A hyperspectral imager based on a Fabry-Perot interferometer with dielectric mirrors

Massimo Zucco; Marco Pisani; Valentina Caricato; Andrea Egidi

doi:10.1364/OE.22.001824

1. Introduction

Hyperspectral imaging (HSI) is a technique where a spectrophotometer is integrated in an imaging device and the spectral content of each pixel of a scene is measured. Many applications of HSI have been developed and described in literature, to name but a few: in colorimetry [1], in cultural heritage to analyze historical documents [2] or to carry out noninvasive analysis of painting [3], in thermal imaging to measure land surface emissivity and temperature [4], to measure spectra of earth surface and atmosphere from a satellite [5], in the UV range for mapping the spectral irradiance of the whole sky in the 300 nm - 400 nm range in order to improve the traceability of spectral solar ultraviolet radiation measurements [6], in order to characterize vegetation spectral features [7], in fluorescence microscopy to identify fluorescent beads with a coded aperture snapshot spectral imager [8], in food analysis to discriminate in a NIR image differences between types of lamb muscles and fat [9].

The core of a HSI device is a spectrophotometer which is based either on a dispersive element like a prism [10] or a grating [11], on an optical bandpass filter [12] or on an interferometer (Sagnac [13], Mach-Zender [14], Michelson [15] or Fabry-Perot [16]). There is an intrinsic advantage (also known as Fellgett advantage) using interferometer-based spectrometers when the noise is dominated by detector noise: in this case the improvement of the signal to noise ratio is proportional to the square root of the number of bins when compared to the other spectrometers [13, 17]. The spectrum is calculated by applying algorithms based on the Fourier transform. The intrinsic resolution of the Michelson interferometer is related to the maximal distance L travelled by the mirrors with respect to zero retardation, the resolution in wavenumber is ½ L [18]. By windowing the interferogram with different apodization functions the resolving power could be improved at the expense of the accuracy of amplitude measurement [19]. In a similar way in a Fabry-Perot (F-P) interferometer the spectrum is calculated by applying the Fourier transform but, due to the intrinsic non linearity of the interferogram (the F-P interferogram is a sum of Airy functions), the harmonics of the spectrum are present. With D as the distance between the mirrors, the resolution of the base spectrum is the same as in the Michelson interferometer (D = L), but considering the N^th harmonic of the spectrum the resolution gets N times smaller: the reason is that the associated beam has travelled a distance equal to ND in the cavity. For example in [20], considering the 20^th harmonic the resolution improved by 20, realizing therefore a more compact spectrophotometer with respect to a Michelson interferometer with the same resolution. Each spectrophotometer requires a different set-up and the measured spectrum has a different format, therefore each spectrophotometer has to be integrated differently in the imaging device in order to obtain the hyperspectral image: different techniques are whiskbroom, pushbroom, windowing, framing described in [21].

In our application we use a spectrophotometer based on a F-P interferometer and the spectral content of each pixel is recorded in the interferogram acquired scanning the retardation with mechanical actuators. The spectrum is eventually calculated with a Fourier based algorithm. It should be stressed that in our set-up the F-P is not used as a tunable optical bandpass filter as described in the review [12], but as an interferometer [18]. Using metallic mirrors with low reflectivity we have realized two different prototypes in the 380 nm - 720 nm range with a reference laser at 410 nm [22] and in the 900 nm – 1700 nm range with a reference laser at 980 nm [23], where the range of the measured spectrum covers nearly an octave. The final resolution is limited by the maximal displacement of the PZT actuators, which is about 40 μm. These prototypes use metallic mirrors with reflectivity of about 20% and with negligible dispersion allowing us to calculate the spectrum with a cosine Fourier transform based algorithm. Nevertheless, metallic mirrors, with respect to dielectric mirrors, have the drawback of having high losses, with a fragile layer of few tens of nanometers of metal, not easily reproducible and difficult to find in the market. In this paper we present our new prototype based on dielectric mirrors. Since dielectric mirrors have a significant dispersion, the spectrum is calculated using an algorithm which takes into account a phase correction which we have measured with different techniques. The prototype has been validated measuring the spectral image of scenes with emitting bodies like LEDs and scenes with reflecting surface in the 520 nm - 720 nm band. In this set-up the movement of the mirrors is obtained with a peristaltic pump which evacuates the air between the mirrors allowing in this way a much higher mirror distance and a much better final attainable resolution, limited only by the optical throughput [16].

2. Discussion about Fourier transform spectroscopy

In principle, for spectrometers based on Michelson interferometers, like FTIR, the spectrum $S (\tilde{ν})$ of a continuum and broadband source is obtained with a cosine Fourier transform integral from the acquired single-sided interferogram $I (δ)$ where $\tilde{ν}$ is the wavenumber and δ is the retardation or optical path delay [18],

S (\tilde{ν}) = 2 \int_{0}^{\infty} I (δ) \cos (2 π \tilde{ν} δ) d δ

In practice, the interferogram is sampled at finite sampling interval Δs and consists of N discrete, equidistant points and Eq. (1) transforms in Eq. (2), where all the constants have been discarded. The discrete version of the cosine Fourier transform is

S [m Δ \tilde{ν}] = \sum_{n = 0}^{N - 1} I [n Δ s] \cos (2 π m Δ \tilde{ν} n Δ s)

The maximum retardation N Δs, the number of points N and the resolution $Δ \tilde{ν}$ are all interrelated through the formula

Δ \tilde{ν} \approx \frac{1}{δ_{\max}} = \frac{1}{N Δ s}

moreover the sampling rate has to satisfy Nyquist criterion: at least two points per fringe of the highest wavenumber are necessary to avoid the foldover of the alias.

Nevertheless, this simple model does not take into account the presence of dispersive effects: the retardation δ is a function of wavenumber due to dispersive components in the optical path like beam splitters or mirrors, to non-symmetric sampling with respect to zero retardation and to electronic filtering of the acquired signal. In presence of dispersive effects, the interferogram is no longer symmetric around the zero retardation. The dispersive effect is contained in a phase correction $Θ (\tilde{ν})$ through the equation $2 π \tilde{ν} δ (\tilde{ν}) = 2 π \tilde{ν} δ + Θ (\tilde{ν})$ , the argument of the cosine in Eq. (1). The solution of Eq. (1) requires the calculation of a complex spectrum $S' (\tilde{ν})$ with a complex Fourier transform

S' (\tilde{ν}) = \int_{- \infty}^{\infty} I (δ) e^{- i 2 π \tilde{ν} δ (\tilde{ν})} d δ .

In the case of slow varying phase

Θ (\tilde{ν})

, Eq. (1) reduces to Eq. (5) [18]:

S (\tilde{ν}) = Re (\tilde{ν}) \cos (Θ (\tilde{ν})) + Im (\tilde{ν}) \sin (Θ (\tilde{ν}))

where

Re (\tilde{ν})

and

Im (\tilde{ν})

are the real and imaginary part of

S' (\tilde{ν})

.

The discrete counterpart of Eq. (5) becomes the series:

\begin{array}{l} S [m Δ \tilde{ν}] = [\sum_{n = - N + 1}^{N - 1} I [n Δ s] c o s (2 π m Δ \tilde{ν} n Δ s)] \cos (Θ (m Δ \tilde{ν})) \\ + [\sum_{n = - N + 1}^{N - 1} I [n Δ s] \sin (2 π m Δ \tilde{ν} n Δ s)] \sin (Θ (m Δ \tilde{ν})) \end{array}

The phase correction

Θ (\tilde{ν})

is calculated by acquiring a double-sided interferogram and applying the following formula

Θ (\tilde{ν}) = arc \tan (\frac{Im (\tilde{ν})}{Re (\tilde{ν})})

At the end, when the phase correction $Θ$ is finally measured, the resulting spectrum is obtained with the Mertz method [24], applying the discrete Fourier series in Eq. (6). Alternatively, with the Forman method [25] the phase correction is convolved with the asymmetric interferogram giving a symmetric interferogram, and the spectrum is calculated with a Fourier series as in Eq. (2). The theory we have presented in this section is valid for a Michelson interferometer, the application to our HSI prototype based on a Fabry-Perot interferometer requires the modification of the technique as described in the next section.

3. Description of the HSI based on metallic mirror F-P interferometers

In the first prototype of HSI that we developed and described in details in [22, 23], we used a photometer based on a F-P interferometer where mirrors are coated with a thin layer of aluminum giving a reflectivity of about 20% and are moved to contact with three PZT actuators. The F-P interferometer is inserted in an optical system where the image of the scene is formed on the CCD camera, and the interferograms for each pixel are extracted from the acquired video. The calibration of the retardation is obtained for each image by shining the image with a laser radiation of known wavelength. We measured the dispersion of the metallic mirrors and verified that there is no need to apply a phase correction.

With respect to Michelson interferometers where the retardation could go to negative values therefore producing double-sided interferogram, this F-P interferometer produces evidently single-sided interferograms and the retardation does not start from zero when the mirrors comes into contact because of the penetration depth in the metallic layer.

Moreover, F-P interferometers produce interferograms based on the Airy function [26] and in the spectrum, calculated with the Fourier transform, harmonics of the original spectrum are present. The harmonics decrease at the rate Rⁿ/n where R is the reflectivity and n the order of the harmonic [20]. The alias then eventually foldover and overlap the base spectrum. The solution we adopted in our prototype is to use mirrors with low reflectivity (R ≈ 20%) in such a way that the Airy function could be approximated by a cosine function plus a constant term, as in Eq. (8), and oversample the interferogram with a sampling rate with a sufficient number of points per fringe: in our application eight samples for each fringe of the reference laser are used.

I (δ) = \frac{1}{1 + (\frac{4 R}{{(1 - R)}^{2}}) \sin^{2} (2 π \tilde{ν} d)} \approx \frac{1}{2 R} + 2 R \cos (4 π \tilde{ν} d)

The incompleteness of the interferogram is solved by inserting a band pass filter in the optical setup with a spectral width slightly less than an octave. The information that certain regions of the spectrum are zero is used to find the missing points, as described in [22]. The apodization is applied to the interferogram and the single-side cosine discrete Fast Fourier transform equivalent to Eq. (6) is applied. This HSI prototype with metallic mirrors is used to acquire spectra of scene in the 380 nm - 720 nm with a reference laser at 410 nm [22] and in the 900 nm – 1700 nm with a reference laser at 980 nm [23].

4. Description of the new dielectric mirror prototype

The new HSI prototype we have developed is based on dielectric mirrors having the reflectivity of about 20% which are mounted in the metallic structure visible in the set-up in Fig. 1 and schematized in Fig. 2.

Fig. 1 HSI acquisition set-up. The target is a Cameratrax standard that is illuminated by a xenon discharge lamp. The reference used to calibrate the retardation is a laser at 532 nm. A photographic objective is used to create the image on the monochromatic camera. The F-P interferometer is inserted in a metallic structure. The retardation is varied by evacuating air with a peristaltic pump (not visible in the image). The flux is controlled with a variable leak valve.

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Fig. 2 Scheme of the set-up to measure the phase correction of the dielectric HSI. The incandescent lamp shines through the F-P interferometer and the transmitted light is collected by an optical fiber and sent to a grating spectrophotometer. The F-P is mounted in a metallic structure and the mirrors come to contact evacuating air with a peristaltic pump. The tightness is assured by an O-ring fixed on the outer part of the mirrors.

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The retardation is varied by evacuating the air in the small gap between the mirrors with a peristaltic pump. The tightness is assured by an O-ring fixed on the outer part of the mirrors. In this way the mirrors come into contact in the central region. The scanning retardation speed is controlled with a variable leak valve. In Fig. 1 a xenon lamp illuminates the scene and a laser at 532 nm shining the scene is used to calibrate the retardation. The optical band pass filter (520 nm – 720 nm) is attached to the objective and it is not visible in the picture. A video is acquired by a Si-CCD camera while the retardation is scanned until the mirrors are in contact. Then the video is downloaded in the PC and the calibrated interferogram for each pixel is obtained. The phase correction $Θ (\tilde{ν})$ of this dielectric mirror configuration is measured with the setup represented in Fig. 2, where a broadband lamp, like an incandescent lamp or a xenon lamp, shines through the F-P interferometer and the transmitted light is collected by an optical fiber. The fiber has a flat endface, a core diameter of 400 μm and is positioned as close as possible to the interferometer. The radiation is sent through the fiber to a grating spectrophotometer while the retardation is scanned by evacuating in a controlled way the air in the mirror gap. The phase correction depends on the angle of incidence and in this set-up the dispersion is measured only for incident beams perpendicular to the mirrors.

As an example to explain the technique used to measure the dispersion, four spectra in Fig. 3 are acquired with the grating spectrophotometer at four different retardations of the F-P interferometer (mirrors in contact, δ = 6.1 μm, δ = 13.5 μm and δ = 31.1 μm). The colored points on the graphs represent the value of the spectra in different wavelength bins (λ = 400 nm, 500 nm, 600 nm, 700 nm and 800 nm). From the full set of spectra acquired for retardations ranging from 0 μm (mirrors in contact) to 31.1 μm, we have extracted the interferograms for each wavelength bin, ranging from 400 nm to 900 nm. To go into details of the technique, in Fig. 3 we have reported on the graph the values of the spectra at five different wavelengths (λ = 400 nm, 500 nm, 600 nm, 700 nm and 800 nm). By grouping and ordering all the values belonging to a wavelength bin, we have finally obtained the interferograms associated to each wavelength bin, obtaining the interferogram as the F-P was illuminated with a quasi-monochromatic source, as reported in Fig. 4(a). The dispersion is measured with respect to the reference wavelength bin at 532 nm which is the same used for the HSI set-up. All the interferograms are then calibrated and resampled using the reference interferogram at 532 nm in Fig. 4(b).

Fig. 3 Four spectra measured with the grating spectrophotometer for different retardations. The first spectra is measured with the mirrors in contact (δ = 0 μm). The other three are calculated for retardation of 6.1 μm, 13.5 μm and 31.1 μm. The colored points represent the value of the spectra in different wavelength bins.

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Fig. 4 (a). A detail of the interferograms obtained ordering the value in the five wavelength bins in Fig. 3. Figure 4(b). The five interferograms are calibrated and resampled with reference to the interferogram at 532 nm.

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The calibrated interferograms in Fig. 4(b) are then fitted to the Airy function in Eq. (9) which takes into account the dispersive term in the phase correction $Θ (\tilde{ν})$

\begin{array}{l} I (δ) = \frac{A}{1 + (\frac{4 R}{{(1 - R)}^{2}}) \sin^{2} (2 π \tilde{ν} d + Θ (\tilde{ν}))} \end{array}

The estimated phase correction

Θ

as a function of wavelength λ is relative to the phase at the reference wavelength of 532 nm and is presented with a black line for the dielectric mirror F-P in Fig. 5. Only the beams with incidence angle equal to zero are taken into account with this set-up.

Fig. 5 The black line is the measured phase correction $Θ$ as a function of wavelength for dielectric mirrors obtained by fitting with Airy functions the measured interferograms measured with the set-up in Fig. 2. The 13 red squares are the phase $Θ$ measured with HSI with the set-up in Fig. 1 with 13 different sources.

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The red squares in Fig. 5 are the results of a second type of measurement of $Θ$ carried out directly with the dielectric mirror F-P inserted in the HSI set-up, as in Fig. 1. In this measurement, we have measured the spectra with the HSI, applying Eq. (6), of 13 sources with narrowband radiation, like LEDS and lasers, using as a reference the laser at 532 nm. Following the example given in Fig. 6 with a radiation from a laser at 633 nm, the spectrum (with Hanning apodization) appears strongly asymmetric without phase correction (black line in the figure). By inserting the right phase correction value $Θ (633 n m) = - 0.15 r a d$ in Eq. (6) the spectrum becomes symmetric (red line) and the value is reported as a red square in the graph in Fig. 5. The uncertainty bar is found considering that each pixel is associated to a different angle of incidence of the beam on the dielectric mirrors [22], and therefore the phase correction depends on the position of the pixel on the image. For the radiation at 633 nm and 532 nm, we have measured the phase correction for different pixels covering all the image and the pooled standard deviation of the scattered data gives 0.05 radians. This is the value of the uncertainty bar for the points reported in Fig. 5.

Fig. 6 The spectra of the radiation of a laser at 633 nm measured with the dielectric HSI for two different phase correction $Θ$ values. The black line is the distorted spectrum calculated with $Θ = 0 r a d$ in Eq. (6), the red line is the symmetric spectrum calculated with $Θ = - 0.15 rad$ in Eq. (6).

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5. Results and discussion

After having measured the phase correction $Θ$ of the dielectric HSI system, we have tested the HSI system by measuring the spectra of scenes containing sources with known spectra. Firstly, we measured the spectra of a scene containing four LEDs with different nominal wavelength peaks. The result is presented in Fig. 7 where the black line is the spectrum extracted from the measured hypercube and the red line is the same spectrum measured with the reference spectrophotometer. For this set of measurements the maximal cavity length of the F-P interferometer is about 15 μm, apodized with Hanning function corresponding to an instrument line shape ILS with a width of about 10 THz. The limited resolution causes the HSI spectra to be slightly larger than the reference spectra.

Fig. 7 In black the spectra of radiation from different LEDs extracted from the HSI hypercube measured by applying Eq. (6). In red the same LED radiation measured with a reference spectrophotometer. In this measurement the ILS of F-P HSI is about 10 THz.

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The dielectric HSI has been tested with a reflective reference standard (Macbeth Color Checker (TM)). The results are in Fig. 8 where a small portion of the Macbeth standard has been considered, the spectra are extracted from the hypercube and are normalized with respect to the white (at the top left of the standard) to obtain the absolute reflectivity spectrum, independent from the transmissivity of the optical components and the spectral content of the lamp. Finally the spectra are compared to the reference spectra acquired by the grating spectrophotometer. The spectra are in the range 520 nm – 720 nm. A good agreement between the two sets of spectra is evident.

Fig. 8 Spectra of a scene containing Macbeth reference. The solid lines are the spectra of radiation extracted from the hypercube normalized with respect to the white at the top left of the standard in the range 520 nm – 720 nm. The dashed lines are the reference spectra measured with a grating spectrophotometer.

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From the analysis of Figs. 7 and 8 we can state that the new HSI prototype with dielectric mirrors has been validated with success for measuring hyperspectra of scenes containing emitting and reflective surfaces. In principle this set-up could be applied to measurement for other applications, in other regions of the electromagnetic spectrum, using a different reference laser and a different band pass filter. The phase dispersion of the dielectric mirrors in the other regions of the electromagnetic spectrum can be measured with the HSI using the validated technique described in section 4.

This last paragraph is devoted to the comparison of the characteristics of this F-P technique with respect to the other hyperspectral imagers. Thanks to its compactness and high numerical aperture F-P technique could be easily integrated in existing imaging set-ups, like in microscopes or telescopes, to transform the output image in a full hyperspectral image by applying the presented algorithm to the acquired video. The same integration cannot be done easily with systems based on dispersive means since the spectrum is recorded by analyzing the image line by line. On the other hand, F-P technique requires a laser radiation to calibrate the mirror distance for each video acquisition and this complicates the set-up. Nevertheless, like all the hyperspectral imaging techniques, the resolution is finally limited by the optical throughput of the optical system and by the full acquisition time and in this sense the attainable resolution of this F-P technique is of the same order of magnitude of the other hyperspectral imagers.

6. Conclusions

We have validated a new prototype of hyperspectral imaging device based on F-P interferometer with low reflectivity dielectric mirrors. This is an upgrade with respect to the former prototype with metallic mirrors we developed. Since this new set-up is based on low reflectivity dielectric mirrors which are available off-the-shelf at low price, with low losses and reproducible characteristics, this opens the possibility of having a low-cost commercial product. On the other hand, the dispersion of dielectric mirrors cannot be neglected in the spectra calculation and the phase correction has been measured with two alternative methods. The spectra are finally calculated with a Fourier transform based algorithm. The hypercube of scenes containing emitting and reflective bodies has been measured showing a good agreement with respect to the reference spectra measured with a calibrated spectrophotometer and this paves the way to implement the new HSI device in different applications.

Acknowledgments

This work was partly funded by the Project “Traceability for surface spectral solar ultraviolet radiation” EMRP ENV03. The EMRP is jointly funded by the EMRP participating countries within EURAMET and the European Union.

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A hyperspectral imager based on a Fabry-Perot interferometer with dielectric mirrors

Abstract

1. Introduction

2. Discussion about Fourier transform spectroscopy

3. Description of the HSI based on metallic mirror F-P interferometers

4. Description of the new dielectric mirror prototype

5. Results and discussion

6. Conclusions

Acknowledgments

References and links

Cited By

Figures (8)

Equations (9)

Optics Express