Zernike: Zernike Polynomials

In matwey/Rfringe: Rfringe

Description

Routines for creating and manipulating Zernike polynomials.

Usage

Zernike(rho, theta, n, m, t)

Arguments

rho	normalized radius, 0 <= rho <= 1
theta	angular coordinate
n	radial polynomial order
m	azimuthal order
t	character for trig function: one of c("n", "c", "s")

Details

The arguments n and m must be relatively even.

Value

The value of the Zernike polynomial of order (n, m) at polar coordinates (rho, theta). The arguments rho and theta may be vectors, matrices, or higher order arrays, in which case the returned value is a vector or array of the same dimension.

Note

This function returns Zernikes scaled such that they form an orthonormal basis set for the space of functions defined on the unit circle. Note that this is not the most commonly used definition (as given e.g. in Born and Wolf). The definition I use is often associated with Noll (1976).

The otherwise unused function zmult can be used to convert between normalized and conventionally defined vectors of Zernike coefficients.

Author(s)

M.L. Peck [e-mail]

References

Born, M. and Wolf, E. 1999, Principles of Optics, 7th Edition, Cambridge University Press, chapter 9 and appendix VII.

Noll, R.J. 1976, Zernike polynomials and atmospheric turbulence, J. Opt. Soc. Am., Vol. 66, No. 3, p. 207.

http://wyant.opt-sci.arizona.edu/zernikes/zernikes.htm

http://mathworld.wolfram.com/ZernikePolynomial.html

See Also

rzernike, makezlist, zlist.qf, zmult, fillzm, pupil, pupilrms, pupilpv, strehlratio.

Examples

Zernike(1, 0, 4, 0, "n")	# == sqrt(5)

# A slightly more complex example

rho <- seq(0, 1, length = 101)
theta <- rep(0, 101)

plot(rho, Zernike(rho, theta, 6, 0, "n"), type="l", 
  ylim=c(-3.5,3.5), main="Some 6th order Zernike Polynomials")
lines(rho, Zernike(rho, theta, 5, 1, "c"), lty=2)
lines(rho, Zernike(rho, theta, 4, 2, "c"), lty=3)
lines(rho, Zernike(rho, theta, 3, 3, "c"), lty=4)

matwey/Rfringe documentation built on May 12, 2019, 8:43 a.m.