Fourier Transform Imaging Spectrometer at Visible Wavelengths

Transcription

1 Fourier Transform Imaging Spectrometer at Visible Wavelengths Noah R. Block Advisor: Dr. Roger Easton Chester F. Carlson Center for Imaging Science Rochester Institute of Technology May 20,

2 Abstract The purpose of the experiment is to construct a device to calculate the spectrum of all the pixels in the scene. The experimental setup is a Michelson interferometer that has been modified to add a second reference beam illuminated with a known wavelength. The additional light beam is used to compensate for errors in the motion of the movable mirror due to the imprecise and unrepeatable motor. The reference beams error is used to correct the object beams spectrum through error analysis. The results for a two-dimensional scene are given as a three-dimensional graph with the intensity of the spectrum displayed along the third axis. This method gives the spectrum of an object at every location in it with up to nanometer resolution. Table of Contents Abstract Objective Background Design of Experiment Data Processing Conclusions Advancement Appendix A

3 Objective The purpose of the experiment is to construct a device to measure the spectrum of a twodimensional object. The Fourier transform imaging spectrometer previously constructed by Eric Sztanko was modified and extended from a proof of concept to the point where the spectrum of a high-intensity multi wavelength object may be measured. Background Michelson interferometry is a method for measuring the spectrum of a single source. A basic Michelson interferometer can be seen in figure 1. Figure 1: The object beam goes through the beam splitter and the amplitude gets halved in each arm (graphs on L 1 and L 2 ), and then recombines. Constructive and destructive interference result at the image plane (the two graphs next to the camera represent constructive and destructive interference). 3

4 It is often necessary or desirable to measure the spectrum of every pixel in a scene simultaneously. The way a Fourier transform imaging spectrometer (FTIS) works is by capturing the fringe patterns created by the source going through the Michelson interferometer. As M 2 moves, the fringe pattern move (thus a single pixel locations gray value oscillates from light to dark) at a speed proportional to the motors movement, the wavelength of the fringe pattern can be calculated if the step size of the motor is known. As the FTIS captures more images, a longer window records more oscillations giving a better resolution (resolution is discussed later). Taking the Fourier transform of the oscillations (also called interferogram) calculates the frequencies present in the interferogram. The frequencies are then turned into their corresponding wavelengths and the spectrum of the source is calculated. The Fourier transform calculates the frequencies present in the signal. When a multicolor light is used, the wavelengths will overlap creating a different fringe pattern than a single wavelength fringe pattern creates. Sinusoids add according to equation 1. cos(k 1 z - w 1 t) + cos(k 2 z - w 2 t) = 2 cos( k 1 - k 2 2 z) cos( k 1 + k 2 2 z - wt) (1) As can be seen in Figure 1, the beam splitter divides the amplitude of the light into (often equal) parts. The beams that travel paths L 1 and L 2 (path lengths may be different) before being recombined at the beam splitter. If the relative path lengths differ, then fringe patterns are produced by constructive and destructive interference. Mirror M 2 on the motorized stage is set to move in the longitudinal direction (left and right in Figure 1). The CCD camera captures images of the interference pattern at preset time intervals as M 2 is moved. The change in the relative optical path lengths as M 2 is moved produces 4

5 different interference patterns. The ensemble of recorded interferograms forms an interferogram cube f(x,y,t). If zµt, then f(x,y,z) may be inferred. Each captured image is stacked (Figure 2) to make an image cube, f(x,y,t). Figure 2: Image cube, the number of samples represents the time as the motor takes the images, then the x and y coordinate of the image is shown. The gray value of a specific pixel in every layer in the interferogram cube (figure 2) is the interferogram of that specific scene pixel, and is the Fourier transform (FT) of the spectrum. The Fourier transform of the interferogram is the spectrum of the object as can be seen in figure 3. 5

6 Figure 3: The single pixel location of every image in the cube graphed, then the Fourier transform of the graph is taken to find the frequencies present. An interferogram is the oscillations caused by the wavelength of the source. Figure 3 is a nice sinusoid because that is from a single wavelength laser. Since the interferogram represents the signal from the source, the Fourier transform represents the frequencies present from the source. Use of an interferometer (instead of a spectrometer) benefits from Fellgett s Advantage (also known as the multiplex advantage) where there is an increase in the accuracy of interferometry over spectrometry by a factor of (N/2) 1/2, where N is the number of samples taken. Fellgett s advantage is only true if all other errors are the same for the interferometer and spectrometer. (Thorne, 1989). Finding the spectrum of a scene by interferometry is more computationally intensive than finding the spectrum of a single source from a spectrometer because the Fourier transform of each interferogram must be calculated, while a spectrometer finds the spectrum directly Two types of interferograms that can be made with this apparatus are a one-sided and two-sided interferogram. The advantages of using a two-sided interferogram are that the zero point distance (ZPD) does not have to be exact because the interferogram is symmetrical. This means that there will not be phase errors due to an inaccurate ZPD. The other benefits are that thermal or electronic drift will not affect the results, and if the 6

7 alignment of the interferometer is lost, the interferogram will not be symmetric, thus it can be seen immediately if the experiment has to be redone. The disadvantage of using a two-sided interferogram is that the maximum optical path difference (OPD) is halved and hence, the resolution is reduced. (Bell, 1972). Due to the motor that is available, a twosided interferogram will be used and doubling the number of images in the image cube will solve the loss of resolution. One limitation of the resolution is the amount of oscillations that are recorded (the length of the RECT function). The longer the total distance L 2 -L 1, the better the spectral resolution, i.e., wavelengths that are very close together are more easily resolved. One method to reduce noise, take longer exposures, that is usually employed cannot be used because of limitations in the performance of the stepper motor. The minimum step size that the manufacturer says the motor can do with precision is 200nm, which corresponds to an optical path difference (OPD) of 400nm. Therefore, the available sampling interval would let the spectrometer resolve wavelengths of up to 800nm due to the nyquest frequency. To bypass this problem, the motor is sub stepped to the manufacturers specifications, this decreases the precision in the motor movements. Another problem is that the motor drifts with respect to time. The motor was found to drift at an almost constant velocity with minor variations that do not increase as time increases but seems to be related to a periodic pattern. The average velocity is about 3.6nm/sec. The graph of the velocity of the motor when power was given to the motor, but no move command was given can be seen in appendix A. The velocity can be determined because the signal is from the reference beam, thus the peak-to-peak distance is nm and each interval is one second apart. 7

8 Therefore, experiments that require long times to complete are less accurate. The exposure time of the CCD must be relatively short so that the fringes do not move significantly during the exposure. This required that the object be sufficiently bright to ensure an adequate signal-to-noise ratio. The ultimate goal of the experiment is to have the object beam of the spectrometer a white-light (tungsten) source because produces the full visual spectrum. However due to the fact that the spectrometer acts like a bandpass filter (in the wavelengths ranging from nm), the coherence length of a visual fringe pattern decreases as the width of a bandpass filter increases. A RECT tunrs into a Sinc function in Fourier space, as the RECT gets longer, the Sinc gets narrower. At a RECT of infinite length, the Sinc turns into a delta function. This means that the coherence length of a white-light source is small, it is approximately 1 micron. Experimental Designs and Methods Part One: Designing the Experiment The original design with a single source was deficient because of the inaccuracies of the motor. It was found that the movements were not repeatable, thus the original concept of creating an index that could be referred to for motor step size was not possible. The solution used is to calibrate the system during every experimental run. The system was calibrated by adding a second source that acts as a reference beam at a known wavelength, 632.8nm for a red HeNe laser. 8

9 Figure 4:The experimental setup with the reference beam and object beam Both reference and object beams reflect from the moving mirror and are incident on the CCD array at the same time during the same run. This removes the position error from the calculation. The object beam is calibrated using the signal error from the reference beam. The only apparatus traversed by the beams that are not identical are the stationary beam splitters and mirrors. Since the positions are constant, they contribute no additional position errors. To decrease the size of the images that were taken, the CCD array had to be binned. Without binning, each image was approximately 2 MBytes, with binning (4 pixels x 4 pixels binned down to1 pixel), the size went down to about 125 kbytes. Then with cropping part of the array, the size of the image was further decreased to 77 kbytes. With the approximately 8000 images that are needed for nanometer resolution, the 9

10 equation to get the resolution of this system from the number of images in the cube, as can be seen in figure 7, is *(#images) (1) Thus with a depth of 8000 images, a 2 MByte size image would come to about 16 GBytes worth of data for a single experimental run. That is just not practical, by binning it down, the size for a run was about 600 MBytes. Figure 5: The smaller pixels on the left diagram are grouped to adjacent pixels making them act as one single larger pixel that is shown on the right. The object that was used for the final result was a four-quadrant color square with red, green, blue, and yellow quarters. It was printed on a transparency and placed between the collimating lens and beam splitter. The object can be seen in figure 6. 10

11 Figure 6: The four-color object that the experiment was conducted with. The object beam illuminated the object (printed on a transparency) and the resulting fringe patterns were recorded. From upper left corner going in clockwise order: green, red, yellow, and blue. Part Two: Data Processing A simple percent error method is used to find the error of the reference source. The step size of the motor must be known to determine the distance of each data point. Equation 2 shows how to find the step size. Dx = l Reference (nm) Period(# images) (2) Where l Reference is known and Period(# images) is the number of images in the image cube that make up one full period of the reference beam. To try and reduce the amount of error, the length of every period was found and averaged together to try and minimize the effects of the drift and off imprecise stepping. Next, equation 3 calculates the resolution of the system, this tells the how small of a difference in frequencies can be found. Dn = 1 N Dx (3) 11

12 As can be seen, as N (the depth of the interferogram cube) increases, Dn decreases, thus better resolution. The actual frequency is then calculated by multiplying change in frequency above by the k index (the location on the frequency axis) in the frequency domain as seen in equation 4. n = k Dn The wavelength is then found by equation 5. (4) (5) l = 1 n n has units of nm -1, thus the inverse is nm, which is the wavelength. The percent error of the reference beam is then calculated using equation 6. (6) % error = l Reference - l experimental l Reference 100 Then using the error of the reference beam, the wavelengths originally calculated for the object beam are then corrected by rearranging equation 6 to get equation 7. (7) l corrected = -100 l experimental % error -100 The results that were calculated by this method were either very close (within one or two nanometers) or dead on. The only wavelengths used to test this were the same (632.8 nm), and a green HeNe with wavelengths at and nm. The relationship between the size of the image cube (depth, proportional to number of images) and resolution can be seen in figure 7. 12

13 Figure 7: The relationship between the depth of the image cube and the resolution the system will have (at 50 nm step size intervals) Figure 7 shows that approximately 8000 images were needed in this setup to obtain a spectral resolution of 1nm where the average OPD was fifty nanometers. Thus, a translation distance of approximately 400 mm was necessary to obtain sufficient resolution. To find the spectrum of a two-dimensional object, the Fourier transform of each interferogram was evaluated at each pixel location in the image. 13

14 Figure 8: The diagram shows how the object relates to the interferogram, and then how the interferogram relates to the spectrum of the signal visualized in a two-dimensional graph. For visualization purposes, the spectrum is separated into three graphs, blue (400 l 500 nm), green (501 l 600 nm), and red (601 l 700 nm). The spectrum could be displayed in any number of ways if a specific wavelength or band of wavelengths was desired to be observed by making minor changes in the program. The three figures (9-11) show the presence of a signal in the lower right corner, this is noise of unknown source. Figure 9 shows no signal in the blue region, as expected. The only signal in figure11 is upper left and is due to the reference beam. Figure 10 shows a 14

15 pattern that correlates to the object. This noise in figure 10 and 11 could be attributed to stray light from reflections in the optics. Figure 9: Blue graph ( nm) Figure 10: Green graph ( nm) Figure 11: Red graph ( nm) The reason why the only detected signal is in the green region is because the object was illuminated by a two-color laser (green and yellow), so all wavelengths will be in the green spectrum ( nm). The target attenuated the amount of green light. As can be seen in figure 11, only a small segment of the reference beam is incident upon the image 15

16 plane. However a space (approximately five millimeters in this experiment) is needed between the reference and object beams so that scattered light will not interfere with each other. Any signal detected in the dead area can be attributed to stray light, as can be seen in figure 10. It should be noted that figures 9-11 have a resolution of about seven nanometers because the interferogram cube had a depth of 1024 images. In the current form, the thickness of the a two-dimensional interferogram cube cannot have more than a depth of 1024 images because the computer system (Sun Blade 1000 was used) cannot create arrays large enough for a depth of 8000 (8192 actually used so that a fast Fourier transform could be applied). The one-dimensional case can calculate image cubes with depths of 8192 giving resolutions of under a nanometer. Conclusions It was found that the correct spectrum of an object could be found using a sub-standard stepper motor (one that does not have repeatability at such small step sizes). The only negative aspect of adding the second beam is that the size of the object that can be imaged has to be decreased so that both beams can fit into the image plane (CCD array). The addition of the reference beam to calibrate the apparatus during every experimental run solved the problem of motor drift. It was found that windowing actually makes the results worse when there are numerous sinusoids in a sample. Since there are so many sinusoids toward the edge where the value goes either to zero, or close to zero (depending on the type of windowing technique that is used), it fails to read some of the oscillations, thus gives a false frequency. To try and decrease the prevalence of noise from the results, a thresholding method was used. When the data was processed, 16

17 only signals that had a high enough intensity were used, through refining the data processing, it is probably possible to use the noise from the reference beam to subtract from the object beam. This is probably possible because the noise is systemic noise, thus both signals should have the same noise. Advancement The experiments to date have demonstrated the successful implementation of the imaging spectrometer. However, several aspects exist that could be improved during further research. The program used to process the data could be upgraded, e.g. to support larger arrays. The program can be changed by performing the calculations on an entire row or column at once, continuing until it goes through the entire two-dimensional image. Another way would be to lower the threshold that was being used, the way that might done would be to find all of the spectral bands in the reference beam other than the reference wavelength and subtract them from the object wavelengths. To determine the full spectrum of an object, a white-light source is necessary. A tungsten light is usually used for this because its spectral curve lies in the visible region. The problem with a white-light fringe pattern is that it has a coherence length of only about one micron. For nanometer resolution in this system, the minimum OPD is approximately 400 microns. A possible way to bypass this problem would be to make the mirror go back and forth every two 250 microns (OPD = 500 microns), thus keeping the fringe pattern coherent. If the OPD goes past 1 micron then the fringe pattern will disappear making it impossible to find the spectrum. That will most likely cause a lot of other frequencies in the reference and object beams, but like mentioned above, if the added frequency is systemic, then hopefully it will be the same for the reference and 17

18 object beams, thus using the same method aforementioned, find the superfluous frequencies in the reference beam, then subtract them from the object. If the above suggestions can be implemented, a full functioning Fourier transform spectrometer will be a reality. The experiment up to this point has created a foundation that solved some problems that were encountered, but others need to be overcome to bring it to fruition. 18

19 Appendix A Drift Velocity of Motor Power applied to the motor when no movement command was issued. The drift velocity of the motor seems to be fairly constant. 19

20 Resources Albergotti, J. C. Fourier Transform Spectroscopy Using a Michelson Interferometer. American Journal of Physics Vol 40 (1972): Bell, Robert J. Introductory Fourier Transform Spectroscopy. New York, NY: Academic Press, Berkey, Donald Kieth. An Undergraduate Experiment in Fourier-Transform Spectrometry. American Journal of Physics Vol 40 (1972): Dorrer, C., N. Belabas, J-P. Likforman, and M. Joffre. Experimental implementation of Fourier-transform spectral interferometry and its application to the study of spectrometers. Applied Physics B, Lasers and Optics Vol B70 (2000): S99-S107. Gingras, D.J. Spectrum Estimation of FT-IR Data with Sampling Errors. SPIE Vol (1989): Mertz, Lawrence. Transformations in Optics. New York: John Wiley & Sons, Thorne, Anne. High resolution Fourier transform spectroscopy in the ultra-violet. SPIE Vol (1989):