Fourier Transform of Light Using a Constructed Spatial Light Modulator

Ryan Frink and Sean Geldert

University of Minnesota

Methods of Experimental Physics Spring 2013

Abstract

We constructed a low-cost spatial light modulator used to create two-dimensional optical Fourier transformations in the far field; a comparison of observed transformations to theoretical models generated computationally was conducted for several transmission patterns. Observed optical Fourier transformations of circular and square apertures matched theory with reduced chi² values of 5.78 and 10.43 respectively.

Introduction

A spatial light modulator (SLM) is an optoelectronic device capable of generating diffraction masks by controlling a liquid crystal display (LCD). These SLMs have a number of commercial applications beyond the pedagogical ability to generate diffraction masks for demonstrational experiments; applications such as: ultrafast pulse shaping used in industrial research, optical computing, microscopic laser surgery¹. An SLM operates by managing which cells composing the LCD panel are allowing light to be transmitted or stopped completely by inducing a polarization shift on incident light which is then selected by a crossed polarizer – referred to as an analyzing polarizer, which can be seen in Fig. 1. In general, a commercial SLM is an expensive device for one to acquire, costing several thousand dollars for lower end models. The SLM that we have constructed was composed from portions of an Infocus LP1000 LCD projector originally released in 2001 and purchased online for less than one hundred dollars – much cheaper than the commercial alternative.

If one considers the cells allowing light to pass through the analyzing polarizer completely unperturbed to act as point light sources at the polarizer, generating light of spherical wave fronts, it is the constructive and destructive interference in the far field that generates these optical Fourier transformations. To compare the optical Fourier transformations we have observed to theory, the computer program MatLab was utilized to compute two-dimensional fast Fourier transformations of the corresponding transmission patterns. The study of optical Fourier transformations is instructive as it allows one to realize some subtleties of what the Fourier transformation actually is, and the wealth of knowledge that is contained within².

FIG. 1. (a) Shows the 90^o general polarization shift across a TN cell. (b) Illustration of voltage settings allowing light to either pass through the cell, or be stopped. Voltage V₁ corresponds to no applied voltage, meaning the light will become further polarized and pass through unaffected. Voltage V₂ will cause the light to travel through the TN cell unpolarized, therefore no light will pass through the second polarizer, and the light will be blocked. [1]

Theory

To characterize the diffraction patterns from the diffraction mask, it is necessary to note that the observed phenomena in our experiment occurred in the far field; this allows the use of Fourier transformations to mathematically describe the manner in which the waves of light propagating from the diffraction mask interfere destructively and constructively to create optical transformations. Thus, the Fourier transformation of light that has passed through the diffraction mask can be thought of as a summation of spherical waves that eventually will form the Fourier transformation in the far field. This can be mathematically represented by the Fresnel integral, which deals with the configuration of the electric field at the point of the transmission pattern, and how it develops as the distance changes. The Fresnel integral becomes increasingly difficult to solve analytically as the complexity of transmission patterns escalates, thus the two-dimensional Fourier transformation pair equation was used, and can be found as Eqn.1:

(1)

where is a two-dimensional function essentially describing the binary representation of the desired diffraction mask, and the variables k_x and k_y are the x- and y-components of the wave vector describing the propagation of the spherical waves. Eqn. 1 can be written more compactly:

(2)

Due to the fact that the spatial light modulator’s LCD panel is composed of individual pixels, the Fourier transform of a function describing the diffraction mask itself will not accurately represent what we observe. Ideally, the distance between one pixel to another will be infinitesimally small, but as the LCD panel is not a perfect device this is not the case, meaning the distance between the pixels will have to be taken into account as they are on the same order of magnitude as the wavelength of the laser (4-5 mm). As these spaces in between the pixels do not pass light (they are filled with wires and transistors necessary for the panel’s operation), a two-dimensional binary function g(x,y) can be introduced to represent the pixel pattern. Thus, the actual function describing the observed Fourier transformation G(k_x,k_y) is a transformation of the pixelized function f(x,y), or rather:

(3)

By rearranging some of the terms present in Eqn. 3, we arrive at a simplified version:

(4)

Using the convolution principle of Fourier mathematics, Eqn. 4 can be rewritten in a more succinct form:

(5)

where G(k_x,k_y) is the convolution of the Fourier transformation of f(x,y) with the Fourier transformation of g(x,y). As our eyes do not directly observe intensity, but instead power, the observed transformation pattern is not actually the function G(k_x,k_y), but rather:

(6)

here G’(k_x,k_y) represents the observed optical Fourier transformation.

It should be noted at this point, that the considerations taken above are that of a limited case; as the intensity profile of the laser beam used is Gaussian in its form, the function f(x,y) has to be placed within the center of this beam and must be small enough to experience negligible intensity differences across its geometry. The intensity of light across the transmission pattern can then be considered homogenous, allowing one to exclude the

FIG. 2. An illustration of the general Gaussian intensity profile of the laser beam. As long as the intensity values of the green points are close to the intensity value of the central peak, shown here in cyan, the intensity across the transmission pattern can be considered constant; thus the geometry of the transmission pattern must remain within the red lines. Note: all axis values in this plot are arbitrary as the figure is intended to be purely instructional.

factoring in of a Gaussian function to account for the intensity differences into the above calculations – Fig. 2 illustrates this. However, if the diffraction mask was not small enough

to allow for the exclusion of this Gaussian intensity profile factor, the function describing the observed Fourier transformation would become:

(7)

where H(k_x,k_y) is the observed Fourier transformation, f(x,y) and g(x,y) are as stated previously, and h(x,y) is a two-dimensional Gaussian function describing the intensity profile of the laser beam. Clearly Eqn. 7 is a more complicated function than the one described in Eqn. 6; for this reason we kept the geometry of the transmission pattern small enough to allow for Eqn. 6 to hold.

Experimental Setup and Technique

To construct our SLM, it was necessary for us to first completely deconstruct the LCD projector. To do so, the casing and all other optics found within were removed. The only components that were left intact were: the power source, lamp ballast (electrical component in which the projection lamp originally connected to), LCD panels – the Infocus LP1000 contained three of them, one for each of the RBG colors, and the main circuit board with the attached user controls. A simple schematic diagram along with an actual image of our experimental setup can be found in Fig. 3. Removing the outer casing and several of the projection components caused safety interlocks to become activated within the projector, stopping the projector from powering up altogether. To restore the projectors original functionality, several of the lamp ballasts’ output wires controlling reference voltages were rewired and the safety locks were permanently tied down. The LCD panels found within the projector came with a polarizing film glued to one side; however, since the film was of poor optical quality, we removed it for the purposes of our experiment and replaced it with a standard linear polarizer.

The light source used in our experiment was a 632.8 nm wavelength HeNe laser. To cover a larger portion of the LCD panel, the beam was first passed through a beam expander to enlarge the diameter to approximately 1 cm. In our experimental setup, the beam expander and the laser are combined into a single device, which is reflected in both images of Fig. 3. Upon exiting the beam expander, the light passed through a polarizer responsible for controlling the intensity of light incident on the LCD panel, as observed transformations would “wash out” if the intensity of light was exceedingly great. Following the first polarizer was the LCD panel connected to our constructed SLM, which bore the diffraction mask of our choosing. Upon passing through the LCD panel, light would strike the second polarizer acting as the analyzer to the LCD panel. Upon passing through this second polarizer, light would either be stopped completely or pass through unperturbed and the optical Fourier transformations would form in the far field. As the far field effects started becoming apparent at around 1.5 meters from our second polarizer – which was beyond the reach of our optics table, the Canon EOS Digital SLR camera that was used to collect data was mounted on a table across the room from our SLM, and Fourier transformations were observed on the wall approximately 2.3 meters away from the second polarizer.

There were several important aspects of our experimental setup that had to be taken into consideration for our SLM to produce optical Fourier transforms that matched theory. The first of which is the Gaussian intensity profile of the laser beam previously described; instead of modifying the theoretical transformation to account for this, we decided to make the dimensions of the transmission pattern present on the LCD panel small enough (in our case, covering areas of approximately 1 mm²), so the Gaussian roll-off of the intensity profile of the beam would have little effect, and could ultimately be ignored. The second aspect of our setup that needed to be taken into consideration was the previously mentioned pixelization of the LCD panel. The transmission pattern was thought of as the desired shape composed of small pixels with space in between one another; this is more clearly illustrated in Fig. 4, which shows the ideal transmission pattern of a circle, along with its true representation of that pattern for our experimental setup.

FIG. 4.Transmission pattern that would actually appear on SLM’s LCD panel due to pixelization (right). To generate theoretical optical Fourier transformations that accurately resembled the ones we have observed, this pixelization effect had to be taken into consideration.

With both of these ideas taken into consideration, we were able to observe optical Fourier transformations for two transmission patterns: a square and a circle. The observed Fourier transformation of the circle transmission pattern can be found in Fig. 5, along with its corresponding theoretical transformation. A comparison of our observations to expected theoretical results was then carried out using MatLab.

Data Analysis and Results

To perform a quantitative analysis we used MatLab to simulate the expected Fourier transform. To do so, a binary function describing the diffraction mask was declared. Next a binary function describing the LCD’s pixelized geometry was declared. The product of these two functions was then Fourier transformed using the matlab algorithm fft2, which itself is the Cooley-Tukey algorithm below:

(8)

As stated above, the experimental Fourier transforms were photographed using a digital camera; digital photos can be represented by three-dimensional arrays in MatLab. The arrays describe position in the x-y plane and an arbitrary unit of intensity corresponding to each point in each of the RGB colors. For each diffraction mask, a comparison between the expected Fourier transformation and the observed Fourier transformation was carried out by taking the difference of the two arrays. As the helium neon laser primarily emits red light, only the array corresponding to red light was used in the analysis.

To simplify the analysis, a quantitative assessment of the structure between the primary maxima of the Fourier transformations was performed. We carried this out by comparing intensity profiles bisecting the maximum, and quantitatively comparing the spatial distribution of the expected and experimental results. To perform the spatial analysis, the intensities in the three-dimensional array were either set to 1 or 0 depending on a threshold set to keep the spatial distribution of the intensities intact. From there, a difference between the experimental array and the theoretical array was carried out. This resulted in an array of c_I values that was used in calculating a reduced chi² value for the spatial distribution of the observed Fourier transformation of the 1 mm diameter pixelized circular aperture. The intensity profiles of the experimental and theoretical transform of a 1mm circular aperture can be seen below.

FIG. 5. The left plot is that of the expected intensity profile for the pixelized circular aperture, while the right plot is the observed intensity profile. Both intensity profiles were taken from a cross section running through the maximum of the expected and experimental two-dimensional intensity profiles.

As can be seen in Fig. 5, a discrepancy between the expected and experimental intensity profiles exists. Possible causes for this discrepancy will be addressed later in the paper. After adjusting the intensity profiles according to a threshold that retains structure, the resulting plots of the structure can be seen below as Fig. 6. A subtraction between these two arrays was taken, which resulted in an array of chi_I values. As the spatial uncertainties on each intensity point in the experimental profile correspond to 0.1333 mm, a resultant reduced chi² of 5.78 was found.

FIG. 6. As can be seen in the figure above, the spatial distribution of the adjusted intensities are qualitatively similar to one another; the expected distribution of intensities for a circular aperture of radius 0.5 mm can be seen on the left while the observed distribution is found on the right.

Conclusion and Acknowledgments

Our research group was able to construct a low-cost SLM by dissecting an old LCD projector. The SLM was used to generate optical Fourier transformations in the far field for a variety of transmission patterns. Theoretical Fourier transformations were generated computationally using a two-dimensional fast Fourier transformation in MatLab, and a comparison was made to observations by subtracting the two images from one another and generating a reduced chi² value from the subtraction data. The constructed SLM produced fourier transforms that qualitatively were similar when compared to the theoretical models, but quantitatively the fourier transforms were greatly lacking in clarity compared to their theoretical counterparts; in the cases of the circle and square transmission patterns, we obtained reduced chi² values of 5.78 and 10.43 respectively. Probable causes for these discrepancies from theory are: incorrect dimensions in pixel geometry used to calculate theoretical expectancies, an overwhelming lack of clarity in captured data, and the small contribution of the previously mentioned Gaussian intensity profile of the laser light source used. It's important to note at this time the discrepancy between the cross sectional intensities between the theoretical values and the experimental values. This is most likely due to the digital camera used in our experiment not having the resolution and intensity range needed to properly capture the intensity profile. The experiment could be further improved upon by spending more time calibrating the pixel geometry in generating theoretical Fourier transformations, taking into account the intensity profile of the laser beam, and devising a better method of capturing images that produces clearer results.

We would like to thank both Professor Clem Pryke for his time spent in helping us understand the theory associated with our experiment, and Kurt Wick for the help he has offered in constructing our experimental apparatus.

References

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