Ординатура / Офтальмология / Английские материалы / Optics of the Human Eye_Atchison, Smith_2000

If no Stiles-Crawford apodization is present, i.e, /3 =0,

2.Volume under PSF =effective flux passing through pupil

If no Stiles-Crawford apodization is present, i.e, /3 = 0,

3.For an off-axis calculation on a rotationally symmetric system and a single wavelength, the peak of the point spread function should occur at the angle in equation (A4.l0), i.e,

The optical transfer function (OTF)

A sinusoidal pattern with a real spatial frequency (Jand an orientation", is shown in Figure A4.1. The spatial frequency components in the X and Y directions are

The two dimensional OTF,which we write as G«(Jx,(Jy)' is the Fourier transform of the PSF

x

o

Object (or image) plane

Figure A4.1. Sinusoidal pattern of grating inclined at an angle IJI.

g(0x,Oy)' that is

However, calculation of the OTF by this direct relationship is not necessarily the best approach, because the PSF has no bounds and the integral should therefore be carried out over an infinite range. In practice we must use finite bounds but, depending upon how much of the PSF is outside these bounds, there will be errors in the final results. A better alternative is to use a Fourier transform identity (Bracewell, 1986), that states that the Fourier transform of the product of the transforms of

262 Appendices

two functions is the convolution of two functions. In this case, equation (A4.16)

reduces to a convolution of the pupil function P(X, Y) with its complex conjugate, as follows:

replace o"x and O"y in G and the integrand. Since the pupil function P(X, Y) is zero

outside the pupil, the limits of integration reduce to the region 'c' common to the two sheared pupils shown shaded in Figure A4.2a,

and we do not have to write the integral limits from -00 to +00, A useful alternative and

equivalent form of this integral is

G(lOx'lOy) =IfcP(X + lO)2, Y+ lOyl2)P*(X

- lO)2, Y- lOyl2)dXdY (A4.18)

with the region of integration now being that

shown in Figure A4.2b.

Macdonald (1971) has described such a method for the routine calculation of the OTF based upon the above auto-correlation integral, but with the amplitude function

A(X, Y) = 1.

It is common practice to normalize the OTF so that G(O, 0) =1.

Summary of main symbols

Point spread function (PSF) angles in X and Ydirections cartesian co-ordinates in the actual

pupil

f3 Stiles-Crawford parameter to base e

optical transfer function (OTF) spatial frequency (c/rad) components of 0" in the X and Y

References

Bracewell, R. (1986). Tile Fourier Transform and its Applications, revised edn, p. 112. McGraw-Hill.

Krakau, C. E. T. (1974). On the Stiles-Crawford phenomenon and resolution power. Acta Oplllilal., 52,

581-3.

Macdonald, J. (1971). The calculation of the optical transfer function. Optica Acta, 18, 269-90.

Smith, G. and Atchison, D. A. (1997). Image quality criteria. In The Eye and Visllal Optical Instrllmellts.

Cambridge University Press pp. 647-72.

Abbe V-value, 189

Aberrations and age, 227

and depth-of-field, 217, 220 higher order, 151 longitudinal, 137, 242 primary (or third order), 244-8 Seidel, 245-8

transverse, 137, 242 wave, 137, 242

Aberrations, chromatic longitudinal, 180

effect of accommodation, 185, 191 effect on accommodation, 186

effect of age, 228

effect of refractive error, 185, 190-2 effect on spatial vision, 186,205 magnitude, 186

measurement techniques, 183 and refraction, 71

and wavelength in focus, 185 Seidel, 190,245

transverse, 181 magnitude, 186

measurement techniques, 185 Aberrations, monochromatic

astigmatism, 147-9, 164 coma, 146, 162

ocular component contributions, 154 distortion, ISO, 165

field curvature, 149, 165 higher order, 151

measurement techniques, 138-42 sagittal and tangential errors, 148, 163 schematic eyes, 161-6, 173-6,254,257

Seidel, 161-5, 167, 170, 171, 173,245-8 spherical aberration, 143-6, 161

Aberrations, ophthalmic devices, ISS, 188 Aberrations, pupil, 155

Aberroscope technique, 141 Absorption

fundus, 129, 131 ocular media, 107-11

Accommodation, 18, 223 amplitude, 19

and age, 224

effect of longitudinal chromatic aberration, 186

effect on longitudinal chromatic aberration, 185, 191

mechanism, 223 Accommodation demand

ocular, 93 spectacle, 93

Achromatizing correcting lenses, 189

Age

and aberrations, 227

and amplitude of accommodation, 224 and cornea, 221

and lens, 222, 230 and photometry, 228 and pupil size, 227

and refractive errors, 61, 223 and schematic eyes, 229

and Stiles-Crawford effect, 229 and visual performance, 228

Airy disc, 196 Aliasing, 203

Ametropia see Refractive errors Amplitude of accommodation, 19,224

264 Index

Angle

alpha (a), 35 gamma ()1, 37

kappa (K"), 36 lambda (it), 35

psi (If/), 37 Aniseikonia, 54, 92

Anisometropia, 55, 61 Annulus method, 140 Anterior chamber, 4 Aperture stop, 21 Aphakia, 57 Aqueous, 4

transmittance, 110

Artificial pupils and defocus, 82 Aspheric surface

conicoid, 13, 166 figured conicoid, 167

Asphericity

of cornea, 13-6,168 of lens, 17, 168

Astigmatism (refractive error), 60 Axis

achromatic, 32, 37 fixation, 33, 37 foveal achromatic, 32 keratometric, 33, 34

line of sight, 31, 34, 35 optical, 8, 30, 35

pupillary, 32-33, 35 visual, 8, 32, 34, 35, 37

Badal optometer, 68-70 Binocular overlap, 9 Binocular vision, 8-9 Birefringence, 114

cornea, 114 form, 114 intrinsic, 114

lens, 115

nerve fibre layer, 132 Black body source, 122

Blind spot, 7

Blur disc see Defocus blur disc Bowman's membrane, 12

Capsule of lens, 16

Cardinal points, 7, 240

of finite schematic eyes, 257

focal, 7, 240

nodal, 7, 52-4, 80, 82, 240

of paraxial schematic eyes, 41, 253

principal, 7, 240 Centre-of-rotation, 8, 33

Choroid, 4, 7

absorption, reflectance, and transmission 130,131

Chromatic aberration see Aberration, chromatic

Chromatic difference of magnification, 182 Chromatic difference of position, 182 Chromatic difference of power, 181, 190 Chromatic difference of refraction, 181, 190 Chromatic dispersion, 189 Chromo-retinoscopy,184

Chromostereopsis, 187 Ciliary body, 4, 223

Circle of least confusion, 150

Component ametropia, 62 Cones,S, 127

Conicoid, 14-6, 166 ellipsoid, 14 hyperboloid, 14 paraboloid, 14

Contact lenses aberrations, 156

Contrast sensitivity function (CSF), 199,200, 203,204

effect of defocus, 205 Cornea, 3, 11-6

and age, 221

asphericity, 13-6, 168 birefringence, 114 power, 12

radii of curvature, 12, 15

refractive index, 12

sagittal (axial) radius of curvature, 15 scatter, 113

surface powers, 12

tangential (instantaneous) radius of curvature, 15

transmittance, 108 toricity, 13

Corneal astigmatism, 60, 223 Corneal sighting centre, 31

Correlation ametropia, 61 coso! law, 119

Crystal, biaxial, 114 uniaxial, 115

Cylindrical lens and axis, 60, 63

Defocus

and alignment of objects at different distances, 85

effect on visual acuity, 85

and wave aberration, 244, 260 Defocus blur disc, 82-5, 214, 217

Depth-of-field, 213-20 and aberrations, 217 and pupil size, 214 and target size, 215, 218

effect of luminance, 215

geometrical approximation, 217 Depth-of-focus see Depth-of-field

Descemet's membrane, 12 and age, 221

Detection task, 203 Diffraction, 194, 195, 198

and depth-of-field, 217 laser speckle, 70, 183

Diplopia with ophthalmic lenses, 94 Double pass techniques, 194, 200

Duochrome test, 71

Eccentric fixation, 32, 36 Eccentricity, 14 Eikonometer, 55 Emmetropia, 57, 58 Emmetropization, 61

Entrance and exit pupils, 21 and accommodation, 22

of paraxial schematic eyes, 42 Equivalent focal length, 7 Equivalent power, 7, 240 Equivalent sphere (mean sphere), 60

Equivalent veiling luminance, 121, 122 and age, 229

Far point, 19, 57, 58, 59, 62, 92

Field-of-view of ophthalmic lenses, 94 Field-of-vision, 8

of ophthalmic lenses, 94

Figured conicoids, 167 Finite (real) ray, 237

Fluorescence lens, 113

Fluorogens,113 Focal points, 7, 240 Fovea, 6

image quality, 204-7 Foveola, 6

Fundus, 67, 129

absorption, 129, 131 reflectance, 130

scatter, 131

Index 265

Gaussian optics, 241

Gradient refractive index of the ocular lens 17,39,41,168-71

and age, 222, 229 Grating focus, 74

Halation, 131

Half-width of point spread function, 197 Heterotropia (tropia), 36

Hippus, 24 Hirschberg test, 36

Hypermetropia (hyperopia), 59

Illuminance, 102 retinal, 117-22

Image quality criteria

line spread function, 195, 198

modulation transfer function, 198,200,204, 205,207,208

optical transfer function, 198-202,261 phase transfer function, 199, 204

point spread function, 195-98, 201, 204, 259-61

wave aberration function, 138, 243-5

Inter-pupillary distance (PO), 9 Intraocular lenses

aberrations, 156, 188

Inverse square law, 103 Iris, 4, 21

Iseikonic lenses, 55 Isogonal lenses, 55

Knapp's law, 91

Knife edge tests, 141

Lambertian sources, 102 Laser speckle, 70

and measuring chromatic aberration, 183

and measuring depth-of-field, 216 Lens equation, 240

Lens of the eye, 16-8 and age, 222 asphericity, 17, 168 birefringence, 115 diameter, 18

fluorescence, 113-4 power, 18, 169-71

principal points, 18 radii of curvature, 17

266 lndcx

Lens of the eve, (cont.)

refractive i~dex distribution, 17, 168-71, 222

scatter, 113

thickness, 17 transmittance, 110

Lenses

thick, 62, 63 Lenses, ophthalmic

contact, 156, 188 intraocular, 156, 188 iseikonic, 55

isogonal, 55

spectacle, 60, 63, 155, 188

Line spread function (LSF), 195, 198 Luminance, 102

Luminous efficacy, 100, 101 Luminous flux, 101 Luminous intensity, 101 Luminous transmittance, 111

Macula, 7 Magnification

pupil,90

relative spectacle, 91 rotational,93

spectacle, 88

transverse (lateral), 239, 241 Maximum spectral luminous efficacies of

radiation

for photopic vision, 100 for scotopic vision, 101

Maxwellian view, 123

Mean sphere (equivalent sphere), 60 Mesopic vision, 100, 101

Mie theory, 112

Modulation transfer function (MTF), 198,200, 204,205,207,208

Myopia, 58

Near point, 19,58,59,92

Nerve fibre layer (retinal), 5, 132 Nodal points, 7, 52-4, 80, 82, 240

of schematic eyes, 41, 45, 253, 257 Nyquist limit, 203

Ophthalmic lenses see Lenses, ophthalmic Ophthalmometric pole, 32, 34

Ophthalmophakometer, 35 Optic disc, 7

Optical invariant, 240, 246 Optical path length, 242 Optical radiation, 99

Optical transfer function (OTF), 194, 198-200, 261

calculation from diffraction and aberrations, 261

diffraction limited, 199 effect of defocus, 200-2

Optorneters Badal, 68-70

incorporating Scheiner principle, 72 laser speckle, 70

polarizing, 72 telescopic, 70

Parallax movement between object and image, 74

Paraxial approximations, 238 Paraxial marginal ray, 22, 43, 246 Paraxial optics, 237-41

Paraxial pupil ray, 22, 243, 246 Paraxial pupil ray angle ratio, 43, 53 Paraxial refraction equation, 238 Paraxial transfer equation, 239

PO (inter-pupillary distance), 9

Phase transfer function (PTF), 199, 204 Photography (photorefraction), 75

Photometric efficiency and Stiles-Crawford effect, 125

Photometry and age, 228

equivalent veiling luminance, 121, 122 illuminance, 101

inverse square law, 103 luminance, 102

luminous efficacy, 100, 101 luminous flux, 101 luminous intensity, 101

photon density levels at retina, 122 reflectance, 105-7

relation between photometric quantities, 103-4

seaHer, 111 transmittance, 107-11 troland, 118

units, 101-3

Photon density levels at retina, 122 Photopic vision, 100

Photorefraction, 75 Piper's law, 103

Point spread function (PSF), 121, 195-8,201,

204,259-61 diffraction limited, 195

half-width, 197

polychromatic sources, 260 and Rayleigh criterion, 197

and Strehl intensity ratio, 197 Polarization

retina, 131, 132 Polymegathism, 221 Posterior chamber, 4 Power

equivalent, 240 of a surface, 238 thick lens, 62, 88 thin lens, 62 vertex, 89

Presbyopia,57,59,224-7 theories, 225-7

Principal points, 7, 240 of lens, 18

Pupil

and accommodation, 22 artificial, 28, 82

auto-correlation of function of, 260 centration, 23

entrance, 21 exit, 21 function, 259

magnification, 90 senile miosis, 227 shape, 25-7

size, 23-5

and age, 24, 227, 228 and binocular vision and

accommodation, 24 and depth-of-field, 214-8 and drugs, 24

and illumination level, 23 measurement, 28

and retinal image quality, 204, 208

and retinal light level, 118, 123 significance, 27

Pupillometry, 28

Purkinje images, 106 Purkinje shift, 101

Radiant power (radiant flux), 99 Ray

extra-ordinary, 114 finite (real), 237 ordinary, 114

paraxial, 238

Index 267

paraxial marginal, 22, 43, 246 paraxial pupil, 22, 43, 246

pupil nodal, 32 Rayleigh criterion, 197

Rayleigh-Gans (Rayleigh-Debye) theory, 112 Rayleigh scattering, 112

Ray tracing finite, 237 paraxial, 238

Reduced aperture model and StilesCrawford effect, 125

Reference sphere, 245 Reflectance, fundus, 130

Reflection, specular, 105-7 Refraction

discrepancies between subjective and objective refraction, 77

factors influencing, 75-7

in presence of spherical aberration, 152 ocular, 63

spectacle, 62 Refraction techniques

objective-only, 73-5 grating focus, 74

parallax movement between object and image, 74

photography, 74 retinoscopy, 73

visual evoked response (VER),75 subjective-objective, 71-3

remote refraction and relay systems, 71 Scheiner principle, 72

split image and vernier acuity

(coincidence method), 72 subjective only, 67-71

laser speckle, 70

longitudinal chromatic aberration, 71

simple perception of blur, 67-70 Refractive errors

and age,61 anisometropia, 61 astigmatism, 60 component ametropia, 62

axial,62 refractive, 62

correlation ametropia, 61 distribution, 61

effect of ocular parameter change, 64 emmetropia, 57, 58

hypermetropia (hyperopia), 59

and longitudinal chromatic aberration, 185, 190-2

myopia, 58

population distribution, 61

268 Index

Refractive index of the cornea, 12 of the lens, 17

continous refractive index model, 168-70 shell model, 170

Relative spectacle magnification, 91 and axial ametropia, 91

and Knapp's law, 91

and refractive ametropia, 91

Remote refraction and relay systems, 71 Resolution limit, 199, 203

Resolution task, 203 Retina, 5-7

birefringence, 132 fovea, 6-7 macula, 7

polarization, 131, 132 shape, modelling, 171 Retinal illuminance, 117-22

Retinal image quality, 194-208 fovea, 204-7

effect of defocus, 205

effect of polychromatic light, 205 effect of pupil decentration, 206

peripheral vision, 207 Retinal image size

defocussed image, 79-86 defocus blur disc, 82-5

defocus ratio, 84-6

focused image, 52-4

eye focused at infinity, 54

Retinoscopy, 73, 153 Ricco's law, 103 Rods, 5-6

Rotational magnification, 93-94

Sagittal power error, 148, 163,247 Sagittal section, 147

Scatter, 111-3

cornea, 113 fundus, 131

lens, 113

Mie theory, 112 Rayleigh, 112

Scheiner principle, 72

and measuring monochromatic aberration,

140

Schematic eyes, finite (wide angle) aberrations, 173-6, 257 Chromatic eye, 173, 192

Drasdo and Fowler, 172, 173-6 Indiana eye, 173, 174, 192

Kooijman, 172, 174-6,255,257

Liou and Brennan, 172-173, 174-7, 256, 257 Lotmar, 171-172, 174-6, 255, 257

Navarro et al., 172, 174-6,255-7 retinal illuminance, 176

Schematic eyes, paraxial, 40-6 aberrations, 161-6, 190, 254

and age, 229

Bennett and Rabbetts reduced, 45 Bennett and Rabbetts simplified, 45, 250,

252-4 development, 40

Emsley reduced, 45, 252-4 entrance and exit pupils, 42-43

Gaussian properties and cardinal points, 41-42

Gullstrand number 1 (exact), 44,161-6, 190,251-4

Gullstrand-Emsley (simplified), 45, 251, 253-4

Le Grand full theoretical, 44, 251, 253-4 Le Grand simplified, 45

variable accommodating, 45 Sclera, 3

seatter, 130

Scotoma with ophthalmic lenses, 94 Scotopic vision, 100, 101

Seidel aberrations, 245-8

influence of asphericity, 167 influence of gradient index, 170, 248

influence of retinal curvature, 163-5, 171, 247

of finite schematic eyes, 173,257

of paraxial schematic eyes, 161-165, 190-191,254

Seidel approximation, 161, 162, 164 Senile miosis, 227

Simple perception of blur, 67-70 Snell's law, 237

Spectacle lenses aberrations, 155-6, 188

Spectacle magnification, 88-90

Spectral luminous efficiency function for photopic vision, 100

for scotopic vision, 101

Spherical aberration, 143-6, 161, 244 effect on depth-of-field, 220 effect on refractive error, 152 effect on retinoscopy, 153

Split image and vernier acuity (coincidence method), 72

Spurious resolution, 200

Stereopsis, 9, 54

Stiles-Crawford effect, 124-7, 260, 261 and accommodation, 126

and age, 229

and eccentricity, 126

and influence on image quality, 206 and luminance, 126

measurement, 126

and photometric efficiency, 125

and pupil function, 260

and reduced aperture model, 125 role, 127

theory, 126

and wavelength, 126 Strehl intensity ratio, 197 Symbols, Greek, ix

Tangential section, 147

Tangential power error, 148, 163, 247

Thick lenses, 62, 63 Transmittance of eye, 107-11

aqueous, 110 cornea, 108 direct, 108 lens, 110 luminous, 111 total,108 vitreous, 110 whole eye, 107

Index 269

Transverse aberrations, 137, 242 Troland, 118

Tropia (heterotropia), 36 Tryptophan, 113

V-value (or Abbe V-Value), 189 Vernier alignment

and chromatic aberration, 183, 185 and monochromatic aberration, 139 and refraction, 72

Vertex distance, 62

Vertex normal of cornea, 33 Visual centre of the cornea, 31

V~sual Evoked Response (VER),75 VIsual performance, 151,203

and age, 228 Vitreous chamber, 4 Vitreous

transmittance, 110

Wave aberration, 137

fun~tion (polynomial), 138, 243-5 vanance, 152

Wave-front sensor, 142