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

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

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

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

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. f > 0 If f < 0, the lens deflects rays away from the axis. f < 0 R 1 > 0 R 2 < 0 R 1 < 0 R 2 > 0

A lens focuses parallel rays to a point one focal length away. f f At the focal plane, all rays converge to the z axis ( x out = 0 ) independent of input position. Parallel rays at a different angle focus at a different x out. A lens followed by propagation by one focal length: Assume all input rays have  in = 0 For all rays x out = 0 ! Looking from right to left, rays diverging from a point are made parallel.

Consecutive lenses f1f1 f2f2 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. When B = 0, all rays from a point x in arrive at a point x out, independent of angle. x out = A x in When B = 0, A is the magnification.

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

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

magnification Linear or transverse magnification — For real images, such as images projected on a screen, size means a linear dimensionreal images Angular magnification — For optical instruments with an eyepiece, the linear dimension of the image seen in the eyepiece (virtual image in infinite distance) cannot be given, thus size means the angle subtended by the object at the focal point (angular size). Strictly speaking, one should take the tangent of that angle (in practice, this makes a difference only if the angle is larger than a few degrees). Thus, angular magnification is defined asoptical instrumentseyepiecevirtual imageangular sizetangent

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. Out-of-focus plane Focal plane Object Image Size of blur in out-of-focus plane Aperture The smaller the aperture, the more the depth of field. Depth of Field

Gaussian Beams

The F-number, “ f / # ”, of a lens is the ratio of its focal length and its diameter. f / # = f / d f f d1d1 f f d2d2 f / # = 1f / # = 2 Large f-number lenses collect more light but are harder to engineer. F-number

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(  ) High-numerical-aperture lenses are bigger. f  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.

Numerical Aperture The maximum angle at which the incident light is collected depends on NA Remember diffraction

Numerical Aperture The maximum angle at which the incident light is collected depends on NA Remember diffraction

Scattering of Light What happens when particle size becomes less than the wavelength? Rayleigh scattering : –The intensity I of light scattered by a single small particle from a beam of unpolarized light of wavelength λ and intensity I 0 is given by: – –where R is the distance to the particle, θ is the scattering angle, n is the refractive index of the particle, and d is the diameter of the particle.refractive index

Scattering of Light What happens when particle size becomes less than the wavelength? MIE scattering : –Use a series sum to calculate scattered intensity at arbitrary particle diameter Sphere Diameter0.10microns Refractive Index of Medium1.0 Real Refractive Index of Sphere1.5 Imaginary Refractive Index of Sphere0 Wavelength in Vacuum0.6328microns Concentration0.1spheres/micron 3

MIE scattering Sphere Diameter0.2microns Refractive Index of Medium1.0 Real Refractive Index of Sphere1.5 Imaginary Refractive Index of Sphere0 Wavelength in Vacuum0.6328microns Concentration0.1spheres/micron 3

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 = - d i / d o So use d i << d o for both lenses. Note that this is easy for the first lens, as the object is really far away! M1M1 M2M2 Image plane #1 Image plane #2 Keplerian telescope

Telescope Terminology

Micro- scopes M1M1 M2M2 Image plane #1 Image plane #2 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| = d i / d o = (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 Object To eyepiece

Example: Magnifying Glass