Aberration Theory

Geunyoung Yoon, Ph.D.

Assistant Professor Department of Ophthalmology Center for Visual Science University of Rochester Optics

Quantum Optics

Coherent Optics

Diffractive Optics (Fourier Optics)

Geometrical Optics (Aberration theory)

Paraxial Optics (First Order Optics) (Gaussian Optics) Outline What Is Wavefront? Huygens’s principle, Snell’s law, Fermat’s principle Paraxial (first order) approximation

What Kind Of Wavefront Aberrations Are There? Monochromatic aberration (Seidel and wave aberrations) Chromatic aberration (Longitudinal, Transverse)

Why Are These Aberrations Important? Relationship between aberrations and image quality (Pupil function, PSF, MTF, Image convolution…)

How Can We Measure These Aberrations Of The Eye? Different types of wavefront sensors Outline What Is Wavefront? Huygens’s principle, Snell’s law, Fermat’s principle Paraxial (first order) approximation

What Kind Of Wavefront Aberrations Are There? Monochromatic aberration (Seidel and wave aberrations) Chromatic aberration (Longitudinal, Transverse)

Why Are These Aberrations Important? Relationship between aberrations and image quality (Pupil function, PSF, MTF, Image convolution…)

What Kind Of Wavefront Aberrations Are There? Monochromatic aberration (Seidel and wave aberrations) Chromatic aberration (Longitudinal, Transverse)

Why Are These Aberrations Important? Relationship between aberrations and image quality (Pupil function, PSF, MTF, Image convolution…)

What Kind Of Wavefront Aberrations Are There? Monochromatic aberration (Seidel and wave aberrations) Chromatic aberration (Longitudinal, Transverse)

Why Are These Aberrations Important? Relationship between aberrations and image quality (Pupil function, PSF, MTF, Image convolution…)

How Can We Measure These Aberrations Of The Eye? Different types of wavefront sensors What Is Wavefront?

Huygens’s principle Snell’s law Fermat’s principle Paraxial (first order) approximation Wavefront vs Ray

“A wavefront is a surface over which an optical disturbance has a constant phase.”

Harmonic wave function ψ (x,t) = Asin(kx −ωt)

Phase

Spherical wavefront Plane wavefront

x x

t=0 t Wavefront vs Ray

“Rays are lines normal to the wavefronts at every point of intersection.”

Plane wavefront Spherical wavefront

Ray Huygens’s Principle “Every point on a primary wavefront serves as the source of spherical secondary wavelets, such that the primary wavefront at some later time is the envelope of these wavelets.”

Secondary spherical wavefront Primary spherical wavefront Snell’s law

normal

Fast medium θ θ i r (smaller refractive index, ni)

Slow medium θ t (larger refractive index, nt)

n sin(θ ) i = t nt sin(θi ) θ θ Reflection: i = r

θ θ Refraction: i > t when ni < nt Fermat’s Principle

“The path actually taken by light in going from some point S to a point P is the shortest optical path length (OPL).”

S OPL = n ⋅ SO + n ⋅OP P i t 2 2 2 2 = ni ⋅ h + x + nt ⋅ b + (a − x) h θi θr

ni dOPL = 0 to minimize OPL O dx nt

b θ x − (a − x) t ni ⋅ + nt ⋅ = 0 h2 + x2 b2 + (a − x)2 OPL = n ⋅ SO + n ⋅OP i t P xa-x a n sin(θ ) i = t nt sin(θi ) Paraxial Optics (First order optics)

p p o R i θ P S

so si n n1 2 θ n R(s + R)sin n R(s − R)sinθ 1 o = 2 i po pi

Approximation θ θ θ 3 θ 5 θ 7 sin = − + − + ⋅⋅⋅ sinθ ≈ θ 3! 5! 7!

n1 n2 n2 − n1 + = Lens maker’s formula so si R Paraxial Optics (First order optics)

“The emerging wavefront segment corresponding to these paraxial rays is essentially spherical and will form a ”perfect” image at its center P.”

P S

n n1 2 Third Order Optics

“The paraxial approximation, sin_ ≈ _, is somewhat unsatisfactory if rays from the periphery of a lens are considered.”

θ θ 3 sin = θ − 3! 2 2 n n n − n  n  1 1  n  1 1   1 + 2 = 2 1 + h2  1  +  + 2  −   s s R 2s  s R  2s  R s  o i  o  o  i  i  

Paraxial rays

P S

n n1 2 Perpheral rays Aberrations What Kinds Of Wavefront Aberrations Are There?

Monochromatic aberration (Seidel and wave aberrations)

Chromatic aberration (Longitudinal, Transverse) Monochromatic aberrations (Seidel aberrations)

Spherical Aberration

Coma

Astigmatism

Field Curvature

Distortion Monochromatic aberrations (Seidel aberrations)

Spherical Aberration

Marginal rays Longitudinal SA Coma

Transverse SA Astigmatism

Paraxial rays Field Curvature Lens

Spot diagram Distortion Monochromatic aberrations (Seidel aberrations)

Spherical Aberration

Coma

Astigmatism

Field Curvature

Distortion Monochromatic aberrations (Seidel aberrations)

Spherical Aberration

Coma

Astigmatism

Field Curvature

Distortion Monochromatic aberrations (Seidel aberrations)

Spherical Aberration

Coma

Image

Astigmatism Object

lens Field Curvature

Distortion Monochromatic aberrations (Seidel aberrations)

Spherical Aberration

Coma

Pin-cushion distortion Astigmatism Image Object Field Curvature

Barrel distortion Distortion Monochromatic aberrations (wave aberrations)

“The optical deviations of the wavefront from a reference plane or spherical wavefront.”

Reference Reference spherical wavefront plane wavefront

Aberrated wavefront

Wave aberration Wave aberrations (defocus)

Myopic (near sighted) eye

Perfect eye

Defocused wavefront Wave aberrations (higher order)

Eye with higher order aberrations

Perfect eye

Aberrated wavefront Wave Aberration of a Surface

Wavefront Aberration

-1 mm (superior-inferior) -2

-3

-3 -2 -1 0 1 2 3 mm (right-left) Mathematical description of the aberration. Zernike circle polynomials

W(_,θ) = ∑ Cm Z m(_,θ) n n

Wavefront Zernike Zernike polynomials aberration coefficient (wavefront mode)

m: angular frequency

2 2 astigmatism Z (_,θ) = _ cos2θ 2

n: radial order Zernike polynomials

Second order aberrations

Higher order aberrations Wavefront mode for each Zernike polynomial m n -5 -4 -3 -2 -1 0 1 2 3 4 5 astigmatism defocus astigmatism

trefoil coma coma trefoil

quadrafoil2nd astigmatism spherical 2nd astigmatism quadrafoil

pentafoil2nd trefoil2nd coma 2nd coma 2nd trefoil pentafoil

5 Wavefront aberration and Zernike coefficients

Wave aberration Zernike Zernike polynomial coefficients (mode)

= 0.3°ø - 0.2°ø + 0.4°ø +

0.3°ø - 0.5°ø - 0.2°ø +

0.2°ø + ……………

Wave Zernike aberration coefficients Wavefront rms error

2 2  2π  m Strehl ≈1− rms 2 rms = ∑()Cn   when rms is small.  λ 

Diffraction-limited 0.8 (Strehl ≥ 0.8)

0.6 Rayleigh’s _/4 rule 0.4 λ Strehl ratio W ≤ λ p−v rms ≤ 4 0.2 14

0 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

rms (_) Chromatic Aberration

1  1 1  Lensmaker’s formula: = ()n −1 ⋅ −  f  R1 R2  n = n(_) f = f (_) for polychromatic light

1.54 98 BK7 Schott glass R , R = 50mm 1.535 1 2 97 Focal length(mm)

1.53 96

1.525 95 1.52 Refractive index 94 1.515

1.51 93 0.4 0.45 0.5 0.55 0.6 0.65 0.7 Wavelength (mm) Longitudinal (axial) chromatic aberration (LCA)

Short wavelength White light Long wavelength

LCA LCA of the human eye

Bennet and Rabbetts (1989) LCA (D)

~2.2D

Wavelength (nm) Spectral sensitivity of the human eye

Optical effect of eye’s LCA on image quality ≈ 0.2D defocus for monochromatic light

0.8 0.2D

0.6 -0.3D 0.4 Sensitivity

0.2 0.4D -0.8D

0 400 450 500 550 600 650 700 Wavelength (nm) Transverse (lateral) chromatic aberration (TCA)

TCA

White light

Long wavelength Short wavelength Why Are These Aberrations Important?

Relationship between aberrations and image quality (Pupil function, Rms, PSF, SR, OTF, MTF, PTF, Image convolution…) Image quality

Image plane Wave aberrations PSF, SR, MTF…

object ?

image How well can an optical system form image? We can improve image quality by correcting the aberrations. Aberration vs Image quality

Pupil function  2π  P(x, y) = A()x, y expi W ()x, y  (aberration)  λ  FT

Point Spread Function Strehl Ratio autocorrelation (PSF) Image convolution FT

Optical Transfer Function (OTF)

Modulation Transfer Function Phase Transfer Function (MTF) (PTF) Point Spread Function (PSF)

PSF = FT()P()x, y 2

The Point Spread Function, or PSF, is the image that an optical system forms of a point source.

The point source is the most fundamental object, and forms the basis for any complex object. Point Spread Function (PSF)

Geometrical optics Real world

perfect optics Diffraction

The PSF for a perfect optical system is the Airy disc, which is the Fraunhofer diffraction pattern for a circular pupil. Airy Disc Point Spread Function vs. Pupil Size Perfect Eye

1 mm 2 mm 3 mm 4 mm

5 mm 6 mm 7 mm Point Spread Function vs. Pupil Size Typical Eye

1 mm 2 mm 3 mm 4 mm

pupil images

followed by

psfs for changing pupil size 5 mm 6 mm 7 mm Strehl Ratio diffraction-limited PSF (with no aberrations) Strehl Ratio eyedlHH

Hdl

actual PSF (with aberrations)

Heye Image Convolution (,) (,) (,)PSFxyOxyIxy⊗

FT-1 [ FT( PSF(x,y) ) °ø FT( O(x,y) ) ] = I(x,y) Modulation Transfer Function (MTF)

MTF ()f x , f y = Re[FT ()PSF()x, y ]

The Modulation Transfer Function, or MTF, is a measure of the reduction in contrast from object to image.

The ratio of the image modulation to the object modulation at all spatial frequencies. Modulation Transfer Function (MTF)

Object 100% contrast

Image diffraction only

Image diffraction + 0.25D defocus

1 MTF

0 Spatial Frequency

Optics Simulations - Have fun!!! -

Fundamental Optics

http://webphysics.ph.msstate.edu/javamirror/java/light.htm

Wavefront Theory and Fourier Optics

http://www.optics.arizona.edu/jcwyant/math.htm How Can We Measure These Aberrations Of The Eye?

Different types of wavefront sensors to measure ocular aberrations Wavefront sensors for the eye

Subjective method Objective method

Ingoing light Ingoing light Outcoming light

Spatially Resolved Tcherning Shack-Hartmann Refractometer Laser Ray Tracing Measurement principle of wavefront sensors for the eye

Measurement of wavefront slope (1st derivative of wavefront) averaged over each subaperture on the pupil

subaperture

Original wavefront W(x,y)

Spot displacement ɺdx, _dy

∂W (x, y) = k ⋅∆d Averaged ∂x x wavefront slope ∂W (x, y) = k ⋅∆d ∂y y Spatially Resolved Refractometer Webb, Penney and Thompson (1992)

reference

Subject adjusts the incident angle of light until retinal spot intersects reference spot.

ɺd , _d x y reference Laser Ray Tracing Navarro & Losada (1997), Molebny et al. (1997)

ɺdx, _dy reference

CCD Scanning different locations of the pupil

reference ɺdx, _dy

CCD Tcherning Aberroscope Tscherning (1894)

Dot pattern mask

CCD

ɺdx, _dy Shack-Hartmann wavefront sensor Liang, Grimm, Goelz, and Bille (1994), Liang and Williams (1997)

Laser beacon ɺdx, _dy

CCD

Perfect eye Real eye Comparison of wavefront measurements using different wavefront sensors Shack-Hartmann vs Spatially Resolved Refractometer (Objective vs Subjective methods)

Salmon et al., J. Opt. Soc. Am. A, 15 (1998)

S-H

SRR Shack-Hartmann vs Laser Ray Tracing (Outcoming vs Ingoing light)

Moreno-Barriuso and Navarro, J. Opt. Soc. Am. A, 17 (2000) Shack-Hartmann vs Laser Ray Tracing vs Spatially Resolved Refractometer

Moreno-Barriuso et al., Optometry and Vision Science, 78 (2001) Thank you!