1.1

Imaging snapshot hyperspectral Fourier transform spectrometer

The birefringent hyperspectral snapshot optical system is composed of an afocal lens adjusting the focal length of the system, a lenslet array dividing the image of the scene into several subimages on a single focal plane array (FPA), and, between the FPA and the lenslet array, an interferometer. This latter generates interference fringes at an optical path difference (OPD) varying continuously along the plane of the FPA. (Fig. 1)

Fig. 1

Schematic representation of the proposed snapshot hyperspectral imaging concept.

By dividing the scene into several subimages, the system instantly generates an interferogram for each pixel in the scene (Fig. 2). Thanks to the Fourier transform relation between the interferogram and the spectrum, the spectrum for each point of the scene can be reconstructed. Such an optical system can acquire a three-dimensional data cube (x,y,ʎ) image within a single detector acquisition.

Fig. 2

Reconstructed interferogram for 1 point of the scene, from the single frame shown on the top-right insert. Note that the sampling of the interferogram depends on the position of the pixel into the subimage, but with linear and equidistant fringes, the sampling pitch is the same for all the pixels.

1.2

Limitation of the concept

Nevertheless, such an architecture presents limitations and require some trade-off.

For a given spectral band [λ_тіп : λ_тах] in wavelength or [σ_min : σ_max] in wavenumber, spectral resolution Δσ in wavenumber, and a detector of Μ × N pixels of p_pix pitch size, the following parameters are determined:

The three ways to increase the number of pixels per subimages, i.e. the number of spatial pixel of the hyperspectral cube are:

There is a clear trade-off between the number of spectral bands and the number of spatial pixels of the hyperspectral cube.

1.3

Scenarios

Here are some examples of scenarios this architecture could achieve with commercial detector (Tab. 1). Those are defined by either spatial or spectral needs.

Tab. 1

Imaging snapshot hyperspectral Fourier transform scenarios.

These scenarios highlight the previous point concerning dimensioning limiting points. Between two scenarios targeting the same spectral region, either the spectral bandwidth, the spectral resolution or the number of pixels per subimages are traded to fit the detector dimensions. The first visible/NIR scenario targets a large band to acquire this whole spectral domain, while the second scenario focuses on a smaller spectral bandwidth to study chlorophyll rapid change of reflection (red edge [3]). In the same way, the first SWIR scenario covers the whole [0.95 ; 1.65] μm range, while the second is dedicated to a smaller spectral band, for instance to detect significant CO₂ plumes[4] (despite the lack of the 2 μm band). The first MWIR scenario refers to mineral studies (MIRS [5]) while the second (2) refers to the measurement of jet plumes [6]. The LWIR scenarios may be useful for gas detection ([7]), and differ from their detector, with a cooled HgCdTe FPA and a microbolometer FPA.

However, these scenarios need further studies because the interferometer and optical system limitations are not taken into account.

2.1

Interferometers design

In this configuration, the interferometer is placed after the lenslet array, the three mains points to comply with are:

2.2

Localization plane in imaging configuration

The plane of fringes localization is a key point in the design of the interferometer. One can define it by the plane where the chief rays of the two paths of the interferometer intersect for each lenslet (Fig. 3). It is then independent of the position of the lenslet array.

Fig. 3

Plane of fringes localization of the interferometer.

On an imaging configuration, the interferometer will produce two images that need to overlap in order to maximize the fringes visibility. These two images may have a transversal displacement dz and a lateral one dy. The fig. 4 describes the influence of the images position in respect to the fringes localization. The closer the images are from the plane of the fringes localization, the closer the lateral gap dy is. Hence, the visibility is maximize when the image plane are on the same plane as the plane of fringes localization. Nevertheless, a longitudinal displacement dz can remain and lead to geometrical spatial default.

Fig. 4

Influence of the image position (directly linked to the lenslet position) compared to the plane of fringes localization. (a) Images are far from the fringes localization; (b) Images are close to the fringes localization.

As a result, it is important to study the position of the fringes localization plane in order to make sure the optical system can focus into it.

2.3

Birefringent interferometer

Using a birefringent interferometer is an advantageous choice thanks to its compactness and robustness to vibration as it is a common path interferometer. The interferometer is made of a combination of prisms, placed between two polarizers (Fig. 5). The light is therefore divided into two paths of two linear polarizations corresponding to the eigenmodes of the medium. Throughout the interferometer, each path see either the ordinary index n_o or an extraordinary n_e. It results into an optical path difference between the two paths.

Fig. 5

Normaski type interferometer dividing an incident ray into two rays of crossed polarizations. The polarizers (P & A) are oriented at 45° with respect of prisms polarization eigenmodes.

The three points announced in Section 2.1 needs to be respected. The first point is achieved by using prisms. One prism at first order will produce an optical path difference OPD = (n_e – n_o)e(y), with e(y) the thickness of the prism at height y. A small rotation of the interferometer about the z axis, enables to create a varying OPD on the y et x direction. The second and third points are achieved by using combination of birefringent prisms with different optical axis orientations. Here are some combinations that enable those conditions (Fig. 6):

Fig. 6

2 combinations of prisms leading to a plane of fringes localization on the image plane. (a) Tilted Nomarski prism [8], (b) Two Nomarski prisms separated by an half-wave plate with axis oriented 45° about the z axis (This plate swaps the polarization along x and y) [2].

3.1

Simulation setup

The goal of the next section is to simulate and compare the 2 previous birefringent interferometers (Fig. 6). A Matlab program has been developed in order to evaluate the image quality of the two paths and to evaluate the exact optical path difference seen by a wavefront going through the interferometer. This program was initially developed to simulate plane wavefront propagating through interferometer where the plane of localization is at infinity [9]. As we are using an interferometer that use spherical wavefront, an adjustment has been done in order to propagate an array of rays through it.

The parameters of the two interferometers are taken in order to generate an OPD ranging from 0 to δ_max = 0.85 mm of the scenario “visible (2)” (Tab. 2) and the lenslet is placed in order to focus on the plane of fringes localization.:

Tab. 2

Geometrical parameters of the two birefringent interferometers simulated.

3.2

Fringes visibility

For one point of the field coming from one lenslet, each ray composing the spherical wavefront will see different thickness and optical indexes. This will lead to a decrease of the fringes visibility. Moreover, rays between the two paths of the interferometer does not intersect properly on the same plane. It makes difficult to estimate the real optical path difference, i.e. the interference state, on the image plane.

One way to estimate it is to study the monochromatic wavefront of the two paths on the exit pupil. On this plane, the coherent sum of the two wavefronts leads to interference. As the energy is conserved between the exit pupil plane and the image plane, the interference state is unchanged. Then, the normalized interference state for one point of the field can be estimated from there : with δ₁(x, y) and δ₂(x, y) the optical path of the wavefront at the position (x,y) in the exit pupil plane for respectively the first and second path of the interferometer, and “< >” the spatial mean over the pupil plane(Fig. 7).

Fig. 7

Interference state for one point of the field of one lenslet estimated within the exit pupil plane.

In order to compute it, the optical path on the exit pupil plane is interpolated with the optical path of each ray. For an ideal interferometer, the difference between these two would be constant. Nevertheless, the two paths experienced different aberrations, leading to a non-constant interference state over the pupil (Fig. 8&9).

Fig. 8

(a) Optical path difference δ₁(x, y) − δ₂(x, y) on the exit pupil plane and (b) corresponding normalized intensity defined by for 2 Nomarski Prisms + ʎ/2; The incident wavefront comes from one lenslet centered at a height y = 7.5 mm (destructive state) and focusing on axis.

Fig. 9

(a) Optical path difference δ₁ (x, y) − δ₂ (x, y) on the exit pupil plane and (b) corresponding normalized intensity defined by for a tilted Nomaski prism; The incident wavefront comes from one lenslet centered at a height y = 7.5 mm (destructive state) and focusing on axis.

The interferogram of one lenslet is simulated in the image plane by evaluating the interference state I_interference for each point of the field, i.e. by simulating a wavefront converging on each point of the field. As we can see on Fig. 8&9, the variation of the difference of optical path on the exit pupil plane is bigger for the tilted Nomarski prism. The resulting interferogram has a lower visibility (Fig. 10). One can observe bigger loss of visibility along the field of view, reflecting field aberrations.

Fig. 10

Interferogram in the image plane of one lenslet simulated with the interference state for each converging wavefront on the image field at [x_im, y_im]. (a) Tilted Nomarski prism (b) 2 Nomarski + ʎ/2; The lenslet is centered at y=7.5 mm; @λ = 800nm.

For this scenario, the interferometer using a half-wave plate produces an interferogram with a better visibility. Nevertheless, this method only takes into account the difference between the two wavefronts in the exit pupil plane without directly looking into the image spot. Those wavefronts will suffer from optical aberrations and/or diffraction limitation that will lead into a non-stigmatic image.

3.3

Spatial limitation

The spot diagram on the image plane can be simulated in order to look into the spatial quality produced by the interferometer. On Fig. 11, the spot diagram on axis of the two interferometers has been simulated. In this case, both produce spot diagrams with a lower size than the diffraction spot : .

Fig. 11

Spot diagrams (Squared aperture @ 800nm) on the image plane produced by the (a) Tilted Nomarski prism (b) 2 Nomarski Prisms + ʎ/2.

However, we notice that the tilted Nomarksi prism produces bigger spot diagrams. This difference comes from the natural focusing shift between the two paths produced by the thickness of the birefringent prisms [10], and due to the anisotropy of the media, wavefront going through extraordinary eigenmodes suffer from astigmatism. This does not affect the spatial quality because the diffraction effect are more important, but they are responsible of the loose of fringes visibility of the previous section. As for the 2^nd configuration, using a half-wave plate enables to partially compensate those unwanted effects.

In this scenario, the interferometer using a half-wave plate produces an interferogram with a better fringes visibility because optical aberrations such as difference of focus of the two paths and astigmatism are not significant. On the other hand, those aberrations affect the tilted Nomarski prism. However, it is important to note that those aberrations depend of several parameters such as aperture size, thickness of the interferometer, n₀, n_e. Moreover, the latter is easier to manufacture because the half-wave plate can be difficult to produce for some large bandwidth. It can be necessary to favour this design in some conditions.