1.

Introduction

Fourier transform spectrometers (FTSs) are well known for their high spectral resolving power.^1,2 Typically, high spectral resolution measurements are based on Michelson interferometers. Although these offer measurement flexibility, high throughput, and spectral resolution, disadvantages include vibration sensitivity, temporal and mechanical scanning requirements, and alignment sensitivity. Fortunately, vibration sensitivity and the need for mechanical scanning can be overcome with common-path interferometers.³ These sensors map optical path difference (OPD) across a two-dimensional (2-D) focal plane array (FPA) or line array (LA) camera such that

One issue with conventional FTS implementations, including common-path systems, is sampling inefficiency. For instance, a conventional nonaliased FTS must resolve all wave numbers to fully measure a spectrum.⁴ This limitation is best observed with an example. Consider a conventional FTS, designed to measure the atmospheric oxygen emission within the spectral band of $558\pm 3\mathrm{nm}$ (e.g., 17,825 to $\mathrm{18,018}{\mathrm{cm}}^{-1}$) to infer atmospheric wind speed and temperature.^5–8 Assuming that a complete interferogram is required (i.e., not the partial interferograms described in Refs. 56.7.–8), then the required spectral resolution $\mathrm{\Delta }\sigma$ of the system is approximately $1.0{\mathrm{cm}}^{-1}$. If the spectrometer’s cutoff frequency were 540 nm ($\mathrm{18,519}{\mathrm{cm}}^{-1}$), then the FTS must measure all spectral points spanning $0{\mathrm{cm}}^{-1}$ to the Nyquist frequency. Therefore, while only 193 samples within the region of the emission line are actually of interest, approximately 18,519 samples, assuming a single-sided interferogram, are required to acquire the region of interest. Ultimately, this value would be slightly greater, since a small double-sided region is required for phase correction.¹

Although sampling inefficiencies can be overcome in a conventional Michelson interferometer by aliasing the interferogram’s measurement,⁹ this is not a viable option with most FPA- or LA-based FTSs. Given a pixel width of $d_w$, the interference’s spatial frequency ${\eta }_I$ in units of cycles/pixel, generated by the OPD defined in Eq. (1), can be expressed as

Fig. 1

Fringe visibility versus the interferogram’s spatial frequency.

Revisiting the previous oxygen emission example demonstrates the impracticality of the aliasing technique when using detector arrays. Assuming a linear array is used with a pixel pitch of $d_w$ means that the required slope of the OPD can be expressed as

Given $N_s=\mathrm{18,519}$ yields ${\eta }_I=1.0$ cycles/pixel, resulting in zero fringe visibility per Fig. 1. Ideally the spatial frequency must be decreased in order to increase the detector’s response. However, the issue is only worsened when using the aliasing technique. For instance, if aliasing is implemented on a 1000 pixels array, then ${\eta }_I=17.9$ cycles/pixel. Thus, while the spatial frequency is aliased, its visibility is too low to remain detectable.

Two techniques exist to increase the spectral resolution in these common-path instruments while minimizing the number of samples required: (1) modifying the offset $b$ while maintaining a resolvable slope $a$ or (2) heterodyning the interference to lower spatial frequencies.^4,10,11 Since modifying $b$ does little to negate the sampling limitation, we consider spatial heterodyning in the current approach. A spatial heterodyne interferometer (SHI) is able to conserve the high spectral resolution of conventional FTS systems.^5–8 They do so by replacing the retro-reflecting mirrors of a Michelson interferometer with Littrow-configured blazed diffraction gratings.^5,11 This enables any arbitrary wave number, ${\sigma }_h$, to generate a 0 cycle/m interference fringe, where ${\sigma }_h$ is the heterodyne wave number (note that in the conventional nonaliased FTS, ${\sigma }_h=0{\mathrm{cm}}^{-1}$).

The implementation of SHI has primarily leveraged the Michelson interferometer (MI) architecture.^7,8 In these systems, the MIs vibration and thermal errors have been addressed by creating monolithic all-glass systems. Alternatives to this technique, which aim to create common-path vibration insensitive designs, are based on Sagnac interferometers.^12,13 However, to minimize both size and vibration sensitivity, SHIs based on fiber-optic Mach–Zehnder interferometers¹⁴ and an SHI system based on birefringent prisms and polarization gratings (PGs)^15,16 have also been introduced. With these implementations, the interferometer’s size, weight, and alignment complexity can be significantly reduced over reflective free-space Sagnac interferometers or uncommon-path monolithic Michelson interferometer designs.

In this paper, we focus on advancing the modeling and calibration procedures of a polarization spatial heterodyne interferometer (PSHI) that was originally presented in Ref. 15. Advantages of this architecture lie in its potential compactness, due to the use of monolithic birefringent crystals, and its vibration insensitivity, stemming from its common-path design.^17,18 In Sec. 2, we discuss a Jones matrix model of the PSHI and investigate its nonideal properties. In Sec. 3, we present our experimental system while Sec. 4 contains the experimental measurements that are used to validate the model. Finally, in Sec. 5, we discuss the methods of calibration and experimentally demonstrate the calibration procedure, validating it against a separate spectral measurement.

2.

Theoretical Model

The goal of our theoretical model is to understand the nonideal characteristics that arise when performing spatial heterodyning with a PG.¹⁶ In the PSHI, interference cross talk was caused by the PG’s unwanted diffraction orders.¹⁵ These orders are presented in Fig. 2, superimposed on the schematic of the original PSHI system. Polarized light initially transmits through a Wollaston prism (WP), with an apex angle $\alpha$, which causes the two incident polarization states to split at an angle ${\theta }_{\mathrm{WP}}$. These beams are then incident on a quarter wave plate (QWP). The QWP has a fast axis orientation of 45 deg with respect to the $x$, such that the orthogonal linear eigenpolarizations of the WP are converted into orthogonal circular polarization states. These states interact with the PG and are subsequently diffracted into either the $m=0$ or $m=\pm 1$ diffraction orders. A linear analyzer (A) unifies the polarization state, enabling interference to be measured on the FPA.

Fig. 2

Schematic of the beams exiting the PSHI.

Ideally, after the WP’s light transmits through the PG, only the two $m=-1$ order beams, with angular separation ${\theta }_d$ (e.g., an angularly reduced component compared to ${\theta }_{\mathrm{WP}}$), should exist. However, the PG’s zero-order light leakage allows two unwanted beams to propagate with an angular separation of ${\theta }_0$, such that ${\theta }_0={\theta }_{\mathrm{WP}}$. Furthermore, error in the QWP’s retardance will cause some light to be coupled into the $m=+1$ order, creating two additional unwanted beams propagating at an angle ${\theta }_u$ with respect to each other. Finally, the cross terms, denoted as ${\theta }_{c1}$, ${\theta }_{c2}$, and ${\theta }_{c3}$, also create observable interference effects. Under the small angle approximation, the angle of diffraction from the PG can be expressed as

The Jones matrix formalism¹⁹ was used to model the interference created by the beams exiting the PG. For a WP, the Jones matrix can be expressed as

Meanwhile, the QWP can be expressed using Eq. (10) with $\varphi ={\varphi }_{\mathrm{QWP}}$ and $\theta ={\theta }_{\mathrm{QWP}}$, where ${\theta }_{\mathrm{QWP}}$ and ${\varphi }_{\mathrm{QWP}}$ are the QWPs orientation and retardance, respectively. The Jones matrix for a linear polarizer, with a transmission axis oriented at 0 deg with respect to the $x$-axis, is

Calculating the electric field incident onto the detector can be accomplished by

Using a 45-deg linearly polarized input makes

It should be mentioned that Eqs. (25)–(29) assume negligible $\pm 2\mathrm{nd}\text{-}\mathrm{order}$ diffraction, which was measured at 0.21% in our experimental PG. Expanding the discussion to continuous spectral distributions means that Eq. (24) can be spectrally band-integrated such that

2.1.

Simulated Results

Numerical simulations were performed, using Eq. (24), to investigate the systematic effects caused by retardance errors in the PG and QWP. Double-sided interferograms were created for monochromatic spectra at wavelengths ${\lambda }_0$ of 460, 560, and 700 nm. A WP, with a wedge angle of $\alpha =6.2$ deg and a PG with a period $\mathrm{\Lambda }=453\mu \text{m}$, were used to maintain consistency with our previous experimental setup per Ref. 15. A fast Fourier transform (FFT) was applied to the simulated 1280 pixel element interferograms, along the $y$-dimension, such that

Eq. (37)

$${\mathrm{FF}}_0(y_j,{\sigma }_0)=|\mathrm{FFT}(H(y)I_0(y_j,{\sigma }_0))|,$$

Eq. (38)

$${\mathrm{FF}}_{c1}(y_j,{\sigma }_0)=|\mathrm{FFT}(H(y_j)I_{c1}(y_j,{\sigma }_0))|,$$

Eq. (39)

$${\mathrm{FF}}_d(y_j,{\sigma }_0)=|\mathrm{FFT}(H(y_j)I_d(y_j,{\sigma }_0))|,$$

Eq. (40)

$${\mathrm{FF}}_{c3}(y_j,{\sigma }_0)=|\mathrm{FFT}(H(y_j)I_{c3}(y_j,{\sigma }_0))|,$$

and

Eq. (41)

$${\mathrm{FF}}_u(y_j,{\sigma }_0)=|\mathrm{FFT}(H(y_j)I_u(y_j,{\sigma }_0))|,$$

where the constant term in Eq. (24) has been removed, the $x$-dimension has been suppressed because interference only occurs along $y$, ${\sigma }_0=1/{\lambda }_0$, $E({\sigma }_0)=1$ for all ${\sigma }_0$, and the subscript $j$ denotes $y$ as a discrete quantity. A Hanning window, $H(y_j)$, was used to apodize the interferograms and is defined as

Eq. (42)

$$H(y_j)=0.5(1+\cos (\pi y_j/w)),$$

where $2w$ is the Hanning window’s full-width.

Equations (37)–(41) were calculated for various values of ${\varphi }_{\mathrm{PG}}$ and ${\varphi }_{\mathrm{QWP}}$. First, the WP’s spatial frequency power spectrum was calculated without the PG by setting ${\varphi }_{\mathrm{PG}}={\varphi }_{\mathrm{QWP}}=0\mathrm{deg}$ for all ${\sigma }_0$. This yielded the spatial frequency spectrum depicted in Fig. 3(a) in which ${\mathrm{FF}}_0$ is the only nonzero component. For an ideal PG and QWP, ${\varphi }_{\mathrm{PG}}=180\mathrm{deg}$ and ${\varphi }_{\mathrm{QWP}}=90\mathrm{deg}$ for all ${\sigma }_0$. For these values, Figs. 3(b) and 3(c) depict the spectra that are heterodyned to both high (${\theta }_{\mathrm{QWP}}=+45\mathrm{deg}$) and low (${\theta }_{\mathrm{QWP}}=-45\mathrm{deg}$) spatial frequencies, respectively. Meanwhile, the spatial frequency spectrum that is obtained with error in the PG’s retardance is depicted in Fig. 3(d), where ${\varphi }_{\mathrm{PG}}=135\mathrm{deg}$ and ${\varphi }_{\mathrm{QWP}}=90\mathrm{deg}$. In this case, zero-order light leakage is manifested by the presence of the ${\mathrm{FF}}_0$ component. Conversely, the result of error in the QWP’s retardance is shown in Fig. 3(e), where ${\varphi }_{\mathrm{PG}}=180\mathrm{deg}$ and ${\varphi }_{\mathrm{QWP}}=45\mathrm{deg}$. In this case, both the high and low spatial frequency components, corresponding to ${\mathrm{FF}}_u$ and ${\mathrm{FF}}_d$, respectively, are present. Finally, simultaneous error in both the PG and QWP retardance values is shown in Fig. 3(f), where ${\varphi }_{\mathrm{PG}}=135\mathrm{deg}$ and ${\varphi }_{\mathrm{QWP}}=45\mathrm{deg}$. Frequency components ${\mathrm{FF}}_0$, ${\mathrm{FF}}_u$, and ${\mathrm{FF}}_d$ are present, as well as the cross-interference terms corresponding to ${\mathrm{FF}}_{c1}$ and ${\mathrm{FF}}_{c3}$. Notable is that the ${\mathrm{FF}}_{c1}$ component is multiplexed with the desired ${\mathrm{FF}}_d$ component.

Fig. 3

Relative magnitude of the Fourier components versus spatial frequency. The three values of ${\lambda }_0$ are shown alongside the channel magnitudes to indicate the approximate mapping of wavelength to spatial frequency within each channel. Simulation results for (a) Wollaston prism only with ${\varphi }_{\mathrm{PG}}=0\mathrm{deg}$, ${\varphi }_{\mathrm{QWP}}=0\mathrm{deg}$. (b) Ideal up-shifted spectra with ${\varphi }_{\mathrm{PG}}=180\mathrm{deg}$, ${\varphi }_{\mathrm{QWP}}=90\mathrm{deg}$, and ${\theta }_{\mathrm{QWP}}=+45\mathrm{deg}$. (c) Ideal down-shifted spectra with ${\varphi }_{\mathrm{PG}}=180\mathrm{deg}$, ${\varphi }_{\mathrm{QWP}}=90\mathrm{deg}$, and ${\theta }_{\mathrm{QWP}}=-45\mathrm{deg}$. (d) Error in only the PGs retardance with ${\varphi }_{\mathrm{PG}}=135\mathrm{deg}$, ${\varphi }_{\mathrm{QWP}}=90\mathrm{deg}$, and ${\theta }_{\mathrm{QWP}}=-45\mathrm{deg}$. (e) Error in only the QWPs retardance with ${\varphi }_{\mathrm{PG}}=180\mathrm{deg}$, ${\varphi }_{\mathrm{QWP}}=45\mathrm{deg}$, and ${\theta }_{\mathrm{QWP}}=-45\mathrm{deg}$. (f) Error in both the PGs and QWPs retardance values with ${\varphi }_{\mathrm{PG}}=135\mathrm{deg}$, ${\varphi }_{\mathrm{QWP}}=45\mathrm{deg}$, and ${\theta }_{\mathrm{QWP}}=-45\mathrm{deg}$.

Finally, a simulation was performed using a PG retardance spectrum that was calculated per Eq. (17) with $d_{\mathrm{LC}}=2.05\mu \text{m}$. A QWP, based on quartz crystal, was also simulated using

Eq. (43)

$${\varphi }_{\mathrm{QWP}}=2\pi \sigma d_{\mathrm{QWP}}\mathrm{\Delta }{n}_{\mathrm{quartz}},$$

where $d_{\text{QWP}}=15.27\mu \text{m}$ was used for the quartz crystal’s thickness. The birefringence of quartz was simulated using

Eq. (44)

$$\mathrm{\Delta }{n}_{\mathrm{quartz}}=L{\sigma }^3+G{\sigma }^2+J\sigma +K,$$

where $L=7.655\mathrm{E}-23$, $G=-3.243\mathrm{E}-16$, $J=9.101\mathrm{E}-10$, and $K=8.137\mathrm{E}-3$ and $\sigma$ has units ${\mathrm{m}}^{-1}$. The simulated retardance spectra for the PG and QWP are presented in Fig. 4(a) while the spatial frequency content is depicted in Fig. 4(b). Of particular importance is that the maximum magnitude of the ${\mathrm{FF}}_d$ component corresponds to ${\varphi }_{\mathrm{PG}}=180\mathrm{deg}$ and that the multiplexing between ${\mathrm{FF}}_d$ and ${\mathrm{FF}}_{c1}$ is present due to simultaneous error in both ${\varphi }_{\mathrm{PG}}$ and ${\varphi }_{\mathrm{QWP}}$.

Fig. 4

(a) Simulated retardance spectra for a PG, with a layer thickness $d=2.05\mu \text{m}$, and a quartz QWP with a thickness of 15.27 μm. (b) Simulated frequency content of each component.

3.

Experimental Setup

A view of the system that was used to validate the theoretical model is depicted in Fig. 5. It consists of a diffuse source that illuminates a linear polarization generator (LPG). The LPG consisted of a linear polarizer with a transmission axis nominally oriented at ${\theta }_G=135\mathrm{deg}$, where the angle is measured with respect to the $x$-axis. Light from the LPG was then incident onto a quartz WP that has an apex angle $\alpha$ of 6.2 deg. The two linear eigenpolarizations of the WP are oriented at 0 and 90 deg. An achromatic quarter wave plate (AQWP) follows the WP with a nominal orientation of ${\theta }_{\mathrm{QWP}}=\pm 45\mathrm{deg}$ such that the linear eigenpolarizations from the WP are converted into circular polarization states. A singlet, with a focal length $f=100\mathrm{mm}$, was used to relay an image of the WP’s wedge onto a PG with a lateral magnification $m_g=-1$. A variable iris was used to stop the lens down to reduce aberrations. It should be mentioned that this low power singlet was incorporated, as opposed to a multiple-element lens, to minimize polarization aberrations between the AQWP and the PG. The PG was patterned with polymerized liquid crystal with a spatial period of $\mathrm{\Lambda }=453\mu \text{m}$ and a peak first-order diffraction efficiency at $\lambda =610\mathrm{nm}$. Light from the PG was then transmitted through a linear polarization analyzer, which consisted of a linear polarizer with a transmission axis nominally oriented at ${\theta }_A=45\mathrm{deg}$. Finally, a 50 mm focal length c-mount objective lens imaged the interference pattern onto an 8-bit $1280\times 960$ pixel element FPA.

Fig. 5

Experimental setup of the PSHI. The PG’s grating vector is indicated by the arrow, as are the WP’s fast axis orientations.

The measured retardance spectrum of the AQWP is provided in Fig. 6(a). This measurement was taken by rotating the AQWP between crossed linear polarizers while measuring the transmitted spectrum using an Ocean Optics HR4000 spectrometer. The accuracy of this measurement is estimated at $\pm 0.25\%$. Meanwhile, the PG’s retardance was calculated from the measured zero-order diffraction efficiency by fitting it to the theoretical closed-form expression

Fig. 6

Measured (a) achromatic QWP retardance spectrum and (b) PG zero-order diffraction efficiency and retardance spectra.

4.

Experimental Results

The experimental procedure for validating the model consisted of aligning the AQWP at ${\theta }_{\mathrm{QWP}}=+45\mathrm{deg}$ and ${\theta }_{\mathrm{QWP}}=-45\mathrm{deg}$ to up- and down-shift the prism’s interference frequency, respectively. Light from a monochromator was used to illuminate the PSHI. The source’s wavelength, ${\lambda }_s$, was varied from 460 to 700 nm in 20 nm increments and at each of these wavelengths a 2-D interference pattern was recorded. Since the interference components amplitude modulate the spectrum of the light source used in the monochromator, the spectrum’s amplitude was removed by calibrating to a flat field at each wavelength.

In a procedure similar to Ref. 17, a flat field measurement was calculated at each monochromatic wavelength by acquiring two interferograms that are out-of-phase by 180 deg. The nominal interferogram, $I_{G45}$, was acquired with the LPG at ${\theta }_G=45\mathrm{deg}$ while the phase-shifted interferogram, $I_{G135}$, was acquired at an LPG orientation of ${\theta }_G=135\mathrm{deg}$. These two interferograms can be expressed as

A flat field measurement was calculated at both orientations of the QWP (${\theta }_{\mathrm{QWP}}=\pm 45\mathrm{deg}$) in order to account for polarization dependencies in the optical transmission. Interferograms of monochromatic spectra were then acquired by the PSHI such that

Normalization of Eq. (49) by (48) produces an interferogram without responsivity, optical transmission, or spectral dependencies.

After all monochromator interferograms were normalized to the flat field, an average across the FPA’s $x$-dimension was taken in order to obtain a one-dimensional interferogram along $y$. Fourier spectra for each interferogram were then calculated per Eqs. (37)–(41). A summary of the experimental data is provided in Fig. 7 for both the (a) down-shifted (b) up-shifted cases. In each plot, the magnitude of the Fourier components for the experimental measurements ($\text{M}$) is plotted alongside the results from the theoretical model ($\text{T}$) for each channel. Due to the use of an achromatic QWP, minimal multiplexing is observed between ${\mathrm{FF}}_d$ and ${\mathrm{FF}}_{c1}$. Hence, the most significant error is a result of zero-order light leakage through the PG, which can be reduced by incorporating an achromatic PG.²² It should be mentioned that this was not implemented in the current paper so that we could study the effects of zero-order light leakage.

Fig. 7

Comparison between the theoretical model ($T$) and measurements ($M$) for QWP orientations of (a) ${\theta }_{\mathrm{QWP}}=+45\mathrm{deg}$ and (b) ${\theta }_{\mathrm{QWP}}=-45\mathrm{deg}$. There is an excellent agreement between $T$ and $M$ for all wavelengths.

The root-mean-square (RMS) error between the measured and theoretical Fourier components was calculated as

Table 1

Percent RMS error between theoretical and measured data for each Fourier component.

5.

Calibration Results

For most applications, the down-shifted spectra of Fig. 7(a) are of greatest utility. To account for linear mixing between the ${\mathrm{FF}}_d$ and ${\mathrm{FF}}_{c1}$ components in this configuration, a linear system model was used to spectrally calibrate the PSHI.^14,23 A matrix $\mathbf{H}$ was created such that

Using the experimental setup, depicted previously in Fig. 5, the H-matrix’s calibration performance was evaluated for calculating the output spectrum. This was conducted with the experimental setup depicted in Fig. 8, which depicts the inclusion of a 100-mm diameter integrating sphere (IS). An Ocean Optics HR4000 spectrometer, connected to the sphere via a 200-μm diameter fiber, was used to measure spectra for independent validation. The sphere was illuminated by either a white light tungsten halogen lamp or a monochromator. Optionally, the white light source could be filtered with removable gelatin filters. H-matrix characterization was accomplished using the monochromator. The monochromator was incremented from 420 to 720 nm in $N=19$ increments with a uniform wave number spacing of $\mathrm{\Delta }\sigma =551{\mathrm{cm}}^{-1}$. This spectral resolution corresponds closely to the spectral resolution of the WP.

Fig. 8

(a) Schematic of the experimental setup for calibrating the PSHI and (b) photo of the system on the benchtop.

The measured H-matrix is depicted in Fig. 9. Notable is the presence of both the $I_d$ and $I_0$ interference components that correspond to the high and low spatial frequency modulations versus $y$, respectively. As observed in the simulations, $I_0$ has increasing contrast for values of ${\varphi }_{\mathrm{PG}}$ away from 180 deg per Fig. 6(b). Calculating the pseudo-inverse of $\mathbf{H}$ enables the input spectrum to be calculated using

Fig. 9

Measured H-matrix.

Fig. 10

Measured transmission data comparison from the PSHI and the OO spectrometer for filters F1 to F4.

The RMS error, averaged for all four transmission measurements, was calculated between the PSHI and OO data to be approximately 1.2% for these data.

6.

Conclusion

In this paper, we successfully demonstrated a theoretical model and experimental calibration procedure for a PSHI. The interferometer was based on a WP that was heterodyned using a PG. As demonstrated by the model and subsequent experiments, nonideal frequency components are caused by the PG’s zero-order light leakage and error in the QWP’s retardance. Fortunately, low zero-order diffraction efficiencies ($<3\%$) in the PGs are achievable using broadband multilayer achromatic PG designs.²² Additionally, as demonstrated in the experiment, AQWPs can be leveraged to further reduce the undesirable frequency components. Finally, the H-matrix calibration procedure was implemented to calibrate the PSHI. This yielded an experimentally observed average RMS error of 1.2% when compared to an Ocean Optics spectrometer.