Fourier transform spectroscopy can increase the spectral resolution by prolonging the maximum optical path difference in the interferometer. However, in our previous setup, long scanning of the spherical mirror induces the deviation of the center of curvature of the spherical mirror from the focused point on the grating and causes error in the optical path difference [3
3. A. Hirai, L. Zeng, and H. Matsumoto, “Heterodyne Fourier transform spectroscopy using moving diffraction grating,” Jpn. J. Appl. Phys. 40, 6138–6142 (2001). [CrossRef]
]. In short-wavelength region (visible), precisely controlled optical path difference is required. Furthermore, longer scanning is required in long-wavelength region (NIR and IR) if the same relative resolution is required and the error becomes large. Therefore, the error in the optical path difference is a severe problem in both short-wavelength and long-wavelength regions. In the present work, we improved the optical setup aiming at long-path scanning necessary for obtaining higher spectral resolution in the long-wavelength region. A collimating mirror such as a spherical mirror with a long radius of curvature or a paraboloidal mirror is used instead of the conventional spherical mirrors. The beams diffracted by the grating are collimated by the collimating mirror and then reflected by a pair of plane mirrors. The optical path difference between the +1st and -1st diffracted beams are scanned by moving one of the plane mirrors. This novel setup can provide a long optical path difference without any errors, therefore it can achieve high spectral resolution in measurement.
2. Heterodyne Fourier transform spectrometer
The optical setup for the newly developed heterodyne Fourier transform spectrometer is shown in Fig. 1
. The broad-band light emitted, reflected or transmitted from the object under measurement is focused on the diffraction grating through an achromatic lens. The diffracted light beams of +1st and -1st orders are collimated by a collimating mirror and reflected by a pair of plane mirrors, and then are focused on the grating again. After that, the light is diffracted in the original direction. The dependence of the diffraction angles on the wavelength is cancelled out by the diffraction effect in the original direction. Finally, the +1st and -1st diffracted light beams interfere on a photodetector.
Fig. 1 Optical setup for heterodyne Fourier transform spectrometer suitable for long-range scanning. PZT; Piezoelectric transducer.
It is assumed that the diffraction grating moves linearly perpendicular to both optical axis and the direction of the grooves of the grating. The optical frequencies of diffracted light undergo Doppler shifts[4
4. T. Suzuki and R. Hioki, “Translation of light frequency by a moving grating,” J. Opt. Soc. Am. 57, 1551 (1967). [CrossRef]
]. The optical frequency shift f
, where M
is the spatial frequency and V
is the moving speed of the grating. Because the light is diffracted twice in the setup, the total optical frequency shifts of +1st and -1st diffracted light beams are +2f
, respectively. A beat signal with the frequency of 4f
is generated on the photodetector and transmitted to a phase-sensitive amplifier. The phase-sensitive amplifier demodulates and amplifies the interferogram. By scanning one of the plane mirrors linearly along the optical axis using a piezoelectric transducer (PZT), the interferogram of the incident light is detected with high sensitivity by the heterodyne effect. As conventional Fourier transform spectroscopy, the optical spectrum of the incident light is accurately obtained when the interferogram is Fourier transformed according to the optical path difference.
The maximum wavenumber and resolution of the measured spectrum are determined by the sampling interval of the optical path difference and the number of sampling points, the same as in the conventional Fourier transform spectroscopy. To obtain spectra with high resolution, long-range scanning of the path difference is required, particularly in long-wavelength regions, such as the NIR and IR regions. In the previous optical setup, two spherical mirrors were used to compensate the difference between diffraction angles. However, scanning of the spherical mirror induces defocusing error. The error was analytically evaluated in reference 3. The extent of the optical path difference when the spherical mirror is displaced along the optical axis by z
where, r is the radius of curvature of the spherical mirrors, θ
max and θ
min is the maximum and minimum diffraction angle, respectively, Δθ=θ
min, and θ is the maximum incident angle of the diffracted beam on the spherical mirror. The maximum optical path difference where the extent of the optical path difference is less than half of wavelength λ under measurement was evaluated. We assumed that λ was 0.8 µm, r was 60 mm, pitch of the grating is 1/110 mm, and the light beam with diameter of 10 mm was focused on the grating by a lens with focal length of 150 mm. The maximum optical path difference was 0.112 µm and it limited the wavenumber resolution to 89.3 cm-1. In the novel setup, there is no defocusing error even with long-range scanning.
Spherical mirrors with long focal-length, a paraboloidal mirror, or an achromatic lens can be used as the collimating mirror. The reflective elements are better than the refractive ones for white light, because there is less wavelength dependence.
3. Experimental results
3.1 Experimental conditions
In the experiment, a transmissive-type diffraction grating with a spatial frequency of 110 lines/mm was moved by a linear motor stage (JASCO, LO-10-50) at the speed of 0.9 mm/s. The obtained beat frequency was approximately 410 Hz. The time constant of the phase-sensitive amplifier was 10 ms. The beat frequency was rather low at present, since the speed of the movement of the grating is limited by the size of the grating and the measurement time, and can be increased. The beat frequency fluctuated slightly. This fluctuation was due to the nonuniformities of the speed of the stage in the slow speed region and the spatial frequency of the grating. A laser light beam, which irradiated the grating from behind, as shown in Fig. 2
was used to compensate for the fluctuation. The beat frequency of the interferogram of the laser beam was used as the reference signal of the phase-sensitive amplifier.
Fig. 2. Experimental configuration.
The light under measurement was focused by an achromatic lens whose focal length was 150 mm. The +1st and -1st orders of diffracted beams were collimated by a spherical mirror whose focal length was 200 mm and then reflected by a pair of plane mirrors. The optical path difference of the interferometer was scanned by moving one of the plane mirrors along the optical axis, using the PZT (PI polytech, P-765.1L). Here, the optical path difference was scanned during the signal acquisition. The optical frequency shift by the mirror scanning was negligible, since the scanning was slow. The sampling interval of the optical path difference was 94.4 nm, and the number of sampling points was 4096, therefore, the wavenumber resolution was 25.8 cm-1. Consequently, the wavelength resolution was about 7 nm at 800 nm and 10 nm at 1000 nm. The maximum optical path difference was limited by the maximum scanning range of the PZT used and can be improved by means of other stages that can travel longer or push-pull configuration, in which both plane mirrors move in opposite directions.
The light source used is a halogen lamp which radiates only NIR light (Hayashi Watch-Works, LA-100IR). Its spectrum expands from 0.8 µm to 1.3 µm. The detector used (New Focus, 2011) has a sensitivity ranging from 0.9 to 1.7 µm.
3.2 Improvement of the signal-to noise ratio
First, to confirm the improvement of the signal-to-noise ratio by the proposed method, the spectrum of the light source was measured with and without the heterodyne detection technique. In the measurement without heterodyne modulation, the signal from the photodetector was recorded while one of the plane mirror was scanned but the grating rested. The acquisition time of the interferogram was the same for the two techniques and about 40 s. Figures 3(a)
show the spectra obtained. The red and blue plots indicate the spectra obtained when the incident light under measurement was strong (67.9 mW/mm2
at the entrance of the setup) and weak (4.47 mW/mm2
), respectively. The intensity of the light source was changed by controlling the applied voltage, therefore, the difference between the spectra of strong and weak light was caused by the temperature change of the filament of the halogen lamp. The ratio of the average to the standard deviation of the spectra obtained by three measurements was calculated at every wavenumber. The SNR is defined as the average ratio over the spectral range from 0.8 to 1.3 µm. The SNR of the spectra were improved by about 1.7 times and 6.8 times by the heterodyne detection when the light under measurement is strong and weak, respectively. At present, the value of SNR is not large, since the throughput of the optical setup is very low (8.3×10-5
) and the power of the light at the photodetector is of the order of µW or nW. The causes of such low throughput are the low diffraction efficiency of the diffraction grating and the use of other optical elements to align the optical setup. The SNR will be improved by using optimally suited elements.
Fig. 3. Comparison of the signal-to-noise ratios of the spectra obtained with and without heterodyne modulation. Spectra of strong and weak light obtained (a) without modulation and (b) with modulation.
3.3 Quantitative calibration and qualitative discrimination of organic solutions
Second, the spectra of organic solutions were measured using the proposed interferometer. Figure 4(a)
shows the spectral absorbance of ethanol of different concentrations in a optical cell with 10 mm optical path length. In the figure, the averages of five spectra for each concentration were displayed. All data are analyzed by chemometrics[5
5. B.R. Kowalski, Chemometrics -Mathematics and Statistics in Chemistry (D. Reidel, Dordrecht, 1984).
6. D.L. Massart, B.G.N. Vandeginste, S.N. Deming, Y. Michotte, and L. Kaufman, Chemometrics: a textbook (Elsevier, Amsterdam, 1988).
], which is a chemical discipline that uses mathematical, statistical, and other methods employing formal logic for chemical data analysis to provide maximum relevant chemical information. The pattern recognition software (GL Sciences, Pirouette) for chemometrics was used for quantitative calibration of the ethanol solutions. Figure 4(b)
shows the results of the quantitative calibration by partial least squares regression with five factors. The horizontal and vertical axes indicate the real and predicted concentrations, respectively. Good calibration was achieved.
Next, the spectra of propanol solutions of different concentrations were measured in the same manner. All spectra of ethanol and propanol solutions were analyzed by principle components analysis using the same software. Figures 5(a)
show the spectral absorbances of solutions and the results of discrimination, respectively. Red and blue plots in Fig. 5(a)
indicate the spectra of ethanol and propanol solutions, respectively. The structures around a wavelength of 0.9 µm are clearly different. In Fig. 5 (b)
, the axes Factor 1 and 2 are the normalized projections onto the first and second principle components calculated from the spectra of ethanol and propanol solutions. The red and blue rhombuses indicate the projections of each spectrum of ethanol and propanol solutions onto two-dimensional space spanned by the first and second principal-component vectors, respectively. It is clear that the projections of the different compounds fall into distinct regions defined by the contributions of the principle components and are well clustered. The qualitative discrimination of ethanol and propanol solutions through the use of their spectra was successful.
Fig. 4. (a) The spectral absorbance of ethanol solutions with different concentrations. (b) The results of the quantitative calibration by partial least squares regression with five factors.
Fig. 5. (a) The spectral absorbances of ethanol and propanol solutions of different concentrations. (b) The results of discrimination by principal components analysis.
3.4 Demonstration of transmission spectra of a grape
Lastly, for demonstration, transmission spectra of a grape were measured. NIR spectra are widely used in food analysis, such as the nondestructive determination of the sugar content of fruit[7
7. S. Kawano, H. Watanabe, and M. Iwamoto, “Determination of sugar content in intact peaches by near infrared spectroscopy with fiber optics in interactance mode,” J. Japan. Soc. Hort. Sci. 61, 445–451 (1992). [CrossRef]
8. S. Kawano, T. Fujiwara, and M. Iwamoto, “Nondestructive determination of sugar content in satsuma mandarin using Near Infrared (NIR) transmittance,” J. Japan. Soc. Hort. Sci. 62, 465–470 (1993). [CrossRef]
]. The grape being measured was a rotational ellipsoid with a long diameter of 30.7 mm and short diameter of 21.9 mm. The light illuminated the point on the circumference where the diameter was 15.9 mm. The results of five measurements of the transmission spectral intensity of the grape with heterodyne modulation are shown in Fig. 6
. The spectrum could not be measured without heterodyne modulation because the intensity of the transmitted light was extremely weak and the interferogram was buried within noise. The improvement of the SNR by heterodyne modulation was also demonstrated.
Recently, several kinds of micro-optical electro-mechanical systems with moving parts fabricated using micromachining technology are reported[9
9. O. Manzardo, H.P. Herzig, C.R. Marxer, and N.F. de Rooij, “Miniaturized time-scanning Fourier transform spectrometer based on silicon technology,” Opt. Lett. 24, 1705–1707 (1999). [CrossRef]
10. K. Hane, T. Endo, M. Ishimori, Y. Ito, and M. Sasaki, “Integration of grating-image-type encoder using Si micromachining,” Sens. Actuators A 97–98, 139–146 (2002).
]. If the proposed setup is fabricated by the micromachining technology, the low-cost, compact, sensitive spectrometer can be realized. Such devices are useful for in-situ measurement such as environmental monitoring and quality control of food.
Fig. 6 Results of five measurements of transmission spectral intensity of a grape.
4. Discussions and conclusions
The configuration for heterodyne Fourier transform spectroscopy was improved for high spectral resolution in the long-wavelength regions, such as NIR and IR regions. As in the previous configuration, the novel configuration has advantageous of high sensitivity by frequency modulation and ease of fabrication of the important optical elements in the interferometer. The improvement of the SNR in the NIR region was experimentally demonstrated. Furthermore, the proposed configuration was used for quantitative calibration and discrimination between organic solutions with different concentrations. At present, the spectral resolution is limited by the scanning range of the PZT stage used, however, it can be improved by means of other stages with longer scanning ranges or push-pull configuration. As an example, the measurement of transmission spectra of a grape has been demonstrated. The transmitted light from the grape was extremely weak and the interferogram was buried within noise without modulation. The heterodyne modulation enabled us to measure the spectrum of such weak light. The proposed technique can be a powerful tool in the examination of food and health monitoring.
The stability of the movement of the grating also limits the spectral resolution because of the limitation of the measurement time. The one-dimensional grating must change its direction of motion, therefore the measurement time is limited. To achieve a continuous frequency shift, the use of a radial grating which can be rotated about its axis [11
11. W.H. Stevenson, “Optical frequency shifting by means of a rotating diffraction grating,” Appl. Opt. 9, 649–652 (1970). [CrossRef] [PubMed]
] is preferable. In the case of a radial grating, the grating spacing varies with the radius, however the amount of frequency shift is constant. The change of diffraction angles due to the different spacing of the grating can also be compensated by employing collimating mirrors.