# Point Spread and Modulation Transfer Functions of Zernike Wavefronts

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This Demonstration shows an aberration density plot, and profiles and a density plot of the point spread function, and and profiles of the modulation transfer function for a wavefront described by orthonormal Zernike circular polynomials . You can analyze any of 45 Zernike polynomials and select a Zernike coefficient between 0 and 1.

Contributed by: James C. Wyant (January 2013)

Open content licensed under CC BY-NC-SA

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## Details

Wavefront aberrations are very important in studying the design, fabrication, and testing of all optical systems. Zernike polynomials, named after Nobel Prize winner Frits Zernike, are often used to describe these aberrations. The density plot shows what an interferogram testing an optical system with a given Zernike polynomial wavefront error would look like.

The point spread function (PSF), which is given by the square of the absolute value of the Fourier transform of the wavefront, shows what the intensity image of a point source would look like. The axis of the PSF plot shows angular distance in units of the wavelength, , divided by the optical system's pupil diameter, .

The modulation transfer function (MTF), which is given by the magnitude of the Fourier transform of the point spread function, gives the spatial frequency transfer function of an incoherent optical system. The cutoff frequency is given by , where is the wavelength, and is the number (or focal ratio) of the optical system given by the focal length divided by the pupil diameter.

References

[1] C-J. Kim and R. R. Shannon, "Catalog of Zernike Polynomials," in *Applied Optics and Optical Engineering*, Vol. X (R. R. Shannon and J. C. Wyant, eds.), San Diego: Academic Press, 1987 pp. 193–221.

[2] J. W. Goodman, *Introduction to Fourier Optics*, 3rd ed., Greenwood Village, CO: Roberts & Company, 2004.

[3] V. N. Mahajan, "Zernike Polynomial and Wavefront Fitting," in *Optical Shop Testing* (D. Malacara, ed.), 3rd ed., Hoboken, NJ: Wiley, 2007 pp. 498–546.

[4] V. N. Mahajan, *Optical Imaging and Aberrations, Part II: Wave Diffraction Optics*, Bellingham, WA: SPIE Press, 2001.

[5] J. C. Wyant and K. Creath, "Basic Wavefront Aberration Theory for Optical Metrology," in *Applied Optics and Optical Engineering*, Vol. XI (R. R. Shannon and J. C. Wyant, eds.), New York: Academic Press, 1992 pp. 11–53.

[6] J. C. Wyant. "*webMathematica* and *LiveGraphics3D*." (Jan 15, 2013) wyant.optics.arizona.edu/math.htm.

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