Optics II Ivo Ihrke / Summer 2011
Aperture: Stops and Pupils • Principal effect: changes exposure • Side effect: depth of field Ivo Ihrke / Summer 2011
Aperture � Irradiance on sensor is proportional to � square of aperture diameter A � inverse square of sensor distance (~ focal length) � Aperture N therefore specified relative to focal length f f-number = f /# A Ivo Ihrke / Summer 2011
Aperture � How to read the f/#: � numbers like “f/1.4” – for 50mm lens, aperture is ~35mm � exposure proportional to square of F-number, and independent of actual focal length of lens! � Doubling series is traditional for exposure � therefore the familiar (rounded) sqrt(2) series � 1.4, 2.0, 2.8, 4.0, 5.6, 8.0, 11, 16, 22, 32, … Ivo Ihrke / Summer 2011
How low can N be? ������������������ �������������� � Principal planes are the paraxial approximation of a spherical “equivalent refracting surface” 1 = f /# θ 2 sin ' � Lowest N (in air) is f/0.5 � Lowest N in SLR lenses is f/1.0 image: Kingslake 1992 Ivo Ihrke / Summer 2011
Depth of Field images: London and Upton Ivo Ihrke / Summer 2011
Depth of focus (in image space) � tolerance for placing the focus plane C’ - circle of confusion � Note that distance from (in-focus) film plane to front versus back of depth of focus differ image: Kingslake 1992 Ivo Ihrke / Summer 2011
Depth of Field (in object space) � the range of depths where the object will be in focus www.cambridgeincolour.com Ivo Ihrke / Summer 2011
Depth of field (in object space) � total depth of field (i.e. both sides of in-focus plane) 2 2 f /# C U D tot = 2 f (from Goldberg) � where � f/# = F-number of lens � C = size of circle of confusion (on image) � U = distance to focused plane (in object space) � f = focal length of lens � hyperfocal distance � back focal depth becomes infinite when U = f 2 / C f/# Ivo Ihrke / Summer 2011
Numerical Aperture NA = θ n sin � The size of the finest detail that can be resolved is proportional to λ /NA. � larger numerical aperture � resolve finer detail Ivo Ihrke / Summer 2011
Numerical Aperture vs. F-Number � low magnification 1 /# ≈ f 2 NA � working f-number: f /# w � distance-related magnification: m 1 = ≈ + f /# ( 1 m ) f /# w 2 NA � relevant for systems with high magnification (microscopes or macro lenses) Ivo Ihrke / Summer 2011
Examples 2 2 f /# C U D tot = 2 f � f/# = f/4, C = 8 � , U = 1m, f = 50mm � D tot = 80mm � f/# = f/16, C = 8 � , U = 9mm, f = 65mm � Canon MP-E at 5:1 (macro lens) � use at short distances (M=5 here) image: Charles Chien � D tot = 0.075mm ! Ivo Ihrke / Summer 2011
Tilt and Shift Lens � Lens shift simply moves the optical axis with regard to the film. � change of perspective (sheared perspective) � Tilt allows for applying Scheimpflug principle � all points on a tilted plane in focus image: wikipedia Ivo Ihrke / Summer 2011
Diffraction Limit � Diameter d of 70% radius of the Airy disc f = λ = λ d 1 . 22 N 1 . 22 A single spot barely resolved no longer resolved Ivo Ihrke / Summer 2011
Resolution image of a lens focusing as a wave optical picture Ernst Abbe (1840-1905) Ivo Ihrke / Summer 2011
Describing Sharpness � Point spread function (PSF) image: Smith 2000 Ivo Ihrke / Summer 2011
Describing Sharpness � Modulation transfer function (MTF) � Modulus of Fourier transform of PSF image: Smith 2000 Ivo Ihrke / Summer 2011
Camera Exposure = × H E T � � Exposure can be varied in two ways: � Aperture: f-stop - 1 stop doubles H Interaction with depth of field � Shutter: Doubling the effective time doubles H Interaction with motion blur Ivo Ihrke / Summer 2011
Aperture vs Shutter f/16 f/4 f/2 1/8s 1/125s 1/500s images: London and Upton Ivo Ihrke / Summer 2011
Imperfections in Imaging Ivo Ihrke / Summer 2011
Lens Aberrations � Spherical aberration � Coma � Astigmatism � Curvature of field � Distortion Ivo Ihrke / Summer 2011
Sharpness Related Aberrations Ivo Ihrke / Summer 2011
Chromatic Aberration � Index of refraction varies with wavelength � For convex lens, blue focal length is shorter � Can correct using a two-element “achromatic doublet”, with a different glass (different n’) for the second lens � Achromatic doublets only correct at two wavelengths… � Why don’t humans see chromatic aberration? Ivo Ihrke / Summer 2011
Chromatic Aberrations � Longitudinal chromatic aberration (change in focus with wavelength) image: Smith 2000 Ivo Ihrke / Summer 2011
Chromatic Aberrations � Lateral color (change in magnification with wavelength) image: Smith 2000 Ivo Ihrke / Summer 2011
Spherical Aberration � Focus varies with position on lens. images: Forsyth&Ponce and Hecht 1987 • Depends on shape of lens • Can correct using an aspherical lens • Can correct for this and chromatic aberration by combining with a concave lens of a different n’ Ivo Ihrke / Summer 2011
Oblique Aberrations � Spherical and chromatic aberrations occur on the lens axis. They appear everywhere on image. � Oblique aberrations do not appear in center of field and get worse with increasing distance from axis. Ivo Ihrke / Summer 2011
Aberrations � Coma � off-axis will focus to different locations depending on lens region � (magnification varies with ray height) images: Smith 2000 and Hecht 1987 Ivo Ihrke / Summer 2011
Coma Ivo Ihrke / Summer 2011
Astigmatism � The shape of the lens for an of center point might look distorted, e.g. elliptical � different focus for tangential and sagittal rays image: Smith 2000 Hardy&Perrin Ivo Ihrke / Summer 2011
Astigmatism � (Video) Ivo Ihrke / Summer 2011
Astigmatism red - unsharp Ivo Ihrke / Summer 2011
Curvature of Field � focus “plane” is actually curved Object Image Ivo Ihrke / Summer 2011
Field Curvature Ivo Ihrke / Summer 2011
Field Curvature different image distance Ivo Ihrke / Summer 2011
Bad Optics curvature of field, coma, chromatic aberration Ivo Ihrke / Summer 2011
Distortion Ivo Ihrke / Summer 2011
Distortion � Ratios of lengths are no longer preserved. Object Image Ivo Ihrke / Summer 2011
Geometric distortion � Change in magnification with image position image: Smith 2000 Ivo Ihrke / Summer 2011
Radial Distortion image: Kingslake Ivo Ihrke / Summer 2011
Contrast Issues Ivo Ihrke / Summer 2011
Radial Falloff � Vignetting – your lens is basically a long tube. � Cos^4 falloff – “rule of thumb”. � At an angle, area of aperture reduced by cos(a) � 1/r^2: Falls off as 1/cos(a)^2 (due to increased distance to lens) � Light falls on film plane at an angle, another cos(a) reduction. Ivo Ihrke / Summer 2011
Vignetting - Example � a white diffuse target � actual photograph Ivo Ihrke / Summer 2011
Flare � Artifacts and contrast reduction caused by stray reflections image: Curless notes Ivo Ihrke / Summer 2011
Flare � Artifacts and contrast reduction caused by stray reflections � Can be reduced by antireflection coating (now universal) images: Curless notes Ivo Ihrke / Summer 2011
Ghost Images � Minimize artifacts, maximize flexibility � Artifacts � Spherical Aberration � Chromatic Aberration � Distortions � Lens Flare image: Kingslake 1992 Ivo Ihrke / Summer 2011
Ghost Images image: Kingslake 1992 Ivo Ihrke / Summer 2011
Lens Flare – Effect of Coating coating Ivo Ihrke / Summer 2011
Other Optical Elements Ivo Ihrke / Summer 2011
Beam Splitters � splits ray into two � can be polarizing or not � 50/50 most common � other ratios available e.g. 90/10 � two types [Zetterling/KTH] � beam splitter cube – no ray offset � semi-transparent mirror – ray offsets cause of ray offsets incoming ray Ivo Ihrke / Summer 2011
Polarizers Ivo Ihrke / Summer 2011
unpolarized with polarizing filter [victorvonsalza/Flickr] Ivo Ihrke / Summer 2011
Neutral Density Filters � pieces of dark glass ─ flat spectral response � Use: generate different exposure at same lens settings and exposure time ─ i.e. preserve motion blur and depth-of-field characteristics of photograph while making it darker � graduated versions exist Ivo Ihrke / Summer 2011
Neutral Density Filter Ivo Ihrke / Summer 2011
Spectral Filters � are attenuating different wavelengths differently � attenuation in mathematical terms: multiplication spectrum filtered spectrum filter response Ivo Ihrke / Summer 2011
Variable Spectral Filters � interference-based filters � can be designed almost arbitrarily for specific incidence angle Ivo Ihrke / Summer 2011
Prisms � several types of prisms exist � dispersing ─ e.g. spectroscope � reflecting ─ e.g. SLR, binoculars � polarizing Ivo Ihrke / Summer 2011
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