Snapshot channeled imaging spectrometer using geometric phase holograms

: In this paper, we present the design and experimental demonstration of a snapshot imaging spectrometer based on channeled imaging spectrometry (CIS) and channeled imaging polarimetry (CIP). Using a geometric phase microlens array (GPMLA) with multiple focal lengths, the proposed spectrometer selects wavelength components within its designed operating waveband of 450–700 nm. Compared to other snapshot spectral imagers, its key components are especially suitable for roll-to-roll (R2R) rapid fabrication, which gives the spectrometer potential for low-cost mass production. The principles and proof-of-concept experimental system of the sensor are described in detail, followed by lab validation and outdoor measurement results which demonstrate the sensor’s ability to resolve spectral and spatial contents under both experimental and natural illumination conditions.


Introduction
An imaging spectrometer samples the irradiance across a two-dimensional (2D) scene at multiple wavelengths. The acquired information forms a three-dimensional (3D) datacube denoted by coordinates x, y and λ. Compared to common broadband imaging, the additional spectral information can be utilized for applications in various domains such as remote sensing, biomedical diagnosis, and astronomical imaging [1][2][3]. Depending on the approaches of acquiring both spatial and spectral contents, imaging spectrometers can be divided into two categories: scanning imaging spectrometers and snapshot imaging spectrometers [4,5].
Scanning imaging spectrometers obtain the 3D datacube through spatial or spectral scanning in a time-sequential fashion. For instance, a whiskbroom or pushbroom imaging spectrometer records selected spectral components of a pixel or a line at a time and completes the 2D scene through physical scanning [6,7], while an interferometric spectral imager, such as the Infrared hyperspectral imaging polarimeter (IHIP) scans through different wavelengths and captures a 2D image at each individual wavelength [8,9]. Different from spectral imaging scanners, snapshot imaging spectrometers acquire the 3D datacube with a single measurement, which often involves trade-offs between spectral resolution and spatial resolution, spectral resolution and signal-to-noise-ratio (SNR), etc. The acquisition time is only limited by the exposure time of the camera. In the recent years, with new optical materials and components becoming available, various snapshot imaging spectrometer designs have been developed. However, compared to scan-based imaging spectrometry technologies, snapshot imaging spectrometry technologies are still scarce [10]. Some of the notable snapshot imaging spectrometers developed in the past 30 years include: the computed tomography imaging spectrometry (CTIS) technique first invented in 1991/1992, which simultaneously projects the image of the 2D scene onto a spatial plane with different directions and reconstruct the spectral information using computed tomography algorithms [11,12]. The image slicing/mapping spectrometer (ISS/IMS) that was first invented for astronomical applications and recently adapted for hyperspectral fluorescence microscopy [13,14]. The coded aperture snapshot spectral imager (CASSI) which combines coded aperture spectrometry with compressed sensing technique [15]. The image replicating imaging spect hyperspectral Kudenov et. [17].
In this pa records both geometric ph focal plane components i spectrometer due to the r Section 2 and models, and concept expe outdoor meas acquired from

Principle
The Geometr onto a spatial components b Light from th and a collima adjacent lensl a pair of PG frequencies a beams have o to enable inte polarization s focal position fabricated suc the GPMLA. (MTF) of ea modulation e processing, si where S 0 , S 1 , and S 2 are the Stokes parameters corresponding to intensity, linear horizontal over vertical, and linear 45° over 135° polarizations, respectively, and ξ 0 is the spatial carrier frequency. This is defined as where m is the PG's diffraction order (m = 0, ± 1), t is the separation between the two PGs, f 0 is the lens's focal length, and Λ is the PG's spatial period.
For this system, LP 1 modulates the incident S 0 component into S 1 such that S 1 = S 0 and S 2 = 0. Furthermore, the spectral content of S 0 is also modulated onto the carrier frequency by the GPMLA's defocus, as modeled by its modulation transfer function versus wavelength. The spectral resolution of each lenslet depends upon a given lenslet's aperture size, focal length, back focal distance, and the spatial frequency of the carrier. A schematic, describing the pertinent parameters of each lenslet, is illustrated in Fig. 2. Two linear systems models were produced based on: (1) geometrical optics; and (2) scalar diffraction theory. We note that, while the geometrical model offers greater error (approximately 10%, as will be discussed shortly), it enables a faster calculation and conceptual understanding of the filtering mechanisms involved with the concept.
As provided in Eq. (1), the shear creates a spatial carrier frequency across the FPA. This is filtered by the lenslet's MTF response versus wavelength, such that a spectral passband is created via. spatial filtering. The spatial contrast of the carrier was first calculated by geometrical optics in one dimension. The focal length of each wavelength can be calculated as From Fig. 2, given Δz = f 0 -f λ , the width of the geometrical point spread function (PSF) w for an FPA located at plane f 0 imaging a wavelength λ can be calculated as where rect is a rectangular function as defined by Gaskill [22]. Fourier transformation of this yields the MTF as where ξ is the transform variable of y. The transmission of the lenslet versus wavelength is equal to the MTF after substituting the wavelength-dependent width w as The full width at half maximum (FWHM) spectral bandwidth of each lenslet is thus In addition to the 1D geometrical model, we also produced a more rigorous 2D model based on scalar diffraction theory and the transfer function of free space [22]. The defocus wavefront error, across the pupil, is defined as where ρ is the normalized pupil radius. The normalized radius is defined as where R = D/2 and x p , y p are the Cartesian pupil coordinates. The wavefront coefficient [23] for defocus can be calculated as which has units of waves at the analysis wavelength λ. The focal shift can be calculated by and finally, the focal ratio used for the simulation is calculated as The PSF was calculated at each wavelength, using the wavefront aberrations defined by Eq. (9) at each value of Δz, corresponding to a given ( ) f λ λ . The MTF was then calculated as the absolute value of the PSF's Fourier transformation to yield a 3D function where Δz is implicitly dependent on wavelength. The relative spectral transmission function of the lenslet was calculated as The geometric 1D model was used to calculate the FWHM spectral bandwidth as depicted in Fig. 3. This quantity scales inversely with respect to the factor Dξ 0 and it scales linearly with respect to the design wavelength. While not explicitly presented here, the scalar diffraction model follows a similar trend with an additional weak dependence on the focal length f 0 given dependence th scalar diffract From our res FWHM by ap    Computationally, this amounts to applying two fast Fourier transforms and a matrix multiplication to apply the bandpass filter. For a datacube on a 5 megapixel image, this can be achieved in 0.18 seconds in Matlab on a Xeon E5-26XX v2 2.4 GHz CPU.

Radiometric and spatial resolution tradespace
A radiometric model was created to assess the spectrometer's signal to noise ratio (SNR) tradespace as related to the carrier frequency and FWHM spectral bandwidth. This tradeoff is more stringent when compared to a multi-aperture filtered camera, since the out-of-band light remains coincident with the in-band light. Thus, the shot noise present in the C 0 channel will be distributed into the C 1 channel, reducing the overall SNR. A Monte Carlo simulation was produced in which a spatially and spectrally uniform scene was generated. The scene consisted of 100 spectral bands spanning 400 to 800 nm with a signal consisting of 25 e -/nm. Equation (15) was used to simulate the modulated intensity across a 500 × 500 pixel subimage containing 2 µm 2 pixels. All calculations were conducted at λ 0 = 550 nm for FWHM spectral bandwidths spanning 5-50 nm in 5 nm increments for three spatial carrier frequencies ξ 0 of 30, 50, and 80 cycles/mm. Finally, shot noise was simulated across 100 trials and a spatial filter, based on a Hanning window, was used to extract the channel data. The window was defined as where w h is the window width and α, are the window's positions within the Fourier plane. For our simulation, w h = ξ 0 /2, α = ξ 0 , and = 0 such that the C 1 channel is extracted with a maximum spatial resolution while maintaining low cross-talk from the C 0 channel. It should be noted that, while the C 0 channel contains all residual unmodulated light, it is highly defocused. Unlike channeled spatial polarimetry systems, where there is an equally intense but in-focus S 0 component that causes significant cross-talk into the channel containing S 1 and S 2 , the inherent defocus minimizes cross-talk by band-limiting its spatial frequency content [24]. Finally, the SNR was calculated for each case and averaged across all trials, the results of which are depicted in Fig. 5    To verify spectralon tile the prototype were obtained as described corresponding the spectral im frequency con individual sp reference, the registration a applied to spe The reflec spectral sub-im measured by t The measured can be seen, standard curv designated wa curves, is prov Fig. 8(a) Fig. 8(b). R models was c ometrical mod the spectral p ns, while sim ltering mechan versus wavelengt ted FWHM spectr ite, blue and f the spectralon y a xenon lam ng and inverse m the proof-o be applied to fficients, a 2D and the full im ub-image at 5 lculated by ap ages [16]. The by normalizin Figure 9

Outdoor imaging and measurement results
In addition to the lab validation, the prototype was also used to image an outdoor environment to test its ability of resolving spatial and spectral content, of a 2D scene, under typical sunlight illumination. An outdoor scene that contained a blue car, plants, and red brick, as depicted in Fig. 10(a), was measured and processed with the aforementioned procedures. The 28 lenslets on the GPMLA were arranged as illustrated in Fig. 10(b), in which each lenslet is denoted by the wavelength of primary focus. The raw data image, as captured by the FPA, is depicted in Fig. 10(c) and a magnified view of one sub-image is presented in Fig. 10(d). This demonstrates the carrier frequency associated with the scene's 650 nm light. Finally, Fig.  10(e) depicts the spatially filtered spectral sub-images. After image registration, the spectra of three locations in the scene, which belong to the (1) car; (2) brick; and (3) plants, as illustrated in Fig. 11(a), were generated by normalizing their spectral intensities to that of the scene's asphalt. The asphalt's reflectivity was separately calibrated against a NIST-traceable white spectralon tile under the same illumination conditions. Spectral responses from these three locations were also measured by an Ocean Optics USB2000 modular spectrometer as references. The normalized spectral curves from the two spectrometers are drawn together in Figs. 11(b)-11(d). The RMS error, calculated for the car's, brick's, and plant's reflectivity was calculated to be 9.2%, 6.1%, and 9.2%, respectively, for wavelengths spanning 475-700 nm. These errors are generally higher than the indoor testing, by a factor of 2.5 on average, which is likely due to issues related to the use of 10 × 10 pixel averaging that was performed in the laboratory data as opposed to the 1 × 1 pixel sampling demonstrated here. We also attribute some issues due to the use of off-theshelf optomechanics, configured using free-space holders, which could be the cause of some artifacts in the data -most notably around 550 and 620 nm. These peaks appear in both the brick and leaf reflectivity and may be indicative of a systematic error in the setup. Future work will focus on developing a higher order model of these errors within the framework of a more robust prototype sensor.

Conclusio
A Geometric together with diffraction m results, the sim framework fo built and cali spectral chara the experimen outdoor scene USB2000 spe proposed spec inspection, an our system is potential adva