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Design of photographic lens Shinsaku Hiura Osaka University Design of photographic lens What is photographic lens? Artifacts of the lens Aberration Photometric issues Basic lens calculation Paraxial analysis Lens design Wide angle / telephoto Zoom lens Intro. of optics design software What is ideal lens? The image is similar to the object on a plane perpendicular to the optical axis No distortion No blur Pinhole camera Common model of ideal lens in Computer Vision area Amount of the incoming light is very limited Diffraction limit Artifacts of actual lens h Failure of the similarity of the image Geometric degradation (aberration) • Distortion • Blur aberration Photometric degradation (vignetting) • Non-uniform sensitivity Definition of focal length Definition of focal length must be constant to Aperture settings aberration Elements of the optics Optical system Typically 3-10 elements, 20 for zoom For aberration correction, + function Aperture For the trade-off of the both amount of light and defocus Nikkor Example of a zoom lens Combination of various types of glasses, shape of the surfaces Why we need many elements? For correcting monochromatic aberration Thin lens with high index glass Thick lens with low index glass There are some differences for aberration even if the focal length is same For correcting chromatic aberration White light Longitudinal chromatic aberration of single lens C spectrum(red) d spectrum(green) F spectrum(blue) Parameters of optical glass Optical glass chart New glass After WW-II Dispersion Jena glass 1890- Extra-low dispersion Fluorite 1970- Early optical glass Abbe number Basically two parameters (index, dispersion) Correction of chromatic aberration Red light White light Blue light flint Crown High disp. Low disp. Balanced out by using two different dispersion Non-linear dispersion can not be corrected Definition of focal length effective diameter h Intersection of optical axis and exiting light Thin lens h0 Focal length Intersection of incoming and exiting light Curve of intersection Focal length is determined by the limit of very small ray height. Paraxial optics : small ray height and angle Aperture and F no. Aperture Aperture is not an actual diameter of diaphragm, but the diameter of incoming light F no = focal length / aperture diameter Smaller F no, higher speed lens Twice F no = quarter incoming light 1, 1.4, 2, 2.8, 4, 5.6, 8, 11, 16, 22, 32, 45, 64, … Image circle lens Image plane Diameter of permissible quality of image is limited Limit of vignetting Limit caused by aberration Focusing Focused at infinity Focused at close range Light from a point is always focused at one point Focused plane Image plane focused plane Image plane Focused surface misunderstanding For ideal lens, focused plane is always flat plane Thin lens law(1) 1 1 1 f a b a Focal length = ｆ b Front and rear focal plane 1 1 1 f a b a Focal length = ｆ Rear focal plane and rear principal point b Focal length = ｆ Front focal plane and front principal point Thin lens equation and principal points Rear p.p. a Front p.p. b Thin lens law still works for complex lens Magnification ratio a Magnification ratio b M a b Ratio of the size of object and image M=1 : life size Newton’s lens formula 1 1 1 f a b ⇔ xy f x y a b f 2 f Definition from the focal plane Practice (0) Prove Newton’s lens formula. a x f ,b y f to 1 1 1 f x f y f 1 1 1 f a b then ( x f )( y f ) f ( x f ) f ( y f ) Practice(0) ( x f )( y f ) f ( x f ) f ( y f ) And expand it, we have xy xf yf f 2 xf f 2 yf f 2 Finally, xy f 2 Caution : some textbooks define x, y as signed values, So it is described as xy f 2 Practice (1) Lens with focal length 50mm. If the lens is moved 5mm forward, how is the focused distance? Practice (1) answer 1/50 = 1/b + 1/55 For usual camera, object distance is defined from the film to the object. b = 550 (mm) Object distance is 550 + 55 = 605mm If we can not ignore the distance between two principal points, we must add it. Practice(2) If we want to focus the 10mm lens to the object with distance 1m, how is the lens movement forward? If the focal length is 20mm, how? Practice (2) answer Focal length is much smaller than the object distance, so let us ignore it. 1/10 = 1/1000 + 1/a a = 10.101 • Therefore, the lens motion is 0.101mm 1/20 = 1/1000 + 1/a a = 20.408 • Therefore 0.408mm Lens motion is proportional to the square of focal length Practice (3) If the photo is life size (M=1), how is the lens movement forward? Practice (3) answer 1/f = 1/b + 1/a, a=b so a = b = 2f Lens movement is as same as the focal length The distance from the object to the film is 4 times as long as of focal length This is the minimum distance between film and the object Tilt (swing) technique Principal plane of the lens Object Image Scheimflüg law Normal setting : S at infinity Examples of tilting (1) Focused to the whole of the object Examples of tilting (2) Pseudo shallow depth of field View camera ARCA-SWISS M-line 4x5 Lenses for 35mm SLR Canon Nikon Proof of Scheimflüg law b x h a In the figure above, all values are positive 1 1 1 a b f Thin lens law x h b a Magnification a h a0 Flat image Proof From the Eq. of planar image to substitute h of a x h b a h a0 , ,and we have On the other hand, from the thin lens law h a a0 is used x 1 a a0 1 a0 b a a 1 1 1 a b f , bf a b f x 1 a0 b f . b bf fa0 f Solve this equation for b , finally we have b x f a0 f a0 is used to substitute a of eq.(x), then It is the linear equation of b and x . . ---(x) Scheimflüg law fa0 f b x f a0 f a0 In case α=0 (plane perpendicular to the optical axis), fa0 b a0 f . Of course, this is the thin lens law. Solve this equation for b=0 (intersection of object and lens plane), Solve a h a0 x a0 . for a=0 (intersection of image and lens plane), h a0 Therefore, two intersections get together. . Perspective effect x 20 10 0 0 50 100 150 200 250 300 - 10 - 20 - 30 - 40 - 50 - 60 By the perspective effect, the projection is nonlinear 350