Transformation of the Diffraction Spectrum at Gradients of the Object

Gregory S. Tymchik, Irina O. Yakovenko


Background. In modern information measuring systems for the study and processing of spatial signals of various physical nature, the Fourier Optics is widely used, which actualizes the problems associated with the formation of diffraction spectra by coherent optical systems. By transforming the spatial-frequency coordinates of the Fourier spectrum of a normally-illuminated transparency, one can obtain a solution to the problem of studying light diffraction on a transparency that has an inclination to the axis of the optical system.

Objective. The aim of the paper is to establish a connection of the diffractive Fourier spectrum of the transparency, which has an inclination to the optical axis of the Fourier spectrum of a normally illuminated transparency, by means of the coordinates’ transformation in the space-frequency domain, expressed through the transparency rotation matrix elements.

Methods. The technique is based on the transformation of the Fourier spectrum of a normal-illuminated transparency (which simulated both the passing wave and the reflected wave) at its arbitrary angular slopes in space and various special cases of the two-dimensional spectrum degeneracy.

Results. The obtained results and substantiated practical recommendations can be used for the realization of practical problems using coherent optical spectral analyzers.

Conclusions. The well-known fundamental studies of light diffraction on a transparency that has an inclination to the axis of the optical system have a solution to this problem by transforming the spatial-frequency coordinates of the Fourier spectrum of a normally illuminated transparency. This result is generalized with the help of the Van Cittert–Zernike theorem in the case of a partially coherent illumination. However, in these studies only a partial case of the flat transparency slope is considered in relation to only one of the coordinate axes, and in the practical applications two-coordinate inclinations and longitudinal shifts take place, that have not been investigated at the present time. The optical system that implements the Fourier transform of the transmittance (reflection) function of a transparency that has an inclination to the optical axis is frequency-non-invariant, since at the linear displacement of spatial frequencies at the input there is a nonlinear shift and the impulse response shape distortion due to the transparency slope, and expressed through the elements of the rotation matrix.


Fourier spectrum; Transparency rotation matrix; Fourier transform; Angular inclination; Coherent optical system


K. Patorsky, “Fraunhofer diffraction patterns of titled planar objects”, Optica Acta, vol. 30, no. 5, pp. 673–679, 1983. doi: 10.1080/713821241

H.J. Rabal et al., “Diffraction by a tatled aperture. Coherent and partially coherent cases”, Optica Acta, vol. 32, no. 11, pp. 1309–1311, 1985. doi: 10.1080/713821668

V. Kolobrodov and G. Tymchik, Design of Diffraction Optical Elements and Systems. Kyiv, Ukraine: NTUU KPI, 2012.

V. Kolobrodov and G. Tymchik, Applied Diffraction Optics. Kyiv, Ukraine: NTUU KPI, 2014.

V.V. Bogatyireva and A.L. Dmitriev, Optical Methods of Information Processing. St. Petersburg, Russia: ITMO University, 2009.

K.E. Okan, Diffraction, Fourier Optics and Imaging. Hoboken: Wiley & Sons, 2007.

G. Curatu and J.E. Harvey, “Analysis and design of wide-angle foveated optical system based on transmissive liquid crystal spatial modulators”, Optical Eng., vol. 48, no. 4, pp. 043001-1–043001-10, 2009. doi: 10.1117/1.3122006

Y. Sheng and P. Ambs, “Diffractive optics”, Optical Eng., vol. 43, no. 11, pp. 2503–2504, 2004. doi:10.1117/1.1813443

V.G. Kolobrodov et al., “The diffraction limit of an optical spectrum analyzer”, in Proc. SPIE 9809, 12th Int. Conf. Correlation Optics, Nov. 30, 2015, Chernivsti, Ukraine. doi: 10.1117/12.2228534

Z. Lei et al., “Design of high resolution Fourier transform lens”, in Proc. SPIE 6722, 3rd Int. Symposium on Advanced Optical Manufacturing and Testing Technologies: Advanced Optical Manufacturing Technologies, Nov. 16, 2007, Chengdu, China. doi: 10.1117/12.782942

V.G. Kolobrodov et al., “The problems of designing coherent spectrum analyzers”, in Proc. SPIE 9066, 12th Int. Conf. Correlation Optics, Dec. 17, 2013. doi: 10.1117/12.2049587

M.S. Kuz’min and M.S. Rogov, “Optical Fourier processor with a liquid-crystal information-input device”, J. Optical Technol., vol. 83, no. 3, pp. 147–152, 2015. doi: 10.1364/JOT.82.000147

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