ECE 5368/6358 - HW 4
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1. Transmission and reflection of a laser beam

Suppose you do an experiment, using a 1-W laser beam to probe a sample as illustrated in the left figure below.

The laser beam polarization is TE. The glass slab has a refractive index=1.5 with no absorption. You can consider the beam as 1-D Gaussian as in LectSet 1, or just treat it as a generic parallel beam with no diffraction. (It doesn’t matter what the beam width is, as long as it is >> the beam wavelength). 
Calculate and plot the transmitted power and reflected power as a function of incident angle from 0 to 70 degree. What is the beam linear intensity on the experimental sample (the beam is 1-D Gaussian)?

Note: the beam has 1 W power. Plot your power in unit of W also. Use the right unit. Hint: use this APP also for clue.

2A. Transmission and reflection of a Gaussian laser beam

Refer to the geometry of Problem 1 above, are the transmitted and reflected beams Gaussian? One can quickly prove that the reflected beam is Gaussian with the beam waist exactly the “mirror” image of the incident beam. 
More challenging is the transmitted beam. If it is or approximately so, what is the approximated beam waist (Subscript[w, t]) and location? (You can try doing this for an easier answer: let θ=0 and see what the transmitted beam is).

Important notes

Answer the question in anyway you like to the best of your capability and go as far as you can do. Remember, let’s stick to the Maxwell’s equations and whatever derived results we have so far in the lectures - If you use some equation from somewhere, make sure including the proof of that result from Maxwell’s theory.

Example: in the lecture, we derive rigorously that:
Given TE incident plane wave:         (Subscript[E, inc])=  (e^(i (β z+(Subscript[k, 1]) x-ω t)))  
Then:                             (Subscript[E, trans])= ((2 (Subscript[k, 1]))/((Subscript[k, 1])+(Subscript[k, 2]))) (e^(i (β z+(Subscript[k, 2]) x-ω t)))  ;     (Subscript[E, refl])= (((Subscript[k, 1])-(Subscript[k, 2]))/((Subscript[k, 1])+(Subscript[k, 2]))) (e^(i (β z-(Subscript[k, 1]) x-ω t)))   
(LectSet 4, Eqs. 2.2.12, 2.2.13, 2.3.5 and 2.2.14). You can use these results because they are proven.
Remember also the time-averaged intensity from LectSet1. 
                                          I=|⟨(Overscript[S, →])⟩|=|(c/(8 π))Re[ (Overscript[E, ->]) ×(Overscript[(SuperStar[H]), ->])]|   
For plane wave, it is:        I=|⟨(Overscript[S, →])⟩|=(c/(8 π))(Sqrt[(ϵ/μ)]) ((|E|)^2)  ( LectSet1, Eq. 6.2.2c)

Another example: You can be theoretically rigorous and use approximation at the same time as long as the approximation is justifiable. You should recall that the 1-D Gaussian laser beam is:
                           E[x,z,t]=(OverHat[y]) A (( 1)/(Sqrt[1+((2 i z)/((w^2) (Subscript[k, 0])))]))   (e^(i z (Subscript[k, 0])-((x^2)/((w^2)+((2 i z)/(Subscript[k, 0])))))) (e^(-i ω t))    (LectSet1 - Eq. 7.2.6) 
Or (dropping (e^(-i ω t))):        E[x,z]=(OverHat[y]) A (( 1)/(Sqrt[1+i(z/(Subscript[ζ, 0]))]))  (e^(i (Subscript[k, 0])z ))   (e^(-((x^2)/((w^2) (1+i(z/(Subscript[ζ, 0])))))))    
where (Subscript[ζ, 0]) is the Rayleigh range. 
Here, we specifically make it polarized along (OverHat[y]) so that we have TE polarization. Remember also that this formula is for the coordinate with z along the beam axis, and x transverse to the beam axis. Here, the beam is tilted by θ on the glass slab.

Now, you can argue that because w>>λ, the beam can be approximated as plane wave as far as the phase front is concerned. This is a justifiable approximation. However, the beam cannot have infinite power like a plane wave, hence the approximation is not justifiable with regard to this aspect. You must decide and argue rigorously how to deal with this and not “guessing.”

One more note: if you choose θ=0, remember that in our coordinate above, x is the propagation and z is the transverse. Thus, the  θ=0 Gaussian beam is:  (Subscript[E, y])[x,z]= A (( 1)/(Sqrt[1+i(x/(Subscript[ζ, 0]))]))  (e^(i (Subscript[k, 0])x ))   (e^(-((z^2)/((w^2) (1+i(x/(Subscript[ζ, 0])))))))     (swapping “x” and “z”)

2B. Focusing a Gaussian beam in a medium

Consider a focused 1-D Gaussian beam as in LectSet 1. It propagates in air as shown in the left hand side figure along x-axis (top-down), and has the following properties: λ=0.532 μm, beam waist radius w=0.5 μm, at location x=2.5 μm.

E[z,x,t]=  A (( 1)/(Sqrt[1+((2 i x)/((w^2) (Subscript[k, 1])))]))   (e^(i x (Subscript[k, 1])-((z^2)/((w^2)+((2 i x)/(Subscript[k, 1])))))) (e^(-i ω t))               (7.2.6)

Now, a slab of sapphire (dielectric material with index of refraction n=1.77) is inserted perpendicularly to the beam axis as shown on the right hand side. Consider the slab as very large and infinite for x ≥ 0.
Determine what happens to the Gaussian beam. Use appropriate Gaussian paraxial approximation and Taylor’s series approximation if needed.
(50 pts): Prove and describe the beam with all appropriate parameters 
(50 pts): Plot your results.

3. Transmission and reflection through a cube beam splitter

A TE-polarized laser beam that can be approximated as a plane wave is incident at (90^o) on one side of a (45^o) glass prism as shown. The two short sides of the prism are AR-coated (hence, no reflection), the long side is bare.

3.1 (10 pts) Describe what happens to the beam

Do a simple calculation: calculate the critical angle for total internal reflection and decide if the beam would refract into air on the long side or 100%-reflect to the other short side. Use lecture app to plot the beam.

3.2 (10- 30 pts) Cube beam splitter transmission and reflection: for TE

A second identical prism is placed such that the two long sides are parallel to each other with an adjustable air gap d. You can do one of the following (both if you wish for higher total score):

1- (20 pts) Calculate an expression for the TE transmission coefficient t and reflection r as a function of air gap d.  (Assume the laser wavelength is λ). Then, plot ((Abs[r])^2), ((Abs[t])^2)  as a function of d from 0 to 1.5 λ (Hint: use the formula in the lecture with correct substitution, or you can derive the whole thing using transfer matrix method similar to what we did for the soap film).


2- (10 pts) Execute the calculation modules below and slowly vary d (be very patient to allow the computer to calculate). Observe what happens (transmission and reflection), then write a paragraph describing the phenomenon, what you observed and explain in plain English (no equation needed) what you think happening. Do explain scientifically, your score depends on your observation and thinking here. Copy and paste from the Internet WON’T HELP your score here.

Demo

3.3 (10- 30 pts) Waves in cube beam splitter transmission and reflection.

1- (20 pts) Calculate and plot the E field along the CC’ axis. The plane wave is infinite in all dimensions of course, not apertured like what shown in demo for the clarity purpose.

2- (10 pts) If you cannot obtain the answer for 1, just execute the code below and discuss what’s going on.

At the very minimum, identify what each colored curve in the plot represents (what wave in what region?). Then explain “how” the waves behave that way. (“How” here means the mechanism that we learn in this chapter).

3.4 (10 pts) is it possible to do any thin-film coating on the long side of the prism such that the beam in 3.1 will be transmitted and no reflection?

i. e. would AR-coating be possible to eliminate TIR? Must have a theoretically rigorous answer for full credit.

3.5 (10-40 pts) Do question 3.2 for TM mode.

(discussion: do you note any difference between transmission efficiency of TE vs. TM?).
10 pts for using the app below to describe what happens. 30 pts for derive and plot the transmitted power and reflected power as a function of the gap between the prism.

4. TIR of a Gaussian laser beam with Goos-Hanchën shift

Consider the below that we have seen in demonstration.

But now, the laser beam is 1-D Gaussian as in LectSet 1, and its polarization is TE or TM (your choice) (Both are qualitatively similar, the only difference is the phase shift). Assume the beam waist is much larger than the wavelength and the beam has a very long Rayleigh range (the beam is like a parallel beam). You can choose whatever index of refractions of the media for your illustration.
Calculate the reflected beam in the range of incident angle much larger than critical angle, but also smaller than 90 degree. What do you observe? (Goos-Hanchën shift).

6. (HW continued from Lecture ) (50 pts) Fresnel rhomb λ/4-rotator

We discuss Fresnel rhomb and how it works in lecture. If you do a research on various Fresnel rhomb on the market and applications, with technical specification, you can get extra credit.

6.1 Reflectivity of TIR

We obtain from lecture that the reflection coefficient of E field is:
                        (Subscript[r, TE])=(((Subscript[k, 1])-(Subscript[k, 2]))/((Subscript[k, 1])+(Subscript[k, 2])))     for TE  ;   and      (Subscript[r, TM])=-(((Subscript[ϵ, 2])(Subscript[k, 1])-(Subscript[ϵ, 1])(Subscript[k, 2]))/((Subscript[ϵ, 2])(Subscript[k, 1])+(Subscript[ϵ, 1])(Subscript[k, 2])))      for TM
 For TIR, we also show in class that: (Subscript[k, 2])=i (Subscript[κ, 2]) where:
                       (Subscript[k, 1])=(Sqrt[(Subscript[ϵ, 1])((Subscript[k, 0])^2)-(β^2)])=(Subscript[n, 1]) (Subscript[k, 0]) Cos[(Subscript[θ, inc])]   ;    β=(Subscript[n, 1]) (Subscript[k, 0]) Sin[(Subscript[θ, inc])]  ; and (Subscript[κ, 2])=(Sqrt[(β^2)-(Subscript[ϵ, 2])((Subscript[k, 0])^2)])  
Show that the magnitude (absolute value) of (Subscript[r, TE]) and (Subscript[r, TM]) is equal to 1 for TIR.

6.2 Phase shift of the reflection E-field

The phase shift of a reflection field is defined as the the phase of the reflection coefficient. Obtain the phase shift for both TE and TM. (You can use ArcTan, ArcCos, or ArcSin function).

6.3 Are the E-field phase shift in 6.2 the same for both TE and TM?

Explain.

6.4 Calculate for a Fresnel rhomb

Use the following values: incident medium: air, (Subscript[ϵ, 1])=1, Fresnel rhomb medium is a glass with (Subscript[n, 2])=1.494 ((Subscript[ϵ, 2])=((Subscript[n, 2])^2)). Let (Subscript[θ, inc]) = (52.1^o). Show that (Subscript[E, y]) component (transverse) is advanced by 3π/4 (or 3λ/8) relative to (Subscript[E, //]) component. (check your answer with the APP below). 
This is for one reflection. There are two reflections like that in a Fresnel rhomb, hence what is the net shift between (Subscript[E, y]) and (Subscript[E, //])?
Note that you can input a linear polarization beam with either at (45^o) or (135^o) orientation relative to the normal plane of the rhomb (the axis convention is arbitrary). If one orientation gives you output polatization (SuperPlus[σ]), then, what output does the other orientation give you?

8. Design and study thin-film filter

To get full credit, you must:
a- Do a research about applications of thin-film filters and cite at least one example.
b- Design a thin film filter (you can use the app below) with the following characteristics: center wavelength: 1.55 μm; rejection ratio to 50-GHz side channels equal or better than 40 dB. Maximum top ripple < 1 dB. 

Note: design means you must provide exact and sufficient information so that anyone with a thin-film deposition machine can make the filter. Hence, you must tell the exact structure: the number of layers and thickness of each layer. Then, you must provide calculation results that validate your design.
For the app below, the substrate index is 1.5. The two layer indices are 1.8 and 1.3. For the design of resonance cavities, look at lecture notes.

Answer:    M1=M2= 10; Mcouple=20