1. Introduction

AOTF has wide applications in the tuning of dye lasers, optical computing, spectral analysis, hyperspectral imaging [1–3], and etc. TeO₂ is very suitable for the application in the large angular aperture AOTF which works on a noncollinear mode because of its intrinsic crystal structure. As is known, the operation of an AOTF is based on the phenomenon of light diffraction by acoustic wave propagating in the A-O crystal. Light can be diffracted in a narrow spectral band centered on a chosen wavelength. The filtered wavelength is changed by tuning 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. Our previous works had approved that considering the birefringence of the interaction material together with its rotatory property was an effective method of ensuring the accuracy of the design [4, 5]. In the actual applications, especially in the hyperspectral imaging area, the spectral resolution is a key index of evaluating the performance of the designed AOTF. The spectral resolution is related with certain factors such as the optical wavelength, the length of piezoelectric transducer, the fabrication parameters of the TeO₂ crystal and etc. If simply through adjusting the above factors, it can be put out that the enhancement of the spectral resolution is much limited. However higher and higher spectral resolution is demanded to keep up with the fast development of AOTF applications in imaging area. In this letter, we introduce a “double-filtering” method that can obviously enhance the spectral resolution. The main technology of “double-filtering” method is discussed; the effectivity of “double-filtering” on spectral resolution enhancement with good imaging quality is confirmed.

2. The theory of acousto-optic interaction

A design of noncollinear AOTF with TeO₂ is based on A-O interaction in [11̄0] plane. Some designs had introduced certain error either from birefringence approximation or the neglect of the rotatory property of TeO₂ [6, 7]. Here, an A-O theory with high design accuracy is used considering the birefringence together with the rotatory property. There are two eigen wave modes propagating in TeO₂: right-handed elliptical polarized mode and left-handed elliptical polarized mode (the direction of the ellipse’ long axis on these two modes are parallel with main plane and perpendicular to main plane, respectively). If the incident beam is right-handed elliptical polarized, the diffracted one will be left-handed elliptical polarized. Accordingly, the diffracted beam will be left-handed elliptical polarized when the incident one is right-handed elliptical polarized. Figure 1 is a wave vector diagram of A-O interaction. k _i+K _a=k _d, k _i, k _d and K _a are the incident optical wave vector, the diffracted optical wave vector and the acoustic wave vector, respectively. The direction of the acoustic wave propagation satisfies the parallel tangents momentum-matching condition. The incident beam is assumed to be right-handed elliptical polarized, the diffracted one is left-handed elliptical polarized (only one mode beam is detected for common spectral imaging applications).

 

Fig. 1. The wave vector diagram of noncollinear AOTF. [001] axis is the optic axis.

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The refractive indices of the incident beam (n_i) and the diffracted beam (n_d) are

where θ_i and θ_d are the polar angle for incident and diffracted beams. σ is relevant with specific rotation ρ(σ=λρ/2πn_o) and has wavelength dependence (ρ=86.9 deg/mm, σ=6.7607×10^-5 at 0.6328 µm) [8]. n_o and n_e are the ordinary and extraordinary refractive indices in the direction perpendicularly to the optical axis, respectively. They are the function of the optical wavelength λ ₀ in vacuum [9],

where A=2.5844, B=0.1342, C=1.1557, D=0.2638, E=2.8525, F=1.5141, G=0.2631. Under the momentum-matching condition, the wave-vector propagation polar angles are

θ_a is the acoustic wave angle. The relationship between θ_a and θ_i is

3. The study of the spectral resolution equation

The optical bandpass characteristics of AOTF are determined by the momentum mismatch caused by the deviation of wavelength from the exact momentum-matching condition. The common equation of the spectral bandwidth was Δλ=1.8πλ ₀ ²/bLsin² θ_i (L and b denote the length of A-O interaction and the dispersion constant respectively) if the rotatory property is out of consideration [9–11]. In this section, we will deduce the expression of the spectral bandwidth Δλ considering both the birefringence and the rotatory property of TeO₂.

Commonly, the diffraction efficiency η can be expressed by

η ₀ is the peak diffraction efficiency which is commonly related with Raman-Nath parameter υ (in TeO₂ AOTF, η ₀=sin²(υ/2). υ depends on the factors such as photoelastic properties and the geometry of A-O crystal, acoustic amplitude. When these factors are selected properly, η ₀ can reach its maximum of a unit) [2, 9]. L is the length of A-O interaction. δ is the mismatch factor. Δk₁ denotes the momentum mismatch. From Fig. 1, we can get the expression of Δk₁,

We define ϕ_i and ϕ_a as the azimuth angle of the incident optical wave vector and the acoustic wave vector, respectively. Here cosα≈1 for α is small enough. The direction cosine of K and k _i is derived by (sinθ_acosϕ_a, sinθ_asinϕ_a, -cosθ_a) and (sinθ_icosϕ_i, sinθ_isinϕ_i, cosθ_i). Thus,

For the large angular aperture AOTF, 1^st-order derivative of Δk₁ with respect to angular deviations Δθ_i and Δϕ_i is required to be zero. ϕ_i=ϕ_a from Eq. (8). We assume K/k_d=a. Then,

The Taylor series expansion of momentum mismatch Δk₁ near Δk₁=0 is,

It can be got from Eq. (9) that,

b′ is defined as an dispersive constant with wavelength dependence, and b′ can be calculated by differentiation of Eqs. (1)–(2). From Eq. (7), the condition of half-peak diffraction efficiency (η=η ₀/2) occurs when δ=Δk₁ L/2π=±0.45. Thus, the full spectral bandwidth can be expressed,

4. The discussion of double-filtering method

4.1 Principle of double-filtering

AOTF with high spectral resolution is demanded in the hyperspectral imaging application. It means that narrow spectral bandwidth at the center optical wavelength is welcomed for perfect spectral recognition capability. However, the effect of narrowing the spectral bandwidth is restricted if the investigation is simply focused on the design of a single AOTF. So a method called “double-filtering” is presented to narrow the spectral bandwidth [12, 13]. The principle of this method is: in the AOTF based hyperspectral imaging system, two AOTFs are placed fore-and-aft. They are controlled by two independent rf signals [Fig. 2(a)]. For an AOTF, the diffracted beam has a spectral bandwidth at a chosen center wavelength when tuning the frequency of rf signal. As the two AOTFs work synchronously, the incident light will experience twice diffraction process. Through selecting the frequency interval of the two rf signal properly, the common region of the corresponding filtered spectral bandpass can be regulated and the final spectral bandwidth will be made narrow obviously.

 

Fig. 2. (a). The principle picture of double-filtering method. (b) The situation of the spectral bandwidth after double-filtering. η ₀=1 is assumed. λ₁₀ is fixed at 500 nm.

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Figure 2 illuminates the principle of double-filtering method clearly. As shown in Fig. 2(a), the beam normally incidents on the surface of AOTF₁ and is diffracted by AOTF₁. Then the output light from AOTF₁ incidents on AOTF₂ and experiences a second diffraction. (We assume that the incident light of AOTF₁ is right-handed elliptical polarized and that of AOTF₂ is left-handed elliptical polarized; the direction of the acoustic wave vectors is same for AOTF₁ and AOTF₂ with θ_a=80° and L=4 mm). Figure 2(b) gives the change of the spectral bandwidth after double-filtering. In Fig. 2(b), the centered wavelength of AOTF₁ (λ ₁₀) is 500 nm, the spectral bandwidth after double-filtering (Δλ ₁₂) has been narrowed in certain extent. λ ₁₀ and λ ₂₀ are respectively the center wavelength of AOTF₁ and AOTF₂, they have interval of Δ′ (Δ′=λ ₂₀-λ ₁₀). η ₁ and η ₂ is the diffraction efficiency of single AOTF₁ and single AOTF₂. η ₁₂ is the diffraction efficiency after double-filtering, it fulfills η ₁₂=η ₁ η ₂ with wavelength dependence. Δλ ₁₀, Δλ ₂₀ and Δλ ₁₂ are the corresponding spectral bandwidth. From Fig. 2(b), we also see a merit of double-filtering method on out of band rejection, and it is extremely meaningful to improve the spectral purity of the final optical signal (The sidelobes is a factor to corrupt the image quality. When a single AOTF is used, the first of the sidelobes is only 13 dB below the main signal. With double-filtering, the first of the sidelobes is 26 dB below the main signal, and this low level of the sidelobes is welcomed in imaging application) [12].

4.2 The process and the law of double-filtering

The curves of η ₁ and η ₂ can be got from Eq. (6). By η ₁₂=η ₁ η ₂, we can draw out the curve of η ₁₂, from which the corresponding double-filtering spectral bandwidth Δλ ₁₂ can be calculated.

In the study, we change the interval Δ′ when the center wavelength in AOTF₁ is fixed at a series of value. The relationship between double-filtering spectral bandwidth Δλ ₁₂ and the centered wavelength interval Δ′ is given in Fig. 3(a). From Fig. 3(a), Δλ ₁₂ decreases with the interval Δ′ at a series of certain λ ₁₀. And Δλ ₁₂ becomes wider and wider with the increasing of the center wavelength in AOTF₁ λ ₁₀, if the interval Δ′ is selected at a certain value. This illustrates that adjusting the interval Δ′ is effective to narrow the spectral bandwidth by double-filtering method. In addition, the maximum of the double-filtering diffraction efficiency (η ₁₂)_max changes with the common spectral region of the two AOTFs [in Fig. 3(b)]. (η ₁₂)_max decreases with the interval Δ′ at a series of reference center wavelength λ ₁₀, the trend of decrement in longer waveband is gentler than that in shorter waveband. And (η ₁₂)_max becomes biger and biger with the increasing of the reference wavelength λ ₁₀ if Δ′ is selected at a certain value. From Fig. 3 we know that, it is unsuitable to increase the interval Δ′ without restriction although increasing Δ′ in double-filtering can narrow the spectral bandwidth obviously. The reason is that the detection of the weak spectral signal (corresponding to a small (η ₁₂)_max) would be difficult when Δ′ is so big. In double-filtering process, the selection of a proper Δ′ value is necessary in order to keep both Δλ ₁₂ and (η ₁₂)_max in suitable area.

 

Fig. 3. (a). Double-filtering spectral bandwidth Δλ ₁₂ versus the centered wavelength interval Δ′ at a series of reference wavelength λ ₁₀. (b). The maximum diffraction efficiency (η ₁₂)_max versus the centerd wavelength interval Δ′ at a series of reference wavelength λ ₁₀.

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To evaluate the quality of the detected spectral signal, we has defined a factor Q which is the division of the spectral bandwidth Δλ and the maximum of corresponding diffraction efficiency (η)_max. By analysis, Q can represent the spectral sharpness of a detected signal. So a signal with a smaller Q is more easily to be detected by the imaging device (e. g. CCD) in better imaging performance. In Fig. 4(a), the relationship between the factor Q and the optical wavelength interval Δ′ at a series of reference center wavelength λ₁₀ is given. Q₁ and Q₁₂ are the factor Q for AOTF₁ and for double-filtering process, respectively. From Fig. 4(a), at a series of λ ₁₀, the double-filtering Q₁₂ increases with the interval Δ′ and the corresponding increasing curve at longer waveband is smoother than that at shorter one. Q₁ at a certain wavelength λ ₁₀ is constant and it increases with the reference center wavelength λ ₁₀. On the two curves of Q₁ and Q₁₂ versus Δ′ at a certain reference wavelength λ ₁₀, there exists an equal point on which Q₁₂=Q₁ when the interval Δ′ reaches certain value. Q₁₂<Q₁ if Δ′ is smaller than the equal point of Δ′, and Q₁₂>Q₁ if Δ′ is bigger than the equal point of Δ′. Figure 4(b) gives the curves of Δ′ and the corresponding Q factor of the equal point at a series of λ ₁₀. It is shown in Fig. 4(b) that Q and Δ′ at the equal points are bigger in longer waveband.

Finally, we give a whole analysis of Fig. 3 and Fig. 4. Bigger Δ′ at a certain λ ₁₀ is undoubtedly welcome in order to acquire a narrower spectral bandwidth Δλ ₁₂, yet Δ′ should not be too big without any restriction because it can result in a weak maximum diffraction efficiency (η ₁₂)_max and influence the following detection of the signal. It is necessary to choose a suitable Δ′ which can keep both Δλ ₁₂ and (η ₁₂)_max in proper value. From Fig. 4, at equal point (Q₁=Q₁₂), the sharp extent of the detected double-filtering spectral signal is same as the signal from only one AOTF, thus the performance of the detected double-filtering spectral signal will be similarly as good as that of signal from single AOTF. Obviously, the equal point of Q has provided a good way for finding the optimum Δ′. Furthermore, the condition of the enhancement of the spectral resolution at equal points is shown in Fig. 5. We can summarize that, corresponding to the equal point of Q, the average percent decrement for the spectral bandwidth is about 32%, and the ratio of the spectral resolution with double-filtering to that with one AOTF is 1.47 approximately. This illustrates the feasibility and efficiency of double-filtering method in enhancement of the spectral resolution.

 

Fig. 4. At a series of reference wavelength λ ₁₀, the value of factor Q. (a) The relationship between Q and the interval Δ′, Q₁=Δλ₁/(η ₁)_max, Q₁₂=Δλ ₁₂/(η ₁₂)_max, dash line indicates Q₁ and solid line indicates Q₁₂. (b) At the equal point (Q₁=Q₁₂), the relationship between Q and Δ′.

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Fig. 5. The condition of the enhancement of the spectral resolution at the equal points of Q₁=Q₁₂.

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5. Conclusions

The spectral resolution of AOTF is a main index to be considered during the design of AOTF. In this letter, an accurate expression has been obtained to describe the spectral bandwidth by considering the birefringence together with the rotatory property of the interaction materials. Moreover, we have put forward a double-filtering method to enhance the spectral resolution firstly. The principle of the double-filtering method is described in details, and the factors functional in the spectral resolution are analyzed. A factor Q is introduced to indicate the quality of the detected spectral signal. We have found an equal point of factor Q, at which the spectral bandwidth is obviously narrowed with a proper signal performance. The conclusion is that, the method of double-filtering is functional in the enhancement of the spectral resolution. This is meaningful to broaden the application of AOTF in the spectral imaging area.