ECE426F Optical Engineering - University of Torontoqianli/publications/ECE318_2007... · ·...

Edward S. Rogers Sr. Department of Electrical and Computer Engineering

ECE318S Fundamentals of Optics

Final Exam

April 16, 2007

Exam Type: D

(Close-book + two double-sided aid sheets + a non-programmable calculator)

Exam Duration: 2 ½ hours

Marks:

Q1 Q2 Q3 Q4 Total 40 16 24 22 100+2

Name: _________________________

Student Number: __________________________ Note:

(1) Follow the sign conventions established in the lectures given this year. (2) Unless explicitly required in the question, explanations to your answers are optional.

However, partial marks will be awarded for incorrect or incomplete answers when accompanied by correct and relevant explanations. Conversely, marks will be deducted accordingly if a wrong explanation accompanies a correct answer.

ECE318S Fundamentals of Optics Final Exam of 19 Name: _________________________ Student Number: __________________________

Q1 Answer the following short questions:

(a) Match the apertures on the left with their far-field diffraction patterns on the right by linking each pair with a line. (3 marks)

Fig. 1-1 Apertures and far-field diffraction patterns.


(b) A plane wave propagating in vacuum is expressed by its E field as: V/m. Write down the expression for the

accompanying H field and its unit. (Do not introduce new symbols other than physical constants.) (3 marks)

[ ] ([ tkxieEE izoyo ωφ ++= expˆˆ zyE

r )]

(c) For the plane wave given in (b), write down the expression for the Poynting vector and its unit. (Write down the final expression, not as a vector operation.) (3 marks)

(d) Consider a TM plane wave incident on a dielectric boundary formed by two lossless media of refractive indices n1 and n2 (Fig. 1-2). r and t are the Fresnel reflection and transmission coefficients, respectively; R and T are the reflectance and transmittance, respectively; 1→2 means light is incident from n1, while 2→1 means the opposite. Indicate whether the following relations are true or false:

n1

n2

θ1

θ2

n1

n2

θ1

θ2

n1

n2

θ1

θ2

n1

n2

θ1

θ2

n1

n2

θ1

θ2

n1

n2

θ1

θ2

Fig. 1-2 Reflection and refraction on a dielectric boundary.

(d.1) r1→2 = –r2→1 True/False (0.5 mark) (d.2) t1→2 = –t2→1 True/False (0.5 mark) (d.3) T1→2 = T2→1 True/False (0.5 mark) (d.4) (t1→2)(t 2→1) – (r1→2)(r2→1) = 1 True/False (0.5 mark) (d.5) T1→2 = (t1→2)2 True/False (0.5 mark) (d.6) R1→2 = (r1→2)2 True/False (0.5 mark)


(e) A pulse with a central frequency ωo is travelling in a medium that has a dispersion relationship as shown in Fig. 1-3. Does the pulse envelope travel faster or slower than its phase? Briefly explain your answer by graphically indicating the group and phase velocities on the plot. (2 marks)

ω

kkο00

ωο

ω

kkο00

ω

kkο00

ωο

Fig. 1-3

(f) Suppose you are buying a new LCD TV for your living room. The sofa, in which

you always sit to watch TV, is 5 meters away from the TV. Since the TV’s pixel resolution need not be higher than what your eyes can resolve, estimate the pixel resolution (i.e., number of pixels per inch) that suits you. State any assumptions you make in your estimation. (4 marks)


(g) Answer the questions regarding the double refraction phenomenon seen in Fig. 1-4.

Fig. 1-4 Two pencil images made by a calcite crystal (negative uniaxial). The arrow indicates the direction of image shift.

(g.1) For normal incidence at the air-to-calcite interface, graphically find the

transmitted kTM and kTE vectors, as well as the direction of the associated Poynting vectors, ŜTM and ŜTE . Indicate them on Fig. 1-5. (2 marks)

Fig 1-5 Air-to-calcite interface. The lines indicate the optical axis of calcite.

kiair

calcite

kiair

calcite

(g.2) At the calcite-to-air interface, given the transmitted k vector (Fig. 1-6), graphically find the incident kTM and kTE vectors, and the direction of the associated Poynting vectors, ŜTM and ŜTE. Indicate them on Fig. 1-6 and separate TE and TM polarizations for clarity. (4 marks)

calcite

ktair ktair

calcitecalcite

ktair

calcitecalcite

ktairktktair ktair ktktair

calcitecalcite

Fig 1-6 Calcite-to-air interface. The lines indicate the optical axis of calcite.

(g.3) A polarizing sheet is placed on the calcite to remove the lower pencil image

shown in Fig. 1-4. Indicate on Fig. 1-4 the transmission axis of the polarizer. (2 marks)


(h) Answer the questions regarding the 4-f system (Fig. 1-7) used for image processing. The original picture and two spatial filters used to process the image are given in Fig. 1-8.

f f f fLaser f

S1 S2 S3

f f f fLaser f

S1 S2 S3

Fig. 1-7 The 4-f optical image processing system.

Original Picture Filter 1 Filter 2

Fig. 1-8 For the filters, light is blocked by the black area, and transmitted in the white area.

(h.1) Indicate the locations for the original picture, the spatial filter, and the processed image. (1 mark)

(h.2) Which filter is a high-pass filter? (2 mark)

(h.3) There are lines and shades in the original picture. How would these features appear in the processed image if Filter 1 were used? (2 marks)


(j) A green mercury lamp (λ0 = 546 nm) is used in a Michelson interferometer, the same interferometer as the one used in Lab 3 (Fig. 1-9). Answer the following questions.

Knob 1

B

D

A

C

Knob 2

Knob 3

Knob 1

B

D

A

C

Knob 2

Knob 3

Fig 1-9 Earling’s Michelson interferometer. B is the mercury lamp. A and D are mirrors, and C is the housing for the beam splitter and the compensator plate.

(j.1) If the fringe pattern is not centered on the viewing area, as shown in Fig. 1-10, which knob(s) should you adjust to center the fringes? Why? (3 marks)

Fig. 1-10 Fringe pattern not centered on the viewing area

(j.2) If the fringe pattern is centered, calculate the translation distance of mirror D in order to allow 10 new dark fringes to emerge from the center. (2 marks)

(j.3) It is found that fringes are visible only within 1 mm of translation range of mirror D. Estimate the bandwidth of mercury emission. (4 marks)


Q2. You are provided with an optical system composed of two polarizers with a rotating quarter waveplate in between as shown in Fig. 2-1. The transmission axes of first and last polarizers were aligned with the x- and y-axis, respectively. Answer the following questions.

xy

z

xy

x θ

fast axis

input output

xy

z

xy

x θ

fast axis

x θ

fast axis

input output

Figure 2-1: An optical system composed of two ploarizers and a quarter waveplate in between.

(a) Write down the Jones matrices for the two polarizers, T1 and T2, in the

coordinate system shown. (2 marks)

(b) If the Jones matrix for the quarter waveplate is when its fast axis is aligned

with the x-axis, what is its Jones matrix T

⎥⎦

⎤⎢⎣

⎡i001

qwp when the fast axis is rotated by an angle θ with respect to the x-axis, as shown? (4 marks)

(c) Find the transmittance of the system as a function of θ if the input is: (c.1) x-polarized light (4 marks) (c.2) randomly-polarized light (2 marks) (d) For randomly-polarized input, if the quarter waveplate is rotating at 60

RPM (rotations per minute), plot the transmittance of the system as a function of time for at least two cycles. Label the axes and the scale. (4 marks)


Q3. You are provided with a spectrometer (Fig. 3-1) to measure the wavelength separation of the sodium double line.

Input Slit

Output Slit

DiffractionGrating

Lens 1f1=30 cm Beam splitter

Lens 2f2=30 cm

θInput

Slit

Output Slit

DiffractionGrating

Lens 1f1=30 cm Beam splitter

Lens 2f2=30 cm

θ

Fig. 3-1 The internal construction of a spectrometer. The grating is mounted on a rotational stage.

θ is the angle between the grating normal and the horizontal.

Using your newly acquired knowledge in Optics, you estimate that the yellowish sodium emission should be within 500 – 600 nm, and 0.1 nm resolution should be sufficient. In order to make sure the spectrometer will achieve this resolution over the specified wavelength range, answer the following questions. (a) Briefly explain how the spectrometer works and describe the function of

each of its elements. (4 marks) (b) What is the required chromatic resolving power (R)? (1 mark) (c) Given a 1-cm-wide diffraction grating with 400 lines/mm, what is the

minimum diffraction order you need to use? (2 marks) Suppose you choose to use the 5th-order diffraction for your measurement in the given configuration (Fig.3-1), (d) What is the free spectral range you will obtain? (1 mark)

(e) Derive the relationship between θ and the output wavelength λ. (3 marks) (f) Find the range of θ for the operating wavelength range of 500 – 600 nm. (1 mark)

(g) Derive the relationship between a small angle change (∆θ) and the corresponding change in output wavelength (∆λ). (2 marks)

(h) What is the minimum beam transverse coherence length required? (3 marks) (i) What is the maximum input slit width? (3 marks) (j) What is the maximum output slit width? (4 marks)


Q4. In the following set up, a tunable Fabry-Pérot filter allows a user to add/drop any one of the 8 channels (Fig. 4-1) by tuning the angle of the device (Fig. 4-2).

100 GHz

ν (THz)192.0 192.7192.1 192.2 192.3 192.4 192.5 192.6ν1 ν8ν2 ν3 ν4 ν5 ν6 ν7

100 GHz

ν (THz)192.0 192.7192.1 192.2 192.3 192.4 192.5 192.6

100 GHz

ν (THz)192.0 192.7192.1 192.2 192.3 192.4 192.5 192.6ν1 ν8ν2 ν3 ν4 ν5 ν6 ν7

Figure 4-1 Central frequencies of the 8 channels

nf

ν1, ν2, … ν8

ν1, ν2, … ν8 νj Dropνj Add

θj

Input/Output Port Add/Drop Port

d

n=1 n=1nf

ν1, ν2, … ν8


θj


d

nf

ν1, ν2, … ν8


θj


d

n=1 n=1

Figure 4-2 Using angle tuning to selectively add/drop any one of the 8 channels.

Using a dielectric material of refractive index nf .0= 2 , design such a filter to meet the

following specifications: • At the Add/Drop port, the user must be able to select any one of the 8 channels and

reject all the rest by tuning the device to an appropriate angle θj (j = 1, 2, ...8). θj is defined with respect to the vertical, as shown in Fig. 4-2.

• The angle-tuning step, i.e., the angle difference between any two adjacent channels (∆θj, j+1) must not be less than 10-3π (or 0.18o).

• The total tuning range, |θ8−θ1|, must be less than 0.05 π (or 9o). • The filter’s FWHM bandwidth must be greater than 10 GHz. • When a given channel is selected, the filter’s transmittance at all other channel

frequencies must be less than 0.01. (That is, channel cross-talk must be < –20dB.) You may use small angle approximation for θ. Answer the following questions: (a) What is the minimum required free spectral range of this F-P filter? (2 marks) (b) What is the maximum thickness of the filter, dmax, for normal incidence? (2 marks) (c) What is the relationship between the angle θ and the frequency ν at the

add/drop port? (2 marks) (d) For a small angle change, ∆θ, from θ, what is the corresponding frequency

change, ∆ν, from ν? (3 marks) (e) If the minimum angle-tuning step is 10-3π, what is the maximum value θ can

take? Represent this value by θmax. (3 marks) (f) Make θ8 = θmax and d = dmax, calculate θ1. (4 marks) (g) For the parameters used in (f), calculate the minimum finesse. (3 marks) (h) For the parameters used in (f), calculate the maximum finesse. (3 marks)

ECE426F Optical Engineering - University of Torontoqianli/publications/ECE318_2007... · ·...

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