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VanderLugt Filter

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VanderLugt Filter

1. 1. EPL 443 VanderLugt Filter & SCALE INVARIANT OPTICAL CORRELATION USING MELLIN TRANSFORMS Presented by : Ajay Singh 2010PH10821
2. 2. VanderLugt Filter (VLF) The total incident intensity at the level of the film is determined by the interference of two distributions of amplitude(coming from L2 and from P). The inclined wave plane is represented by its complex amplitude in the plane P2 . An optical filter having the remarkable property to permit the efficient control of the amplitude and of the phase of a transfer function at the same time even if they are only formed in purely absorbent figures.
3. 3. The total intensity incident at each point on the recording medium is determined by the interference of the two mutually coherent amplitude distributions present. The tilted plane wave incident from the prism produces a field distribution where the spatial frequency a = sin Ɵ/ λ So the total in intensity distribution may be written as
4. 4. We can write H as an amplitude distribution A and a phase distribution Ψ => As a final step in the synthesis of the frequency-plane mask, the exposed film is developed to produce a transparency which has an amplitude transmittance that is proportional to the intensity distribution that was incident during exposure. Thus the amplitude transmittance of the filter is of the form
5. 5. Now the field strength transmitted by the mask then obeys the proportionality Finally we take FT of the desired response impulse by either Modified Mach- Zehnder interferometer or modified Rayleigh interferometer
6. 6. we see that the third output term yields a convolution of h and g, centred at coordinates (0, -ah f ) in the (x3, y3) plane. Similarly, the fourth term may be rewritten as which is the crosscorrelation of g and h, centered at coordinates (0, ah f ) in the (x3, y3) plane.
7. 7. The Mellin transform of a function g(t) is defined by  SCALE INVARIANT OPTICAL CORRELATION USING MELLIN TRANSFORMS If we take s , an imaginary quantity s = j2πf and substitute ξ = exp(-x) M (j2πf ) = = which we see is nothing but the Fourier transform of the function g(e-x). Means it is possible to perform a Mellin transform with an optical Fourier transforming system provided the input is introduced in a "stretched" coordinate system, in which the natural space variable is logarithmically stretched (x = - In ξ ).
8. 8. Such a stretch can be introduced, for example, by driving the deflection voltage of a cathode ray tube through a logarithmic amplifier and writing onto an SLM with the resulting stretched signal. Here we see that the magnitude is independent of scale-size changes in the input. To prove this we replace M( g (ξ ) ) by M( g (aξ ) ) , and we see the invariance in the output :
9. 9. http://www.optiqueingenieur.org/en/courses/OPI_ang_M02_C02/co/Co ntenu_08.html  introduction to Fourier optics by Joseph W. Goodman .  REFERENCES :