Final Review: some of your questions

1. Final Review: some of your questions • Could you touch on the Cornu spiral and how to use it again? • If possible, could you do a ray diagram for the Aperture stop /field stop/exit pupil/chief ray? • When we did the internal reflection that was the technology behind the large computer board, how did that work again?  Did it depend on the indices of refraction (ie sandwich/pizza) or was it strictly due to the properties of the light that was propagating? • Does beam spreading necessarily relate to the airy disc?  I know the airy disc deals with the central maximum of a Fraunhoefer diffraction pattern, so can the two relate if there is no diffraction pattern, and just a beam of light?

2. Your Questions (continued) • I am confused about how we use the Cornu Spiral in application. Can we do an example?  • How can we tell if light is right or left circularly polarized?  • What does "optical activity" mean and how does it affect polarization? • How does k vary with omega in a medium with a wavelength-dependent refractive index? • Can we do an example or two of finding the aperture stop, entrance window, exit window, entrance pupil, and exit pupil? 

3. Your Questions (continued) • For problem 11-22 on the homework I was wondering why does unit analysis seem to fail? • Could we go over how to effectively use the Cornu Spiral? • Could we work a problem of double refraction and the conditions for getting extraordinary rays and ordinary rays and how the indices of refraction relate to this? • What types of problems do we need to know from earlier tests? (just an overview would be nice)

4. Your Questions (continued) • I have some problems from old review sheets and tests I'd like to see worked out: • The question from review sheet one where you need to prove that for a convex thin lens, the minimum distance is always 4f. • Problem 18-12 from Pedrotti • The question from review sheet 2 where you have to calculate the force exerted on a solar panel, given wavelength, irradiance, and area of the panel. • A question from exam 2 on finding where the electric field drops by a factor 1/e given a complex index. • Another question from exam 2 where given a beam and knowing power, radius, and the mass of a sphere the beam is focused on you must calculate the pressure on the sphere. • The problem from review sheet 3 where you have a monochromatic source of light above a mirror produces an interference pattern and you have to i.d. the location of the virtual source, explain the presence of a dark fringe, state the condition for the path difference to produce the mth maximum, and calculate the wavelength given the location of the 3rd fringe. • A question from Exam 3 where you are underwater and see a circle of light of a given radius and want to know how deep you are. • The first workout from exam 3 where you have an argon laser fired down a column and you have to calculate the index of the unknown gas the column fills up with. • How a wire works with a cornu spiral.  I was able to make very little sense of that from the book. • The problem on the final review sheet where you are given a fringe pattern and have to determine the separation of slits. • Sorry that it's a lot, and I'm sure these won't all get answered, but these are the problems I would most like to see worked out at tonight's review.

5. Geometrical Optics (2) Fermat’s Principle Huygen’s Principle Refraction Stigmatic Imaging Aspheric Lenses Thin Lenses Cylindrical Lenses Paraxial Approx Gaussian Thin Lens Equation Thick Lenses Optical Power Lens Maker’s Equation Focal Length Magnification Real Images/Objects Virtual Images/Objects Magnifying Power Orientation Aberrations Optical Systems f/# and NA Eyepieces Telescopes Stops Marginal Ray Pupils Optical Instruments Microscopes Windows Chief Rays Matrix Methods (18) The Eye Magnifiers Marginal Ray Cameras Reflection Planar Mirrors Spherical Mirror Equation

6. Dispersion 25 Michelson Interferometer Penetration Depth Fringe Visibility Reflection from Metals Lorentz Atom Absorption Coefficient Reflectance Interference 7-8 Skin Depth Plasma Frequency Total internal reflection Young’s Double Slit Phase Changes Brewster’s Angle Thin Films Euler’s relationship Evanescent Wave Fresnel Equations Critical Angle Multilayer Dielectrics Complex analysis Fabry Perot Interferometer TE and TM Frustrated TIR Transmittance Wave Equations Harmonic Waves Diffraction-Limited Resolution Momentum Density Cornu Spiral Maxwell’s Eq EM Waves Conservation of Energy Wave Optics Plane Waves Fresnel Zones Radiation Pressure Intensity Cylindrical Waves Circular Apertures Spherical Waves Fresnel Rayleigh Scattering Fiber Optics Diffraction 11 - 13 Fraunhofer Superposition Malus’ Law Scattering Single Slit Group Velocity Double Slit Standing Waves Beats Dichroism Elliptical Reflection Many Slits Phase Velocity Ordinary Incoherent Sources Diffraction Grating Polarization Coherent Sources Wave plates Unpolarized Babinet’s Principle Linear Circular Fourier Analysis Coherence Extraordinary Birefringence Periodic Functions Optical Activity Temporal Spatial Aperiodic Functions Rotary Dispersion

7. Chapter 8 Review: Polarization • By adding two waves together we can have:

8. Polarization Review • Several producing polarized light • Reflection • Brewster’s Angle • Total Internal reflection (Fresnel Rhomb) • Dichroism • Selective absorption of one of two polarization components: • Wire grids • Polaroids • Birefringence • Some materials have two distinct indices of refraction on orthogonal axes leading to a retardation of one component of E. • Calcite, Quartz etc • Circular birefringence using the optical activity of left or right handed isomers

9. Full, Half and Quarter Waveplates Full wave plate • By choosing the thickness of the crystal, a phase shift can be chosen for a design wavelength • Full Wave Plate • No effective phase shift at the design wavelength • Linear → Linear • Half wave plate • Causes a 180° phase delay in the slower component • Causes an inversion of the polarization about the fast axis • Linear → Linear • Quarter wave plate • Causes a 90° phase delay in the slower component • Linear → Elliptical or Circular ! Half wave plate Quarter wave plate

10. out A Example • Determine the polarization of natural light that passes through a linear polarizer oriented at 45° to the fast axis of a quarter wave plate. • After passing through the polarizer • Slow axis lags by 90 ° y,slow in x ,fast Left Circular Polarization The polarization vector “bends” toward the slow axis

11. y,slow in out A x ,fast Right Circular Polarization • What would we change to get Right Circular polarization instead ? The polarization vector bends toward the slow axis

12. Example • Left Circularly Polarized light is incident on a quarter wave plate with a horizontal fast axis. What is the polarization of the transmitted light? • The y-component already lags by 90 degrees • After the QWP it will lag by 180 degrees • Linearly polarized at -45 degrees • What would happen if we switched the fast and slow axes? • It would be linearly polarized at +45 degrees y,slow in A x ,fast out

13. What happens when left circularly polarized light is passed through a right circular polarizer composed of a linear polarizer oriented at 45 degrees to a QWP? No Light is emitted The emitted light is left circularly polarized The emitted light is right circularly polarized The emitted light is linearly polarized The emitted light is unpolarized Try this one…

14. RCP LCP Antireflection Polarizer Screens Natural light is incident • Assume the polarizer and QWP act as a right circular polarizer for natural light incident on the screen • How does the polarization state change upon reflection? • The electric field continues to rotate in the same direction • The k-vector is inverted, • Looking down the beam, you see a a reversal in the rotation of light • So LCP light is reflected off of the screen • LCP light incident on the QWP produces linearly polarized light again, but the polarization axis is now perpendicular to the transmission axis of the polarizer • No reflected glare, but natural light emitted from the monitor passes slow Reflective Screen No reflection! Linear QWP

15. Circular Birefringence Demonstration Some materials have the ability to rotate the polarization of light without absorbing it: • Examples: Sodium Chlorate, Sugar, Crystallized Quartz • “Dextrorotary” (rotates to the right) • “Levorotatory” (rotates to the left) • Degree of rotation depends on the length of the medium (more rotation with increasing length) • Degree of rotation depends on the wavelength of light (blue light is rotated more than red)

16. Circular Birefringence • The material possess two distinct indices of refraction for left and right polarized light states • By slowing down one component relative to the other, the resulting linear polarization is shifted continuously with length of the material • This only occurs with crystal structures and molecular isomers that have a particular handedness • Right or left handed helices are an example • The rotory power is defined to be the change in angle per length of material The rotary power is greater for smaller wavelengths

17. Review of diffraction • Secondary wavelets leaving the aperture can interfere constructively or destructively to produce a diffraction pattern • Pattern characteristics depend on placement of source, detector, and aperture mask • Far Field or near? • Far = Fraunhofer • Planar wavefonts at aperture • Fourier transform of aperture • Near = Fresnel • Spherical wavefronts • Fresnel half-zones • Cornu Spiral, each term is length of line2 • Off axis points, measure from line of sight in aperture

18. Summary of Fresnel Diffraction • This expression determines the field at a point (0,0) (u=0, v=0) for an aperture extending from umin→ umax, and vmin→ vmax • (Length of line)2 = • To find E or I at xp,yp just shift • Compute E or I using new umin,umax (umax) (umin)

19. Another Fresnel Example Fresnel diffraction is observed behind a a long horizontal slit of width 3.7 mm thick, which is placed 2 m from the light source and 3 m from the screen. If light of wavelength 630 nm is used, compute, using the Cornu Spiral, the irradiance of the diffraction pattern on the axis at the screen. Express the answer as some number times the unobstructed irradiance there. 3 m 2 m 3.7 mm

20. Solution • r’0= 2 m, r0= 3 m • L=1.2 m Horizontal slit

21. The Cornu Spiral Length2=((0.4963+0.4963)2 + (0.6058+0.6058)2=2.453 Length2=((0.5+0.5)2 + (0.5+0.0.5)2=2 • S(3.0)=0.4963 • C(3.0)=0.6058 • S(-3.0)=-0.4963 • C(-3.0)= -0.6058 • Length2=2.453 • S(inf)=0.5 • C(inf)=0.5 • S(-inf)=-0.5 • C(-inf)= -0.5 • Length2=2

22. How would this work for a wire? 3 m 2 m 3.7 mm Compute as before, but Use the sum of these lengths Above the wire Length2=0.0224 Below the wire

23. Babinet’s Principle The diffraction pattern of a hole is the same as that of its opposite! Holes Neglecting the center point: Anti-Holes

24. Focal points of a Zone • A 500 nm plane wave is incident on a 5 mm diameter circular slit which acts a Fresnel Lens. Determine locations on the optical axis where the light diffracted by the slit reaches a maximum value. Zone Radius = Rn When rn Rn r0 so Focal points at odd n There are more zones in a fixed radius aperture as you move closer (r0 decreasing), so the intensity will go through maxima and minima.

25. Example: Fraunhofer • Plane waves illuminate a square ring • Determine an expression for the intensity distribution on distant screen

26. Another random example • No lens can focus light down to a perfect point. Estimate the minimum spot size expected at the focus of lens. Discuss the relationship among the focal length, the lens diameter, and the spot size.

27. The CD A compact disc bears information encoded in the form of pits arranged along a long spiral track on a plastic disc. The spot of light so formed covers only one turn of the spiral track, and can resole the presence or absence of a pit along the track. The wavelength is 0.78 m and the light is focused by a lens into a cone of semi-angle 27 degrees. The track runs from radius 25 mm to 58 mm. Estimate the number of bits that can be held on a CD and be ead by the laser. Check your estimate by finding the playing time of a music CD, given that information is read off it at 4.32 Mbit s-1

28. Example: Fresnel • Plane waves of wavelength 624 nm impinge normally on a circular aperture 2.09 mm in radius. The diffraction pattern is viewed on a screen 1 m from the hole. Is the spot bright or dark? Explain

29. Example: Fresnel • A plane wave (500 nm) of irradiance 20 W/m2 is normally incident on a square aperture 2 mm on a side. The resulting diffraction pattern is measured on a ccd camera place 4 m from the screen. • What irradiance will it measure 0.1 mm to the right of dead center? • How does this compare to the irradiance of the central axial point? • Use the Cornu Spiral, provided to answer this question!

30. target 1 cm Question 3 Exam 3 • A laser beam is centered on a target and the power is measured to be 10 mW (dashed line). When a 1 cm thick glass window (n=1.6) is inserted in the path of the beam such that the angle of incidence is 50 degrees, the beam is observed to shift relative to the target. The front face of the window is 0.5 m from the target. The electric field oscillates in the plane of the paper, as shown below. TM! 1cm θt y L Above target!

31. Question 4, Exam 2 How much power will be in the reflected and transmitted beams? First interface (θi=50, θt=28.6, ni=1,nt=1.6) Second interface (θi=28.6, θt=50, ni=1.6,nt=1) r2=r1

32. Exam 1 Review: Some of your questions • virtual objects • I had some issues getting the right matrix numbers on the homework. I used maple and tried the matrix by hand but couldn't get the same answer as the book. Would it be possible to address the problem in class? • determining location and size of pupils and windows. calculating amount of light transmitted. aperture stops, field stops locations etc... reduction of abberations. • Is it possible to get a table defining the sign conventions so that we can be sure that we have them strait. I am always second guessing myself and it would be nice to put this all to rest. For some reason it is giving me memory problems if you know what I mean. • Thick lens problems (non matrix). Sign conventions and differences for mirrors/lenses. • cameras and matrices • the Matrix. magnification, on the homework I was never able to get it the first time, it seems to take my random guessing to get the answer in the back...must be magic. • Could we specifically do a problem that involves using matrices? • I would like for you to discuss cameras, specifically the many the camera. The Matrices would also be appreciated. • If you could do a multiple lens example. • Review sign conventions for Gaussian equation and for the thick lens/optical system points. A more complicated ray tracing problem like 2-22 in the text. • Cameras and telescopes. • I would like to see a free body diagram involving virtual objects. are we going to get all of our hw back before the test? • can we have an extra credit question dealing with a fable? • can we go over one more time how to determine aperture stops, field stops, entrance exit windows and pupils? • in the last hw assignment 18-14 was confusing for the telescope because it has 3 lenses where the equations only deal with two, how do you change it? • The matrix thing is confusing more then everything else combined. Also, I am interested in exactly what kind of equation sheet we will be allowed to have. Are we able to see it before hand?

33. Review of Chapter 1: Introduction, wave particle duality • The wave and particle aspects of light have both been firmly established from experiments conducted over the last 400 years • Light behaves like a wave when propagating, and with interference and diffraction • When exchanging energy with matter, it behaves a particle • Light is an electromagnetic wave that has a frequency range which the eye can detect • We can characterize the energy content of EM waves through various radiometic quantities such as • Radiant energy (J) • Radiant energy density (J/m3) • Radiant power (W) • Radiant exitance, Irradiance (W/m2) • Radiant intensity (W/sr) • Radiance (W/sr m2)

34. Review of Chapter 2: geometrical optics • Geometrical optics treats light as though it were a particle or a ray (=0). Rays propagate in straight lines • Law of reflection and refraction • Fermat’s principle says that because nature is efficient light will take a path that minimizes the time of transit between two points of interest • Stigmatic imaging = Point to Point • The Cartesian Oval is a perfect imaging surface. The OPL is the same for every point regardless of path between conjugate object and image points • Spherical surfaces also show this property for rays that have small angles of incidence (paraxial approximation) • We developed equations for single spherical interface, and a thin lens (gaussian thin lens equation) that relates conjugate object and images points • Images are describe according to their orientation and magnification. Real images can be viewed on a physical screen, while virtual images must be viewed through a lens • Conjugate points can also be determined by three-ray diagrams • The lensmakers equation describes the focal length of a thin lens • The focal length for a mirror is f = - R / 2 • Remember the sign conventions! • The optical power of an interface, lens or mirror is measured in diopters

35. Review of Chapter 3: Optical instrumentation • The intensity of light passing through an optical system is controlled by physical aperture stops, and the pupils they produce • AS is the limiting aperture for a cone of rays extending from object to image • Entrance pupil is the image of the AS looking in from the object point • Exit pupil is the image of the AS looking from the image point • The field of view of an optical instrument is controlled by physical field stops, and the windows that they produce. • FS is the element that subtends the smallest angle from the center of the Entrance pupil • Entrance window is the image of the field stop formed by preceding elements • Exit window is the image of the field stop by elements following it • The amount of light collected by an optical system depends on the numerical aperture (or inversely with the f-number) • The depth of field describes the axial range of object points that produce “acceptable” circles of confusion on an image plane • Accomodation describes the eyes ability to change its optical power to focus on near objects • The angular magnification is defined according to a 25 cm near point (M = 25/f ) • We considered various simple optical systems (magnifiers, eyepieces, cameras, telescopes, etc) • Spherical aberration occurs anytime the paraxial approximation is not valid • Chromatic aberration occurs because of the wavelength-dependence of the index of refraction – blue and red light will not focus at the same location

36. Review of Chapter 18: Thick lenses and Matrix Methods • A thick lens or lens system is characterized by 6 cardinal planes • 2 principle planes • 2 nodal planes • 2 focal planes • When ray tracing, light travels along straight lines from object point to PP1, it then “hops” to PP2 at the same axial height • The rules of reflection and refraction and propagation can be described by a 2x2 system matrix and a 2x1 ray matrix (ray height and angle) • The system matrix for several elements is obtained by multiplying 2x2 matrices according to the order in which the ray encounters each optical element • All of the cardinal points can be determined from the system matrix

37. 0 Similar geometrical analysis shows Locations of Cardinal Points Consider input plane to PP1 From these distance, all of the cardinal points are located Sign Convention: ( - to left, + to right) of associated plane

38. Group Problem (Pedrotti3 18-12) • A gypsy’s crystal ball has a refractive index of 1.5 and a diameter of 8 in. • By the matrix approach, determine the location of the cardinal points • Where will sunlight be focused?

39. Group Problem An object measures 2 cm high above the axis of an optical system consisting of a 2-cm aperture stop and a thin convex lens of 5-cm focal length and 5-cm aperture. The object is 10 cm in front of the lens and the stop is 2 cm in front of the lens. Determine the position and size of the entrance and exit pupils, as well as the image. Sketch the chief ray and the two extreme rays through the optical system, from the top of the object to its conjugate image point.

40. Exam 2 Review: Some of your questions • The two topics that I have had some difficulty on are 1) what exactly defines something as a traveling wave and 2) how to turn a function into a wave. I know they are both really easy but I some missed them while studying • I forgot to ask for a review of Maxwell's equation and how they apply to this class. Thanks • can we go through an example with the propagation vector. i'm not sure why we really need it. • How do you find the penetration depth of light in absorbing medium? • What do we need to know with Rayleigh Scattering? • Can we do an example of a phasor diagram? • any of those index of refraction graphs would be nice. btw, wednesday is national pi day! (3-14) and at 1:59 we will be in class! (3-14 1:59). so we should do somthing special! ooh! we could do the wave, cuz its like optics ya know. ...or we could just all get A's on the test...or somthin... • differences between different types of waves • General review would be nice, since Spring break tends to eliminate things from my memory • The differences of effects of having a real or imaginary index of refraction. • Using intensity, irradiance and radiation • Chapter 25 in general

41. Review of Chapter 4: Waves and Wave Motion • Any function that satisfies the wave equation can be used to describe a wave: • Harmonic waves are periodic in space and time • Phase velocity describes the speed of the waveform • Euler’s formula can be used to find a complex representation of the wave • Three dimensional ways • Plane • Cylindrical • Spherical

42. Review of Chapter 4: Waves and Wave Motion • Maxwell’s equations can be combined to form a wave equation for E and B, and they show the relationship between E and B. • Light is a transverse electromagnetic wave • Light waves carry energy and momentum

43. Example Which of the equations below that could represent a traveling wave. 1. I ; 2. II. ; 3. III. ; 4. I and II ; 5. I and III; 6. II and III What is the speed and direction of each “genuine” wave?

44. Example • Write an expression for the Electric field of a 500 nm harmonic plane light wave with amplitude 100 V/m propagating in the direction of the vector k, which lies on a line drawn from the origin to the point (2, -1, 0). State any assumptions that you need to make.

45. You wish to design a “radiation sail” for a space craft that harnesses the solar radiation pressure of the sun. Which type of sail would produce result in the greatest force? • A perfectly reflecting sail • A perfectly absorbing sail • A perfectly transparent sail • A sail that is 50% reflecting and 50% absorbing • A sail that is 80% reflecting and 20% absorbing

46. Review of Chapter 5: Wave Superposition • The wave equation is a linear, 2nd order, partial differential equation • Any linear combination of solutions is also a solution • Since the plane wave is the easiest solution, we often represent any other light wave as an (infinite) series of plane waves, • Add the Electric Fields together to find the total Electric Field, then square to find the intensity • This can be shown graphically in a phasor diagram • Interference can result • Constructive (in phase addition) • Destructive (out of phase addition) • Incoherent Coherent sources have different scaling laws

47. Example • Microwaves of frequency 1010 Hz are beamed directly at a metal reflector. A uniform plate of mashed potatoes are placed near the reflector until a striped pattern of cooked and uncooked zones results. What is the distance between two cooked zones?

48. Review of Chapter 5: Wave Superposition • When more than one wave is present we observe two waves • Carrier wave: moves at phase velocity • Envelope wave: moves at group velocity • Some examples: • Standing waves : wave appears to stop moving • Same frequency, opposite direction • Group velocity = 0 • Beats • Different frequency • Group velocity can be either greater or less than the phase velocity of its constituent waves • Normal Dispersion: dn/d < 0, vg<vp • Anomalous Dispersion: dn/d > 0, vg>vp

49. Review of Chapter 25: Optical Properties of Materials + • Simple dielectrics can be modeled as a continuum of Lorentz Atoms • E-Field from the light wave polarizes the material • Atomic “Spring constant” sets the resonance frequency • “Damping” causes energy loss • Electron mass results in a frequency-dependent phase lag • In phase at low frequencies • 180-degrees out of phase at frequencies much greater than the resonance frequency • Model predicts a specific dispersion equation: • n varies with frequency • Imaginary part = absorption • Real part = phase index ω0 -

50. Review of Chapter 25: Optical Properties of Materials • Skin Depth refers to the distance at which the field falls to 1/e • Metals can be treated as a free-electron gas • dielectrics without the resonance frequency • Completely absorbs (and re-radiates) EM waves at low frequencies • Transparent to EM waves above the resonance frequency