Ph.D. Qualifying Examinations Exam details · Ph.D. Qualifying Examinations ... Grading out of the qualifying exam: ... Find the parity of part f) if it is rotating. [1 point] h) - [PDF Document] (2024)

Ph.D.QualifyingExaminations

General rules: All students seeking a Ph.D. are required to attempt the Written QualifyingExaminations after their first complete academic year; they are required to have passed all fourexaminationsbeforebeginningtheirthirdacademicyear.Eachexammaybeattemptedatmosttwice*(see footnotebelow).Part‐timestudentsarenotsubject to the timetablesabove.The lowestpassingscoreis60%.

Examdetails:The fourexamsconsistof twocommonexams (common toall tracks)and two track‐

specificexams(foreachtrack).AllexamsareofferedinAugust;sometrack‐specificexams(e.g.,examsthatareadministeredbytheECEdepartment)mayalsobeofferedinJanuary.

Theexamsthatarecommontoalltracksare:

•Electromagnetics(1.5hours,basedonPHYC511ortheECE555/ECE561sequence)

•GeneralOptics(3hours;basedonPHYC/ECE463andPHYC476L)

Thetrack‐specificexamsare:

OpticalSciencestrack

•AdvancedOptics(1.5hours;basedonPHYC/ECE554.)

•Lasers(1.5hours;basedonPHYC464orECE464)

Photonicstrack

•SemiconductorOpticalMaterials&Devices(basedonECE570;thisexam isadministeredby theECEdepartment)

•SemiconductorPhysics(basedontheECE471/ECE572sequence;thisexamisadministeredby theECEdepartment)

ImagingSciencetrack

•StochasticProcesses(basedonECE541;thisexamisadministeredbytheECEdepartment)

•DigitalImageProcessing(basedonECE533;thisexamisadministeredbytheECEdepartment)

Exit‐examinationrequirementforMSPlan2a:ThePh.D.qualifyingexaminationwillsatisfytheexit‐examinationrequirementforstudentsenrolledinMSPlan2a.

Gradingoutofthequalifyingexam:Theexam iswaivedforstudentswhoearnanaverageGPAof4.0orgreaterononeofthethreesetsoffourOSE‐requiredcorecoursesdependinguponthestudent’sconcentrationofchoice:

OpticalSciencetrack:PHYC/ECE463,PHYC511orECE561,PHYC/ECE464,andPHYC/ECE554

Photonicstrack:PHYC/ECE463,PHYC511orECE561,ECE570,andECE572

ImagingSciencetrack:PHYC/ECE463,PHYC511orECE561,ECE533,andECE541

* PLEASE NOTE THAT THE ELECTROMAGNETICS, GENERAL OPTICS, ADVANCED OPTICS, AND LASEREXAMSAREOFFEREDONLYONCEPERYEAR.THUS, IFYOUWISHTOHAVETWOATTEMPTSATTHESEEXAMS,YOUMUSTSITFORTHEEXAMSBEFOREYOUR2NDYEARBEGINS.

August 13th, 2015

OSE Qualifying Examination 2015 – Electromagnetics Answer any 3 questions. Begin each question on a new sheet of paper. Put your banner ID at the top of each page. Staple all pages for each question together. Be sure to indicate what question # you are answering.

August 13th, 2015

Question #1

August 13th, 2015

August 13th, 2015

Question #2

August 13th, 2015

Question #3

August 13th, 2015

Question #4 Consider a circularly polarized wave at 𝑓 = 10! Hz propagating within a uniform, lossless dielectric medium. The wave encounters a planar interface with air at normal incidence. The functional form of the wave electric field at the interface is given as

𝑬!"# 𝑧 = 𝒂! − 𝑗𝒂! exp −𝑗6𝜋𝑧 where 𝑧 in meters is in the direction normal to the interface. The interface is in the 𝑥 − 𝑦 plane. Assume that the permeability in both regions is 𝜇!. a) Draw a sketch describing the set-up of this problem and label as many things as you can (there should be at least 6 items labeled). [2 points] b) Find the dielectric constant of the dielectric medium from which the circularly polarized wave emerges. [1 point] c) Based on your answer to b) what is the dielectric medium? [1 point] d) Find the reflection coefficients for both the 𝒂! and 𝒂! components. [1 point] e) Find the transmission coefficients for both the 𝒂! and 𝒂! components. [1 point] f) Find the polarization of the reflected field (is it still circular? or is it linear or elliptical? Justify your answer). [1 point] g) Find the parity of part f) if it is rotating. [1 point] h) Find the polarization of the transmitted field (is it still circular? or is it linear or elliptical? Justify your answer). [1 point] i) Find the parity of part h) if it is rotating. [1 point]

August 13th, 2015

Question #5

A dielectric slab of polystyrene (𝜀 = 2.56𝜀!, 𝜇 = 𝜇!) of height 2ℎ is bounded above and below by free space, as shown in Figure 1. Assuming the time-harmonic instantaneous electric field within the slab is given by

𝑬 𝑥, 𝑡 = 𝒂!10+ 𝒂!5 cos 𝜔𝑡 − 𝛽𝑥

where 𝛽 = 𝜔 𝜇!𝜀, determine the: (a) the complex spatial electric field intensity 𝑬 𝑥 within the slab [1 point]; (b) the intrinsic impedance of the polystyrene [1 point]; (c) the time-harmonic instantaneous magnetic field within the slab [2 points]; (d) the corresponding complex spatial magnetic field intensity within the slab [1 point]; (e) the time-average Poynting vector (average power density) 𝑺𝒂𝒗𝒆 within the slab [2 points]; (f) the instantaneous electric and magnetic fields in free space right above and below the slab [2

points]; (g) the complex electric and magnetic fields in free space right above and below the slab [1

point].

Qualifying Exam, Fall 20

Balanis c01.tex V3 - 11/22/2011 3:03 P.M. Page 33

PROBLEMS 33

radius a = 0.1 m, placed on the xy plane atz = 0, is given by

! = az10−12

1 + 25ρcos(1500π t) Wb/m2

where ρ is the radial distance in cylindricalcoordinates. Find the:(a) Total flux in the z direction passing

through the loop.(b) Electric field at any point ρ within

the loop. Check your answer by usingMaxwell’s equation 1-1.

1.13. The instantaneous magnetic flux density infree space is given by

! = ax Bx cos(2y) sin(ωt − πz )

+ ay By cos(2x) cos(ωt − πz )

where Bx and By are constants. Assum-ing there are no sources at the observationpoints x , y , determine the electric displace-ment current density.

1.14. The displacement current density within asource-free ("i = 0) cube centered aboutthe origin is given by

"d = ax yz + ay y2 + az xyz

Each side of the cube is 1 m long and themedium within it is free space. Find the dis-placement current leaving, in the outwarddirection, through the surface of the cube.

1.15. The electric flux density in free spaceproduced by an oscillating electric chargeplaced at the origin is given by

# = ar10−9

4π

1r2 cos(ωt − βr)

where β = ω√

µ0ε0. Find the time-averagecharge that produces this electric flux den-sity.

1.16. The electric field radiated at large distancesin free space by a current-carrying small cir-cular loop of radius a , placed on the xy planeat z = 0, is given by

$ = aφE0 sin θcos(ωt − β0r)

r, r ≫ a

where E0 is a constant, β0 = ω√

µ0ε0, ris the radial distance in spherical coordi-nates, and θ is the spherical angle measuredfrom the z axis that is perpendicular to the

plane of the loop. Determine the correspond-ing radiated magnetic field at large distancesfrom the loop (r ≫ a).

1.17. A time-varying voltage source of v(t) =10 cos(ωt) is connected across a paral-lel plate capacitor with polystyrene (ε =2.56ε0, σ = 3.7×10−4 S/m) between theplates. Assuming a small plate separation of2 cm and no field fringing, determine at:(a) f = 1 MHz(b) f = 100 MHz

the maximum values of the conduction anddisplacement current densities within thepolystyrene and compare them.

1.18. A dielectric slab of polystyrene (ε =2.56ε0, µ = µ0) of height 2h is boundedabove and below by free space, as shownin Figure P1-18. Assuming the electric fieldwithin the slab is given by

$ = (ay 5 + az 10) cos(ωt − βx)

where β = ω√

µ0ε, determine the:(a) Corresponding magnetic field within the

slab.(b) Electric and magnetic fields in free

space right above and below the slab.

x

e0, m0

e0, m0

2.56 e0, m0

2.56 e0, m0

y

z

h

h

Figure P1-18

1.19. A finite conductivity rectangular strip,shown in Figure P1-19, is used to carryelectric current. Because of the strip’s lossynature, the current is nonuniformly dis-tributed over the cross section of the strip.The current density on the upper and lowersides is given by

" = az 104 cos(2π×109t) A/m2

and it rapidly decays in an exponential fash-ion from the lower side toward the center bythe factor e−106y , or

" = az 104e−106y cos(2π×109t) A/m2

August 13th, 2015

Question #6

The electrical constitutive parameters of moist earth at a frequency of 1 MHz are 𝜎 =10!! 𝑆 𝑚, 𝜀! = 4, and 𝜇! = 1. Assuming that the electric field of a uniform plane wave at the interface (on the side of the earth) is 3×10!! 𝑉 𝑚, find the: (a) Distance through which the wave must travel before the magnitude of the electric field

reduces to 1.2×10!! 𝑉 𝑚 [1 point]; (b) Attenuation constant inside the earth (Nepers per meter) [1 point]; (c) Phase constant inside the earth (radians per meter) [1 point]; (d) Explain the physical meaning of propagation constant, attenuation constant and phase

constant [2 points]; (e) Phase velocity inside the earth (in meters per second) [1 point]; (f) Wavelength inside the earth (in meters) [1 point]; (g) Intrinsic impedance of the earth [1 point]; (h) The expressions for the instantaneous electric and magnetic fields inside the earth, assuming

the plane wave propagates in the +𝒛 direction and the electric field is in the +𝒙 direction [2 points].

General Optics PhD Qualifying Examination 2015 Answer all questions. Begin each question on a new sheet of paper. Put your Banner ID at the top of each page, and staple the pages (if more than one) for each question together. 1a. It is possible to measure the rotation speed of a wheel that has reflective tape on its perimeter with the set up at right, taken from the Optics lab manual. The rectangle at the top is a HeNe laser, wavelength 632.8 nm. It is split into two equally intense beams using beamsplitter BS. The beams are made to intersect on the perimeter of the wheel W. Lens L and photodiode PD detect scattered light from the wheel. If the lower beam has an incident angle of 20° above the normal, and the upper beam has an incident angle of 30° above the normal, what is the spacing of the interference pattern formed on the wheel? 1b. For a setup where the fringe spacing is 6 microns (maximum to maximum), a peak in the spectrum analyzer is detected at 3 kHz. The wheel diameter is 14 cm. What is the speed of the perimeter, and the angular speed of the wheel? 1c. A student wishes to make the lower beam impinge on the wheel at normal incidence, so he moves the mirror M very far to the left, a distance of a few meters. He notices that the signal peak moves to a different frequency, but also gets very weak. Does it move up or down in frequency? Why does the signal become very weak? 2. An astronaut is looking at a point source in a training pool (filled with water, n=1.33). The source is 200 cm from the vertex of the transparent hemispherical shield in front of the helmet, Where is the image of the point source that the astronaut sees? (The shield has a radius of 20 cm and is thin and of uniform thickness.)

3. Unpolarized light in air is incident on a flat surface of a diamond. The reflected light is completely polarized. a. To the nearest degree, what are the angles α, β, γ in the figure? b. What is the nature and orientation of the polarization (electric field orientation) for the reflected light? Express in terms of the coordinate system given below.

4. A gas-­‐filled cell of length 5 cm is inserted in one arm of a Michelson interferometer, as shown in the figure below. The interferometer is in vacuum and is illuminated by light of wavelength 500 nm. As the gas is evacuated from the cell, 40 fringes cross a point in the field of view, Estimate the refractive index of this gas (the thickness of the splitting mirror can be ignored).

5. A plano-­‐convex lens with a refractive index of 1.5 and power of 0.1 diopter (in air) is placed, convex surface down at bottom of a glass container (that is optically flat) filled with water (n = 1.33). Using a microscope and a sodium lamp (λ = 590 nm), we observe interference fringes from top. Determine the radius of the first dark ring.

Gas-filled cell

6. Ice is birefringent, with indices 1.309 and 1.313. a. If you want to make a “zero-­‐order” quarter wave plate for a HeNe laser (wavelength 632.8 nm in vacuum), what would be its thickness? b. Suppose you make a quarter wave plate with a nominal thickness of 1 mm. Estimate its “bandwidth.” In other words, what is the vacuum wavelength nearest to 632.8 for which the ice waveplate will give linearly polarized output (rather than the desired circularly polarized output) for input linearly polarized at 45° to the optical axis? An approximate answer is sufficient.

7. You have 2 microscope objectives: Objective A is 100X magnification, with a numerical aperature of 0.4. Objective B is 40X magnification, with a numerical aperture of 0.8. a. Estimate the resolution of each. Which has better resolving power? b. These objectives were designed to be used with a microscope with tube length 25

cm (i.e. they place an image 25 cm from the objective.) Approximating these as simple lenses, what are their focal length and diameters?

8. A solar sail is to be used to suspend a spacecraft at a distance from the Sun equal to Earth’s, by reflecting solar photons. It is 1.5 x 1011 m from Earth to the Sun. The Sun radiates 3.9 x 1026 W. It is proposed to make the sail out of aluminum foil, which has a density of 2.7 g/cm3. How thin does the foil have to be? (The spacecraft is quite far from the Earth, so you may ignore gravitational force from the Earth and other planets.)

9. A laser at 488 nm provides a 100 W TEM00 Gaussian beam with a 1/e2 waist radius of w0 = 1 mm at the output coupler. a. Estimate the approximate size of the spot this laser would shine on the moon (distance = 400,000 km). b. If there is a corner cube on the moon to retro-­‐reflect the light, estimate the number of photons per second detected by a 1 cm-­‐square detector on Earth. 10. a. What is the physical meaning of phase velocity and group velocity? b. Describe two methods for mode-­‐locking a laser c. Describe two methods for Q-­‐switching a laser d. What waveplate would you use to change horizontal polarization to vertical, without loss? e. Explain why Brewster windows are sometimes used in lasers? f. What should be the orientation of the transmission axis on polarized sunglasses?

Advanced Optics PhD Qualifying Examination – August, 2015 Answer all questions. All count equally. Begin each question on a new sheet of paper, and staple all pages for each question together. Put your banner ID# at the top of each page. Q1. Monochromatic linearly polarized light passes through a Faraday rotator, a half-­‐wave plate (HWP), and finally a linear polarizer (LP). The original polarization and the slow axis (SA) of the HWP are vertical (perpendicular to the page) while the transmission axis of the last polarizer is 45 degrees to vertical. The Faraday cell is 10 cm long with a Verdet constant of 0.0161 min/G-­‐cm and a uniform magnetic field B is applied on the Faraday cell. a) Show the orientation of the magnetic field for maximum rotation. What is the polarization state of the rotator output if B=10kG? b) At what magnetic field strength the intensity of the output beam is I0/4 (where I0 is the irradiance of the input beam)?

Q2. Design two Fresnel lenses of a little less than 2 mm radius, for a focal distance of 80 cm, at a wavelength of 0.8 µm.

a) What are the radii of the zones?

b) The first lens is made by blocking selected zones. Which ones should be blocked?

c) The second lens is to be made by applying a phase shift to selected zones. What phase shift and what zones?

d) Calculate the field and intensity at the focus for the two lenses, each illuminated by a plane wave of amplitude E0.

e) Compare your answer to (d) to the intensity at the focus of a perfect lens of the same focal length (80 cm) illuminated by a Gaussian beam of w = 2 mm.

Hint: the optical field generated by the entire unobstructed wavefront is equal to one-­‐half the contribution from the first zone.

Q3. You are asked to select a square flat grating 10 cm wide, for best wavelength selection (in first order) at 600 nm. The clearance is such that the acceptance angle for the incident beam ranges from -­‐89° to +89°.

a) What is the configuration (angle of incidence and diffraction) and groove density that would give you the best resolution?

b) Assuming you can resolve a deviation angle of 1°, what is the corresponding resolution in wavelength Δλ1 ?

c) What is the diffracted angle (first order) at the second harmonic (300 nm) and resolution Δλ2 corresponding to a deviation angle of 1°?

d) Can you achieve the same resolution at 300 nm (rather than 600 nm) in higher order? If so, which order?

Laser Physics PhD Qualifying Examination 2015 Answer all questions. Begin each question on a new sheet. Put your Banner ID on each page. Staple all pages for each question (separately) together. 1.

2.

3.