Phys 322 Lecture 12 Chapter 5
Geometrical Optics Geometrical optics (ray optics) is the simplest version of optics.
Ray optics Ray Optics
axis
We'll define light rays as directions in space, corresponding, roughly, to k-vectors of light waves. We won’t worry about the phase. Each optical system will have an axis, and all light rays will be assumed to propagate at small angles to it. This is called the Paraxial Approximation. Light wavefronts and rays Light wavefronts are to rays (isotropic media) Group of rays form normal congruence if one can find a surface that is perpendicular to every ray. Theorem of Malus and Dupin: a group of light rays will preserve its normal congruence after any number of reflections and refractions Procedure for finding wavefronts: 1. Find paths of all rays from the light source 2. Draw surfaces perpendicular to every ray 3. Time it takes to get from one wavefront (at one phase) to another one is the same along all rays Is geometrical optics the whole story?
No. We neglect the phase. ~0 Also, our ray pictures seem to imply that, if we could just remove all aberrations, we could focus a beam to a point and obtain infinitely good spatial resolution. Not true. The smallest possible focal spot is the > wavelength, . Same for the best spatial resolution of an image. This is due to diffraction, which has not been included in geometrical optics. Focus
Each point on an illuminated or a self- illuminating surface is a source of spherical waves: Rays diverge from that point Spherical wave can also converge to a point
A point from (to) which a portion of spherical wave diverges (converges) is a focus of the bundle of rays
Stigmatic optical system - perfect image
Reversibility: SP P and S are conjugate points object space image space Lens - refractive device that changes the wavefront curvature The wavefront changed from convex to concave OPL should be the same for red and blue rays
Qualitatively: Insert a transparent object with n>1 that is thicker in center and thinner at the edges Aspherical surfaces Change spherical wave to plane wave The shape of the interface: Time to ravel from S to DD’: F A AD 1 vi vt n F A n AD or: i 1 t c c Time of travel from S to DD’ must be the same for any point in plane DD’: ni F1A nt AD constant
nt F1A AD constant ni nti nt/ni > 1 - hyperboloid nti nt/ni < 1 - ellipsoid
rays can be reversed Aspherical surfaces
Convex lenses Converging lenses
Concave lens Diverging lens
virtual image F: Focal points
When a bundle of parallel rays passes through a lens, the point to which they converge (converging lens) or the point from which they appear to diverge (diverging lens) is called focal point Spherical lens
Use Fermat’s Principle Optical OPL=n1lo+ n2li axis Law of cosines for image distance SAC and ACP: object distance
1/ 2 l R2 s R 2 2R s R cos2 Vertex o o o 1/ 2 2 2 2 li R si R 2R si R cos cos180 cos dOPL n Rs R sin n Rs Rsin 1 o 2 i 0 d 2lo 2li
n1 n2 1 n2si n1so For different P will be different l0 li R li l0 Spherical lens n n 1 n s n s 1 2 2 i 1 o l l R l l Optical 0 i i 0 axis Approximate: for small : image distance object distance cos 1 lo s0 Vertex sin li si n n n n 1 2 2 1 s0 si R The position of P is independent of the location of A over small area close to optical axis. Paraxial rays: rays that form small angles with respect to optical axis Paraxial approximation: consider paraxial rays only Spherical lens: focal length
n1 n2 n2 n1 object so si R (or first) Focal point F0 : si = focus n n n 1 0 2 1 fo R
First focal length: n1 fo R (object focal length) n2 n1 image (or second) Reverse: focus
Second focal length: n2 fi R (image focal length) n2 n1
R > 0, n2 > n1 f > 0 - converging lens Spherical lens: focal length What if R is negative?
n1 fi R n2 n1 an image is virtual: it appears on object side
n1 fo R n2 n1 an object is virtual: it appears to be in the image side Sign convention
Optical n1 n2 n2 n1 axis so si R image distance object distance
Vertex Assume light entering from left:
so, fo + left of V si, fi + right of V R + if C is right of V Lens classification
Thinner in Thicker in the middle the middle
(negative) (positive)
R1<0 R2= R1<0 R2>0 R1>0 R2>0 n n n n Thin lens equation 1 2 2 1 s0 si R For the first surface: n n n n m l l m so1 si1 R1 Second surface: n n n n l m m l si1 d si2 R2
Add two eq-ns and simplify using nm=1 (air) and d0: Thin-lens equation (Lensmaker’s formula) 1 1 1 1 nl 1 so si R1 R2 Gaussian lens formula
1 1 1 1 nl 1 s s R R xo xi o i 1 2 ff s0 si 1 1 1 Find focal lengths (so, or si) n 1 f l R R fo = fi f 1 2 Gaussian lens formula:
1 1 1 2 Newtonian form: xo xi f so si f
This is one of the most widely used equations. All one needs to know about the lens is its focal length. Example
R = 50 mm n = 1.5 Plano-convex spherical lens
What is a focal length of this lens? Solution 1 1 1 1 1 nl 1 1.5 1 1/100 mm f R1 R2 50 mm
f 100 mm Example
f= 100 mm
Object is placed at 600 mm, 200 mm, 150 mm, 100 mm, 50 mm. Where would be the image? 1 1 1 s s Solution o i 600 120 so si f 200 200 s f 150 300 s o i 100 so f 80 -400 Focal plane
second, or back focal plane Thin lens + paraxial approximation: All rays that go through center O do not bend
All bundles of parallel rays converge to focal points that lay on one plane: second, or back focal plane
Fo- lies on first, or front focal plane. Imaging with a lens
Each point in object plane is a point source of spherical waves and the lens will image them to respective points in the image plane. Converging lens: principal rays
F i Optical Image Object axis Fo
Principal rays:
1) Rays parallel to principal axis pass through focal point Fi. 2) Rays through center of lens are not refracted.
3) Rays through Fo emerge parallel to principal axis. In this case image is real, inverted and enlarged Assumptions: Since n is function of l, in reality each color has different focal point: • monochromatic light chromatic aberration. Contrast to • thin lens. mirrors: angle of incidence/reflection • rays are all “near” the principal axis not a function of l (paraxial).