1. Overview of existing AOTF system

The Acoustic-Optic Tunable Filter (AOTF) is a new conception of mono-wave-length generator which is different from traditional prism and grating. The wavelength selection is made possible by means of optic wave and acoustic wave reaction inside a birefringence crystal under a phase matching condition. Optic wavelength is changed simply by changing the acoustic frequency applied on the crystal. There are no moving parts involved in the system so a quick scan of spectrum can be realized electronically.

The Acousto-Optic Tunable Filter (AOTF) was first made practice in 1967[1]. But the most usable and useful AOTF device was made after the T_eO₂ was and the noncollinear optic arrangement was put forward by I.C.Chang [2]. Most of the early work was based on the theoretical contributions of I.C.Chang [2] and T.Yano and A. Watanable [3] in which the momentum matching and phase match conditions are commonly accepted. In 1985, Mo Fuqin, and in 1987, Epikhin put forward more general equation set to describe the accurate relations between the acoustic and optic wave reaction for AOTF system [4][5]. In 1999, B. Xue, K. Xu and H. Yamamoto contributed a special “equivalent point design” based on the parallel-tangent phase match condition and pointed out that around the “equivalent point design” of 56° incident angle, the AOTF system may reach its optimum design [6]. Succeeding to [6], this paper is going to discuss another possible optimum parameter in AOTF design: to enlarge the acceptance angle and improve the spectrum resolution.

2. The angular aperture narrowing behavior inside the crystal:

A well known feature of AOTF is it works well with cone beam incidence. But if the cone angle is so large that the going-through incident beam and the filtered beam overlapped, the filtered beam is difficult to be separated. The “separation angle” which is measured between the incident direction and the filter output direction outside the crystal determines the maximum cone angle or “acceptance angular aperture”. Once the crystal cutting angles are determined, the separation angle is fixed and so is the angular aperture. In this paper, we are trying to narrow down the cone angle inside the crystal by off-perpendicularly incidence so that the equivalent acceptance angular aperture is enlarged. The narrowed beam inside the crystal also helps to reduce the spectrum widen due to the angular phase mismatching.

 

Fig. 1. Beam unevenly narrowed after going through a boundary

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Notice that the optic wave vector K is still obey the deflection law inside the birefringence crystal, Fig. 1 describes the idea that a round section beam is unevenly narrowed to an ellipse section beam after going through the air-crystal boundary off-perpendicularly. The degree of narrowing is the function of the incident angle, the polarizing state and the optic wavelength.

 

Fig. 2. Vector diagram of acouctic-optic reaction

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θ1_ the incident off-perpendicular angle of the optic beam in the air. θs _ the incident surface orientation angle with respect to the crystal axis. θie and θio _ the optic vector angles inside the crystal with respect to the crystal axis of e-ray and o-ray respectively. Ki _ the optic incident vector. Kaoe and Kaeo _ the phase matched acoustic vector of o in e out and e in o out. Kio and Kie _ the optic incident wave vector inside the crystal. Kdo and Kde _ the optic deflection vector inside the crystal. Ka _ the acoustic field vector.

To describe the narrowing behavior mathematically, we need to introduce some angles and vectors, as plotted in Fig. 2. In the incident and refraction axes defined plane, using refraction law of sin(θ1)/n1=sin(θ2)/n2 and substitute n2 with n_ie or n_io [6] respectively, refer to Fig. 2, we have:

The numerical computing results of (1) (2) are plotted in Fig. 3 and Fig. 4. The “incident angle” denoted that at the horizontal axis of the figures is the off-perpendicular angle to the incident surface rather than the incident angle inside the crystal. For a cone beam with the divergence or convergence angle of Δθ, the narrowing behaviors are shown in Fig. 3 with an arbitrary selected θ_s=55°. Fix the cone angle of Δθ to 6° and change θ_s at 10°, 35°, 60°, the results are shown in Fig. 4.

Conclusions:

(a) From Fig. 3 we see that the larger the off-perpendicular angle outside the crystal (incident angle), the smaller the cone angle inside the crystal. From Fig. 4 we see that the orientation of the incident surface dose not influence the narrowing behavior very much especially in the range of a large off-perpendicular angle.

 

Fig. 3. Cone angle narrowed inside crystal via the off-perpendicular angle with θ_s=55°.

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Curves plotted with the changes in cone angle Δθ. (b) For the extraordinary ray, there is no problem of the Brewster Angle. A large off-perpendicular incident angle of 45° may narrow the beam to 75% as that of perpendicular incident or enlarge the acceptance angular aperture 1.33 times. With an off-perpendicular angle of 60°, the acceptance angular aperture may be doubled. The reflection loss arisen from the large off-perpendicular incident angle can be reduced simply by a properly designed coating on the incident surface.

 

Fig. 4. Beam with cone angle of 6° narrowed inside the crystal via off-perpendicular angle.

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Curves plotted with changes in incident surface orientation angle θ_s. (c) For the ordinary ray, incident surface coating does not reduce all the reflecting loss because of the Brewster Angle. The Brewster Angle for TeO₂ is about 66.2° (arctan2.27). Making the off-perpendicular incident angle less than 60° would be better for balancing the reflecting loss and the beam narrowing.

 

Fig. 5. Crystal diagram (left) and vector diagram (right) of the normal design. Vp and Vg _ the acoustic phase velocity and group velocity.

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(d) The acoustic vector angle Ka is decided solely by the optic incident angle. If the angle between the optic incident surface and the transducer affix surface is much larger than 90°, as plotted in Fig. 5, the acoustic-optic interaction length is shorten or a larger crystal is required to keep the necessary interaction length because the longer the interaction length the higher spectrum resolution is obtained. The off-perpendicular incident can solve the problem in addition to the acceptance angular aperture improvement. Fig. 6 gives an idea of the improved AOTF cell, where Vp and Vg are the acoustic phase velocity and group velocity inside the crystal respectively.

 

Fig. 6. Crystal diagram (left) and vector diagram (right) of the improved design. Vp and Vg _ the acoustic phase velocity and group velocity.

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(e) The output surface of the crystal has to be cut perpendicularly to the filtered beam or with a “compensation angle” as suggested by T. Yano as angle β [3], otherwise the narrowed cone beam would be widen up after going through the crystal-air boundary.

3. An off-perpendicular incident arrangement for the equivalent point:

If the design is based on the “equivalent point” where θi=55.98°[6], some further calculations are necessary to confirm the equivalence properties are still hold.

The vector diagram is shown in Fig. 2. The optic beam in the air is input to the surface with the angle of θ1 and separated into two beams of o-ray and e-ray according to the polarizing states of the incident beam inside the crystal. The o-ray and e-ray inside the crystal are no longer collinear and they are denoted as Kio and Kie. For the two closed vector triangles of Kdo+Kaeo=Kio and Kio+Kaoe=Kde, another attempt is made to find the “equivalent point of equivalence”. A series of calculations are carried out using equations from [6] :

Where the subscript of “aeo” means the acoustic vector angle or frequency for “e in o out”, “ie” means extraordinary beam incident and “do” means ordinary beam diffracting.

Where the subscript of “aoe” means the acoustic vector angle or frequency for “o in e out”, “io” means ordinary beam incident and “de” means extraordinary beam diffracting.

To investigate the properties of the equivalent point, we make (3)–(5)=0 and (4)–(6)=0 and substitute θ_i with θ_io and θ_ie from (1) (2) accordingly, the computing results are plotted in Fig. 7.

 

Fig. 7. The position differences of the equivalent point via incident surface orientation angle. The differences are compared between acoustic frequency with different optic wavelengths (curve “*” and “+”) And between the acoustic frequency and the acoustic phase match angle (curve “-”).

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In Fig. 7, curve “*” indicates the equivalent point difference via the incident surface orientation angle inside the crystal compared between optic wavelength of 1.0µs and 0.5µs. Curve “+” is compared between 2.5µs and 1.0µs. It is convinced by experiments that in the shorter wavelength range, the difference becomes larger. Curve “-” indicates the equivalent point difference via the incident surface orientation angle inside the crystal compared between acoustic frequency and acoustic phase match angle at optic wavelength 1.5µs. In the real applications, it is not uncommon to have a beam cone angle of 0.2° or larger, even for the laser beams. This would tolerate the AOTF system designed with the conception of “off-perpendicular equivalent point” capable of working in the near infrared range.

If a design requires both the equivalent point and large acceptance angle properties, a compromise is reached by choosing θs=38° where it is well off the Brewster angle and a coating of reducing reflection is easier to be designed. The beam is narrowed to 79% as much as perpendicular incidence.