Geometrical Optics Prof. Rick Trebino, Georgia Tech Geometrical light rays Ray matrices and ray vectors Matrices for various optical components The Lens Makers Formula Imaging and the Lens Law Mapping angle to position Cylindrical lenses Aberrations The Eye Ray optics axis We'll define light rays as directions in space, corresponding, roughly, to k-vectors of light waves. 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. Is geometrical optics the whole story? No. It neglects the phase. Also, the ray picture implies that we could focus a beam to a point with zero diameter and so obtain infinite intensity and infinitely good spatial resolution. Not true. The smallest possible focal spot is actually about the wavelength, . Same for the best spatial resolution of an image. This is fundamentally due to the wave nature of light, which has is not included in geometrical optics. ~0 > ~

Geometrical optics (ray optics) is the simplest version of optics. Ray optics The optic axis A mirror deflects the optic axis into a new direction. This ring laser has an optic axis that scans out a rectangle. Optic axis A ray propagating through this system We define all rays relative to the relevant optic axis. Choosing the optic axis We always try to choose the optic axis to make the problem as simple as possible. Fortunately, we have the freedom to do so. Here, the beam propagates back and forth inside a laser, so we can use two different coordinate systems, one for the beam

propagating to the right with z increasing to the right, and another for the beam propagating to the left with z increasing to the left. z 0 0 z The ray vector xin, in xout, out A light ray can be defined by two co-ordinates: its position, x ra y l a

c i t op its slope, x Optic axis These parameters define a ray vector, which will change with distance and as x Ray matrices For many optical components, we can define 2 x 2 ray matrices. An elements effect on a ray is found by multiplying its ray vector.

Optical system 2 x 2 Ray matrix xin in A C B D Ray matrices can describe simple and complex systems. xout out These matrices are often (uncreatively) called ABCD Matrices.

Ray matrices as derivatives Since the displacements and angles are assumed to be small, we can think in terms of partial derivatives. spatial magnificati on xout xin xout xout xout xin in xin

in out out out xin in xin in xout in xout A B xin C D in out out xin out

in angular magnificatio n We can write these equations in matrix form. For cascaded elements, we simply multiply ray matrices. xin in O1 O2 O3

xout out xin xout xin O3 O2 O1 O3 O2 O1 out in in Notice that the order looks opposite to what it should be, but it makes sense when you think about it. Ray matrix for free space or a medium If xin and in are the position and slope upon entering, let xout and out be the position and slope after propagating from z = 0 to z. xout, out xin, in z

z=0 Ospace 1 z = 0 1 xout xin z in out in Rewriting these expressions in matrix notation: xout 1 z xin 0 1 in out

Ray matrix for an interface At the interface, clearly: out in xout = xin n1 Now calculate out: Snell's Law says: xin xout n2 n1 sin(in) = n2 sin(out) which becomes for small angles: n1 in = n2 out

out = [n1 / n2] in Ointerface 0 1 0 n / n 1 2 Ray matrix for a curved interface At the interface, again: xout = xin. R s

To calculate out, we must calculate 1 and 2. If s is the surface slope at the height xin, then 1 xin in n1 out s 2 s = xin /R n2 z

1 = in+ s and 2 = out+ s 1 = in+ xin / R and 2 = out+ xin / R Snell's Law: n1 1 = n2 2 n1 (in xin / R ) n2 ( out xin / R) out (n1 / n2 )(in xin / R) xin / R out (n1 / n2 )in (n1 / n2 1) xin / R 1 0 Ocurved ( n / n 1)

/ R n / n interface 1 2 1 2 A thin lens is just two curved interfaces. Well neglect the glass in between (its a really thin lens!), and well take n1 = 1. Ocurved interface 1 0 ( n

/ n 1) / R n / n 1 2 1 2 R1 n=1 R2 n1 n=1 1 0

1 0 Othin lens Ocurved Ocurved (1/ n 1) / R 1/ n ( n 1) / R n interface 2 interface 1 2 1 1 0 1 0

( n 1) / R n (1/ n 1) / R n (1/ n )

( n 1) / R (1 n ) / R 1 2 1 2 1 1 0

( n 1)(1/ R 1/ R ) 1 2 1 This can be written: where: 1/ f ( n 1)(1/ R1 1/ R2 ) 1

1/ f 0 1 The Lens-Makers Formula Ray matrix for a lens 1/ f ( n 1)(1/ R1 1/ R2 ) Olens 1 = -1/f 0 1 The quantity, f, is the focal length of the lens. Its the single most important parameter of a lens. It can be positive

or negative. In a homework problem, youll extend the Lens Makers Formula to lenses of greater thickness. R1 > 0 R2 < 0 f>0 If f > 0, the lens deflects rays toward the axis. R1 < 0 R2 > 0 f<0 If f < 0, the lens deflects rays away from the axis. Types of lenses Lens nomenclature

Which type of lens to use (and how to orient it) depends on the aberrations and application. A lens focuses parallel rays to a point one focal length away. For all rays A lens followed by propagation by one focal length: xout = 0! f xin 0 xout 1 f 1 0 xin 0 0 1 1/ f 1 0 1/ f 1 0 x / f in out f f Assume all input rays have in = 0

At the focal plane, all rays converge to the z axis (xout = 0) independent of input position. Parallel rays at a different angle focus at a different xout. Looking from right to left, rays diverging from a point are made parallel. Lenses can also map angle to position. From the object to the image, we have: 1) A distance f 2) A lens of focal length f 3) A distance f xout 1 0 out 1 0

f 1 0 1 f xin 1 1/ f 1 0 1 in f 1 f xin 1 1/ f 1 in f xin f in 0 x / f 1/ f

0 in in So xout in And this arrangement maps position to angle: out xin Spectrometers To best distinguish different wavelengths, a slit confines the beam to the optic axis. A lens collimates the Camera beam, and a diffraction grating disperses the colors. A second lens focuses the beam to a

f f Entrance slit f f point that depends on its beam input angle (i.e., the wavelength). There are many types of spectrometers. But Diffraction theyre all grating based on the same principle.

Lenses and phase delay Ordinarily phase isnt considered in geometrical optics, but its worth computing the phase delay vs. x and y for a lens. All paths through a lens to its focus have the same phase delay, and hence yield constructive interference there. Equal phase delays Focus f f Lenses and phase delay ( x, y ) d First consider variation (the x and y dependence) in the

path through the lens. ( x, y ) R12 x 2 y 2 d lens ( x, y ) (n 1) k ( x, y ) lens ( x, y ) (n 1)k R12 ( x 2 y 2 ) d But: x2 y 2 R x y R1 1 ( x y ) / R R1 2 R1 2 1 2 2 2

2 2 1 lens ( x, y ) (n 1)(k / 2 R1 )( x 2 y 2 ) neglecting constant phase delays. x,y Lenses and phase delay (x,y) Now compute the phase delay in the air after the lens: Focus 0

air ( x, y ) k x 2 y 2 z 2 If z >> x, y: 2 x y x y z z 2z 2 2 2 air ( x, y ) (k / 2 z )( x 2 y 2 ) 2 z neglecting constant phase delays. lens ( x, y ) air ( x, y ) (n 1)(k / 2 R1 )( x 2 y 2 ) (k / 2 z )( x 2 y 2 )

= 0 if 1 1 (n 1) z R1 that is, if z = f ! Ray matrix for a curved mirror Consider a mirror with radius of curvature, R, with its optic axis perpendicular to the mirror: 1 in s s xin / R R out s

out 1 s (in s ) s in 2 xin / R 1 1 in xin = xout z 0 1 Omirror = 2 / R 1

Like a lens, a curved mirror will focus a beam. Its focal length is R/2. Note that a flat mirror has R = and hence an identity ray matrix. Laser cavities Mirror curvatures matter in lasers. Two flat mirrors, the flat-flat laser cavity, is difficult to align and maintain aligned. Two concave curved mirrors, the usually stable laser cavity, is generally easy to align and maintain aligned. Two convex mirrors, the unstable laser cavity, is impossible to align! Unstable resonators

The mirror curvatures determine the beam size, which, for a stable resonator, is small (100 m to 1 mm). But an unstable cavity (or unstable resonator) can be useful! In fact, it produces a large beam, useful for high-power lasers, which must have large beams. An unstable resonator can have a very large beam. But the gain must be high. And the beam has a hole in it. Consecutive lenses Suppose we have two lenses right next to each other (with no space in between). 1 Otot = 1/f 2 f1

f2 0 1 0 1 = 1 1/f1 1 1/f1 1/ f 2 0 1 1/f tot = 1/f1 + 1/f 2 So two consecutive lenses act as one whose focal length is computed by the resistive sum. As a result, we define a measure of inverse lens focal length, the diopter. 1 diopter = 1 m-1

A system images an object when B = 0. xout A 0 xin A xin C D C x D in in in out When B = 0, all rays from a point xin arrive at a point xout, independent of angle. xout = A xin When B = 0, A is the magnification. Lens Image Object f do di

The Lens Law From the object to the image, we have: Lens Image Object 1) A distance do 2) A lens of focal length f 3) A distance di 0 1 do 1 di 1 O 1/ f 1 0 1 0 1

do 1 di 1 1/ f 1 d / f 0 1 o 1 di / f do di d o di / f 1/ f 1 d / f o

f do di B d o di d o di / f d o di 1/ d o 1/ di 1/ f 0 if 1 1 1 d o di f This is the Lens Law. Imaging magnification Lens

Image Object f If the imaging condition, 1 1 1 d o di f do is satisfied, then: 1 di / f O 1/ f 0 1 do /

So: M O 1/ f 0 1/ M f di 1 1 A 1 di / f 1 di do di di M do 1 1 D 1 d o / f 1 d o

d o di do 1/ M di Magnification power Often, positive lenses are rated with a single magnification, such as 4x. Object under observation In principle, any positive lens can be used at an infinite number of possible magnifications. However, when a viewer adjusts the object distance so that the image appears to be essentially at infinity (which is a comfortable viewing distance for most individuals), the magnification is given by the relationship: Magnification = 250 mm / f Thus, a 25-mm focal-length positive lens would be a 10x magnifier.

Virtual images A virtual image occurs when the outgoing rays from a point on the object never actually intersect at a point but can be traced backwards to one. Negative-f lenses have virtual images, and positive-f lenses do also if the object is less than one focal length away. Virtual image Virtual image Object infinitely far away Object f<0 Simply looking at a flat mirror yields a virtual image.

f>0 F-number The F-number, f / #, of a lens is the ratio of its focal length and its diameter. f/# = f/d f d1 f f/# =1 f d2 f f/# =2

Large f-number lenses collect more light but are harder to engineer. Depth of field Only one plane is imaged (i.e., is in focus) at a time. But wed like objects near this plane to at least be almost in focus. The range of distances in acceptable focus is called the depth of field. It depends on how much of the lens is used, that is, the aperture. Object Image f Depth of field Only one plane is imaged (i.e., is in focus) at a time. But wed like objects near this plane to at least be almost in focus. The range of distances in acceptable focus is called the depth of field. It depends on how much of the lens is used, that is, the aperture. Out-offocus Object

f Out-of-focus plane Correct image plane In-focus object Aperture Film plane The smaller the aperture, the more the depth of field. Size of blur in out-of-focus plane Depth of field example

A large depth of field isnt always desirable. f/32 (very small aperture; large depth of field) f/5 (relatively large aperture; small depth of field) A small depth of field is also desirable for portraits. In 1932, Ansel Adams and seven photographers from the San Francisco Bay Area, including Edward Weston, formed Group f/64. The name refers to the smallest camera lens opening available on most cameras and which produces maximum depth of field. Group f/64 Ansel Adams

Rose and Driftwood San Francisco 1932 Bokeh Bokeh is the rendition of out-of-focus points of light. Something deliberately out of focus should not distract. Poor Bokeh. Edge is sharply defined. Neutral Bokeh. Evenly illuminated blur circle. Still bad because the edge is still well defined. Good Bokeh. Edge is completely undefined. Bokeh is where art and engineering diverge, since better bokeh is due to an imperfection (spherical aberration). Perfect (most appealing) bokeh is a Gaussian blur, but lenses are usually designed for neutral bokeh! Very bad bokeh The pinhole

camera If all light rays are directed through a pinhole, it forms an image with an infinite depth of field. Pinhole Image Object The concept of the focal length is inappropriate for a pinhole lens. The magnification is still di/do. The first person to mention this idea was Aristotle. With their low cost, small size and huge depth of field,

theyre useful in security applications. The Camera Obscura A nice view of a camera obscura is in the movie, Addicted to Love, starring Matthew Broderick (who plays an astronomer) and Meg Ryan, who set one up to spy on their former lovers. A dark room with a small hole in a wall. The term camera obscura means dark room in Latin. Renaissance painters used them to paint realistic paintings. Vermeer painted The Girl with a Pearl Earring (1665-7) using one.

Numerical aperture Another measure of a lens size is the numerical aperture. Its the product of the medium refractive index and the marginal ray angle. NA = n sin() f High-numerical-aperture lenses are bigger. Why this definition? Because the magnification can be shown to be the ratio of the NA on the two sides of the lens. Telescopes

Keplerian telescope Image plane #1 Image plane #2 M1 M2 A telescope should image an object, but, because the object will have a very small solid angle, it should also increase its solid angle significantly, so it looks bigger. So wed like D to be large. And use two lenses to square the effect. Oimaging Otelescope M 1/ f

0 1/ M where M = - di / do 0 M1 0 M2 1/ f 1/ M 1/ f 1/ M 2 2 1 1 M 1M 2 0

M / f M / f 1/ M M 1 2 2 1 1 2

Note that this is easy for the first lens, as the object is really far away! So use di << do for both lenses. Telescope terminology Telescopes (contd) The Galilean Telescope f1 < 0 f2 > 0 The analysis of this telescope is a homework problem! The Cassegrain telescope Telescopes must collect as much light as possible from the generally very dim objects many light-years away. Its easier to create large mirrors than

large lenses (only the surface needs to be very precise). Object It may seem like the image will have a hole in it, but only if its out of focus. The Cassegrain telescope If a 45-mirror reflects the beam to the side before the smaller mirror, its called a Newtonian telescope. No discussion of telescopes would be

complete without a few pretty pictures. Galaxy Messier 81 Uranus is surrounded by its four major rings and by 10 of its 17 known satellites NGC 6543-Cat's Eye Nebula-one of the most complex planetary nebulae ever seen Microscopes Image plane #1 Objective M1 Eyepiece Image plane #2

M2 Microscopes work on the same principle as telescopes, except that the object is really close and we wish to magnify it. When two lenses are used, its called a compound microscope. Standard distances are s = 250 mm for the eyepiece and s = 160 mm for the objective, where s is the image distance beyond one focal length. In terms of s, the magnification of each lens is given by: |M| = di / do = (f + s) [1/f 1/(f+s)] = (f + s) / f 1 = s / f Many creative designs exist for microscope objectives. Example: the Burch reflecting microscope objective: Object To eyepiece Microscope terminology If an optical system lacks cylindrical

symmetry, we must analyze its x- and ydirections separately: Cylindrical lenses A spherical lens focuses in both transverse directions. A cylindrical lens focuses in only one transverse direction. When using cylindrical lenses, we must perform two separate ray-matrix analyses, one for each transverse direction. Large-angle reflection off a curved mirror also destroys cylindrical symmetry. The optic axis makes a large angle with the mirror normal, and rays make an angle with respect to it. Optic axis before reflection tangential ray Optic axis after reflection Rays that deviate from the optic axis in the plane of incidence are called tangential. Rays that deviate from the optic axis to the plane of incidence are

called sagittal. (We need a 3D display to show one of these.) Ray matrix for off-axis reflection from a curved mirror If the beam is incident at a large angle, , on a mirror with radius of curvature, R: tangential ray Optic axis R where Re = R cos for tangential rays and Re = R / cos for sagittal rays 1 2/ R e 0

1 Aberrations Aberrations are distortions that occur in images, usually due to imperfections in lenses, some unavoidable, some avoidable. They include: Chromatic aberration Spherical aberration Astigmatism Coma Curvature of field Pincushion and Barrel distortion Most aberrations cant be modeled with ray matrices. Designers beat them with lenses of multiple elements, that is, several lenses in a row. Some zoom lenses can have as many as a dozen or more elements. Chromatic aberration Because the lens material has a different refractive index for each wavelength, the lens will have a different focal length for each wavelength. Recall the lens-makers formula: 1/ f ( ) (n( ) 1)(1/ R1 1/ R2 )

Here, the refractive index is larger for blue than red, so the focal length is less for blue than red. You can model spherical aberration using ray matrices, but only one color at a time. Chromatic aberration can be minimized using additional lenses In an Achromat, the second lens cancels the dispersion of the first. Achromats use two different materials, and one has a negative focal length. Spherical aberration in mirrors For all rays to converge to a point a distance f away from a curved mirror requires a paraboloidal surface.

Spherical surface But this only works for rays with in = 0. Paraboloidal surface Spherical aberration in lenses So we use spherical surfaces, which work better for a wider range of input angles. Nevertheless, off-axis rays see a different focal length, so lenses have spherical aberration, too. Focusing a short pulse using a lens with spherical and chromatic aberrations and group-velocity dispersion. Minimizing spherical aberration in a focus R2 R1 q R2 R1 R1 = Front surface

radius of curvature R2 = Back surface radius of curvature Plano-convex lenses (with their flat surface facing the focus) are best for minimizing spherical aberration when focusing. One-to-one imaging works best with a symmetrical lens (q = ). Spherical aberration can be also minimized using additional lenses The additional lenses cancel the spherical aberration of the first. Astigmatism

When the optical system lacks perfect cylindrical symmetry, we say it has astigmatism. A simple cylindrical lens or off-axis curved-mirror reflection will cause this problem. Model astigmatism by separate x and y analyses. Cure astigmatism with another cylindrical lens or off-axis curved mirror. Coma Coma causes rays

from an off-axis point of light in the object plane to create a trailing "comet-like" blur directed away from the optic axis. A lens with considerable coma may produce a sharp image in the center of the field, but it becomes increasingly blurred toward the edges. For a single lens, coma can be caused or partially corrected by tilting the lens. Curvature of field Curvature of field causes a planar object to project a curved (nonplanar) image. Rays at a large angle see the lens as having an effectively smaller diameter and an effectively smaller focal length, forming the image of the off axis points closer to the lens. Pincushion and barrel distortion These distortions are fixed by an

orthoscopic doublet or a Zeiss orthometer. Barrel and pincushion distortion Pincushion Barrel Photography lenses Photography lenses are complex! Especially zoom lenses. Double Gauss These are older designs. Petzval Photography lenses Modern lenses can have up to 20 elements! Canon 17-85mm

f/3.5-4.5 zoom Canon EF 600mm f/4L IS USM Super Telephoto Lens 17 elements in 13 groups $12,000 Geometrical optics terms Anatomy of the eye Incoming light Eye slides courtesy of Prasad Krishna, Optics I student 2003. The cornea, iris, and lens The cornea is a thin membrane that has an index of refraction of around 1.38. It protects the eye and refracts light (more than the lens does!) as it enters the eye. Some light leaks through the cornea, especially when its blue.

The iris controls the size of the pupil, an opening that allows light to enter through. The lens is jelly-like with an index of refraction of about 1.44. This lens bends so that the vision process can be fine tuned. When you look at an object, youre squeezing this lens, changing its focal length. The ciliary muscles bend and adjust the lens. Near-sightedness (myopia) In nearsightedness, a person can see nearby objects well, but has difficulty seeing distant objects. Objects focus before the retina. This is usually caused by an eye that is too long or a lens system that has too much power to focus. Myopia is corrected with a negative-focal-length lens. This lens causes the light to diverge slightly before it enters the eye. Near-sightedness

Far-sightedness (hyperopia) Far-sightedness (hyperopia) occurs when the focal point is beyond the retina. Such a person can see distant objects well, but has difficulty seeing nearby objects. This is caused by an eye that is too short, or a lens system that has too little focusing power. Hyperopia is corrected with a positive-focallength lens. The lens slightly converges the light before it enters the eye. Far-sightedness As we age, our lens hardens, so were less able to adjust and more likely to experience far-sightedness. Hence bifocals. Astigmatism is a common problem in the eye. How tightly can we focus a beam?

Geometrical optics predicts a focused spot of width zero. ~0 But it neglects the wave nature of light. Lets reconsider this problem in view of our knowledge of light waves. Well consider the rays in pairs of symmetrically propagating directions and add up all the fields at the focus. Beams crossing at an angle k2 x z k1

k1 k cos z k sin x k2 k cos z k sin x E1 E2 E0 exp i (t kz cos kx sin exp i (t kz cos kx sin E0 exp i (t kz cos ) exp( ikx sin ) exp(ikx sin ) E0 exp i (t kz cos ) cos(kx sin ) Etot ( x, z , t ) cos(kx sin ) I tot ( x, z , t ) cos 2 (kx sin ) Notice that the fringes are finest when /2, and the beams counterpropagate. In this case, their field has a fringe spacing of 2/k = . Fringes from the various crossed beams So lets add up all the sinusoidal electric fields from every angle : E

x E ( x, ) cos(kx sin ) cos( ax) where a = k sin The best we can do is to focus rays from all angles ( = 0 to /2), so the fringe spacing will vary from to . Or a = 0 to k. = /2 will require an infinitely big lens, but, hey, why not? The tightest focus possible So, to find the focused field in this ideal case, we integrate from a = 0 to k : a k sin(kx) sin(ax) E ( x) E ( x, a ) da cos(ax) da

0 0 x x a 0 k The focused field k E(x) Width (between zeros) = 2/k = = 0 when x = /k = /2 x The focused irradiance (|E|2) can have a width (from the half max

on one side to the half max on the other) as small as ~ /2.