Mar 2

Concepts

9.6.0 Thin film

·        Focus on just after eq 9.50 to 9.53.  We add all the infinite number of reflections with the following series sum: .  Works for |x|<1, even complex.

·        Phasor picture 9.40 as vector sum of the same series.

·        Fig 9.38 as special case of half-wavelength resonance: r goes to ____.

·        Finesse expresses contrast between max and min transmissions (see Fig 9.41).  Requires many waves to interfere to get sharp contrast, so r must be close to ____ for high finesse.

·        Fig 9.38 as special case of half-wavelength resonance: r goes to ____.

·        Two-ray approximation of last lecture is only good to find max, min conditions, not R, T intensities, as all the little amplitude waves series addition is crucial!

9.7.1  Multilayers

• Instead of adding infinite reflections, we can include them with just two waves moving in opposite directions, and boundary conditions on E, B.
• Boundary conditions (same as Fresnel coefficients):  components _____  to film plane must be continuous.
• Boundary conditions and phase changes due to thickness connect E, B at different interfaces.  The connection is represented by the matrix ___, eq 9.91.  Each matrix has its own index n and effective thickness .
• Notation: incident medium (0), layers (1,2,3…), finally substrate  (s)   is for example thick glass (but could be any medium including air).
• 9.97 and 9.98 give r,t.

9.7.2,3 Antireflection and multilayer coatings

• Single quarter-wavelength coatings for broadband antireflection coatings. Layer index should be ____ higher/lower than glass index.
• Multilayer quarter-wavelength coatings:  Lecture: anti reflection coatings  start air-LH…glass.  High reflectors start air-HL…glass.  Rationalize this with reflection phase shift.

Skills  Find R and T for multilayer films

Derivations

Eq. 9.51 reflected r from a thin film with series . 

Mar 4 9.8 Applications of interference

Concepts

9.8.1 Skim this; it would make a nice personal choice lab.

9.8.2

·        How a Michelson interferometer can be modified to test optical elements.

·        Follow the wavefronts through to understand how interference fringes arise that correspond to deviations of the tested optics from the ideal.  Where are the pathlength differences that the fringes display?

9.8.3

• In the Sagnac interferometer (ring gyro) he uses a “classical” argument, essentially invoking changes in speed of light vs direction of rotation: i.e. the ether.   Lecture: you get the same result from looking at the difference in distances the light travels to the screen.

9.8.3

• Know how synthetic aperture radar interferometry works (there are other SAR techniques such as 2-D imaging which we won’t treat).  
• Lecture: why longer baseline in SAR increases the resolution (in this case how rapid a change in height can be represented).

• Lecture: binomial expansion  

  Skills   

Derivations

Time differences for light in circular Sagnac interferometer in terms of rotating w

Mar 6

10.1, 10.3.1-2 (Save 10.3.3-4 for next time’s reading)

Diffraction intro, Fresnel diffraction

Concepts

10.1.0

·        Physically, diffraction is an example of interference of waves

·        Huygens-Fresnel principle: when light encounters an obstacle, add spherical wavelets from every point in the wavefront that is not blocked to get the diffracted wave

_·          Fig 10.1,2 and discussion: why a wider aperture causes light to be more intense in forward direction, while  for very small apertures, all angles are about equal probability. Phasor explanation.   

_·          It’s essentially impossible to use Maxwell’s equations to solve for diffraction…have to use approximations lilke Huygen’s-Fresel.

10.1.1- interesting, but can skip

10.1.2

·        Fraunhofer  vs Fresnel  diffraction

Which is for screen close to aperture and looks a little like the aperture?

Which bears no resemblance to aperture?

Which has linear dependence on aperture (and screen) variables? (Remember that both binomial approx and small angle approx yield linear forms).  These will combine to give dependence only on  the angle to the screen.

Which do we get on a screen  if we put a lens after the aperture (screen at focal length from lens)?

R>a²/l as boundary between the two regimes (applies only when lenses are not used).

10.1.3

·        Adding fields far from spaced, phased oscillators.  Truncation of series. 

·        As N gets large, interference peaks get very sharp

·        How do we change the direction of emission from a line of oscillators (antennas) without moving the line?

·        Lecture: what’s different if we observe at a screen close to oscillators:  strengths (1/r) matter, many angles to each screen point.

·        Integrals over line of uniformly distributed oscillators.  Lecture: over area

10.3.1 Fresnel

·        Obliquity factor to improve Hugyens-Fresnel approximation.

·        Each Fresnel zone  edge differs by a difference of ____ in distance to the observation point, so the light from each zone differs by a phase difference of ____ on average.

·        stop after equ 10.77

10.3.2 Vibration curve

·        Phasor addition of light from each zone

·        If light from all infinite zones is added at P, the phasor spirals in to the center.  Then what E-field does the center represent?

·        The first zone’s field at P is about twice the light field of all zones’ light together at P (twice the incident field).  Show it from the phasor curve.

·        Why does it spiral?  Each zone has weaker light because of obliquity factor.

Save 10.3.3-4 for next time’s reading

Mar 9  

10.3.3-5 Circular Fresnel, Zone plates 10.3.11 Babinet

Concepts

10.3.3

• Circular apertures let only certain zones (or parts of them) of the wavefront through to interfere at the screen
• Fig 10.43:  As you move off center of screen, lose symmetry in zone pattern filling aperture.  Look at area of opposite kinds of zone: if they balance, region is dark.  If one kind of area outweighs the other, it gets brighter. You get rings on the screen.
• Far enough off axis, you have equal areas of both kinds and it goes dark always in geometrical shadow region.
• If screen is far enough away, you go to Fraunhofer regime, in which less than one zone fills the aperture (so center is always bright)
• Can increase the number of zones in aperture by opening aperture or moving screen _____ closer/farther or moving point source ______ closer/farther.  When source is infinitely far away, get plane waves at aperture, so only screen distance determines zones.

10.3.4

• Circular obstacles: allow light from all zones beyond some radius to interfere at the screen. 
• Blocking the first few zones still has a vibration curve that circles forever and spirals into the center.   
• So at distances large enough (that the obliquity factor doesn’t kill it by being just behind the obstacle), there will always be a bright spot at the center that is almost the intensity of the unobstructed wave, called __________ spot

10.3.5  Fresnel zone plates

• If you can block alternate zones (doesn’t matter which), center is always bright.  
• Get an imaging equation.  Can use zone plate to focus light, even image where lenses can’t be mafe
• Lecture: Resolution is the thickness of the last zone included in mask

10.3.11

• Complementary apertures
• Babinet’s principle: By superposition, E due to an aperture is the incident unobstructed E minus the E due to the complementary aperture.  
• In __________  (Fresnel/Fraunhofer) diffraction the intensity patterns of complementary  apertures are identical.
•  

Skills   

Mar 11

7.3,4 11.1Fourier Analysis

Concepts

7.3  This should be mostly a review.

• Fourier series add multiples of a fundamental frequency, so result is periodic with that frequency.
• Always need to specify two variables at each frequency: cos, sin weights, or amplitude, phase
• zero frequency weight A_o is simply ________  of the function.
• Odd, functions are all sine, even : cos.  What does that do to complex form?
•  spatial frequency, angular frequency.  
• Convention in bold pg 308

7.4.1-2 

• It takes frequencies distributed continuously (an integral) to get something nonperiodic: a pulse.  Fig 7.31: spacing in time is _______ (proportional/inversely proportional) to spacing in frequency.
• FT of a square pulse is a __________ function
• Bandwidths: frequency spreads.  Uncertainty principle:  
• Fig 7.34 vs 7.35.  Difference between FT of  a pulse alone  vs  of a pulse envelope multiplying a sinusoid (carrier frequency): same FT function, but shifted to the carrier frequency.

7.4.3 –review

7.4.4 skip to pg 319 FT and Diffraction section

• There are two ways to do a FT to represent spatial variations of light due to aperture :  1) add waves of same direction, but different , (with  high k needed for sharp features) or 2)  add plane waves of same ,  but different angles (with  high angles needed for sharp features).
• Fraunhofer diffraction is the FT (in angles) of the aperture.  Each angle on screen represents a different spatial frequency that’s part of the aperture function

11.1   

• Complex FT
• Phase of FT is phase shift of wave before it’s added
• FT of a sum of functions is ______________________ (superposition, as FT is a linear operation)
• FT of a  Gaussian is a _____________

Skills

Fourier integrals

Derivations

Mar 13

11.2.2-2.3 Fourier optics (and 11.2.1 if you didn’t read it last time)

11.2.2-3 

• Note  is angle in real (x,y) space  (you can think of it as an angle on the aperture plane).   is angle in FT  space, in the plane (you can think of it as an angle on the diffraction viewing screen).   is the same as .
• FT of circular aperture (shutter) function is a circular sync-like function J_o.
• Lens at focal length f from aperture, makes light at screen at f display the FT of aperture.  Or remove lens, put screen very far away.
• Dirac delta function:  its integral, sifting properties. 
• eq 11.38 delta function as integral over complex exponential
• shifting a function in space (or time) causes a linear phase shift  (or ) in the FT.  (Similar to wave propagation)

11.3.

• Imaging systems are linear and to first approximation spatially invariant.
• Point spread function Fig 11.16: image of a delta function (tiny point) of light (“impulse” response of imaging system.  It is not a diffraction pattern (though it can be affected by diffraction).  Integral that relates PSF to the total image.  
• What is different between using coherent vs incoherent light with a point spread function?

Skills

Use delta functions in integrals and FTs.

Derivations

 11.38 delta function as integral over complex exponential

Mar 16

11.3.2 Convolution theorem

Concepts

• A convolution of two functions is the area under the product of the two functions, which changes as the one of the functions is shifted vs the other.  
• If the functions are not symmetric, we have to invert the shifted one.  
• Convolution results in a smearing of the two functions. The width of the two convolution is approx. the sum of the widths of the two functions.
• The convolution of a function with a delta function puts a copy of the function at the delta function location.  
• An image is the convolution of the object intensity function with the point spread function.
• Convolution theorems:  The FT of the product of two functions is a convolution of the transforms of each.   Likewise, the FT of a convolution of two functions is the product of the transforms of each.

Skills

Sketch convolutions, compute simple convolutions.

Use the convolution theorem in FTs and inverse FTs, including using the convolution with a delta function to represent repeated functions.

Derivations

Mar 18

10.2.4, 11.3.3 Fraunhofer as FT

Concepts

• 2-D apertures need and 2-D FT.
• To get Fraunhofer diffraction patterns as function of angle we use , etc.  ( I will  instead of in text).
• Single slit: ____ function
• Double slit: _____ function  ______ function.  Relation to convolution theorem.
• array theorem
• apodization:  Lobes come from FTs of functions that cut off sharply as in slits.  These can overlap other images. To reduce lobes, we put in a mask that makes the aperture function go to zero more smoothly, more like a _____ function, for which there are no lobes in the FT

Skills

Find diffraction patterns in 1 and 2 dimensions from FT

Derivations

Array theorem from convolution theorem

Given the steps in lecture that lead to the FT in Fraunhofer diffraction, explain each step.  

Mar 20

10.2.3-4,8 Multiple slits, gratings

Concepts

• Grating viewed as N slits in a line, so is a convolution of a slit with N deltas.   Review sec. 10.1 for N oscillators (deltas).
• Intensity for N deltas involves .  This has small peaks (order 1) when the numerator peaks, and large peaks (order N²) when the denominator and numerator both go to zero.  
• Lecture: By conservation of energy (power), width of peak must decrease by N if height increases by N²
• As N gets large, the diffraction peaks get very strong and narrow.  Ability to resolve different l at different q increases.  Resolution also increases as m increases.
• Finite width sources means that we multiply repeating N-delta pattern by a single slit pattern envelope.  
• Blazed gratings concentrate light more in one diffraction order.
• Both random-position and ordered arrays of large N obey the array theorem, but the results are very different. Ordered arrays give interference fringes across the single aperture pattern, while random arrays add incoherently to give N times the intensity of a single aperture pattern.

Skills

Find peak and zero locations for N sources.  Sketch patterns including single slit envelope.

Derivations

 Intensity function for of N deltas in a line, separated by d.

Mar 23

10.2.5,6  Fraunhofer circular apertures

Concepts

10.2.5

• Just as a lens can be used behind an aperture to put Fraunhofer regime at the focal plane, a lens in aperture of camera or telescope creates Fraunhofer diffraction of aperture at focal plane (narrowest possible point spread function)
• You can skim derivation leading up to 10.56.
• Airy disc is:
• J₁(u)/u  looks like a sinc but has circular symmetry.  Has first zero just beyond p at 1.22p.   Text: 2a = D
• It’s best to remember simply the half angular width of the airy disc:   (similar to half width of vertical slit central max: )
• Rayleigh’s criterion for minimally resolved objects/images
• Lecture: if  why does this all break down?  Why can’t we form images of objects of size less than l?

Skills

Derivations

Resolution of a telescope from first minimum of Fraunhofer diffraction pattern of a circular aperture.

 

Mar 25

12.1-3 Coherence

Concepts

·        Interference term (product of fields, averaged) measures coherence.  In partially coherent light this average diminishes due to lack of long-time spatial or temporal correlation between fields.

·        Fig 12.2 : why does a glass plate in one case cause fringes to completely disappear?

·        Fig 12.3 .  Illustration of what we went over already on spatial coherence: that a wide angle source causes fringe patterns to overlap and reduce visibility.

·        Visibility is essentially a measure of contrast between light and dark regions, and is a measure of coherence.

·        in sec 12.3, don’t stress over the K’s... think of them as scaling constants that will divide out later in eq 12.17.

·        compare eq 12.18 with 9.14.   We can now treat partially coherent light.

·        “Degree of coherence” function is a convolution between the two interfering E-fields (12.17). 

·        Fringes that are farther away from center will involve longer and longer time differences between two spots…they are the first fade.

·        By the convolution theorem, because the self-coherence function is a convolution with itself, its FT is FT(E(t)) * FT(E(t)) = {FT(E(t))}² or the power spectrum.  Many frequencies in the FT of E then yield a short temporal coherence time or length.

·        Likewise the spatial coherence at between two points on an aperture is related to a FT of the source intensity profile.  So a broad source makes the light at two points on the aperture less coherent if they are far apart.

 

Skills

     Predict visibilities from FT of power spectra (temporal coherence) or  source widths (spatial coherence)

 

Derivations

 

Mar 27

12.4 Stellar & Correlation interferometry

Concepts

• Fig 12.3, when we use mirrors of baseline h to interfere from mirrors separated by a,  the fringe separation depends on ____ but the visibility depends on ____. 
• Can find diameter of a star by increasing ____ to find the first zero in the visibility (sinc or jinc function zero).
• eq 12.29.  Has the same angular resolution as a telescope with diameter ___.

12.4.2

• E(t) amplitude ( or envelope) fluctuates due to the presence of many frequencies.  We can’t record E(t) fast oscillations, but we can record the slower amplitude (intensity)  variations.
• If we correlate I(t) fluctuations, they depend on the same degree of coherence function as a Michelson interferometer, but only its envelope (amplitude).  I(t) fluctuates over times of about the coherence time.
• Why is it so much easier on the optical requirements to correlate I(t) rather than look at interference fringes?
• Intensity correlation interferometry gets the same angular resolution as the fringe experiment.
• Lecture: this combines experiments of both temporal and spatial coherence.  If we correlate I(t) of one recording station with itself, can measure the coherence time of the source light from the correlation.  If we correlate I(t) of one recording station with the others’, we get less correlation, depending on the width of the source (and h).  So we can measure the diameter of the star by looking at the loss of correlation vs h.  
• The way the correlation function is changed by a convolved with the source intensity profile function. 
• Can skip pg 578.

Mar 30

13.1.0, 13.1.2-3 (stop before “Gaussian Laser beams”)  Lasers, Stimulated emission

Concepts

• Stimulated absorption is just normal absorption
• Stimulated emission has the same probability as absorption per atom (in excited vs ground state respectively) and per photon.  This is why we must have a population _________ for a laser. 
• Spontenous emission probability depends on number of excited atoms and the electromagnetic properties of the vacuum (density of states).
• Einstein A/B relation 13.14 contains the density of states in a vacuum
• Be able to write eqs 13.7-9
• Three-level laser
• Longitudinal laser modes:  integer number of ____ in cavity
• Q-factor of a resonator. Q-switching to build up population inversion
• Transverse modes

Skills

• Calculate relative probability of stimulated to spontaneous emission
• Find cavity modes and how they match with the gain curve to find which ones are populated in a laser.

Derivations

·        Almost trivial: Longitudinal laser modes frequencies

Apr 1

13.1.3 (Gaussian to end) -13.1.4 (through speckle effect)

Concepts

• Gaussian laser beam profile is the _______ mode
• The  beam _______ is the smallest radius of the beam at focus.  At a distance of the _______ range away from focus, the beam area has doubled.
• Any finite width beam must diverge, in essence due to self-diffraction (uncertainty principle).   
• Divergence increases for tighter focus; likewise to get a tight focus, have to use wide beams and large angles
• Lecture: focusing with lenses. Radius of curvature of the beam vs z
• Semiconductor diode lasers: recombination of electrically injected electrons and _______.
• What property of a screen or wall causes speckle pattern?  What property of the light is necessary?

Skills

• Analyze  laser beam divergence, focusing

Derivations

·         Find divergence angle from equation of w(z) at large z.

·         Find beamwaist from f and d in focusing

Apr 3

13.2.3-4 Spatial filtering, phase contrast imaging

Concepts

•  Lecture: spatial filtering of a laser beam to achieve mostly TEM00 mode

13.2.3

• What parts of the FT plane correspond to the high spatial frequences? 
• What does the k=0 central part of the FT plane represent?
• Blocking part of the transform plane manipulates the image by removing parts of its FT

13.2.3

• If the object is transparent, it will still shift phase.
• If there were no object there at all, how would the incident plane wave light appear in the FT plane (at focal plane of lens)?              If we block this plane wave light , we can see what part of the FT is caused by the object shifting the phases of light differently from the incident light.
• Or we can block part of FT plane and see phase contrast

Apr 6

13.3 Holography

Concepts

• Read through p 630 carefully
• Recording phase and amplitude information allows reconstruction of the light field itself from an object
• Each point on object creates its own “zone plate” in the film; these all superpose to form the hologram. 
•  Or, if instead we break the light into plane waves of different angles, each angle interferes to make a diffraction grating in the film, which yields the same angle (for m=1) upon reconstruction.  So angle of light  is encoded in the fringe spacing.
• Film requirements: need film grains to be as small as about l. 
• Phase is encoded in the ____________ of the fringes, amplitude in the __________ of the fringes.
• The ______ (real/virtual) image is the one we usually look at.  The ______ (real/virtual) image is “inside out” in its 3-D depth, in the sense of having stereo glasses lenses interchanged.
• After p 630,  skim content, until p 638, and read Holographic Optical Elements.   Can create in a flat film hologram the effects of  many optical elements such as:                 
  …used in:
• Lecture: temporal and coherence requirements

13.4 Nonlinear optics

Concepts

• For strong laser light, electron motion is no longer simple harmonic motion.  Electrons are driven farther from their equilibrium positions and no longer feel a parabolic potential. Polarization magnitude depends on power expansion in E (eq 13.32).    
• Crystals that have inversion symmetry have only even powers of E in the P expansion
• Efficient second harmonic generation requires index matching (phase matching) of fundamental and harmonic light.  Lecture: conservation of momentum).  Achieved by having  w and  2w  as a o-ray and  e-ray (or vice versa) at a certain ray angle vs optic axis.
• Difference and sum frequency generation requires nonlinear crystal and phase matching , too.
• Self focusing (or defocusing) can be understood as light intensity changing the index in the beam path, most at the center, like a GRIN rod.

Skills

Analyze harmonic and sum/difference frequency generation in terms of energy and momentum conservation of photons.