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Phys 322 Lecture 12 Chapter 5

Geometrical Geometrical optics ( optics) is the simplest version of optics.

Ray optics Ray Optics

axis

We'll define 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 and rays Light wavefronts are  to rays (isotropic media) Group of rays form congruence if one can find a surface that is to every ray. Theorem of Malus and Dupin: a group of light rays will preserve its normal congruence after any number of reflections and 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 (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 a beam to a point and obtain infinitely good spatial resolution. Not true. The smallest possible focal spot is the > , . Same for the best spatial resolution of an image. This is due to , 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: SP P and S are conjugate points object space image space - 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 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 Converging lenses

Concave lens Diverging lens

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  cos180   cos dOPL n Rs  R sin n Rs  Rsin  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 d0: 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 . Contrast to • thin lens. : angle of incidence/ • rays are all “near” the principal axis not a function of l (paraxial).