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Wave Optics Experiment 1: Microwave Standing Waves

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Wave Optics Experiment 1: Microwave Standing Waves Wave Optics This project includes four independent experiments. The first three, I think, are fun and surprising. The final experiment is tedious but teaches you about useful optical devices (quarter-wave plates and half-wave plates); half-wave plates are used in the quantum entanglement experiment, for example. Experiment 1: Microwave Standing Waves Objective: To infer that standing waves are created by the superposition of incident and reflected waves between a microwave receiver and transmitter. Introduction: As students varied the distance between a microwave receiver and transmitter, they recorded the data shown in Figure 1.1. Figure 1.1. Microwave intensity at the receiver as a function of distance between the receiver and the transmitter. (Data taken by N. Cuccia.) Why in the world should the intensity behave like this? It's certainly not a "snapshot" of the traveling wave emitted by the transmitter. How did we derive an equation to fit the data? I'll get you started. Suppose that a (sector of a) spherical wave is emitted by the transmitter. Then, the electric field along the line between the transmitter and the receiver is A E cos(kx t) , (1.1) 1 x where A is a constant, and you probably know what everything else is. A little reflection (ha ha!) will convince you that the wave reflected off the receiver is rA E cos[k(2L x) t] , (1.2) 2 2L x where r is the amplitude reflection coefficient, and L is the distance between the transmitter and the receiver. What's the significance of (2L-x)? It's the total distance traveled by the wave. (If the reflected wave has gone halfway back, to x = L/2, it's traveled a total distance of 2L-x = 3L/2.) Your job is to determine E3, the first wave reflected back off the transmitter toward the receiver. Then, find the superposition of E1 and E3. (If you want to add additional terms for additional reflections, have fun! I used only E1 and E3. When is this approximation valid?) If E(L,t) = E1(L,t) + E3(L,t) (1.3) is (approximately) the electric field at the receiver, then the detected intensity is proportional to the time average of [E(L,t)]2. Here's a suggestion: To vastly simplify the algebra, replace every cos(something) with ei(something). Then, the detected intensity will be proportional to E(L,t) × [E(L,t)]*. You don't even need to take a time average because time won't be in the result. (Why does this give the same result as the harder-math, real-number-only calculation? We can prove it if you want.) Experiment: Collect data, such as what's shown in Figure 1.1. There are a couple challenges. First, if the transmitter and receiver have the same polarization angle, the intensity will be way off scale. You need to adjust the relative polarization angle until it's close to 90. Second, the needle on the receiver sometimes fluctuates. I think this happens more when the intensity is low. One solution is to try to choose a range of L and a polarization angle so that the intensity is always fairly high (on the 30X scale). Another, fancier solution is to use Malus's Law to enable you to adjust the polarization as needed to keep the intensity high. To measure distance between the transmitter and receiver, use the positions of the T marking on the base of the transmitter, and the R marking on the base of the receiver. Curve fitting: We can do the curve fitting by running the Python script, standing_microwaves_fit.py. Open this file in Spyder (the software that runs Python programs). First, after the word “return” in the indented line, type the expression you derived for intensity (use x for distance, and name your three fitting parameters k, R, and a). If you found that intensity simply equals 1/x, line 20 would read “return 1/x”. The symbol ** represents exponentiation, so 1/x**2 means 1/x2. You will probably need to narrow the bounds on the fitting parameters in the line after “return.” The initial range is 0 through infinity, but we can help the program by narrowing the range. What should be the maximum allowed value of R? Can you estimate k from your data? Instead of typing your data into Spyder, I think it’s easier if you enter it first in Excel, in two columns (distance and intensity). Run the Python script and follow the onscreen instructions to copy your data from Excel. If you don’t get a good fit, try adjusting the bounds. Questions to answer in your lab report: What values do you find for k and r? Are these reasonable? Does the value of r2 justify our exclusion of additional reflections? What is the microwave wavelength? Conceptually, what causes maxima and minima in your data? Quantitatively, does your equation make sense? Look at some limiting cases. If r = 0, does the dependence of intensity on L make sense? If r = 1, does it make sense? In this case, 1/L2 is multiplied by a term that varies from 4/9 to 16/9. Why do these fractions make sense? Experiment 2: Interference in a Glass Slide Objective: To determine the thickness of a glass slide by counting interference fringes. Theory: If you shine light on a glass slide, some will reflect off the front surface, and some will reflect off the back surface (after traveling through the slide). The interference between these two reflected beams produces dark fringes. These fringes appear as dark lines that "slide" through the reflected red spot as you vary the angle of incidence (i). You will show that the condition for destructive interference is as follows: 2dncost m , (2.1) where d and n are the thickness and refractive index of the slide, m is any whole number, is the wavelength of illumination, and t is the angle of transmission. Equation (2.1) can be rewritten as cos m . (2.2) t 2dn Equation (2.2) says that a new fringe line appears every time cost changes by /(2dn). So if you plot cost vs. fringe number, you should obtain a line with slope -/(2dn). (The minus sign occurs because I counted fringes while increasing t; if you counted fringes while lowering t, there would be no minus sign). This is shown in Figure 2.1. Thus if and n are known, the slope can be used to determine d. We know that = 632.8 nm, but n must be determined. One of the simplest ways to determine n is to find the Brewster angle (also called polarization angle): the angle at which there is zero reflection of light polarized in the plane of incidence (the plane of incidence is the plane containing the light ray and the normal to the interface). At this angle, the Fresnel coefficient rP is 0. According to the standard derivation that you can study online, tani t rP . (2.3) tani t According to Equation (2.3), rP is 0 when i + t = 90. Thus if you experimentally determine the i that minimizes reflected light, you can combine i + t = 90 with Snell's Law to find 1.000 -4 0.996 slope = -2.076 x 10 = -/(2dn) ) 0.992 t cos( 0.988 0.984 0.980 0 20 40 60 80 Fringe number Figure 2.1. Linear relationship between cost and fringe number. Instructions: Align the glass slide so that the light is reflected directly back into the laser aperture. This is i = 0. Keeping the glass slide in this position, set the vernier scale to 0. Now vary the angle of incidence and observe the reflected beam. You should see dark interference fringes "sliding" through the reflected red spot as you vary i. At some arbitrary but small i, center the dark fringe in the reflected spot and record i. Call this fringe number 0. Now increase i and count fringes. Record i as a function of fringe number at least every five fringes over a minimum range of 80 fringes. To determine n, shine the incident light through a linear polarizer so that the light incident on the slide is polarized parallel with the plane of incidence. Record i at which the reflected intensity is minimized. Directly measure the thickness of the glass slide. In your lab report: Determine n from the measured angle at which rP is minimized. You recorded i as a function of fringe number. Compute cos(t) for each i to generate a plot like Figure 2.1. From the slope, determine d. Compare with direct measurement (with micrometer). Derive Equation (2.1). Hints: o Referring to Figure 2.2, write AD and AC+CD in terms of d and 2. o Next write AB in terms of d, 2, and 3. Eliminate 3 by first writing it terms of 1 and then using Snell’s law. o Next determine the optical path length difference between the two rays, remembering that the wavelength depends on the index of refraction. o Remember to think about phase change due to reflection. 1 2 1 B A 3 D d 2 C Figure 2.2. Diagram used to derive Equation (2.1). Experiment 3: Fresnel Reflection of a Light Cone Objective: To investigate the refraction and Fresnel reflection of a polarized light cone, through direct observation and analytical calculation. Theory: Figure 3.1. Schematic diagram of the experiment. The small arrows sticking out of the light rays represent polarization (vertical, horizontal, or both).
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