An imaging spectrometer based on a Fabry-Perot interferometer is presented. The Fabry-Perot interferometer scans the mirror distance up to contact and the intensity modulated light signal is transformed using a Fourier Transform based algorithm, as the Michelson based Fourier Transform Spectrometers does. The resulting instrument has the advantage of a compact, high numerical aperture, high luminosity hyperspectral imaging device. Theory of operation is described along with one experimental realization and preliminary results.
©2009 Optical Society of America
A hyperspectral imaging system (HI, also called imaging spectrometer) is a combination of an imaging device (a digital camera) and a spectrophotometer. The obtained data set, known as “hyperspectral cube”, is a 3D matrix formed by a 2D image combined with a third dimension that is the spectral composition of each pixel of the image.
HI devices are finding an increasing number of applications in different fields; just to mention a few, space missions (e.g. Earth survey for environment or security), fluorescence microscopy, cultural heritages, chemistry (infrared spectroscopic analysis), thermal imaging.
Hyperspectral devices are commonly made by integrating a dispersive means (a prism or a grating) in an optical system, with the drawback of having the image analyzed per lines and some mechanics integrated in the optical train. Alternatively, HI devices are based on optical band-pass filters either tuneable or fixed and the spectrum has to be scanned in steps. A comprehensive overview on classic HI devices can be found in . Both techniques lead to good quality results but have very low efficiency (i.e. long integration times are necessary to obtain a full hyperspectral cube).
A third method to obtain the spectrum of a light source, is the so called Fourier transform (FT) spectroscopy ; here the FT is applied to the interferogram acquired by a scanning interferometer. It is known that by analysing the interferogram obtained from a two beam interferometer with a FT based algorithm we obtain the spectral composition of the light entering in the interferometer. In fact this technique has been used for decades by spectroscopists to obtain high resolution absorption spectra by using a Michelson interferometer. An HI device based on a scanning Michelson interferometer having large numerical aperture is described in . In  the FT has been applied to a high finesse Fabry-Perot (F-P) interferometer to obtain high resolution spectra of small portions of sidereal light sources. An HI based on a Sagnac interferometer used for space applications is described in .
In this paper we propose a HI based on a low finesse F-P interferometer. We demonstrate that it is possible to apply an FT based algorithm with the advantages of a high luminosity imaging spectrometer together with a compact and lightweight structure.
The HI device we have developed is based on a F-P interferometer inserted in the optical system. Fabry and Perot realized the multiple-beam interferometer at the end of the 19th century and used it for different applications like spectroscopy and astrophysics . Considering the cavity formed by two partial reflecting mirrors, the formula for the transmitted irradiance is obtained by summing the contribution of multiple beams that are reflected and transmitted in the cavity. The formula gives an Airy function :
where R is the reflectivity of both mirrors and d is distance between the mirrors. In the formula the mirrors are considered to be ideal: the reflection occurs at the coated surface of the mirror and the optical path difference (OPD) between the ray transmitted by the cavity without reflections and the ray reflected twice before exiting the cavity is exactly 2d. In the real world we have to consider the penetration depth experienced by the wave when reflected by the mirror coating, being of the order of tens of nanometres. As a consequence the OPD is greater than 2d and even when the mirrors are in contact (d = 0) OPD is not zero. We will discuss later the importance of that.
2.1. Michelson and Fabry-Perot interferometer
Having in mind to use the FT technique to obtain the spectrum of a light source, we have to deal with two important differences between Michelson and F-P interferometers. Figure 1 illustrates the two interferometers having a scanning mirror when illuminated by a monochromatic source and when illuminated by a broadband source. In the four schemes the scanning mirror (in blue) translates from position 1 to position 2 with a travel d (OPD ≈ 2d). In a Michelson interferometer without dispersion the interferogram of a monochromatic source is a cosine and the FT gives a single line. In general, the interferogram of a broadband source is a sum of cosines and the FT gives the spectrum of the source. In the F-P interferometer the interferogram of a monochromatic source is an Airy function, Eq. (1), and since Fourier series decomposes periodic functions, like Airy function, into a sum of cosines, the FT gives a series of lines: the spectrum of the monochromatic source plus the harmonics. This means that in the case of a broadband source the spectrum and its harmonics sum up, causing ambiguities in the spectrum obtained from the F-P interferometer. Secondly the interferogram obtained with the F-P interferometer is evidently single-sided and it does not start from the central or zero fringe when the mirrors come in contact because of the penetration depth of the metallic coating. The latter implies that the interferogram is incomplete (i.e. it does not contain the data corresponding to the OPD = 0 condition), therefore it is not possible to apply directly the FT to the interferogram as in two beam interferometers.
The two aforementioned problems are solved by introducing an optical bandpass filter in the optical system transmitting slightly less that one octave of the electromagnetic spectrum. In fact, since the FT of the F-P interferogram gives the spectrum summed up with its harmonics, by filtering and limiting the spectrum before the F-P cavity to one octave we are sure that the FT spectrum is not mixed with harmonics. Moreover, the optical band pass filter transmits slightly less than one octave, hence a small region of the spectrum between the first and the second harmonic is zero and we can use this information to compensate for the missing information due to the incomplete interferogram.
The use of a F-P interferometer in an HI system has two main advantages with respect to other HI systems:
1. I t is known that the use of a scanning interferometer instead of a dispersing means or a bandpass filter has much higher luminosity. Indeed the whole image is observed at the same time (no line scanning) and all the components of the spectrum are observed at the same time (no spectral scanning). These two are known as Jacquinot (or throughput) advantage and Felgett (or multiplex) advantage, respectively and allow us to obtain hyperspectral images with shorter exposure times.
2. With respect to known scanning interferometer based HI systems [3, 5], the F-P cavity allows larger numerical aperture systems to be realized. Indeed, the geometry of both Michelson and Sagnac interferometer limits the angle of the light entering the system, whereas a F-P interferometer can accept larger entrance angles. Furthermore its compactness allows easier integration in compact optical systems.
In the next sections we will show how the scanning mirror F-P interferometer can be practically used to obtain hyperspectral images.
3. The experimental realization
In this section the practical realization of our experiment is described starting from the construction of the scanning F-P interferometer, continuing with the optical set-up and ending with the acquisition system.
3.1. The scanning F-P interferometer
The core of the device is the scanning F-P cavity. In order to minimize the OPD, the distance between mirrors d should be as close as possible to zero. In practice the device is designed to push the mirrors up to physical contact (d = 0). Further, the contact must happen along the whole aperture of the optical system, which, in order to get a high luminosity, must be of the order of squared centimeters. This is not an easy task indeed: if we use a couple of optically flat mirrors (say λ/10 flat) when placed in contact they will strongly attach each other because of surface forces. The method found to avoid this effect is to make one of the two mirror slightly convex. The contact will thus start from a small area and will become larger when the two mirrors are pressed one against the other, until it reaches the required area (Fig. 2).
The two mirrors are coated with a thin aluminum layer and have a reflectivity around 25%. The choice of the mirror coating was driven by the effect that this has on the phase delay (or the penetration depth) of the reflected light. Dielectric coatings show a strongly wavelength dependent penetration depth. The penetration depth of a metallic coated mirror has been verified to be constant in the considered spectral interval (400-720 nm): this allows the use of simpler algorithm for the data analysis. The mirrors are mounted in aluminum frames and the distance is scanned by means of three piezo actuators allowing a maximum displacement of 60 μm at 100 V. A system made by three elastic hinges and three screws allows the optimal alignment and working distance of the mirrors to be found, so that, when the actuators are completely retracted (maximum voltage applied), the mirrors are in contact and the contact area is sufficiently large and centered. Figure 3 shows the 3D rendering of the F-P and a section of it.
3.2. Experimental layout
The HI system is made of a photographic objective coupled with a 12-bit CCD camera. The F-P is placed as close as possible to the camera sensor. Between the objective and the F-P is placed the optical band pass filter (370-720 nm) needed to select the wanted portion of the spectrum as explained before. The layout of the experiment used to obtain the data described in section 5.1 is schematized in Fig. 4. Other setups used in this work are variations of this one. The sample is illuminated by a xenon discharge lamp focused on a grounded glass acting as a diffuser which creates a uniform illumination area.
The sample is a Gretag Macbeth ColorChecker®. A blue laser having 410 nm wavelength is diffused by a holographic pattern generator creating light spots on a semitransparent screen placed behind the sample. In the latter we made small holes (one for each color tab) in order to make the light spots visible by the camera through the holes. So, in the image focused on the CCD sensor we can see the 24 color tabs, each of them including a small blue spot which will be used as a calibration reference for the same tab. Figure 5 shows a picture of the experimental setup.
3.3. Acquisition software
The system is controlled by two PCs, one used to drive the F-P and the other to drive the camera. The first, equipped with a 16 bit digital-to-analog converter, generates a triangular voltage ramp sent to a HV amplifier which generates the 0-100 V signal to drive the three piezo actuators (connected in parallel). Through the same board a trigger signal is generated synchronously with the ramp. The second PC, through the camera acquisition board, acquires a video starting from the trigger signal (corresponding to the maximum mirror distance condition) and ending with complete contact. In order to have a sufficient sampling rate to respect Nyquist criterion, about 500 frames each acquisition are required for a 10 μm scan, as in the experiment of section 5.1. This figure, in combination with the maximum frame rate of the camera sets the maximum ramp speed. The video is saved in TIFF format.
4. OPD calibration
The distance between the F-P mirrors is varied with three piezo actuators. Unfortunately the voltage/OPD transfer function is highly non linear and not repetitive, therefore it is not possible to calibrate it once for all. Moreover since the mirrors are slightly curved every pixel would have a different transfer function. Therefore we have to calibrate the OPD for every scene acquisition and for each region of the image. In this section the method used for the calibration is described.
The calibration makes use of a monochromatic light (a laser or a lamp) having sufficiently narrow linewidth to have coherence length longer that the maximum OPD. During the calibration procedure the light illuminates the scene while the OPD is scanned and a video is captured. In the present implementation we have used a holographic grating to generate a pattern of laser spots from a single laser beam in order to have a number of references distributed along the scene. The calibration scene (illuminated with the laser spots) and the measurement scene (illuminated with the white light) can be either recorded in the same video (as in the case of the measurement described in section 5.1), or in two consecutive videos (as in the case of the measurement described in section 5.4).
In Fig. 6a the interferogram extracted from the video of one of the blue laser spot is reported, the value is the pixel intensity in bit depth measured by the camera for each frame. The fringes are not equi-spaced due to the non linearity of the piezo actuator and even when the two mirrors are in contact (pressed by the piezo actuators, left side of the graph) the OPD does not go to zero (indeed a zero OPD should correspond to the in-phase condition, thus to a maximum of the interferogram).
It can be observed also that the fringe amplitude diminishes when the OPD increases (right side of the graph). This is due to an integration effect in the solid angle seen by each single pixel in combination with the reduction of the thickness of the ring fringes depicted in Fig. 2 for larger OPDs. Although in the presented application this effect is not an issue, it must be carefully evaluated if higher spectral resolution (larger OPDs) is required.
In Fig. 6(b) the zoom of the interferogram from video frame 80 to 130 is presented, the black dots represent the original pixel intensity for each frame. The non linearity of the piezo actuator is evident. The maxima and minima of the fringes are calculated (see red dots in the picture) and are interpolated (see green dots) in order to obtain a reference scale used to re-sample the video frames. The re-sampling interval in this case is equal to one eighth of the laser lambda: 51.25 nm.
In this way, we have obtained the OPD calibration only for the pixels corresponding to the laser spots, the OPD calibration for the remaining image pixels are calculated by interpolation. At the end a file containing the OPD calibration for each pixel of the scene is obtained. This will be used to re-sample the video and obtain the interferogram for each pixel (see sect. 5.1).
As discussed in section 2.1, due to the presence of the penetration depth, the interferogram does not start from zero OPD and the first useful reference point is the first minimum of the blue (corresponding in the example to video frame 98) which occurs at OPD 205 nm. The ambiguity in the spectrum calculation could be explained simply by saying that there are four points missing in the interferogram: corresponding to OPD = 0, 51.25, 102.5 and 153.75 nm, and that imposing that the spectrum has to be zero in the blocked regions of the optical band pass filter is equivalent to find the value of the four missing points. This process will be clearer with the example in 5.1.
5. Results and discussion
This section shows some results obtained on a calibrated reflective target to test the accuracy of the system, on a selectively absorptive target to test the potentialities as a spectroscopic analytic instrument, with laser sources to test the spectral resolution and on a natural target to demonstrate the potentialities as a color imaging device.
5.1. Reflection spectra
The first target is a ColorChecker used in the set up described in Figs. 4 and 5. The image covers a rectangle of about 670×460 pixels on the CCD area. Each colour tab is illuminated by a laser spot in order to calibrate the optical path difference (OPD) for each frame in the video. In Fig. 7 a video frame is presented.
The magenta square in Fig. 7 contains the pixels used for the calculation of the spectrum of the magenta coloured tab. The blue square on the bottom right hand side contains the pixels illuminated by the blue laser used to calibrate the OPD for the magenta. The same applies for the white tab (bottom left) used for normalization purposes.
Figure 8(a) reports the interferogram of the pixel set in the magenta square in Fig. 7 with the x-axis represented in video frames. In Fig. 8(b) the black dots are the first part of the same interferogram after the re-sampling using the reference scale from Fig. 6(b) (as explained in section 4). The OPD sampling interval is 51.25 nm and the first record corresponds to an OPD of 205 nm (half wavelength of the blue laser).
As explained it is not possible to obtain the spectrum from this incomplete interferogram by simply applying FT since the first four points are missing. Considering the Fourier series decomposition, the first missing point gives an indetermination of the spectrum DC background, the second missing point gives an indetermination on the lowest frequency cosine component amplitude and so on. The four component amplitudes are found by imposing that the resulting spectrum has to be zero in the blocked regions of the optical band pass filter, using a least square method. In the presented setup the filter blocks the spectrum below 370 nm and beyond 720 nm. To impose the spectrum to be equal to zero in these known areas is equivalent to find the value of the four missing points, shown in red in Fig. 8(b).
The spectra obtained with this process are presented in Fig. 9(a) (in black and magenta respectively the spectra of the white and magenta tabs). The resolution is about 14 THz corresponding to about 12 nm at a wavelength of 500 nm, (by applying the zero padding method the number of points is artificially increased in order to have one point each nanometer for practical computational reasons). The absolute reflectivity spectrum is given by the ratio of the coloured tab spectrum and the white tab spectrum used as a reference. In this way the effect of the non uniform spectral responses of the optical system, of the camera and of the light source is cancelled. In Fig. 9(b) the reflectivity spectrum is shown and compared with the same spectrum measured with a commercial spectrometer (thin black line); relatively large differences are evident at the extremes of the spectrum mainly due to the reduced intensity of the reference spectrum in Fig. 9(a).
Figure 10 shows the 24 reflection spectra of the ColorChecker calculated as described above. Each colour has been normalized with respect to the white tab. The spectra are compared with the value obtained with a classical dispersion spectrometer (thin lines). The average difference between the measurement of the two instruments is within few part percent in most cases. The differences are more evident in the extremes of the spectra and are mainly due to the lower absolute values of the spectra before normalization (see Fig. 9(a)) leading to larger errors in the calculated ratio. Further non perfect uniformity of the illuminant in terms of intensity and spectral distribution must be considered.
As a second application we have measured the transmission of an optical didymium oxide filter, and the results are presented in Fig. 11. A white screen placed in front of the camera is illuminated with the xenon discharge lamp. A portion of the field of view of the camera is covered with the filter so that the light reflected by the screen passes through it before entering the objective. Another portion is illuminated with the blue laser again used for the OPD calibration. The transmittance spectrum is obtained by the ratio between the spectrum of the filtered portion and the spectrum of the white portion of the same image. Again the result is compared with the measurement done with a spectrometer (thin line). This was done to demonstrate the potentialities of the instrument to detect complex absorption spectra, thus to be used in spectroscopy based chemical analysis.
5.3. Spectral resolution
In order to test the spectral resolution of the device a white target has been illuminated with lasers at five different wavelengths: a blue diode laser at 410 nm (used as a reference), a green duplicated Nd:YAG laser at 532.4 nm, a red He-Ne laser at 633 nm and two red diode lasers with nominal wavelength 637.5 and 674 nm. The scan applied to the mirrors is 50 μm corresponding to about 240 entire fringes in the blue interferogram. Figure 12 shows the obtained spectrum using Welch windowing. As expected from the FT theory the resolution is constant in the frequency domain and inversely proportional to the wavelength in the wavelength domain. The obtained FWHM is 2 THz in the whole spectrum corresponding to about 2 nm @ 532 nm. The maximum difference of the measured wavelengths with respect to the nominal values is 1 nm. The resolution can be appreciated in the pair 633 and 637.5 nm whose peaks are well distinguishable.
5.4. Colour imaging
The last test was made using some flowers as a target. In this case the exercise was made to demonstrate the potentialities of the technique to implement a colour photo-camera by using a monochromatic sensor. This could allow high spatial resolution colour imaging by taking the advantage of the higher spatial resolution of monochromatic with respect to colour CCDs. A relatively short scan has been applied to the mirrors in order to obtain low spectral resolution for each pixel. The OPD calibration has been made by recording a pattern of violet dots on a white screen just before the recording of the scene. As a normalization white a spectrum calculated in a previous record has been used. The spectra are eventually converted to a RGB space and the resulting image is presented in Fig. 13.
We have demonstrated a technique and presented a device to obtain the hyperspectral data cube from a scene. The device is based on a scanning Fabry-Perot interferometer where the mirrors move from contact to tens of micrometer distance by means of piezo actuators. The interferogram obtained for each pixel is mathematically elaborated to give the spectral composition of the light hitting the same. With respect to commercially available hyperspectral devices the luminosity of our device is higher because it is based on an interferometer instead of a monochromator (Jacquinot and Felgett advantages) allowing shorter exposure times. Furthermore the F-P interferometer can in principle be realized in a very compact form, thus it can be easily integrated in optical imaging set up. On the other hand the present realization has a spectral range limited to slightly less than one octave and the disadvantage of needing frequent calibration of the mirrors displacement.
The potentiality of the instrument for spectral analysis has been demonstrated in the visible region (400-720 nm) with a resolution of about 2 nm @ 532 nm, but the technique can be used in the infrared (simply changing the CCD technology, e.g. by using InGaAs instead of silicon) where the absorption spectra of chemical substances is rich in information. Eventually a possible application in colour imaging has been demonstrated: the technique could be used to obtain colour images from a monochromatic CCD with the advantage of the higher resolution of the latter with respect to colour sensors.
Besides classical spectroscopic analysis for remote recognition of chemical compounds, other applications are foreseen in the field of fluorescence microscopy where more fluorescent markers can be observed and discriminated on the same biological sample, in thermal imaging where the temperature can be accurately calculated from the shape of the emission spectrum, and in preservation of cultural heritages.
The authors would like to thank Maria Luisa Rastello for helpful discussions, Wilbert Samuel Rossi for the work he carried out during his bachelor degree thesis, Franco Alasia and Marco Santiano for the design and realization of the F-P cavity.
References and links
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