photoelastic coefficients p*

_{fs}_{11}and p

_{12}without considering the coefficient p

_{44}). From Eq. (7) and Eq. (6), the relationship between the acousto-optic figure of merit

*M*

_{2}(

*θ*

*,*

_{a}*θ*

*) and the incident polar angle*

_{i}*θ*

*can be got as shown in Fig. 3(a). From Fig. 3(a), we can see that the acousto-optic figure of merit decreases with the increasing of the incident polar angle. In the application of AOTF based on TeO*

_{i}_{2}crystal, big acousto-optic figure of merit is welcomed and needed, so the selected incident polar angle should be relatively small.

Besides, the frequency of acoustic wave at a certain optical wavelength is dependent on the incident polar angle *θ*
* _{i}*. Figure 3(b) has shown the specific relationship.

At a certain optical wavelength, the acoustic frequency gets a peak corresponding to an incident polar angle of about 56°. Except for the peak, there are two different incident polar angles at the same acoustic frequency. In actual design, only the smaller incident polar angle is usually selected not only for saving the material but also for obtaining higher magnitude of the AO figure of merit and lower level of rf driving power. In the left range of the peak point (the range of smaller *θ*
* _{i}*), the needed acoustic frequency increases with

*θ*

*at a certain optical wavelength and decreases with the filtered center optical wavelength at a fixed*

_{i}*θ*

*. Here, actual selected incident polar angle should not be relatively big in the left range of the peak (the range of smaller*

_{i}*θ*

*), otherwise the needed acoustic frequency at relatively big*

_{i}*θ*

*will be much high especially in short visible wavelength range. Moreover, high frequency of acoustic wave will increase the attenuation of sound which is a major factor limiting the performance of the designed AOTF. Considering the influence of the acoustic frequency, actually designed incident polar angle should be selected at a relatively small value. From Fig. 3(b), we think it better when*

_{i}*θ*

*is smaller than 30° with*

_{i}*f*

*under 300 MHz (corresponding AO figure of merit, rf driving power and the attenuation of sound can hold in a proper range) in VIS-NIR.*

_{a}## 3.2 The influence of the optical bandwidth

For AOTF, the spectral bandwidth is calculated by Δ*λ*=1.8*πλ*
^{2}/*bL*sin^{2}
*θ*
* _{i}* [11].

*L*and

*b*denote AO interaction length and the dispersion constant, respectively. At certain

*b*,

*L*and

*λ*, the spectral bandwidth Δ

*λ*decreases with

*θ*

*. At practice, imaging AOTF is hoped to have a high spectral resolution. Thus the incident polar angle selected should not be too small, because the spectral bandwidth would be too wide. This is worth being paid attention extremely at the longer wavelength range (NIR). From Fig. 4, the incident angle selected should be bigger than 20° to make the corresponding spectral bandwidth narrower than 10 nm in 0.4–1.0 µm.*

_{i}## 3.3 The influence of the spread of the diffracted beam

It is implied in Eq. (3) that the diffracted polar angle *θ*
* _{d}* is the function of the wavelength

*λ*at a fixed incident polar angle. So the diffracted beam would spread with the bandpass of the spectral (it is measured by the spectral bandwidth around a center wavelength). We have shown the relationship between the spread angle of the diffracted beam with the unit bandpass (in crystal) and the incident polar angle in Fig. 5.

In Fig. 5, the relations are compared under a series of optical wavelength. Δ*θ*
* _{d}*/Δ

*λ*has a maximum at an incident polar angle of about 56°. At the same incident polar angle, Δ

*θ*

*/Δ*

_{d}*λ*is relatively big under the shorter waveband. In the left range of the peak, Δ

*θ*

*/Δ*

_{d}*λ*increases with

*θ*

*at a certain wavelength. Thus we should avoid the extreme point and select a smaller incident polar angle. It can keep the value of Δ*

_{i}*θ*

*/Δ*

_{d}*λ*smaller in VIS-NIR, which is very favorable for improving the imaging resolution of AOTF.

## 3.4 The wavelength dependence of the incident polar angle

It can be also known from Eq. (6) that, the incident polar angle has wavelength dependence at a fixed acoustic angle. We have discussed the variations of the incident polar angle *θ*
* _{i}* with the optical wavelength

*λ*and have shown the relationships between

*θ*

*and*

_{i}*λ*at a selected acoustic angle of 80° in Fig. 6. It is indicated that, the optimum incident polar angle is different (although this difference is not too obvious) at a fixed acoustic angle. The incident polar angle of the actually manufactured AOTF must be at a fixed value. Thus, the wavelength dependence must be considered together with the range of AOTF applications, when we select the value of the optimum

*θ*

*in the design of AOTF.*

_{i}From the above analysis, we realized that the value of the incident polar angle must be restricted within a certain range in order to keep good performance of AOTF. We would think it better that the optimum incident polar angle is selected at a value with 20–30°. The AOTF designed in this range of optimum incident polar angle would be fit for the imaging applications in VIS-NIS. For instance, at an incident polar angle of 23.804°, AOTF shows good performance: the acoustic angle is 80°; the acoustic frequency is in 71.5–224.2 MHz within the wavelength range of 0.4–1.0 µm; the spectral resolution in 0.4–1.0 µm is in the range of 1.22–7.61 nm; Δ*θ*
* _{d}*/Δ

*λ*is also in a proper range (e.g. it is 0.16 rad/µm at 0.6328 µm).

## 4. Conclusion

In summary, we have presented a relationship between the incident polar angle and the acoustic angle through considering the rotatory property of the interaction material. The relationship is justified to make a foundation for accurate design of AOTF. We have analyzed the effects of many factors on the selection of the optimum optical incident polar angle in VIS-NIR. The conclusion is that the optimum optical incident polar angle should be restricted in a certain range to make the AOTF satisfy its applications with good performance. Besides, we point out that a range of optimum incident polar angle for applications in VIS-NIR is in 20–30°. At a particular incident polar angle of 23.804°, we have given the parameters of the performance as an example, which is confirmed to be suitable for applications in VIS-NIR. Our work will stimulate the development of imaging AOTF with high accuracy and perfect performance.

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