Zernike polynomials

Why does anyone care about Zernike polynomials?

A little history about their development.

Definitions and math - what are they?

How do they make certain questions easy to answer?

A couple of practical applications

What will Zernikes do for me?

• Widely used in industry outside of lens design• Easy to estimate image quality from coefficients• Continuous & orthogonal on unit circle, Seidels are not

– Can fit one at a time, discrete data not necessarily orthogonal– ZP’s will give misleading, erroneous results if not circular aperture

• Balance aberrations as a user of an optical device would• Formalism makes calculations easy for many problems

– Good cross check on lens design programs

• Applicable to slope and curvature measurement as well as wavefront or phase measurement

History of Zernikes

• Frits Zernike wrote paper in 1934 defining them– Used to explain phase contrast microscopy– He got a Nobel Prize in Physics in 1953 for above

• E. Wolf, et. al., got interested in 1956 & in his book• Noll (1976) used them to describe turbulent air • My interest started about 1975 at Itek with a report• Shannon brought to OSC, John Loomis wrote FRINGE• J. Schwiegerling used in corneal shape research • Incorporated in ISO 24157 with double subscript

Practical historical note

• In 1934 there were no computers – stuff hard to calculate• In 1965 computers starting to be used in lens design• Still using mainframe computers in 1974

– Personal calculators just becoming available at $5-10K each

• People needed quick way to get answers– 36 coefficients described surface of hundreds of fringe centers– Could manipulate surfaces without need to interpolate

• Same sort of reason for use of FFT, computationally fast• Early 1980’s CNC grinder has 32K of memory• Less computational need for ZP’s these days but they give

insight into operations with surfaces and wavefronts

What are Zernike polynomials?

• Set of basis shapes or topographies of a surface– Similar to modes of a circular drum head

• Real surface is constructed of linear combination of basis shapes or modes

• Polynomials are a product of a radial and azimuthal part– Radial orders are positive, integers (n), 0, 1,2, 3, 4, ……– Azimuthal indices (m) go from –n to +n with m – n even

The only proper way to refer to the polynomials is with two indices

Some Zernike details

Zernike Triangle

m = -4 -3 -2 -1 0 1 2 3 4

n =

0

1

2

3

4

Rigid body or alignment terms

Tilt y and x

Focus z

For these terms n + m = 2

Location of a point has 3 degrees of freedom, x, y and z

Alignment refers to object under test relative to test instrument

Third order aberrations

Astigmatism

n = 2, m = +/- 2

Coma

n = 3, m = +/- 1

Spherical aberration

n = 4, m = 0

For 3rd order aberrations, n + m = 4

These are dominant errors due to mis-alignment and mounting

Zernike nomenclature

• Originally, Zernike polynomials defined by double indices• More easily handled serially in computer code• FRINGE order, standard order, Zygo order (confusing)• Also, peak to valley and normalized

– PV, if coefficient is 1 unit, PV contour map is 2 units– Normalized, coefficient equals rms departure from a plane

• Units, initially waves, but what wavelength? • Now, generally, micrometers. Still in transition• For class, use double indices, upper case coeff for PV

– lower case coefficient for normalized or rms

Examples of the problem

Z 1 1Z 2 (p) * COS (A)Z 3 (p) * SIN (A)Z 4 (2p^2 - 1)Z 5 (p^2) * COS (2A)Z 6 (p^2) * SIN (2A)Z 7 (3p^2 - 2) p * COS (A)Z 8 (3p^2 - 2) p * SIN (A)Z 9 (6p^4 - 6p^2 + 1)

Z 1 1Z 2 4^(1/2) (p) * COS (A)Z 3 4^(1/2) (p) * SIN (A)Z 4 3^(1/2) (2p^2 - 1)Z 5 6^(1/2) (p^2) * SIN (2A)Z 6 6^(1/2) (p^2) * COS (2A)Z 7 8^(1/2) (3p^3 - 2p) * SIN (A)Z 8 8^(1/2) (3p^3 - 2p) * COS (A)Z 9 8^(1/2) (p^3) * SIN (3A)

FRINGE order, P-V Standard order, normalized

Normalization coefficient is the ratio between P-V and normalized

One unit of P-V coefficient will give an rms equal normalization factor

Zernike coefficients

Addition (subtraction) of wavefronts

Rotation of wavefronts

These equations look familiar

Derived from multi-angle formulas

Work in pairs like coord. rotation

021.)866.14(.)5.2.()2sin(14.)2cos(2.'

243.)866.20(.)5.14(.2sin20.2cos14.'2

2

a

a kl

Rotation matrix in code1 0 0 0 0 0 0 0 a0

0 b00

0 cos sin 0 0 0 0 0 a1-1 b1

-1

0 -sin cos 0 0 0 0 0 a11 b1

1

0 0 0 cos2 0 sin2 0 0 a2-2 b2

-2

0 0 0 0 1 0 0 0 a20 b2

0

0 0 0 -sin2 0 cos2 0 0 a22 b2

2

0 0 0 0 0 0 cos3 0 a3-3 b3

-3

0 0 0 0 0 0 0 cos a3-1 b3

-1

Rot a b

Aperture scaling

1212 22202 c'rr),r(Z 112 222 cc)'r(

Aperture scaling matrix

1 c^2-1

c 2c^2(c^2-1)

c 2c^2(c^2-1)

c^2

c^2

c^2

c^3

c^3

c^3

AMB c

Aperture shifting

1 h 2h^2 h^2

1 2h 3h^2 3h^2

1 4h 2h

1 3h 3h^2

1

1

1

1AMB x

Useful example of shift and scalingZernike coefficients over an off-axis aperture

Symmetry properties

Determining arbitrary symmetry

Flip by changing sign of appropriate coefficients

Symmetry of arbitrary surface

For alignment situations, symmetry may be all you need

This is a simple way of finding the components

Symmetry properties of Zernikes

o-o e-o o-o e-o rot o-e e-e o-e e-e

If radial order is odd, then e-o or o-e, if even the e-e or o-o

e-e even-even

o-o odd-odd

e-o even-odd

o-e odd-even

n =

1

2

3

4

Symmetry applied to images

Same idea applied to slopes

References

Born & Wolf, Principles of Optics – but notation is dense

Malacara, Optical Shop Testing, Ch 13, V. Mahajan, “Zernike Polynomials

and Wavefront Fitting” – includes annular pupils

Zemax and CodeV manuals have relevant information for their applications

http://www.gb.nrao.edu/~bnikolic/oof/zernikes.html

http://wyant.optics.arizona.edu/zernikes/zernikes.htm

http://en.wikipedia.org/wiki/Zernike_polynomials