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Abstract
The paper aims to show the advantages of the infrared-optimised quasi-collinear AOTF (acousto-optic tunable filter) for the spatio-spectral hyperspectral imaging system. The optimisation process is presented based on the selected tetragonal anisotropic materials with exceptional optical and acousto-optical properties in IR (infrared) spectral region. These materials are further compared in terms of their features and suitability for AOTF design. The spectral resolution is considered as the main optimising parameter. Resulting from the analysis, the mercurous chloride (Hg2Cl2) single crystal is selected as a representative of the mercurous halide family for the presentation of the quasi-collinear AOTF model operating in LWIR (long-wave infrared) spectral band. The overall parameters of the AOTF model such as spectral resolution, chromatic field of view, acoustic frequency, and operational power requirements are estimated and discussed in results.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Hyperspectral imaging has become considerably popular in several areas of interest, such as remote sensing [1], medicine [2], forensics [3], food quality control [4], chemistry [5], and many others [6]. The main part of the hyperspectral systems form diverse dispersive components such as diffraction gratings, prisms, liquid crystals, and AOTFs (Acousto-optic Tunable Filters) [7].
The hyperspectral imaging in IR (infrared) spectral region ($0.75-20\mathrm{\mu} m$) has recently also increased its popularity in the mentioned applications. Several remote sensing hyperspectral systems [8] operate within this spectral region, but usually in a narrower spectral band, depending on the atmospheric windows. Frequently, two main spectral bands MWIR (Mid-wave-infrared $3-5\mathrm{\mu} m$) and LWIR ($8-10\mathrm{\mu} m$) are selected [8–10]. The key parameters of hyperspectral systems are the spectral resolution and the number of resolvable spectral channels. Commonly, the systems offer in the MWIR and LWIR bands spectral resolution about tens of $nm$. Concerning the spectral resolution, the exploitation of the AOTF as a main disperse device for the IR hyperspectral instruments has been more frequently studied in previous years. Compared to standard dispersive devices such as diffraction gratings, the AOTF offers no mechanical parts, fast tuning, better spectral resolution, and a higher number of resolvable spectral channels in the operational spectral band. These features offer faster capturing of the scene and its spectral decomposition compared to standard push-broom systems.
There have been several approaches of AOTF designs suitable for the hyperspectral imaging in a IR spectral region [11–17]. These AOTFs are frequently based on the non-collinear interaction [18] between the RF (radio-frequency) acoustic wave and the input light. In some cases, the standard collinear interaction is exploited [19,20]. The main problem of the non-collinear AOTF design is the interaction length between the input light and the acoustic wave, which is a critical parameter determining the spectral resolution. The collinear design offers a higher spectral resolution but limits the aperture. In addition, some AOTF fabrication materials cannot be used for the collinear interaction [21]. To overcome this issue, Voloshinov introduced an interesting approach of close-to-collinear or quasi-collinear interaction [22]. This approach has also been studied and improved in the following years by several other authors [23–26]. The main advantage of the quasi-collinear approach is the extension of the interaction length, therefore, an improvement in the spectral resolution, and a possibility of the wide aperture.
A new concept of a collinear AOTF-based spatio-spectral hyperspectral system was proposed by Maksimenka in [27]. In the paper, the authors presented their promising idea of the mentioned system operation and built two breadboard imager prototypes based on the $\mathrm {Hg_2Cl_2}$ collinear AOTF operating in MWIR and LWIR bands. The operability has been subsequently demonstrated by captured images of the black body source and a propane flame. To obtain a full hypercube image, the authors also outlined the fundamental principle of their post-processing algorithm based on the AO (acousto-optic) phase matching wavelength dependency through AO material refractive indices. The authors have selected the Earth remote sensing as a possible application of the proposed hyperspectral system. The physical properties of the AOTF were concluded as the main limitation of the presented spatio-spectral hyperspectral system. Nevertheless, the replacement of this type of AOTF by the quasi-collinear-based one could keep the hyperspectral system properties and even improve its performance. The design of this AOTF, however, depends on in-depth optimisation due to many degrees of freedom, which must be addressed during the design process. One of the key decisions is an AOTF material selection. Convenient selection of the fabrication material can highly influence and improve the properties (e.g. spectral resolution) of the final instrument and, therefore, worth focusing on.
This paper presents optimisation criteria for designing a quasi-collinear AOTF based on several optical materials suitable for the IR AOTF operation. The presented optimisation is based on the AOTF spatio-spectral hyperspectral system requirements and aims to maximise the AOTF spectral resolution parameter. The selected materials for this optimisation process are well-known tetragonal crystals with high birefringence and exceptional AO properties. The first material is $\mathrm {TeO_2}$, which is commonly used for the AOTF fabrication [28–30]. The other explored materials are Mercurous Halides ($\mathrm {Hg_2Cl_2}$, $\mathrm {Hg_2Br_2}$, $\mathrm {Hg_2I_2}$), which has been also under research for several optical applications [31–36] and for AOTF design and fabrication [37–43]. In this paper, we assess the quasi-collinear AOTF parameters for several IR spectral bands, but mainly at 4$\mathrm{\mu} m$. This wavelength has been chosen in purpose as a representative of the MWIR spectral band with high atmospheric transmission window, which is exploited for several remote sensing applications [44]. Moreover, the selected materials offer great transparency at this wavelength and can be compared in terms of the AOTF properties. Apart from the comparison of the selected materials, we provide a model of the quasi-collinear AOTF based on one selected tetragonal material with bandwidth operation $8-10\mathrm{\mu} m$ for thermal sensing applications. The presented model parameters are then further discussed and assessed their suitability for the spatio-spectral hyperspectral system.
The paper is structured into six main sections. The Introduction is followed by a description of the selected concept of the spatio-spectral hyperspectral system. Then the paper focuses on the theory of the quasi-collinear AOTF, its parameters, and optimisation criteria. The next section aims to compare the selected fabrication materials for the quasi-collinear AOTF. Apart from the Conclusion, the two last sections present the quasi-collinear AOTF model based on $\mathrm {Hg_2Cl_2}$ and assess the results and discuss its suitability for the spatio-spectral hyperspectral system.
2. Spatio-spectral AOTF-based hyperspectral system
Maksimenka has introduced the concept of the spatio-spectral AOTF-based hyperspectral system in [27]. Despite the traditional hyperspectral systems, this concept exploits the AOTF as the main diffractive instrument. Suppose the acoustic wave of specific RF frequency propagating within the AOTF crystal. The incoming light from a captured scene incident on the input AOTF aperture (facet or window of the single crystal). The input light of specific wavelength and incident angle then interacts within the crystal with the acoustic wave, which results in diffraction. This is given by a well-known Bragg condition [46]. Therefore, for varying incident angles and wavelengths, there are several diffracted beams of the imaging scene. Despite the commonly used concept of AOTF devices, this system relies on encoding the spectral information by an incident light angle. Thus, the spatio-spectral image can be obtained. By tuning the acoustic wave frequency, the Bragg condition is changed. Therefore, different spectral information (wavelengths) will be diffracted and captured at the same incident angle. The final hyperspectral cube of the captured scene might be retrieved by post-processing of the detected image data. The introduced concept of the described hyperspectral system is outlined in Fig. 1.
Fig. 1. The concept of the spatio-spectral hyperspectral imagining system, with the AOTF. The input light wavefronts are decomposed and diffracted into several spectral lines according to the incidence input angle, which fulfils the Bragg condition at a specific wavelength and tuning frequency of the AOTF. To obtain the full hyperspectral 3D cube, the RF tuning frequency must be changed. Subsequently, all the spatio-spectral images have to be post-processed. The system is also composed of several other necessary devices such as front and back optics, polarisers detectors, supply electronics and others. For more details see [45].
This concept may be exploited in VIS (visible) spectral region similarly as in UV (ultraviolet) or IR spectral band. Nevertheless, the system operation and even the post-processing of the data would highly depend on the AOTF construction material. Therefore, the detailed and specific design of the AOTF is crucial, and it highly affects the system performance. In this paper, we propose a quasi-collinear AOTF model as a convenient selection for the introduced system. Because of the complexity of the presented system, it is necessary to precisely design and evaluate the properties of such AOTF and identify possible hold-backs that may occur. Therefore, the next sections of the paper focus on the optimisation of the quasi-collinear AOTF and the selection of suitable fabrication materials.
3. Quasi-collinear AOTF
In the paper, the operation of the quasi-collinear AOTF for all selected single crystal materials has been set in a [001], [110] crystallographic plane. This plane offers exceptional AO properties, especially the acoustic slow shear wave phase velocities, which strengths the AO interaction. This means that the transverse acoustic wave propagates perpendicularly to the crystallographic plane along the [110] axis for the selected orientation. This paper deals with the quasi-collinear principle of the AOTF, which exploits S-polarized ordinary input light and diffracts the light beam with the extraordinary P-polarization.
For all selected tetragonal anisotropic materials, the propagation of the acoustic wave in the [001], [110] plane [21] can be described as a slowness ellipse
The $V_{110}$ and $V_{001}$ are the sound velocities in the [110], [001] directions. Nevertheless, in the anisotropic materials, it is necessary to differentiate an acoustic wavefront propagation (phase velocity) and an energy propagation (group velocity), because they do not have to be similarly oriented. The acoustic phase velocity can be represented as a wave vector with a particular direction within the acoustic slowness ellipse and with the acoustic wave velocity according to Eq. (1). However, the direction of the acoustic group velocity vector is represented as a normal vector to the tangent of the slowness ellipse for a given direction of the acoustic phase velocity vector. The angle $\theta _a^g$ between the acoustic group velocity vector and the [110] axis can be described as
The $\tan \theta _a$ stands for the acoustic phase velocity angle measured from the [110] axis. The acoustic group velocity should be oriented in a collinear direction with the input light to obtain an AO quasi-collinear interaction. Contrary to that, the acoustic phase velocity does not have to be collinear. With this collinear approximation, the $\theta _a^g$ can be replaced in Eq. (2) by $\theta _i$, where $\theta _i$ is the incident light angle measured from the [110] axis.
In addition, a standard vector phase matching equation can be obtained for the AO quasi-collinear interaction. Assume the ordinarily polarised light represented as a wave vector $\vec {k_i}$ and an acoustic phase velocity wave vector $\vec {K}$. The extra-ordinarily polarised wave vector of diffracted light $\vec {k_d}$ then can be expressed as
Scalar representation of Eq. (3) can be re-written as
The angle between the wave vector of diffraction $\vec {k_d}$ and the [110] axis is $\theta _d$, and $n_e$ stands for the extraordinary refractive index of the material.
Since the input light vector $\vec {k_i}$ and the acoustic group velocity vector are collinear for the quasi-collinear interaction, the acoustic phase vector direction $\vec {K}$ is also known. Therefore, from the ideal phase matching [Eq. (4)], it is possible to derive the parameters of diffracted wave vector $\vec {k_d}$ and the necessary frequency of the acoustic wave. The wave vector diagram, which describes the presented situation of the phase matching for the quasi-collinear AO interaction, is provided in Fig. 2.
Fig. 2. Phase matching vector diagram of quasi-collinear AO interaction in [001], [110] plane. A blue curve indicates an ordinary slowness circle of the birefringent material with a radius $2\pi n_o/\lambda$. A red curve represent an extra-ordinary slowness ellipse of the birefringent material, with axes $2\pi n_o/\lambda$ and $2\pi n_e/\lambda$. A blue vector $k_i$ represents the ordinary polarised input light beam wave vector. A red vector $k_d$ is the extraordinary polarised diffracted light wave vector, and a black $K$ vector represents the acoustic wave vector. The blue and black dashed lines of the input light and acoustic phase vector indicate the quasi-collinear principle. The input light wave vector has a collinear direction as a black-dashed group velocity vector of the acoustic wave. However, an acoustic phase velocity vector $K$ has different direction with angle $\theta _a$ from [110] axis. The angle of the acoustic group velocity $\theta _a^g$ to the [110] axis has also been shown. An Azure-dashed vector shows the group-velocity direction of the diffracted beam within the material. The red-dash vector indicates the refraction of the diffracted beam from the birefringent material to air.
Assume there is an intersection point of the vector $\vec {k_i}$ and ordinary refractive index of the anisotropic material represented by an optical slowness circle with radius $2\pi n_o/\lambda$. This intersection point has coordinates [$I_x$, $I_y$], where $x$ replaces [110] axis and $y$ replaces [001] axis. Suppose, the acoustic phase wave vector $\vec {K}$ is represented by a line expression $y = mx + c$, where $m = \tan \theta _a$ and $c = I_y - m \cdot I_x$. Additionally, the optical slowness ellipse of the extraordinary refractive index for the uniaxial anisotropic material is represented as $x^2/a^2 + y^2/b^2 = 1$, with $a = 2\pi n_o/\lambda$ and $b = 2\pi n_e/\lambda$. Thus, the intersection coordinates [$D_x$, $D_y$] of the acoustic wave vector $\vec {K}$ and the extraordinary refractive index ellipse may be obtained as
The angle of diffraction $\theta _d$ can be then derived as
Moreover, the particular frequency for the ideal phase matching can be retrieved as
where $K_x$, $K_y$ were obtained from the phase matching condition as $K_x = D_x - I_x$ and $K_x = D_y - I_y$. The non-collinear orientation of the acoustic wave phase velocity allows fulfilling the parallel tangent matching condition in the input angle vicinity. This increases the angular aperture of the AOTF [18]. Therefore, for a fixed magnitude and direction of the acoustic wave vector, the phase matching can be considered as satisfied for several values of the incident light directions.However, by changing the magnitude of the acoustic wave vector, wavelength or input light angle, the phase matching condition may be violated. The deviation from the ideal phase matching can be determined by the phase mismatch factor $\vec {\Delta k}$. The phase mismatch significantly influences one of the main parameters of the AOTF, which is the diffraction efficiency $\eta$. From the phase matching condition, the phase mismatch parameter is equal to
Scalar representation of $\vec {\Delta k}$ may be analytically obtained as a minimal distance to extraordinary slowness ellipse from the acoustic phase vector end-point [$K^{a}_x,~K^{a}_y$]. This can be considered as an optimisation problem and solved, for example, by standard Lagrange multipliers as
The ratio $I_d / I_i$ represents both diffracted and input light intensities. $M_2$ is an AO figure of merit. $P_a$ is an acoustic power. $H$ represents a height of the acoustic beam (usually limited by the transducer size), and L is an interaction length. The AO figure of merit $M_2$ represents the strength of the AO interaction and significantly influences the diffraction efficiency and the AOTF performance. The value of $M_2$ is entirely defined by the material from which the AOTF crystal is fabricated. It depends on the index of refraction of a birefringent material, acoustic slowness, density $\rho$ of the material, and crystallographic orientation, which means the directional dependency. $M_2$ is given by
The effective elasto-optic constant $p_{\mathrm {eff}}$ comes from the mentioned crystallographic orientation of the material and from a geometry of the AO interaction. For tetragonal crystals in the [001], [110] plane the $p_{\mathrm {eff}}$ can be estimated [28] as
The elasto-optic coefficients $p_{44}$, $p_{11}$, $p_{12}$ comes from the fourth-rank elasto-optic tensor. These coefficients are material-dependent and must be assessed separately for each material.
For sufficient diffraction efficiency, the power of the acoustic beam is also essential. Acoustic energy needed for the AO interaction and at least 50% diffraction efficiency during the ideal phase matching can be additionally obtained from Eq. (12) and expressed as
Therefore, the AO figure of merit $M_2$ and interaction length $L$ are critical for the design of the AOTF, especially for the transducer requirements and other additional electronics such as electric supply amplifiers and so on.
In general, the diffraction efficiency is according to Eq. (12) represented as $sinc^2$ function. Taken that into consideration, the expression for the best achievable spectral resolution $\Delta \lambda$ can be derived at FWHM (full width half maximum) of the $sinc^2$ ($\Delta k L \approx 0.9 \pi$) as
where $b$ is a dispersion constant equal toThe expression for the diffraction efficiency also indicates that the phase matching condition fulfilment can be achieved for several wavelengths at different input angles and one particular acoustic beam frequency. It allows the encoding of spectral information into the incident angle. The concept was mentioned in the previous section and outlined in Fig. 1. The acoustic beam energy travels collinearly with the light incident on the AOTF input window because of the quasi-collinear concept. Therefore, the instrument overall field of view will be limited by the width and height of the acoustic beam. These parameters are determined by the transducer size and proportions of the AOTF crystal. Another important parameter of the AOTF for the spatio-spectral hyperspectral system is the chromatic field of view. This term represents the standard angular aperture - maximum full angle deviation from the normal incidence for which the phase matching condition is fulfilled at a given wavelength and acoustic frequency. Therefore, the hyperspectral instrument overall angular resolution is also limited by the chromatic field of view. Similarly, from the diffraction efficiency Eq. (12), for a given wavelength, $\Delta k L \approx \pi$ and $\Delta \lambda \approx 0$ the chromatic field of view $\Delta \theta$ can be approximated [18,47] as
From Eqs. (12), (16), and (17) it is apparent, that the improvement of the AO interaction length $L$ significantly increases the performance of the AOTF and favours the quasi-collinear interaction. However, there is also a trade-off between the high spectral resolution and chromatic field of view. Extension of the interaction length $L$ and increase in birefringence by selecting a suitable anisotropic material improves the spectral resolution significantly but decreases the chromatic field of view. Since the design of the AOTF uses the spectral encoding by the angle of incidence and expects diffraction of several wavelengths at one RF frequency, the maximisation of the spectral resolution is then prioritised over the chromatic field of view.
The presented theory shows that the AOTF properties are highly material-dependent. The next parameters, which influence the AOTF, are the single crystal proportions. Therefore, selecting a suitable material is the first prerequisite for the AOTF design that should be made.
4. Characteristics of selected tetragonal single crystals
Tetragonal anisotropic materials with high birefringence are an excellent choice for the AOTF construction. These uniaxial crystals offer superb optical and elasto-optical properties. From a variety of tetragonal materials suitable for modelling of the quasi-collinear AOTF, we have selected specific types with exceptional optical features, transparency in IR spectral band, high and positive birefringence and extensive AO figure of merit. The materials are $\mathrm {TeO_2}$ (Paratellurite) and Mercurous Halide single crystals such as $\mathrm {Hg_2Cl_2}$ (Calomel), $\mathrm {Hg_2Br_2}$ (Kuzminite), and $\mathrm {Hg_2I_2}$ (Moschelite). This paper compares the fundamental properties of the selected materials and assesses their performance as fabrication optical materials for the quasi-collinear AOTF design.
In general, $\mathrm {TeO_2}$ is the most common material for the AOTF construction. However, compared to Mercurous Halides, $\mathrm {TeO_2}$ is transparent only in a spectral region 0.35 - 5$\mathrm{\mu} m$. Mercurous Halides are slightly less transparent in VIS spectral band, but they are highly transparent in a wide IR spectral region (up to 20$\mathrm{\mu} m$ and even more). The summarised general properties [34,48] of the selected materials are in Table 1. As was mentioned above, these materials offer high birefringence. The evolution of their refractive indexes in the selected bandwidth is shown in Fig. 3. The ordinary and extraordinary refraction indices have been modelled via Cauchy’s dispersion formula [49]
Fig. 3. Ordinary and extraordinary refractive indices for materials of $\mathrm {TeO_2}$ (A), $\mathrm {Hg_2Cl_2}$ (B), $\mathrm {Hg_2Br_2}$ (C), $\mathrm {Hg_2I_2}$ (D), respectively (refractive indexes obtained in [50,51]).
Table 1. General material properties.
The estimated coefficients $A, B, C, D$ for the selected materials are summarised in Table 2. The significant birefringence of the materials is apparent from the presented diagrams. At the selected wavelength $4\mathrm{\mu} m$, Calomel offers approx. birefringence 0.5536, Kuzminite 0.6838, and Moschelite even 0.9831, which is significantly higher than birefringence 0.1392 of Paratellurite. To assess sufficiently all selected materials and compare them in the IR region, the selected wavelength $4\mathrm{\mu} m$ has been retained.
A significant parameter for the AO interaction in the [001], [110] plane, that can be compared, is the acoustic shear wave velocity. For the selected crystallographic plane, the overall velocity can be provided by Eq. (1). Specifically important are the velocities $V_{001}$ and $V_{110}$. The summary of the selected material acoustic velocities are in Table 3. The exceptional low values of $V_{110}$ for Calomel ($V_{110} < 350m/s$), Kuzminite ($V_{110} < 285m/s$) and Moschelite ($V_{110} < 255m/s$) significantly increase the AO figure of merit [Eq. (14)] and, therefore, favour Mercurous Halides over Paratellurite ($V_{110} < 616 m/s$).
Table 3. The acoustic wave velocities in the crystallographic plane [001], [110] for the selected materials.
The AO figure of merit $M_2$ represents the AO interaction parameters and is completely based on the material characteristics. Moreover, $M_2$ is also determined by the effective elasto-optic coefficient, which can be calculated for the plane [001], [110] according to Eq. (15). The known elasto-optic coefficients $p_{11}$, $p_{12}$, $p_{44}$ are listed in Table 4. All the elasto-optic coefficients are currently known only for the $\mathrm {TeO_2}$ material. For Calomel material, one remaining elasto-optic coefficient $p_{44}$ is still unknown and usually considered equal to 0 [24,25]. For Kuzminite and Moschelite, the elasto-optic coefficients have not yet been determined. It is mainly caused by their problematic growth and polishing processes. Therefore, for calculating the $M_2$, we exploited the correct refractive indices of these materials and combined them with the Calomel elasto-optic coefficients, which can be considered as a sufficient approximation [24,25].
Table 4. The elasto-optic coefficients of the selected materials. For $\mathrm {Hg_2Br_2}$ and $\mathrm {Hg_2I_2}$, the coefficients are currently unknown. The elasto-optic coeficient $p_{44}$ of Calomel is also unknown and is commonly considered as equal to 0.
Assuming the general quasi-collinear interaction for the incoming light beam at $4 \mathrm{\mu} m$, the AO figure of merit $M_2$ of the selected materials can be calculated against the input light angle (measured from the [110] axis). The results can be seen in Fig. 4(A). It is clear that Moschelite outperforms significantly the remaining materials mainly due to the low acoustic velocities and high birefringence. However, all Mercurous Halides outperform the Paratellurite in general. It is also possible to approximate the $M_2$ for several values of Calomel elasto-optic coefficient $p_{44}$. Figure 4(B), indicates that for values of $p_{44} > 0$, the $M_2$ rises and reaches higher values than shows the 4(A). The evaluation of Calomel coefficient $p_{44}$ has also been studied in other research [40] and must be addressed in the future.
Fig. 4. The AO figure of merit versus input angle calculated for quasi-collinear AO interaction in [001], [110] with assumed the light wavelength $4\mathrm{\mu} m$. Sub-figure A shows the $M_2$ for all selected materials - $\mathrm {TeO_2}$ (orange line), $\mathrm {Hg_2Cl_2}$ (green dashed line), $\mathrm {Hg_2Br_2}$ (red dashed line), $\mathrm {Hg_2I_2}$ (blue dashed line). Sub-figure B express the estimation of Calomel $M_2$ versus input angle for several values of elasto-optic coefficient $p_{44}$, which is currently unknown and usually estimated as 0. The exploited values of $p_{44}$ varies in interval from $-0.4$ to $+0.4$
In addition, for the varying input angle, other parameters of the general quasi-collinear AOTF can also be estimated based on the material characteristics. The best theoretically achievable values of the quasi-collinear AOTF for the $4\mathrm{\mu} m$ wavelength are summarised for all materials in Table 5. The properties based on the proportions of the AOTF single crystals such as interaction length $L$ and height of the acoustic wave $H$ has been selected for the sake of simplification as $L = 3 cm$, $H = 0.8 cm$. Interesting are the values of spectral resolution, which result from the high birefringence. Mercurous Halides are better in spectral resolution over $2.75 nm$ than Paratellurite, which makes a notable difference.
Table 5. Estimated values of quasi-collinear AOTF properties based on the selected materials. For the estimation the interaction length has been set equal to $L = 3 cm$ and the width of the acoustic wave was set equal to $H = 0.7 cm$. For all parameters, the incident angle of the light has been selected for their maximisation.
Currently, the $\mathrm {TeO_2}$ optical material is the most exploited for the AOTF fabrication. However, as was mentioned above, it has a limited spectral band within the IR spectral region, and above $5 \mathrm{\mu} m$ is not transparent. It is a limitation for devices that want to operate in LWIR (long-wave infrared) spectral band. This bottleneck may be suppressed by Mercurous Halide materials, which offer even better properties and are transparent over $20 \mathrm{\mu} m$ and highly suitable, for example, for the LWIR operation. Concerning Mercurous Halides, currently, only Calomel has been grown to a sufficient size for the AOTF fabrication [33,38]. The ongoing research on the growth of Kuzminite and Moschelite, which ensures sizeable and robust crystals, is still in progress [35,52]. In addition, the bonding of the transducer to the Mercurous Halide crystals is currently also under ongoing research [53].
Considering the analysed characteristics of the selected tetragonal crystals, Calomel has been chosen for a design of the quasi-collinear AOTF model. In general, Calomel can be considered as a valid representative of Mercurous Halides and possess several of their properties. Compared to $\mathrm {TeO}_2$, Calomel also offers wider operational bandwidth due to its transparency and higher spectral resolution, which was selected as the main optimising parameter.
5. Calomel-based quasi-collinear AOTF
In general, the properties of Calomel single crystals allow the operation in the thermal LWIR spectral band. Therefore, in this paper, we propose optimisation criteria and model of the quasi-collinear AOTF operating at wavelengths $8-10\mathrm{\mu} m$. From the previous analysis, it is clear, that the design and performance of such AOTF is highly related to the crystallographic orientation and thus directionally-dependent. Resulting from that, a cut angle of the crystal input window within the selected [001], [110] plane is crucial. In general, the cut angle of the AOTF crystal determines the input angle of the light at normal incidence (measured from the [110] axis) and affects the parameters of the AOTF. For the presented model, assume, the acoustic wave emitting transducer placed on the crystal side. Therefore, the overall cut of the single crystal is also crucial. An example of a convenient crystal shape for the quasi-collinear AOTF and its operation may be seen in Fig. 5(A). For the quasi-collinear AOTF, the transducer may be oriented in a way that the acoustic group velocity vector propagates to the input facet of the crystal. The acoustic group velocity (energy) is then reflected at the crystal input window, and it propagates collinearly with the input light at normal incidence. After the reflection at the input window, the acoustic phase wave velocity vector should be conveniently oriented for the maximisation of AOTF parameters, such as AO figure of merit $M_2$, diffraction efficiency, and chromatic aperture. As was mentioned above, the different directions (walk-off angle) of the acoustic group (energy flow) and phase velocities (wavefront orientation) vectors are caused by the characteristics of the anisotropic medium. In addition, the shape of the presented crystal is constrained by symmetricity. The output facet of the crystal should be parallel to the input window. The sides of the crystal should also be parallel. Thus, the acoustic wave can be reflected at the output window and absorbed by the crystal lateral side by the additional acoustic absorber.
Fig. 5. The example of proposed design of the quasi-collinear AOTF (A) and deviation angle between input and diffracted light beams (B). The transducer bonded to the crystal emits acoustic shear wave in the selected crystallographic plane [001], [110]. The acoustic wave group velocity does not follow the acoustic phase velocity due to the significant walk-off angle. Therefore, the acoustic wave directs to the crystal input window, where the acoustic wave is reflected. Thus, the input light wave and acoustic wave propagates collinearly. The phase velocity vector of the acoustic beam has, however, different orientation. The acoustic wave is then similarly reflected at the crystal output window and can be absorbed in the backside. The polishing of the crystal, therefore, has to be symmetrical.
To maximise the $M_2$, Fig. 4 may be used. Therefore, for the Calomel-based quasi-collinear AOTF, the input window cut angle should be between $-45^{\circ }$ and $-60 ^{\circ }$ (measured from the [110] axis). The separation of the diffracted and input light beam not only by polarisation may also be a critical factor. The spatial separation between the input light beam and the diffracted light beam can be represented as $\mid \theta _d - \theta _i \mid$. The results of separation between the incident and diffracted light beams for Calomel material at 8$\mathrm{\mu} m$ and 10$\mathrm{\mu} m$ can be seen in Fig. 5(B). Due to the low birefringence changes in the IR region, the separation between the diffracted and incident light almost does not vary. Conveniently, for the maximal spatial separation, the input window cut angle should be from $-40^{\circ }$ to $-50 ^{\circ }$.
Because of the directional dependence of the $M_2$, the cut angle of the AOTF input window influence required acoustic power for sufficient diffraction efficiency. This power dependency may be expressed by Fig. 6 and via Eq. (16), where $8\mathrm{\mu} m$ wavelength was selected as dominant. The required power profile corresponds proportionally to the shape of the AO figure of merit $M_2$ and confirms its high impact on the AOTF performance. The interaction length also lowers the power requirements.
Fig. 6. The required power for the 50% diffraction efficiency of the calomel-based quasi-collinear AOTF depending on the input angle of the light and interaction length. For the simplicity, the interaction length has been set equal to $3cm$ and width of the acoustic beam to $0.8cm$. For this approximation, the wavelength equal to $8\mathrm{\mu} m$ has been selected.
From the provided analysis, we can conclude that the recommended cut angle for the Calomel crystal for the selected type of AOTF should be set between $-45^{\circ }$ and $-60 ^{\circ }$ from the [110] crystal axis.
For the model of the quasi-collinear Calomel-based AOTF, suppose, that the input cut angle has been selected equal to $-48^{\circ }$ (measured from the [110] axis). It is also possible to analyse the spectral resolution of such AOTF via Eq. (17). The spectral resolution dependence on the wavelength and interaction length can be seen in Fig. 7. The figure shows that the increase of the interaction length favours the quasi-collinear AO interaction. The difference may be up to tens of $nm$.
Fig. 7. The spectral resolution versus wavelength and interaction length for the calomel-based quasi-collinear AOTF. For this estimation, the input angle of the light has been selected equal to $-48^{\circ }$ form the [110] crystallographic axis.
Selecting the interaction length equal to 3 $cm$ as a typical length for currently grown Calomel crystals [33,34,54] other parameters of the quasi-collinear AOTF may be modelled. Firstly, the frequency dependence on the wavelengths in $8-10 \mathrm{\mu} m$ spectral band can be analysed for the quasi-collinear AOTF with selected parameters. The results of the required frequency for ideal phase matching during the normally incident light may be seen in Fig. 8(A).
Fig. 8. Estimated properties of the Calomel-based quasi-collinear AOTF operating in $8-10\mathrm{\mu} m$ spectral band. The first diagram (A) represents the tuning acoustic frequency dependence on the wavelength. The next diagram (B) shows the required power for at least 50% diffraction efficiency versus the input wavelength. The diagram (C) shows estimated spectral-resolution of modelled AOTF against the input light wavelength. The last diagram (D) represent approximated chromatic aperture versus the input wavelength. For the sake of simplicity, the interaction length has been set equal to $3cm$ and width of the acoustic beam to $0.8cm$. The cut angle of input window of the calomel crystal has been selected equal to $-48^{\circ }$, measured from the [110] crystallographic axis.
The necessary power requirements for at least 50% diffraction efficiency for the modelled quasi-collinear AOTF may also be analysed. The results are plotted in Fig. 8(B). The spectral resolution for the selected wavelengths and the chromatic aperture may be seen in Figs. 8(C), 8(D), respectively.
Fig. 9. Estimated phase mismatch $\Delta k$ for the calomel-based quasi-collinear AOTF with a cut angle of the input window equal to $-48^{\circ }$ form the [110] axis. The values of the mismatch are plotted against the wavelength and input angle of the incident light. The frequency of the acoustic wave is $12.87 MHz$, and the input power of the acoustic beam was set to $8.39. W$.
Fig. 10. Diffraction output from the calomel-based quasi-collinear AOTF model. The diffraction output corresponds to the phase mismatch obtained in Fig. 9. The acoustic wave frequency was kept at $12.865 MHz$, and the input power of the acoustic beam was set to $8.39 W$. These values are set for the 50% diffraction of the $9\mathrm{\mu} m$ beam at normal incidence on the crystal input window ($-48^{\circ }$ from [110] axis). The results properly correspond to the expected phase matching. The diffraction is observable for several input wavelengths and incident angles. Thus, spatio-spectral imaging through the AOTF may be achieved. For diffraction of other wavelengths, the acoustic frequency should be tuned. The whole spectral information may be obtained after several steps of tuning and post-processing.
Having a specific cut of the AOTF input window in the [001], [110] crystallographic plane, we can also assess the diffracted light output for different incident light angles and wavelengths in the IR region. Therefore, it is possible to evaluate the spatio-spectral output expected from the introduced hyperspectral concept.
Setting the acoustic beam frequency to 12.87$MHz$, the deviation from the phase matching condition may be shown [Eq. (11)]. Only a narrow band of wavelengths and input angles create a small enough mismatch for obtaining the diffracted output beam. For the quasi-collinear AOTF model, the results of the phase mismatch are plotted against the wavelength and incident light angle in Fig. 9.
Based on the obtained phase mismatch and Eq. (12), it is also possible to evaluate the diffraction output. Assume the same acoustic frequency $12.87 MHz$ and the acoustic beam power equal to the $8.39W$. Figure 10 shows the diffraction output for incident wavelengths in $8-10\mathrm{\mu} m$ spectral band and incident light angle varying from $-40^{\circ }$ to $-58^{\circ }$ (measured from the [110] axis). For the sake of simplification, the multispectral light beam is considered to incident at the centre of the input window of the AOTF, similarly as it is shown in Fig. 5. From the observable peaks of the diffraction efficiency, which corresponds to the ideal phase matching condition, it is apparent that the spatio-spectral hyperspectral imaging concept was achieved. It is clear that the diffracted output equal to $9 \mathrm{\mu} m$ properly corresponds to the diffracted output peak with 50% diffraction efficiency for $8.39W$, as was expected. However, it can be seen that lower and higher wavelengths are, due to one selected frequency and level of power, diffracted with varying efficiency levels. This phenomenon must be addressed in the future by the design of the post-processing of the captured data.
6. Results and discussion
The principle of the quasi-collinear AOTF for spatio-spectral hyperspectral imaging was demonstrated in the previous section. It has been shown, the characteristics of the selected tetragonal materials may offer significant benefits for the quasi-collinear AOTF in the IR region. The selection of the material highly depends on the chosen operational spectral band. The material crystallographic plane has been selected as [001], [100] due to the slow shear wave speed of the acoustic wave within this plane. The refractive indices evolution in the VIS and IR bands for all selected materials have been shown in Fig. 3. The primary optimised parameter is the spectral resolution. The spectral resolution of the quasi-collinear AOTF depends mainly on the used crystal proportions and material characteristics [Eq. (17)]. The maximally achievable spectral resolution for the AOTF based on the selected materials, at MWIR wavelength $4\mathrm{\mu} m$ and for selected crystal proportions, was shown in Table 5. Frequently used Paratellurite single crystal offers spectral resolution $3.45 nm$ for the quasi-collinear AOTF at the $4\mathrm{\mu} m$ wavelength. Contrary to that, exceptional parameters were exhibited by Mercurous Halide single crystals. The spectral resolution for quasi-collinear AOTF based on Calomel was estimated as $0.87 nm$ at the same wavelength. Kuzminite and Moschelite offered even narrower spectral resolution $0.7 nm$ and $0.49 nm$ respectively. Comparing the other parameters such as AO figure of merit $M_2$ and related acoustic power requirements, Mercurous Halides provided even better results. The $M_2$ for Paratellurite-based quasi-collinear AOTF was estimated as $246 \cdot 10^{-15} s^3/kg.$ and necessary power ($\eta \approx 50\%$) equal to $2.17 W$. Estimated $M_2$ and acoustic power ($\eta \approx 50\%$) for Calomel, Kuzminite and Moschelite was $327 \cdot 10^{-15} s^3/kg$, $1008 \cdot 10^{-15} s^3/kg$, $2790 \cdot 10^{-15} s^3/kg$ and $1.63 W$, $0.53 W$ and $0.19 W$ respectively. However, due to the nature of this type of the AOTF and high birefringence of Mercurous Halides, the trade-off is the lower value of the maximal chromatic field of view $2.37 mrad$, $2.14 mrad$, $1.78 mrad$, compared to Paratellurite maximal chromatic field of view $4.73 mrad$. Nevertheless, the biggest advantage of Mercurous Halides is the broadband IR spectral operation over $20\mathrm{\mu} m$. Hence, the operation in the LWIR spectral band allow only Mercurous Halides. Concerning the results, Mercurous Halides or more precisely Calomel has been selected as a suitable material for the quasi-collinear AOTF modelling.
The cut angle selection of the Calomel single crystal input window had also been discussed and optimised. The crystal input window cut highly influences the parameters such as $M_2$, power requirements, the deviation between the input and diffracted light, and others. The highest value for the Calomel $M_2$ was according to Fig. 4 between $-45^{\circ }$ and $-60 ^{\circ }$ for the cut angle of the crystal window (measured from the [110] axis). This selection was confirmed by Fig. 6. This figure represented the power requirements of the AOTF for at least 50% diffraction efficiency and showed the importance of the AO interaction length. The extension of the AO interaction length from $1cm$ up to $5cm$ lowered the power requirements almost by an order. Another important benefit of the AO interaction length extension has been shown in Fig. 7. A similar extension of the interaction length up to $5cm$ improved the spectral resolution $\Delta \lambda$ also almost by an order.
For the model of the Calomel-based quasi-collinear AOTF, the input cut angle has been selected equal to $-48^{\circ }$. A shape of the quasi-collinear AOTF, which ensures the collinear interaction between the input light and the acoustic beam, has been proposed in Fig. 5(A). For this model, the energy propagation of the acoustic wave is diverse from the acoustic waterfront orientation. It is caused by the walk-off angle between the phase and group acoustic velocities within the anisotropic medium. Therefore, in the presented model, the acoustic wave is reflected at the crystal input window and propagates collinearly with the light. The obtained parameters of this type of AOTF have been presented in Fig. 8. Introduced AOTF model achieved the spectral resolution and chromatic field of view approximately $\Delta \lambda \approx 6.3nm$, $\Delta \theta \approx 0.65 mrad$ at $8\mathrm{\mu} m$ and $\Delta \lambda \approx 9.95 nm$, $\Delta \theta \approx 0.73 mrad$ at $10 \mathrm{\mu} m$. The tuning acoustic frequency and power ($\eta \approx 50\%$) have been $14.5 MHz$ with $6.6W$ at $8\mathrm{\mu} m$ and $11.5 MHz$ with $10.3 W$ at $10 \mathrm{\mu} m$ respectively.
The presented model also served for the estimation of the phase mismatch and allowed the diffraction output evaluation. The phase mismatch results with the frequency $12.865MHz$ and the power $8.39W$ against the wavelength and the light incident angle have been shown in Fig. 9. The lowest mismatch values were estimated between the incident angles $-35^{\circ }$ and $-60^{\circ }$ (measured from the [110] axis) for operational wavelength interval $8-10 \mathrm{\mu} m$. Subsequently, according to the phase mismatch results, the diffracted output has been modelled. We exploited the same acoustic beam frequency and power. The diffracted output for varying wavelength and light input angles is shown in Fig. 10. The presented diffracted output corresponded to several $sinc^2$ function peaks depending on the phase matching and the acoustic beam power. The expected diffraction peak, which corresponds to the 50% diffraction efficiency at $9\mathrm{\mu} m$ and normal light incidence at the crystal input window ($-48^{\circ }$) have been obtained. The spectral resolution at the FWHM of the $sinc^2$ function at $9\mathrm{\mu} m$ was approximately $10nm$. This spectral resolution is about $2nm$ worse than the overall estimation in Fig. 8(C). This difference resulted from the limited acoustic power selected for 50% diffraction efficiency. The other diffracted peaks, which fulfilled the Bragg condition, have been obtained with varying efficiency due to the unchanged level of acoustic power. In general, there were obtained 18 diffraction peaks within the operational spectral region. By tuning the frequency and the acoustic power, different mismatch and diffraction peaks may be obtained. Thus, the spatio-spectral concept of the AOTF has been demonstrated.
Considering the presented results, the suitability of the quasi-collinear AOTF for the spatio-spectral hyperspectral system has been shown. The future work should focus on the fabrication of the presented quasi-collinear AOTF based on Mercurous Halides and its integration into the introduced hyperspectral system. The fabrication of Mercurous Halides-based AOTF and further test procedures may allow specifying the missing elasto-optic coefficients of the materials precisely. This would also help to describe the AOTF parameters more accurately and improve its design optimisation. In addition, the overall optical modelling of the introduced hyperspectral system should also be a priority. The determination of its other parameters such as PSF, field of view, spatial resolution, and others should be focused on. Apart from the AOTF, these parameters are influenced by the additional system instruments such as fore and back optics, detector, electrical amplifiers, necessary polarisers, and so on. Therefore, the appropriate selection and further optimisation of these instruments would be necessary for the final construction and parameter determination of the presented hyperspectral system.
7. Conclusion
This paper has proposed the analysis of the quasi-collinear AOTF suitable for the spatio-spectral IR hyperspectral imaging system. Apart from the quasi-collinear interaction theory, we have analysed and discussed the properties of suitable anisotropic materials for the AO IR operation and compared them against each other. The materials were selected among the tetragonal single crystals with high positive birefringence and exceptional AO properties. Conveniently, the quasi-collinear AO interaction for the selected materials proceeded in the [001], [110] crystallographic planes. The Calomel single crystal has been chosen as the suitable representative for the quasi-collinear AOTF model from the provided comparison. We have shown and discussed the convenient cut of the input window of the Calomel-based AOTF and showed the undeniable advantages of the quasi-collinear interaction. In addition, the power requirements, the maximal achievable spectral resolution, and the angular aperture of the quasi-collinear AOTF have also been discussed. Finally, the evaluation of the spatio-spectral hyperspectral concept by the AOTF model has been confirmed and presented.
Funding
Grantová Agentura České Republiky (20-10907S); Grant Agency of the Czech Technical University in Prague (SGS20/179/OHK3/3T/13).
Disclosures
The authors declare no conflicts of interest.
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