From Scattering to Wavefronts: What’s in between? Michael Mrochen, PhD
Swiss Federal Institute of Technology Institute of Biomedical Engineering
IROC Institute for Refractive and Oculoplastic Surgery Zürich Switzerland Foreword • This is a report on a work in progress ! • Geometric-optical methods are used to develop an analytical theory for surface scattering on rough surface sturctures (irregular structures). • Parts of this work are published in: – SPIE Proceedings Ophthalmic Technologies (1998,1999) – Laser Physics 12: 1239-1256 (2002) – Laser Physics 12: 1333-1348 (2002) Clinical observation:
Hartmann -Shack and Tscherning wavefront measurements demonstrate an increased background noise when the tearfilm breaks up.
Clinical observation:
Hartmann -Shack and Tscherning wavefront measurements demonstrate an increased background noise before and after LASIK After LASIK
Before LASIK Clinical observation:
Some eyes have irregular surface structures that are not reasonably described in terms of modes (Zernike, Taylor, ...) Striae Cornea after complicated LASIK
BSCVA: 20/50 decentered measurement
Centered measurement Problem !
Clinical observations indicate that there are other sources of optical errors that are not detected or described or measured by the currently used wavefront sensors.
-> Visual symptoms do not correlate well with wavefront data in such complicated cases Question ?!
What are the optical effects of such irregular structures ?
Irregular structures = surface roughness Optical effects of that might detoriate the image quality of the eye
• Elastic light scattering (i.e. Mie - scattering)
• Statistical refraction (roughness / irregularities)
• Geometrical refraction (wavefront aberrations) ALL SURFACES IN NATURE ARE ROUGH Definition of surface structures h(x ) = i S(xi )+ξ (xi ) x xi mean profile hight h h(xi )
Hight at lattice site RMS - surface roughness
The root mean square (rms) surface roughness describes the fluctuations of surface heights around the mean profile height
n 1 2 wrms = ∑()h()xi − h n i=1 Surface hight distribution function
1 h2 p(h) = exp− 2π wrms 2wrms The height distribution function gives the propability of the emergence of height h at a given point of the surface. h Problems of rms - surface roughness and surface hight distribution function • Both surface representations give only the
statistics at individual positions (xi) • They do not reflect correlations
between two different points (x1, x2) • Different rough surfaces can have the
same rms - roughness wrms and hight distribution functions p(h), but different height fluctuation frequencies (correlation radius). Correlation radius (length) The correlation radius is a measure for the averaged width of the grains of a rough surface (irregularity).
The correlation radius can only be determined by analyzing the height - height correlation function for a measured surface structure (irregularity). Correlation radius = 50 a.u. Correlation radius = 25 a.u.
Correlation radius = 15 a.u. Correlation radius = 4 a.u. Correlation radius r0
Small correlation radius Large correlation radius
? The human eye ?
“Light scattering” “wavefront aberrations” statistical surface roughness / modal representation statistical optical effect of wavefront aberrations (Zernike, Taylor, Seidel, ...) Optical effect of rough surfaces f(x,y)
Refractive index n1 Normal - vektor Light incidence α1 2 2 tan(α1) = f x + f y , ∂f (x, y) ∂f (x, y) f = , f = x ∂x y ∂y
2 2 2 σ = f x = f y α2 characteristic surface tilt
Refracted beam Refractive index n2
n1 ⋅sin(α1) = n1 ⋅sin(α 2 ) Surface roughness characteristics
Statistical distribution (Gaussian) of surface tilts
2 2 1 (f + f ) 2 2 p()f , f = exp− x y , σ 2 = f = f x y 2 2 x y 2πσ 2σ
Large diameter a of the light spot that is shined onto the surface
wavelength << correlation radius << light spot diameter
λ << r0 << a Surface heights larger than the wavelength h >> λ Intensity distribution
The intensity distribution in the retinal image plane is directly dependent on the statistical distribution function of the statistical rough surface !
I(xr , yr , feye ) ~ p(σ ,r0 ,a)~ p(h)
1 h2 p(h) = exp− 2π wrms 2wrms
Details are published in: Laser Physics 12: 1239-1256 (2002) and Laser Physics 12: 1333-1348 (2002) Point spread function in the retinal plane of the eye for variouse surface roughnesses
σ 1 = 0 < σ 2 < σ 3 < σ 4 < σ 5 = 0.4 Optical consequences
• Intensity distribution in the retinal plane is a function of the surface roughness / structure • Increased noise during wavefront sensing might be caused by “rough” surface structures. • Rough or uneven structures cause a loss of optical resolution in the retinal plane
? Threshold for detoriation of vision ? Summary • Is modal wavefront (i.e. Zernike) representation adequate for representing the optical properties of the human eye - at least in a few special cases. ? • Should optical diagnosis of the eye include the analysis of statistical optical effects ? • Will the analysis statistical and modal imaging errors help to explain visual symptoms and performance ? Thank you !