Discussion Question 13A P212, Week 13 Electromagnetic Waves

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1 Discussion Question 13A P1, Week 13 Electromagnetic Waves This problem is a continuation of discussion question 1B from last week. Please refer to your work from last week as necessary. A laser beam travels through vacuum. The electric field of the plane electromagnetic wave produced by the laser has the form given below. The wavelength of the beam is = 514 nm, and the amplitude of the electric field is E 0 =.5 x 10 4 N/C. E (x, y,z,t) y ˆ E 0 cos(kz t 45 ) (a) What is the intensity I of the wave? Be sure to indicate the units of your answer. I I 0 4 E.510 rms Z ( W/m or alternatively E B E B 4 rms rms max max W/m (b) A solar cell is an example of a photo-sensitive detector that absorbs the energy of incident electromagnetic radiation and converts it into useful power (with which to operate your solar-cell calculator, for example). Suppose our electromagnetic wave is incident on a small, square photo-sensitive detector of side mm. You can assume that this detector is smaller than the wave itself, and is oriented so that its surface is perpendicular to the incoming beam. You can also assume that the detector is 100% efficient at converting all the radiative energy incident on it into useful power. What, then, is the average power <P> that our detector would generate? As always, units are an enormous help. The units of your answers to previous parts of the problem (including last week s) tell you how to determine the answer to this question! detector The intensity is the power per unit area. 5 P I A W/m 000. m 33. W

2 Discussion Question 13B P1, Week 13B Intensity of Polarized Radiation Unpolarized light of intensity I 0 is incident upon a stack of N+1 ideal polarizing sheets. The sheets are all parallel to the xy plane, and the light wave is traveling in the +z direction. Each sheet is rotated by an angle of /N radians counterclockwise with respect to the preceding sheet, as viewed from the perspective of the light wave. The transmission axis of the first sheet is parallel to the x-axis. (a) Start by making a sketch of this stack of polarizing sheets (just draw the first three polarizers, to get an idea of what is going on). (b) What is the intensity of the light after it passes through the first sheet? And what is its polarization vector? I = I 0 /, polarized in x-direction. One always drops ½ of the intensity when unpolarized light passes through a linear polarizer (c) What is the intensity of the light after it passes through the second sheet? I = I 0 / cos (/(N) (d) Finally, what is the intensity of the light and the polarization vector after it passes through the last sheet? What is the limiting value of this intensity as the number of sheets gets very large? One gets an additional factor of cos for each sheet Hence with N sheets. N I I 0 N cos 1 angles cos cos N In the limit of small p p( 1 p ) 1 Recall 1+x 1 px x 8 N cos LimN 1 N 1 Lim 1 N 8 4 I0 Hence as N I!! Page 1 of Still not convinced! Here is a plot of N cos as a function of N

3 (e) Find the intensity of the final transmitted light if I 0 = 5 W/m and N=3. I I cos W/m

4 Discussion Question 13C P1, Week 13 Quarter Wave Plates A quarter wave plate (QWP) is an example of a birefringent device: its refractive index n depends on the polarization of incoming light. If light is polarized along the fast axis, it will experience a smaller refractive index but if it is polarized along the slow axis, the refractive index it sees will be larger. Recall that the index of refraction describes the degree to which light slows down in a material: v = c / n. So here is how a birefringent element works: if linearly polarized light passes through the element, the component of the light polarized along the slow axis will be slowed down more that the component along the fast axis. Thus, when the light emerges from the element, its two components will be out of phase with each other. A QWP is a special case of such a device, where the phase shift between the fast and slow components is exactly 90. Further, if the incoming light is polarized at 45 to both the fast and slow axes (so that the fast and slow components are of equal amplitude), the outgoing light will be circularly polarized. Consider a single QWP that lies in the xy plane. It is oriented so that its fast axis lies along the y direction and its slow axis lies along the x axis. We will send light of various types at this QWP, but the light will always be traveling in the +z direction. (a) Find the polarization state of the light transmitted by the QWP in each of the following cases. If your answer is linearly polarized, be sure to specify the direction of linear polarization. 1. The incident light is linearly polarized in the y direction. still linearly polarized in y direction. The incident light is linearly polarized in the direction ( ˆ x ˆ y )/ circularly polarized 3. The incident light is unpolarized. still unpolarized 4. The incident light is circularly polarized. polarized with equal components along x & y directions (b) Given your answer to part 4 above, what sort of device would you have if you glued two identical QWP s together? And what if you glued four of them together?

5 Each QWP changes the phase of the slow component relative to the fast component by a If you do this twice you get a phase change of which is equivalent to changing the sign of the slow component relative to the fast component. This means the polarization direction is reflected about either the fast or slow axis. If you do this 4 times the change is which is equivalent to doing nothing to the polarization. (c) Suppose the incident light has intensity I 0 = 5 W/m. Find the intensity I of the transmitted light in each of the following cases: 1. The incident light is linearly polarized in the y direction.. The incident light is linearly polarized in the direction ( ˆ x ˆ y )/ 3. The incident light is unpolarized. 4. The incident light is circularly polarized. all have intensity 5 W/m. Notice a trend? (d) Now for a challenge: let s design a QWP! You will build your QWP from crystalline quartz (SiO ), a common birefringent material. Your task is to deterime how thick your plate of quartz needs to be to make a QWP. As you will quickly discover, you need one more parameter: the frequency of the incident light QWP s only function properly at certain frequencies. So let s use the blue-green line at = 488 nm produced by an Argon- Ion laser. At this wavelength, the fast and slow indices of refraction of crystalline quartz are n f = and n s = respectively. How thick should your QWP be? Note: the given = 488 nm is the laser s wavelength in vacuum, not in the material. additional hint: set = f - s = /, where is the total angle the waves travels through in the material

6 The phase shift that a wave gets going through a slab of thickness d through the two media are: n n s f s d d ; f d s o o Our condition for a QWP is s f dns nf d n n o s f d m (e) I suspect that you came up with a very small thickness. In practice, optical elements cannot be machined too thin, or they will be mechanically unstable (i.e. they break too easily to be practical!). A conservative minimum thickness is 1 mm. Is there some other thickness that you could use for your QWP that is 1 mm but still works at = 488 nm? In fact, there is a whole series of thicknesses that you can use in your design hint: change / to / + *N We can generalize condition for a QWP by adding integral to the phase difference: 1 N 4 s f dns nf d n n where N is 0,1,, o s f