1. Introduction

As an optical filter, an AOTF is widely used in the field of laser tuning, spectral analysis, hyperspectral imaging and etc [1–5]. Generally, AOTF is operated on the acoustic-optic (AO) interaction that the incident light at a particular central wavelength is diffracted by a propagating acoustic wave in fixed direction. The acoustic wave is generated by applying a RF signal to a piezoelectric transducer bonded on AO crystal. Thus, changing the frequency of acoustic wave, i.e. that of the applied RF signal, can realize to tune the filtered optical wavelength accordingly. The relationship is generally called as the frequency tuning (FT) relation. To the best of our knowledge, pervious used AOTFs are all controlled by RF signal [6–15]. And by FT relation, the diffracted wavelength (under a fixed incident angle) is absolutely determined by the applied RF, with the merits of fast and easy operation. Our group have done some relevant studies on RF controlled AOTF design and applications [16–22]. But FT based AOTF could not be suitable in some circumstances: the desired diffracted optical wavelength is out of the wavelength tuning range determined by RF, especially in short-wave area, a proper RF source with high frequencies and power would be difficult to reach for modern electrical technology; or an adjustable spectral bandwidth at a certain diffracted central wavelength is needed. In the previous related studies, the bonded two-transducers design was reported for broadening the working wavelength range of an AOTF [1, 9], yet it would bring defects of complex structure, bigger size and more cost. In this work, the authors have theoretically and experimentally demonstrated a special method that can continually tune the diffracted optical wavelength of AOTF at a fixed RF. On AO interaction principle, the function of the momentum mismatching vector is discussed, and its relationship with diffraction efficiency, incident optical wavelength and direction is analyzed. A new tuning relation between the diffracted central optical wavelength and the incident angle is proposed, and is called as the angle tuning (AT). Experimentally, the optical wavelength of AT based AOTF can be changed in tens of nm at a fixed acoustic frequency, and good performance of narrow spectral bandwidth is also kept (even be narrowed in some conditions). Besides, the bandwidth can also be adjusted during AT process. AT method is confirmed to be effective on providing more practical AOTFs with broadened optical waveband.

2. Theoretical analysis of the angle tuning method

Firstly, the AO interaction of a noncollinear AOTF is demonstrated in Fig. 1. Here, k_i, and K_a represent the incident optical wave vector and the acoustic wave vector respectively. k’ is the polarized wave vector, which is related with AO interaction of k_i and K_a, k_i + K_a = k’. On the Bragg-diffraction condition, k_d represents the diffracted wave vector, which is generated from the polarized wave vector k’. △k represents the momentum mismatching, and △k = k’- k_d in the direction of optic axis [001]. θ_i is the polar angle for incident beams. θ_d is the diffracted polar angle, which is θ_i and optical wavelength λ dependence. θ_a is the polar angle for the acoustic wave. θ’ expresses the direction of the polarized wave vector.

 

Fig. 1 Wave vector diagram of the noncollinear AOTF.

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As for the traditional FT controlled AOTF, it commonly operates near the area of △k = 0. And k_i, K_a and k_d are connected with the parallel tangents momentum-matching condition. We had put forward the FT relation as [21],

n_i and n_d are the refractive indices as the incident wave vector k_i and the diffracted wave vector k_d propagating in AO crystal, they are related with θ_i, θ_d and λ. V_a is the acoustic speed in certain direction. Therefore from Eq. (1), we concluded that the diffracted wavelength λ was related with incident polar angle θ_i and the acoustic frequency f_a. In the applications of this AOTF, the incident light generally propagates in a fix direction, thus the diffracted wavelength λ is only determined by the applied f_a. This simple relation has undoubtedly shown the merits of FT based AOTF. λ can be tuned fast and easily by only changing the value of RF in the μs orders. Such advantage surely has improved the wide development of AOTF in the past studies [1–22]. But the flaw of this FT principle had also been indicated. In some extent, the AOTF wavelength range is limited by the finite work bandwidth of RF signal. Particularly in the shorter wave area, the need for the high frequency radio signal with bigger power would enhance the study difficulty. Actually, AOTF with broader work wavelength range could be welcomed in the applications. From our analysis, it is usually out of the capability of the FT controlled AOTF. A special method called AT is demonstrated to resolve the problem. Vector △k is the key parameter here. And the value of △k is,

n_o and n_e are the ordinary and extraordinary refractive indices in the direction perpendicular to the optical axis. σ indicates the rotation property of AO crystal. On the AO interaction, Δk has direct effect to the diffraction efficiency η of an AOTF. η is expressed as,

Here, δ ² = ζ ² + ξ ², ζ = ΔkL/2, ξ ² = ξ₀ξ₁/4, ξ₀ = −2πΔn₀L/λ, ξ₁ = −2πΔn₁L/λ. L means the length of AO interaction, Δn₀ and Δn₁ are the refractive indices variation derive from acoustic wave. From Eqs. (2) and (3), we know that, η is related with incident optical polar angle θ_i, diffracted wavelength λ, the direction and frequency of the acoustic wave. Under a fix acoustic wave (f_a, θ_a), we have calculated the relationship between η and θ_i (in Fig. 2).

 

Fig. 2 (a) The relationship between incident angle θ_i, optical wavelength λ and diffracted efficiency η (f_a = 120 MHz, θ_a = 99.76°); (b) an expanded by ~15 of the A region in (a).

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From Fig. 2(a), for a fixed acoustic wave, the incident light at a particular wavelength λ and direction θ_i could be 100% diffracted, and the diffracted central wavelength λ₀ changes with the incident polar angle θ_i obviously. Therefore, we can realize to tune λ₀ by changing the corresponding θ_i at the fixed acoustic wave. This is the AT method mentioned above. In the past applications, θ_i of a FT based AOTF system is usually chosen in the region A (in Fig. 2(b)) in pursuit of big incident aperture. But in AT, a well designed frontal optics system can offset the decrease of the aperture, the performance of AOTF is guaranteed. Furthermore, in Fig. 2(a), the diffracted central optical wavelength is very sensitive to θ_i. It means we can tune the λ₀ in a very wide range with smaller variation of θ_i, under any fixed RF signal. By the above analysis, AT method is feasible in the wavelength tuning of AOTF theoretically.

3. Experimental results and discussions

The experimental setup for the AT method is shown in Fig. 3. The collimated wide-band light (Nikon) goes through the TeO₂ based AOTF which is mounted on a precisely rotatable stage (Edmund). Then the incident light is filtered by an AOTF, driven under the RF source. The narrow-band diffracted light is fed into the fiber spectrometer (Avantes). The RF source, rotatable stage and the spectrometer are all under the control of PC. Here, the used AOTF is self-designed with good performance. The A-O material is TeO₂ crystal with a high figure-of-merits of 772 × 10⁻¹⁵ s³/kg at λ = 633 nm & θ_a = 99.76°, operating wavelength range is in 400 nm-850 nm, the finesse of AOTF is 190@651.62 nm, the insertion loss is on the low level of about 0.2 dB, the polarization dependent loss is lower than 2.9 dB, the RF input power limit is 4 W, the power limit of input light is about 4 W/mm², and the tuning voltage limit is 24 V.

 

Fig. 3 Experimental setup of AOTF system based on angular tuning. 1-collimated wide-band light source; 2-AOTF; 3-rotatable stage; 4-RF source; 5-diffracted light; 6- spectrometer; 7-PC

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Experimentally, we have measured the corresponding spectrums at a series of the incident polar angles. In the experiment, RF is fixed at 120 MHz. The PC controlled rotatable stage has a step of 0.1°(with the angular velocity of 45 deg/s). We change the incident angle by rotating AOTF relative to the incident light. At every step, PC records the incident angle on the surface of AO crystal and transforms it into the inner incident polar angle θ_i. θ_i is 20.2° when the incident light runs into AOTF perpendicularly to the incident face of AO crystal.

In Fig. 4(a), we have given the relation between the incident polar angle θ_i, diffraction efficiency η and the diffracted central optical wavelength λ₀. It is obvious that the diffracted central optical wavelength λ₀ changes with the incident polar angle θ_i accordingly. When θ_i = 20.2°, the shortest λ₀ is got, relative to other θ_i. This was called the optimum incident polar angle for FT based AOTF [21]. Away from the optimum θ_i, the value of λ₀ will increase, whether θ_i increases or decreases. Furthermore, in the region of smaller incident polar angle (θ_i <20.2°), the central optical wavelength λ₀ is more sensitive to θ_i than that in the region of bigger incident angle (θ_i >20.2°). In the region of smaller θ_i, it is easier to tune the diffracted central wavelength λ₀ widely (with the maximum wavelength scanning speed of 414 nm/s as rotating AOTF). These regularities are well agree with the theoretical computation in Fig. 2. In our AT experiment, λ₀ has changed in 628.5 nm-602.1 nm, as the θ_i tuned in 16.0° −20.2°. It indicates that AT method is functional in tuning diffracted optical wavelength under any fixed RF. Thus as we carry out AT method in the whole working range of bonded RF source, the working waveband of AOTF would be broaden obviously. We also analysis and contrast the peak diffracted efficiency η₀ at different incident angle with RF power of 2.5 W in Fig. 4(b). When we rotate AOTF, the diffracted lights with different central wavelengths nearly can reach the equally high diffraction efficiency. It means incident polar angle had no obvious influence on the peak diffracted efficiency in our tuning range. It is helpful of getting diffracted optical signal with high intensity. We consider this could owe to consistently high level of figure of merits M₂ in our AT process, which is connected with the peak diffraction efficiency [12, 21]. Wide range tuning of diffracted wavelength on AT method can be feasible.

 

Fig. 4 (a) The relation between the incident polar angle θ_i, diffraction efficiency η and the diffracted optical wavelength λ₀ in AT method (f_a = 120 MHz, θ_a = 99.76°); (b) Relationship between peak diffraction efficiency η₀ and incident polar angle θ_i.

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In order to discuss the performance of the diffracted light deeply, we measure the bandwidth Δλ of the diffracted spectrums got from AT and FT respectively (shown in Fig. 5(a)). In the region of θ_i ≤20.2°, the diffracted spectral bandwidth Δλ increases with the decrease of the θ_i. Corresponding to a certain diffracted central wavelength λ₀, Δλ got from AT method is bigger than that from FT. In contrast, in the region of θ_i ≥20.2°, Δλ decreased with the increase of θ_i. At a certain λ₀, Δλ from AT method is smaller than that from FT. Therefore, in the region of bigger θ_i (≥20.2°), a narrower bandwidth could be easier to acquire as operating on AT method than FT method. In Fig. 5(a), we can also see that, Δλ is very narrow in the whole tuning area. AT method is indicated to be effect to keep well spectral quality. Also we need to mention that, the diffracted light with the same central wavelength λ₀ could be realized under different combination of RF and incident polar angle. For example, in Fig. 5(a), points A (f_a = 120 MHz, θ_i = 15.8°), B (f_a = 110 MHz, θ_i = 20.2°) and C (f_a = 120MHz, θ_i = 26.7°) respond to the same central wavelength of λ₀ = 644.1 nm. But the bandwidth Δλ are 4.68 nm, 3.30 nm and 1.89 nm respectively. AT has made it possible to adjust the diffracted spectral bandwidth Δλ for a fixed central wavelength λ₀. By comparison, in FT, λ₀ and its Δλ are completely determined if the RF has been chosen [1–22]. The AT method can provide a more flexible way for the wavelength tuning of a designed AOTF.

 

Fig. 5 The relationship of the diffracted spectral bandwidth and optical wavelength. (a) Contrast of diffracted spectral bandwidth got from AT and FT at a series of same central wavelengths; (b) Spectrums with the same central wavelength got from AT and FT.

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Figure 5(b) shows the spectrums of three diffracted light beams with the same central wavelength(λ₀ = 610.1 nm) got from AT and FT. The solid line represents the diffracted signal from FT (θ_i = 20.2°, f_a = 118 MHz). The dashed line and dot line represent the diffracted signal from AT at different incident direction (θ_i = 18.0° and 23.7°), they are set at the equal RF of 120 MHz. We can see that the sidelobes of the diffracted spectrums got from AT and FT are both on a very low level. From Figs. 5(a) and 5(b), in the region of θ_i≥20.2°, Δλ got from AT is narrower than that from FT. For points B and C in Fig. 5(a), the spectral bandwidth at 644.1 nm has been narrowed about 42.7% if we operate on AT method instead of FT. In FT, double-filtering technique was the most functional method to narrow the bandwidth of the diffracted light [11, 17–20]. But in double-filtering study, the necessary two-times AO interaction process would bring complicated tuning procedure or higher cost. On this aspect, AT method can be welcomed for narrow-band optical wavelength tuning with a single AOTF.

4. Conclusions

For the traditional AOTF, its optical wavelength can be fast tuned by changing the applied RF value. In this FT method, the optical wavelength range would be limited for the working bandwidth of RF source. In this work, a special method called AT has been presented. The function of AT method in tuning noncollinear AOTF has been studied theoretically and experimentally. In our analysis, on the condition of fixed acoustic wave, the diffracted central optical wavelength of AOTF can be widely tuned by changing the direction of incident light. It indicates AT method is effect to broaden the tuning limit for a FT based AOTF system to a large extent. During the operation of AT process, the diffracted light also keeps its well performances of high diffraction efficiency, low sidelobes and narrow spectral bandwidth. In certain area of the incident polar angle (bigger than the designed optimum incident polar angle), we have acquired narrower diffracted spectrum than that from FT at a fixed central wavelength. It shows that AT method can provide a way to improve the spectral resolution for a single AOTF. Through suitable combination of RF and incident angle, we have got different value of spectral bandwidth at a same central optical wavelength. AT method gives a more flexible selection for the spectral bandwidth. In conclusions, AT is an original and functional method of tuning the filtered optical wavelength of AOTF, and also a necessary supplement of the RF controlled AOTF. This work can be helpful of pushing the noncollinear AOTF applications in the areas of optical filtering, spectral analysis and hyperspectral imaging.