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Transcript
Chapter 5 Geometrical optics January 21,23 Lenses 5.1 Introductory remarks Image: If a cone of rays emitted from a point source S arrives at a certain point P, then P is called the image of S. Diffraction-limited image: The size of the image for a point source is not zero. The limited size of an optical system causes the blur of the image point due to diffraction effect: d l f D Geometrical optics: When l <<D, diffraction effects can be neglected, and light propagates on a straight line in homogeneous media. Physical optics: When l ~ D or l>D, the wave nature of light must be considered. 1 5.2 Lenses Lens: A refracting device that causes each diverging wavelet from an object to converge or diverge and to form the image of the object. Lens terminology: •Convex lens, converging lens, positive lens •Concave lens, diverging lens, negative lens •Focal points •Real image: Rays converge to the image point •Virtual image: Rays diverge from the image point •Real object: Rays diverge from the object point •Virtual object: Rays converge to the object. S P S P P S 2 5.2.1 Aspherical surfaces Determining the shape of the surface of a lens: The optical path length (OPL) from the source to the output wavefront should be a constant. Example: Collimating a point source (image at infinity) OPL ni FA nt AD constant y FA nti AD constant A(x,y) D x 2 y 2 nti (d x) constant F x 2 y 2 nti x c x y n x 2cnti x c 2 2 2 ti 2 d ni 2 x nt (nti2 1) x 2 2cnti x y 2 c 2 0 The surface is a hyperboloid when nti>1, and is an ellipsoid when nti<1. Example: Imaging a point source. The surface is a Cartesian oval. (n 2 ti 1)( x 2 y 2 ) 2nti2dx nti2d 2 c 2 2 4c 2 ( x 2 y 2 ) Aspherical lens can form perfect image, but is hard to manufacture. Spherical lens cannot form perfect image (aberration), but is easy to manufacture. 3 5.2.2 Refraction at spherical surfaces Terminology: vertex, object distance so, image distance si, optical axis. lo S qi A R qt li j V P C so si n n1 2 so R lo sin( q i ) sin j so R si R n1 n2 1 n2 si n1so n1 ( so R) n2 ( si R) si R li l sin q l sin q o i i t l l l l R l l o i o i o i sin q t sin( j ) n1 sin q i n2 sin q t Gaussian (paraxial, first-order) optics: When j is small, cosj ≈1, sinj ≈j, lo ( so R )2 R 2 2( so R ) R cos j so li ( si R )2 R 2 2( si R ) R cos j si Paraxial imaging from a single spherical surface: n1 n2 n2 n1 so si R 4 Paraxial imaging from a single spherical surface: n1 n2 n2 n1 so si R Note: 1) This is the grandfather equation of many other equations in geometrical optics. 2) For a planar surface (fish in water): si 3) Magnification: M T n2 so . (A bear needs to know this.) n1 yi ns 1 i . (P5.6). yo n2 so 5 Object (first) focal length: when si = , Fo n1 f o so R. n2 n1 Image (second) focal length: when so = , f i si fo n2 R. n2 n1 Fi fi Virtual image (si< 0) and virtual object (so< 0): EVERYTHING HAS A SIGN! Sign convention for lenses (light comes from the left): • so, fo + left of vertex • si, fi + right of vertex • xo + left of Fo • xi + right of Fi •R + curved toward left • yo, yi + above axis Fi C V si C Fo V so 6 Read: Ch5: 1-2 Homework: Ch5: 1,5,6 Note: In P5.1 the expression should be (so+si-x)2. Due: January 30 7 January 26, 28 Thin lenses 5.2.3 Thin lenses Thin lens: The lens thickness is negligible compared to object distance and image distance. Thin lens equations: nm Forming an image with two spherical surfaces: P' 2nd surface 1st surface S P' (R1, nm, nl) (R2, nl, nm) V1 S P R1 R2 C2 P V2 nl C1 so1 si1 d so2 si2 nm nl nl nm so1 si1 R1 nl d nl nm nm nl nm nm (nl nm ) 1 1 R R (s d )s For the 2nd surface : s s o 1 i 2 2 i1 i1 1 so 2 si 2 R2 so 2 | si1 | d si1 d For the 1st surface : 8 1 1 nm nm nl d (nl nm ) so1 si 2 R1 R2 ( si1 d ) si1 If the lens is thin enough, d → 0. Assuming nm=1, we have the thin lens equation: 1 1 1 1 (nl 1) so si R1 R2 lim si lim so f so si Lens maker’s equation: 1 1 1 (nl 1) f R1 R2 Gaussian lens formula: 1 1 1 s o si f Remember them together with the sign convention. Question: what if the lens is in water? 9 Optical center: All rays whose emerging directions are parallel to their incident directions pass through one special common point inside the lens. This point is called the optical center of the lens. Proof: incident// emergent θi θt θt 'θi ' sin θi sin θt ' θi arcsin θ ' arcsin t n n C O BC 2 R2 θt ' θi AC1// BC 2 2 OC1 AC1 R1 qi C2 A q t Oq i ' R2 C1 B qt ' R1 O does not depend on A and B. Conversely, rays passing through O refract parallelly. C2O R2 BC 2 C2O OC1 rule of sine Proof : θi ' θt incident// emergent OC1 R1 AC1 BC 2 AC1 For a thin lens, rays passing through the optical center are straight rays. Corollary: For a thin lens, with respect to the optical center, the angle subtended by the image equals the angle subtended by the object. 10 Focal plane: A plane that contains the focal point and is perpendicular to the optical axis. In paraxial optics, a lens focuses any bundle of parallel rays entering in a narrow cone onto a point on the focal plane. Proof: 1st surface, 2nd surface C’ C Focal plane Fi Focal plane S C P Image plane Image plane: In paraxial optics, the image formed by a lens of a small planar object normal to the optical axis will also be a small plane normal to that axis. 11 Read: Ch5: 2 Homework: Ch5: 7,10,11,15 Due: February 6 12 January 30 Ray diagrams Finding an image using ray diagrams: Three key rays in locating an image point: 1) Ray through the optical center: a straight line. 2) Ray parallel to the optical axis: emerging passing through the focal point. 3) Ray passing through the focal point: emerging parallel to the optical axis. S' yo Fo S 2 A 1 O Fi P 3 yi B P' f xo xi f so si Newtonian lens equation: xo so f xi si f yo f x o | yi | xi f Meanings of the signs: xo xi f 2 Transverse magnification: M T Longitudinal magnification: dxi f2 ML 2 M T2 dxo xo yi s i yo so • so • si •f • yo • yi • MT + Real object Real image Converging Erect object Erect image Erect image – Virtual object Virtual image Diverging lens Inverted object Inverted image Inverted image 13 Thin lens combinations 4 I. Locating the final image of L1+L2 using F Fi2 O1 O2 i1 ray diagrams: Fo1 Fo2 1) Constructing the image formed by L1 as if there was no L2. 2) Using the image by L1 as an object (may d<f1, d<f2 so2 d be virtual), locating the final image. The si1 ray through O2 (Ray 4, may be backward) is needed. 1 1 1 II. Analytical calculation s s f of the image position: o1 i1 1 1 1 1 1 1 1 si2 is a function of (so1, f1, f2, d) so 2 si 2 f 2 si 2 f 2 d f1so1 so1 f1 so 2 si1 d Total transverse magnification: M T M T 1M T 2 si1 si 2 f1si 2 so1 so 2 d ( so1 f1 ) so1 f1 14 Back focal length (b.f.l.): Distance from the last surface to the 2nd focal point of the system. Front focal length (f.f.l.): Distance from the first surface to the 1st focal point of the system. 1 1 b.f.l. si 2 1 1 f.f.l. so1 so1 si 2 1 1 f 2 so 2 1 1 f1 si1 so 1 si 2 1 1 f 2 d si1 1 1 f1 d so 2 1 1 f 2 d f1 1 1 f1 d f 2 so1 si 2 Special cases: 1) d = f1+f2: Both f.f.l. and b.f.l. are infinity. Plane wave in, plane wave out (telescope). 1 1 1 1 1 2) d → 0: effective focal length f: f f1 f 2 f.f.l. b.f.l. 3) N lenses in contact: 1 1 1 1 . f f1 f 2 fN 15 Read: Ch5: 2 Homework: Ch5: 20,25,26,32,33 Due: February 6 16 February 2 Mirrors and prisms 5.4 Mirrors 5.4.1 Planar mirrors 1) |so|=|si|. 2) Sign convention for mirrors: so and si are positive when they lie to the left of the vertex. 3) Image inversion (left hand right hand). 5.4.3 Spherical mirrors The paraxial region (y<<R): y ( R x )2 y 2 R 2 x R R 2 y 2 y2 y4 3 2 R 8R y 2 2 Rx 4 fx x 17 The mirror formula: SC CP SA PA (paraxial) A 1 1 2 so R si R so si R so si qi q S f F C P R 1 1 1 fo fi , 2 so si f V f si R so Transverse magnification: 1 yi si 1 2 1 R MT . yo so so R so 2 so R Ray diagrams of mirrors: Four key rays in finding an image point: 1) Ray through the center of curvature. 2) Ray parallel to the optical axis. 3) Ray through the focal point. 4) Ray pointing to the vertex. 2 4 S 1 F 3 V C P 18 5.5 Prisms Functions of prisms: 1)Dispersion devices. 2)Changing the direction of a light beam. 3)Changing the orientation of an image. qi1 qt1 qi2 qt2 5.5.1 Dispersion prisms Apex angle, angular deviation (θi1-θt1 ) (θt 2 -θi 2 ) θi1 θt 2 θt 1 θi 2 θt 2 arcsin( n sin θi 2 ) arcsin n sin( θt1 ) arcsin[ n(sin cos θt1 cos sin θt1 )] arcsin[sin n 2 sin 2 θi1 cos sin θi1 ] θi1 arcsin[sin n 2 sin 2 θi1 cos sin θi1 ] 19 Minimum deviation: θi1 θt 2 qi1 qt 2 dqt 2 d 1 0 dqi1 dqi1 qt1 qi 2 The minimum deviation ray traverses the prism symmetrically. At minimum deviation, θt1 θi 2 m θ t1 sin θt 1 θ i 2 2 sin θi1 2 n sin θt1 m θi1 θt 2 m sin θi1 2 θi1 θt 2 2 This is an accurate method for measuring the refractive indexes of substances. 20 Read: Ch5: 3-5 Homework: Ch5: 54,60,61,64,65,67,68 Due: February 13 21