OpticsI13Geometrical OpticsTheEye1

Geometrical Optics: 

Geometrical Optics Geometrical light rays Ray matrices and ray vectors Matrices for various optical components The Lens Maker’s Formula Imaging and the Lens Law Mapping angle to position Cylindrical lenses Aberrations The Eye

Ray Optics: 

Ray Optics 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.

The Optic Axis: 

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 We define all rays relative to the relevant optic axis.

The Ray Vector: 

The Ray Vector A light ray can be defined by two co-ordinates: xin, qin xout, qout its position, x its slope, q Optical axis optical ray x q These parameters define a ray vector, which will change with distance and as the ray propagates through optics.

Ray Matrices: 

Ray Matrices For many optical components, we can define 2 x 2 ray matrices. An element’s effect on a ray is found by multiplying its ray vector. Ray matrices can describe simple and com- plex systems. These matrices are often called ABCD Matrices. Optical system ↔ 2 x 2 Ray matrix

Ray matrices as derivatives: 

Ray matrices as derivatives We can write these equations in matrix form. Since the displacements and angles are assumed to be small, we can think in terms of partial derivatives.

For cascaded elements, we simply multiply ray matrices.: 

For cascaded elements, we simply multiply ray matrices. Notice that the order looks opposite to what it should be, but it makes sense when you think about it. O1 O3 O2

Ray matrix for free space or a medium: 

Ray matrix for free space or a medium If xin and qin are the position and slope upon entering, let xout and qout be the position and slope after propagating from z = 0 to z. Rewriting these expressions in matrix notation:

Ray Matrix for an Interface: 

Ray Matrix for an Interface At the interface, clearly: xout = xin. Now calculate qout. Snell's Law says: n1 sin(qin) = n2 sin(qout) which becomes for small angles: n1 qin = n2 qout Þ qout = [n1 / n2] qin

Ray matrix for a curved interface: 

Ray matrix for a curved interface At the interface, again: xout = xin. To calculate qout, we must calculate q1 and q2. If qs is the surface slope at the height xin, then q1 = qin+ qs and q2 = qout+ qs If R is the surface radius of curvature, the surface z coordinate will be:

Ray matrix for a curved interface (cont’d): 

Ray matrix for a curved interface (cont’d) Now the output angle depends on the input position, too. Snell's Law: n1 q1 = n2 q2 q1 = qin+ xin / R and q2 = qout+ xin / R

A thin lens is just two curved interfaces.: 

A thin lens is just two curved interfaces. We’ll neglect the glass in between (it’s a really thin lens!), and we’ll take n1 = 1. This can be written: The Lens-Maker’s Formula where:

Ray matrix for a lens: 

Ray matrix for a lens The quantity, f, is the focal length of the lens. It’s the single most important parameter of a lens. It can be positive or negative. In a homework problem, you’ll extend the Lens Maker’s Formula to lenses of greater thickness. If f > 0, the lens deflects rays toward the axis. If f < 0, the lens deflects rays away from the axis. R1 > 0 R2 < 0 R1 < 0 R2 > 0

Types of lenses: 

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.: 

A lens focuses parallel rays to a point one focal length away. 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. A lens followed by propagation by one focal length: Assume all input rays have qin = 0 Looking from right to left, rays diverging from a point are made parallel.

Spectrometers: 

Spectrometers f f Entrance slit Diffraction grating f Camera To best distinguish different wave- lengths, a slit confines the beam to the optic axis. A lens collimates the beam, and a diffraction grating disperses the colors. A second lens focuses the beam to a point that depends on its beam input angle (i.e., the wavelength). q  l-l0 There are many types of spectrom- eters. But they’re all based on the same principle.

Ray Matrix for a Curved Mirror: 

Ray Matrix for a Curved Mirror 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. Consider a mirror with radius of curvature, R, with its optic axis perpendicular to the mirror:

Slide18: 

Laser Cavities 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! Mirror curvatures matter in lasers.

Slide19: 

An unstable cavity (or unstable resonator) can work if you do it properly! In fact, it produces a large beam, useful for high-power lasers, which must have large beams. Unstable Resonators

Consecutive lenses: 

Consecutive lenses Suppose we have two lenses right next to each other (with no space in between). 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.: 

A system images an object when B = 0. 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.

The Lens Law: 

The Lens Law From the object to the image, we have: 1) A distance do 2) A lens of focal length f 3) A distance di This is the Lens Law.

Imaging Magnification: 

Imaging Magnification If the imaging condition, is satisfied, then: So:

Magnification Power: 

Magnification Power Often, positive lenses are rated with a single magnification, such as 4x. 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.

Slide25: 

Virtual Images When the object is less than one focal length away from a lens, no image occurs, but a virtual image is said to occur if you look back through the lens. Object f > 0

Slide26: 

f It depends on how much of the lens is used, that is, the aperture. Only one plane is imaged (i.e., is in focus) at a time. But we’d 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. Object Image Size of blur in out-of-focus plane Aperture The smaller the aperture, the more the depth of field. Depth of Field

Depth of field example: 

Depth of field example f/32 (very small aperture; large depth of field) f/5 (relatively large aperture; small depth of field) A large depth of field isn’t always desirable. A small depth of field is also desirable for portraits.

Bokeh: 

Bokeh Poor Bokeh. Edge is sharply defined. Good Bokeh. Edge is completely undefined. Neutral Bokeh. Evenly illuminated blur circle. Still bad because the edge is still well defined. Bokeh is the rendition of out-of-focus points of light. Something deliberately out of focus should distract. Bokeh is where art and engineering diverge, since better bokeh is due to an imperfection (spherical aberration). Perfect bokeh is a Gaussian blur, but lenses are usually designed for neutral bokeh!

The pinhole camera: 

The pinhole camera You can make an entire room into a camera this way by cutting a small hole in a wall and looking at the opposite wall. This is called the camera obscura. If all light rays are directed through a pinhole, it forms an image with an infinite depth of field. The first person to mention this idea was Aristotle. The concept of the focal length is inappropriate for a pinhole lens. The magnification is still –di/do.

Slide30: 

The F-number, “f / #”, of a lens is the ratio of its focal length and its diameter. f / # = f / d f f d1 f / # = 1 f / # = 2 Large f-number lenses collect more light but are harder to engineer. F-number

Slide31: 

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

Lenses can also map angle to position.: 

So And this arrangement maps position to angle: 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

Telescopes: 

Telescopes 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 we’d like D to be large. And use two lenses to square the effect. where M = - di / do So use di << do for both lenses. Note that this is easy for the first lens, as the object is really far away! Keplerian telescope

Slide34: 

Telescope Terminology

Telescopes (cont’d): 

Telescopes (cont’d) The Galilean Telescope The analysis of this telescope is a homework problem!

The Cassegrain Telescope: 

The Cassegrain Telescope Telescopes must collect as much light as possible from the generally very dim objects many light-years away. It’s easier to create large mirrors than large lenses (only the surface needs to be very precise). It may seem like the image will have a hole in it, but only if it’s out of focus.

The Cassegrain Telescope: 

The Cassegrain Telescope If a 45º-mirror reflects the beam to the side before the smaller mirror, it’s called a Newtonian telescope.

No discussion of telescopes would be complete without a few pretty pictures.: 

No discussion of telescopes would be complete without a few pretty pictures. NGC 6543-Cat's Eye Nebula-one of the most complex planetary nebulae ever seen Galaxy Messier 81 Uranus is surrounded by its four major rings and by 10 of its 17 known satellites

Slide39: 

Micro-scopes M1 M2 Image plane #1 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, it’s 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 Eye- piece Many creative designs exist for microscope objectives. Example: the Burch reflecting microscope objective: Objective

If an optical system lacks cylindrical symmetry, we must analyze its x- and y-directions separately: Cylindrical lenses: 

If an optical system lacks cylindrical symmetry, we must analyze its x- and y-directions 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.: 

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. 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.) tangential ray

Ray Matrix for Off-Axis Reflection from a Curved Mirror: 

Ray Matrix for Off-Axis Reflection from a Curved Mirror If the beam is incident at a large angle, q, on a mirror with radius of curvature, R: where Re = R cosq for tangential rays and Re = R / cosq for sagittal rays tangential ray q

Aberrations: 

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 can’t 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: 

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-maker’s formula: 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: 

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: 

Spherical Aberration in Mirrors For all rays to converge to a point a distance f away from a curved mirror requires a paraboloidal surface. But this only works for rays with qin = 0.

Spherical Aberration in Lenses: 

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.

Minimizing spherical aberration in a focus: 

Minimizing spherical aberration in a focus 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: 

Spherical aberration can be also minimized using additional lenses The additional lenses cancel the spherical aberration of the first.

Astigmatism: 

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. Cure astigmatism with another cylindrical lens or off-axis curved mirror. Model astigmatism by separate x and y analyses.

Coma: 

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 become 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 Curvature of field causes a planar object to project a curved (non-planar) 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: 

Pincushion and Barrel Distortion These distortions are fixed by an “orthoscopic doublet” or a “Zeiss orthometer.”

Barrel and pincushion distortion: 

Barrel and pincushion distortion Barrel Pincushion

Photography lenses: 

Photography lenses Photography lenses are complex! Especially zoom lenses. These are older designs.

Photography lenses: 

Photography lenses Modern lenses can have up to 20 elements! Canon EF 600mm f/4L IS USM Super Telephoto Lens 17 elements in 13 groups $12,000 Canon 17-85mm f/3.5-4.5 zoom

Geometrical Optics terms: 

Geometrical Optics terms

Anatomy of the Eye: 

Anatomy of the Eye Eye slides courtesy of Prasad Krishna, Optics I student 2003. Incoming light

The cornea, iris, and lens: 

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 it’s blue. The iris controls the size of the pupil, an opening that allows light to enter through. The lens is jelly-like lens with an index of refraction of about 1.44. This lens bends so that the vision process can be fine tuned. When you squint, you are bending this lens and changing its properties so that your vision is clearer. The ciliary muscles bend and adjust the lens.

Near-sightedness (myopia): 

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) 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-focal-length lens. The lens slightly converges the light before it enters the eye. Far-sightedness As we age, our lens hardens, so we’re less able to adjust and more likely to experience far-sightedness. Hence “bifocals.”

Astigmatism is a common problem in the eye.: 

Astigmatism is a common problem in the eye.

Is geometrical optics the whole story?: 

Is geometrical optics the whole story? No. We neglected the phase. 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, l. Same for the best spatial resolution of an image. This is due to diffraction, which has not been included in geometrical optics.