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Abstract
The acousto-optic tunable filter (AOTF) with wide wavelength range and high spectral resolution has long crystal and two transducers. A longer crystal length leads to a bigger chromatic focal shift and the double-transducer arrangement induces angular mutation in diffracted beam, which increase difficulty in longitudinal and lateral chromatic aberration correction respectively. In this study, the two chromatic aberrations are analyzed quantitatively based on an AOTF optical model and a novel catadioptric dual-path configuration is proposed to correct both the chromatic aberrations. The test results exhibit effectiveness of the optical configuration for this type of AOTF-based imaging spectrometer.
© 2017 Optical Society of America
1. Introduction
An acousto-optic tunable filter (AOTF) is an electronic dispersive device which can diffract a certain passband of incident light by acoustic wave generated by transducer [1]. The spectral resolution is determined by the length of the transducer and the spectral coverage is limited by the tuning range of the transducer [2,3]. In recent years, imaging spectrometers based on AOTF have been widely used in many fields [4–7], because of completely controllability without sweep mechanism, fast and random tuning over a wide wavelength range and programmability in each band [8]. To improve the quality of the spectral data acquired by this kind of imaging spectrometer, there are three chromatic aberrations need to be corrected, which are longitudinal chromatic aberration, lateral chromatic aberration [9] and image blur [10].
Longitudinal chromatic aberration exists in every refractive imaging system, and commonly, AOTF-based imaging spectrometers use refractive optics in both fore-optics and relay optics in consideration of alignment and cost [11–15]. But spectrometers often work in wider wavelength range compared with panchromatic or color imaging systems and the acousto-optic crystal make this aberration worse especially in confocal optical structure. Therefore, simple lens group have limited capability to correct the aberration [16]. Complex optical structure or various types of glass is needed to ensure image quality [17,18]. In order to simplify the optics, catadioptric structure can be used in AOTF systems. Westinghouse [19] proposed an infrared AOTF spectrometer with three mirrors as afocal telescope and three Silicon and Germanium doublets to re-image, in which several aspheric surfaces were used.
Lateral chromatic aberration in an AOTF system refers to that the output angle of diffracted beam is a function of wavelength, resulting in a scene shift on image plane [20]. Generally, the aberration is compensated by natural dispersion. The most common method is placing a wedge on the output surface, which was proposed by Yano [21]. By doing this, the divergence of diffracted angle was reduced from 0.7° to 0.04° in the wavelength range from 400nm to 700nm, and then to 0.01° [22]. Researchers also try to use additional prism to achieve a better correction, such as correction for + 1 order and −1 order diffracted beam simultaneously by a triple prism [23] or improving correction precision to 0.0003° in the range from 440nm to 780nm by a doublet prism [24]. Besides compensation by refracting surface, there are other methods as well. Shure [25] found that telecentric confocal optics have certain inhibitory effect on scene shift. After analyzing hyperspectral image data, researchers in University of Valencia considered scene shift by using a first order polynomial, so it can be corrected by image processing [26].
Image blur is a specific aberration of AOTF, which is caused by the divergence of ultrasonic beam and results in sidelobe images [27]. Thus, it brings down the spatial resolution seriously in the direction of acousto-optic interaction. Traditionally, this aberration can be suppressed by amplitude apodization of the acoustic excitations [28]. The telecentric confocal optics mentioned above also have advantages in compensating this aberration so that it is widely used in AOTF systems [29]. Other researchers corrected image blur by placing a prism after AOTF [30] or developing image processing methods that minimizes AOTF-related image degradation [20].
So far, the chromatic analyses and correction are generally based on normal AOTF. The requirement of wider spectral coverage and higher spectral resolution need a longer AOTF crystal and two transducers. Consequently, a longer crystal leads to more chromatic focal shift and the polar angle difference of acoustic wave from two transducers causes discontinuous variation of direction of output beam, which makes the chromatic aberrations more complicated and hard to be corrected by traditional methods. Due to the high spectral resolution, the effect of image blur can be neglected [31].
In this work, the longitudinal and the lateral chromatic aberrations of the specific AOTF are analyzed in different conditions quantitatively by using AOTF optical model proposed in our previous work [32]. And then, according to the characteristics of the two aberrations, a novel catadioptric dual-path configuration is proposed to correct both the chromatic aberrations. Finally, a prototype with wavelength range from 400nm to 1000nm and spectral resolution from 0.8nm to 3nm is made, the test results demonstrate effectiveness of this method.
2. Theoretical Background
2.1 Working principle of AOTF
A noncollinear AOTF is mainly composed by acousto-optic (AO) crystal, transducer and absorber, as shown in Fig. 1. A typical transducer structure consists of a metal top electrode, a piezoelectric crystal and one (or more) metal bonding layer which attaches the piezoelectric crystal to the AO substrate and is used as a bottom electrode. The absorber bonded at the end of acoustic propagation route to prevent acoustic wave from reflecting back. When acoustic wave propagates in the AO device, a phase grating is formed in crystal caused by periodic variation of refractive index and the incident light is diffracted at specific wavelength, therefore, realizing spectral filtering. The and are the polar angle of the incident optical beam and acoustic wave respectively, and is the angle of crystal wedge for lateral chromatic correction.
The working principle of AOTF can be explained by momentum matching theory of acousto-optic vectors. Noncollinear AOTF become popular in spectral imaging applications under the noncritical phase matching (NPM) condition proposed by Chang [33], which results in an AOTF with a large angular aperture. Figure 2 illustrates the vectors relationship in AOTF when momentum matching and NPM condition are both satisfied.
Momentum matching requires that
where is the incident optical wave vector, is the acoustic wave vector, and is the diffracted optical wave vector. The magnitudes of the wave vectors are given bywhere is the incident optical index of refraction, is the diffracted optical index of refraction, is the vacuum optical wavelength, and is the acoustic wavelength.2.2 Optical model for ray tracing of AOTF with two transducers
With regard to wide wavelength range and high resolution AOTF, two transducers are bonded to the side face of AO crystal to extend the spectral range. An illustration of AOTF with two transducers is shown in Fig. 3.
The acoustic wave is assumed to be infinite plane wave generally and propagates along the normal direction of transducer facet. But actually, according to the fabrication difficulties and the finite size of transducer, the acoustic wave filed is not perfect and homogeneity, which has an effect on ray trace and tuning relation as well. Furthermore, there is an exceptional issue on AOTF with two transducers that the polar angles of two acoustic vectors ( and ) are not equivalent completely. Thereby, the optical design and aberrations correction become more complicated. To solve this problem, a three-surface model [32] for the ray tracing of an AOTF is proposed in previous work to promote optical design accuracy. This model establishes two refractive surfaces and one interactional surface of acoustic and optic waves, and more importantly, it revises acoustic polar angle by high precision test data to ensure the differences between measured diffracted angles and modeling value below 0.01°.
2.3 Analysis of chromatic aberrations of AOTF with two transducers
The spectrometer proposed in this paper uses an AOTF produced by MolTech and a monochrome EMCCD camera manufactured by Princeton Instruments which the following analysis is based on. Table 1 sketches some important parameters.
Table 1. Parameters of AOTF and camera.
2.3.1 Longitudinal chromatic aberration
In order to simplify the analysis of longitudinal chromatic aberration, an AOTF crystal can be considered as a parallel plate by ignoring the wedge angle and the change of ray in the crystal and assuming that the refractive index is , where and are the refractive indices for the o-polarized and e-polarized light respectively.
As shown in Fig. 4, axial chromatic dispersion leading to focus position shifting in different wavelength occurs when a beam of focus light passing through the AOTF crystal. The incident light enters crystal with an angle of and at height of referring to optical axis. Passing through the crystal with length of , the exit light focus at a point far from exit surface with distance of . According to law of refraction and geometrical relationship, the focus position can be expressed as
Provided that is reference wavelength, then the distance between focus of reference wavelength and others isand in small-angle approximation, the Eq. (4) can be simplified asThus it can be seen that longitudinal chromatic aberration of AOTF is related to wavelength range and crystal length . In this work, is longer than normal AOTF crystal because of double long transducers for high spectral resolution and wide wavelength range. Figure 5 compares of normal AOTF with long AOTF by Eq. (5), the maximum chromatic focal shift of long AOTF is 2.36mm and 2.65 times larger than normal one.
2.3.2 Lateral chromatic aberration
When incident light entering AOTF and interacting with acoustic wave, as depicted in Fig. 6, a narrow band light is diffracted with a certain angle related to wavelength so that causes scene shift.
For vertical incident light, the wave vector diagram is shown in Fig. 7. According to three-surface model, the diffractive angle of exit light in crystal can be calculated by coordinate of point [32]:
Due to e-polarized light diffracting, the refractive index can be expressed as
According to the geometrical relationship shown in Fig. 7 and the law of refraction, deflection angle between exit light and incident light is given byThe lateral chromatic correction in imperfect situations of acoustic wave mentioned in section 2.2 is analyzed by Eq. (8). Figure 8 simulates the ability of wedge correction in acoustic wave deviating from ideal condition.
The curves reveal that the minimum lateral chromatic is 0.02° by optimizing wedge angle, but with the 0.2° deflection of , the compensation ability declines almost 50%. Figure 9 shows the wedge correction ability in condition of different polar angle of two transducers.
In this analysis, 600nm is the critical wavelength between two transducers and it can be clearly seen that there is a mutation at that wavelength which increases along with the value of . By optimizing wedge angle, the lateral chromatic can be controlled within 0.03° while the difference between two acoustic angles is 0.02°. In a practical system, the wedge angle are fixed that means cannot be optimized, and the two situations are existent at the same time making this aberration more complicated.
3. Optical design and chromatic correction
3.1 Influence of optical structure on chromatic aberrations
The optical design of imaging spectrometer is based on ZEMAX and three-surface model of AOTF by which chromatic aberrations are analyzed. Ordinarily, there are two optical structures used in AOTF imaging spectrometers, i.e. collimated optics and confocal optics. To evaluate the influence of the two optics quantitatively, a three-surface model and three paraxial lenses are founded in software, as exhibited in Fig. 10, paraxial lens 1 and paraxial lens 2 constitute confocal optics and the last lens is placed as a re-imager. And by placing the AOTF model in the middle of lens 1 and lens2 or after lens 2, the two optical structures are realized.
Figure 11 shows the chromatic focal shift produced by AOTF in two optical structures. Changing the distance from lens 1 to lens 2, the confocal structure can be broken slightly in order to simulate the imperfect parallel light in collimated optics. For parallel incident light, there is no chromatic focal shift, but with the light becoming slightly convergent or divergent, the focal shift increases. Moreover, the focal shift caused by convergent is small than divergent, when the light diverges to 0.6°, the value of focal shift is about 0.25mm. In confocal optics, the focal shift is more than 7mm which is much greater than it in collimated optics.
Fig. 11 Longitudinal chromatic aberration of AOTF in two optical structures: (a) Collimated optics; (b) Confocal optics.
The lateral chromatic aberration is also calculated in two optical structures, the consequence is displayed in Fig. 12. In order to be more intuitive, the pixels of image moving on CCD is used instead of . Figure 12(b) illustrates that the scene shift will gradually increase when the image gets away from the last surface of AOTF. In the most extreme cases, i.e. the image is at infinity, the value of image movement on CCD is about 11 pixels shown in Fig. 12(a).
Fig. 12 Lateral chromatic aberration of AOTF in two optical structures: (a) Collimated optics; (b) Confocal optics.
Through the analysis above, it can be concluded that any optical structure cannot satisfy the demand of both the longitudinal and lateral chromatic aberration, which should be selected based on correction difficulty of each chromatic aberration. Due to the wide wavelength range, the longitudinal chromatic aberration needs complex lens structure and various glass materials to be compensated in confocal optics. So, the imaging spectrometer employs reflective collimated optics to make chromatic focal shift as small as Fig. 11(a) shows, and then focuses on the correction of lateral chromatic aberration.
3.2 Correction of lateral chromatic aberration
Traditional lateral chromatic aberration correction utilizes dispersion of prism to compensate angular shift of emergent light of AOTF. But with regard to angular mutation caused by different acoustic polar angle of two transducers, the dispersion correction ability declines remarkably. To solve this problem and decrease the scene shift, a method of dual-path correction is been presented, the layout of lateral chromatic aberration correction part is shown in Fig. 13.
Fig. 13 Optical design of lateral chromatic aberration correction part: (a) Layout; (b) actual components.
The diffracted light exiting from AOTF is split into long wave channel and short wave channel by dichroic beam splitter 1 on the basis of critical wavelength between two transducers. Thus, the scene shift of respective channels is continuous and can be corrected by doublet prism respectively. The optical paths are folded by mirrors and then the dichroic beam splitter 2 of which the cutoff wavelength equals to dichroic beam splitter 1 realizes the light overlapping. The mirror mount of mirror 1 can adjust image position of short wave channel on CCD array to ensure that the images of two channels are matched. Spatial separation and merging of the two wave range of diffracted light solve the problem of angular gap and achieve lateral chromatic aberration correction of AOTF with two transducers in all wavelengths. Owing to the transition band of dichroic beam splitter, the diffractive light in these bands will enter the both channels and focus on CCD array at the same time. As a result, there may be double image when incident light is too strong or AOTF works near cutoff band. So, the mirrors are replaced by dielectric mirrors in order to diminish double image and fold optical path as well.
A comparison is made between the correction effect of single doublet prism and dual-path correction concerning the angular shift of central field chief ray. The correction result of optical design is shown in Fig. 14. The deflection angle in 600nm is considered reference and the maximum difference of deflection angle in whole wavelength range is 0.013° when corrected by single doublet prism and is less than 0.00045° with method of dual-path correction, which means the image movement is 1.6 pixels and less than 0.06 pixels respectively. The design result of lateral chromatic correction by more traditional methods and dual-path method is listed on Table 2. The result of 0.00045° is coincident with former research and reflects normal capability of lateral chromatic aberration by doublet prism [24].
Fig. 14 Correction result of lateral chromatic aberration: (a) Single doublet prism method; (b) Dual-path correction method.
Table 2. Correction effect of lateral chromatic aberration by different methods
3.3 Optical system design
The spectrometer adopts catadioptric structure which consists of afocal telescope and imaging lens group as shown in Fig. 15.
The two-mirror afocal system compresses beam angle without chromatic aberrations, which has a diagonal FOV of 8.4° and a magnification of 0.5. For ease in alignment and manufacturing, the primary mirror and secondary mirror are parabolic and use aperture off-axis to avoid obscuring. The output beam from afocal system is not perfect collimated, thus, when passing through the AOTF and prism, it will cause longitudinal chromatic aberration. But as mentioned, the focal shift is so gentle under collimated optical structure that a simple relay optics can compensate well, which includes a doublet lens and a singlet lens with spherical surfaces. The correction result of chromatic focal shift and spot diagram are shown in Fig. 16, residual chromatic focal shift is about 78 μm.
Fig. 16 Analysis results of AOTF spectrometer optical design: (a) Chromatic focal shift; (b) Spot diagram.
4. Experimental results and discussions
4.1 Lateral chromatic aberration test
The integrating sphere, knife edge target and reflecting collimator produced by Electro Optical Industries are used to test the effect of correction as displayed in Fig. 17. The reflecting collimator has a 12 inches optical aperture and the integrating sphere can adjust its luminance up to 25,000 foot-lamberts, so the target has a strong energy to remedy the low signal-to-noise ratio (SNR) caused by high spectral resolution and low quantum efficiency (QE) at both ends of the wavelength range.
In the test, the movement of knife edge in images reflects the magnitude of lateral chromatic aberrations. The spectral images of knife edge target are acquired by AOTF spectrometer in different wavelength, the image at 700nm is shown in Fig. 18(a). Then, the Canny operator is used to detect edge of target shown in Fig. 18(b) and the position of knife edge is calculated by averaging horizontal coordinate of every point on it. After calculation on all spectral images, the scene shift curve can be drawn up as Fig. 18(c). The maximum movement of image is about 1.2 pixels in this test and it is an improvement in angular shift correction relative to traditional methods mentioned in previous analysis. Compared with design result 0.06 pixels, the test result deteriorates seriously. But, in contrast with traditional methods, this new structure is still effective because its test result excels design results of traditional methods. The actual result is influenced by three aspects. The most important factor is the residual lateral chromatic aberration in both channels (400-600 nm, 600-1000nm). From Fig. 18(c), it can be seen that the maximum scene shift of short wave channel and long wave channel are 1.06 and 0.8 pixels respectively, which are much bigger than the value in design. Because the AOTF optical model has limited precision (the angle of diffracted beam of the model is not equal to the test value strictly and the differences between them change with wavelength) [32] and the refractive index of prism cannot equal to the value set in software strictly, the design accuracy of doublet prism is declined, which affects the correction ability. Secondly, the optical alignment error will increase the scene shift. The image movement of long wave channel is small than short wave channel, and then the movement range of long wave channel can be covered by short wave channel theoretically by adjusting the mirrors, as depicted in Fig. 14(b). But, the assembly of the prototype is too rough to reach that situation, so that the total lateral chromatic aberration is increased. Thirdly, low SNR in the short wave part decreases the edge detection precision, which leads to estimation error of aberration.
Fig. 18 Test result of lateral chromatic aberration correction: (a) Image of knife edge target; (b) Edge of target; (c) Scene shift curve.
4.2 Longitudinal chromatic aberration test
The longitudinal chromatic aberration results in changes of focus positions of at different wavelengths, so the spectral images have different spatial resolution accordingly which can be used to evaluate longitudinal chromatic aberrations correction effect. Thereby, the 1951 USAF resolution test chart is used. The spectral images at 450nm, 650nm and 900nm are shown in Fig. 19 and more resolution test results are listed in Table 3. It shows that the resolution changes with wavelength and the spectral images in intermediate bands have higher resolution than in both ends. But the difference of resolution is generally small in the whole wavelength range with a maximum difference of 0.2311mm. It is noteworthy that the result is not only caused by residual longitudinal chromatic aberration, but also affected by low SNR problem as mentioned above, especially in the wavelength shorter than 450nm.
Table 3. Comparison of resolution in different bands.
4.3 Spectrum detection experiment
As shown in Fig. 20, leaves of euonymus japonicus are chosen to be objective in the experiment of spectrum detection. And to test and verify the validity of spectral data of prototype, the leaves are also detected by Analytical Spectral Devices (ASD) FieldSpec pro FR spectroradiometer.
The high intensity discharge lamps and halogen lamps are used for prototype and ASD respectively, because the former is so strong in certain bands that the ASD cannot optimization correctly, and the latter is too weak in short wave range to be detected by prototype. The spectral image is shown in Fig. 21(a). The ASD is non-imaging system and has 30° field of view, thus the leaves are flattened to ensure that the detection range of the two spectrometers is identical as far as possible. According to the working distance in experiment, the detection range of ASD has been signed in Fig. 21(a) by red circle, the red rectangle represents the spectral curve calculation range of images obtained by prototype.
Fig. 21 Spectrum test results: (a) The spectral image at 696nm detected by prototype; (b) Spectral reflectivity of leaves on region of interest obtained by ASD and prototype.
After curve smoothing by Savitzky-Golay filter, it can be found from Fig. 21(b) that the trend of two curves is consistent in most bands, the spectral signature of reflection peak near 550nm and red edge are expressed commendably. But in both ends of the curve, the spectral reflectivity is not accurate due to low signal response. There are three main reasons for this situation: (a) the diffraction efficiency of AOTF is relatively low in both ends; (b) the QE of CCD declines in both ends; (c) the leaves have strongly absorptive property in 400nm-450nm because of chlorophyll, so the spectral data is more inaccurate in short wave end than in long wave end. The spectral relative error can be calculated by
where the is wavelength, is spectral reflectivity detected by ASD, is spectral reflectivity detected by prototype. The relative error is depicted in Fig. 22, and because of the signal energy problem in short wave range, the error in bands small than 450nm is neglected. For other bands, the maximum error from 450nm to 1000nm is under 40%.5. Conclusions
The main chromatic aberrations of AOTF with wide wavelength range and high spectral resolution are longitudinal chromatic aberration and lateral chromatic aberration. The two chromatic aberrations are analyzed by theoretical arithmetic, and owing to the long crystal length and two transducers structure, they are more complicated and serious when compared with traditional AOTF. For this AOTF imaging spectrometer, the difference between collimated optics and confocal optics is simulated by ZEMAX and three-surface model. The results illustrate that the choice of optical structures is to make a tradeoff between longitudinal chromatic aberration and lateral chromatic aberration. Therefore, the prototype adopts afocal mirrors system as fore-optics. Because of non-strict-collimated incident light, there is a little longitudinal chromatic aberration induced by AOTF crystal, and it can be compensated by refractive re-imager easily. To suppress the lateral chromatic aberration related to diffractive angle mutation that results from the difference between polar angles of two transducers, a dual-path optical structure based on doublet prism has been proposed. It reduces the aberration to 0.06 pixels, which is equivalent of angular shift of 0.00045°. Although the residual lateral chromatic aberration of prototype is bigger than design result, which is about 1.2 pixels, it is still better than the best design results using traditional methods (1.6 pixels). This new catadioptric dual-path configuration shows effectiveness in chromatic aberrations correction of AOTF with wide wavelength range and high spectral resolution, and the results can be further optimized by a higher precision alignment and manufacturing. When compared with ASD, the prototype exhibits spectral detection ability and reflects objective spectral signature correctly in experiment. But for some hardware restrictions, the spectral data in both ends is not accurate enough, especially in 400nm to 450nm. For future applications, enhancing light input in both ends and using CCD with high QE in short wavelength can play a role in improving data quality.
Funding
National Natural Science Foundation of China (NSFC) (61227806); National Key R & D Program (2016YFB0500502, 2016YFB0500505).
Acknowledgments
We would like to thank Hongzhi Jiang, Xudong Li and Shaoguang Shi for useful discussions.
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