Solution Set 1 Phys 4510 Optics Fall 2014

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1 Solution Set 1 Phys 4510 Optics Fall 2014 Due date: Tu, September 9, in class Scoring rubric 4 points/sub-problem, total: 40 points 3: Small mistake in calculation or formula 2: Correct formula but calculation error 1: Partial credit depend on the answer Reading: Hecht, skim Ch. 2, thoroughly read Ch 3 introduction, 3.1, 3.2, 3.3.1, 3.3.2, 3.4.1, Tell me in a few words about your main interests in physics, your current or possible plan to do research in optics, your plan to go to graduate school or industry or other after graduating, etc. This will help me fine tune topics we cover. 2. We derived the wave-equation for the electric field E(y, t) which reads for the 1D case: 2 E(y, t) y 2 1 c 2 2 E(y, t) 2 = 0 (1) a) Show that the ansatz E(y, t) = E 0 cos(ωt ky + ϕ) is a solution of the wave equation, with wavevector k, angular frequency ω, and phase ϕ. What are the units of k, ω, ϕ, and c? What is the relationship between k and ω (it is called dispersion relation)? A straightforward substitution demonstrates that the ansatz is valid as long as the dispersion relation k(ω) = ω/c holds. The argument of the cosine function requires as its input radians, hence we have for the dimensions of k, ω, ϕ, and c [k] = radians length, [ω] = radians time, [ϕ] = radians, [c] = length time. b) What is the spatial and temporal evolution of the plane wave? Discuss, whether and how real physical waves can be represented by plane waves. Clearly at a fixed position in space, the wave evolves sinusoidally in time. At a fixed time, the wave has a sinusoidal spatial variation. Due to the infinite extent in space and time, a plane wave carries an infinite amout of energy and extends infinitely in both space and time and is hence non-physical. 1

2 ky Figure 1: Plot of the spatial distribution of the normalized electric field at a time t (blue) and a time t later (red). c) Sketch the wave at t = 0 and t = t for 0 < t < 2π/ω. What can you say about the propagation direction of the wave? What is the velocity with which the crest of the wave propagates along the y-direction? The sketch is shown in Fig. 1. Now, consider a point on the wave: when the time increases by t, the position increases by y. The condition that we stay at the same place on the wave is equivalent to saying that the cosine takes on the same value. This means that (ωt ky) = ω t k y = 0. Hence we have t/ y = ω/k = c. Thus the wave speed is c. Furthermore, since ω and k have diﬀerent signs in the argument of the cosine function, if t > 0 we must have y > 0 so that any point on the wave is traveling to the right or positive direction. 3. Consider a electromagnetic plane wave propagating in the x-direction, and polarized in the ydirection. a) Show from Maxwell s equations that Bz = We can take Faraday s law B Ez Ex Ez Ex = E =( )ˆ x+( )ˆ y+( )ˆ z y z z y Since the direction of propagating wave is x, non-x derivatives are 0. E = 2 Ez yˆ + zˆ (2)

3 In this case, E z must be 0 because polarization direction of propagating wave is y. So E = E y ẑ Now we know that B only have components in z-direction. So E y = B z b) Let us now assume a specific harmonic wave of the form E y (x, t) = E y,0 cos{ω(t x/c)}. What is the corresponding expression for the magnetic field B z (x, t)? Also show that E y = cb z. E y = ω c E y,0sin(ωt ωx/c) = B z So we can derive expression of magnetic field And we know the relationship is B z (x, t) = 1 c E y,0cos(ωt ωx/c) cb z = E y c) Sketch in a single 3D graph E y and B z as they propagate in the x-direction. What is the phase relationship between E y and B z? Lastly calculate the corresponding Poynting vector (component) S x. The electric and magnetic field oscillate in phase, but their direction of oscillation is perpendicular (Sketch is shown in Fig. 2). S = 1 µ 0 E B = 1 µ 0 c (E y,0cos(ωt ωx/c)ŷ) (E y,0 cos(ωt ωx/c)ẑ) = (E y,0) 2 cos 2 (ω(t x/c))ˆx µ 0 c 4. This problem shall give you a feeling for intensities, field amplitudes, and frequencies of light in the visible and infrared. Consider two cw laser sources, (1) a green, e.g., HeNe laser (λ = 543 nm), and (2) an IR, e.g., CO 2 laser (λ 10µm), both with a power of 10 mw each. a) Calculate for each laser light in vacuum the frequency, the wavevector, the photon energy (in ev, and in multiples of kt at room temperature), and the intensity in number of photons per second. Note on units: quantum energy is most conveniently given in ev, wavevector in cm 1 ). The relationship between λ, ν, and c is given by λν = c. From this it is easy to calculate the frequencies resulting in ν He:Ne = Hz, 3

4 y z x Figure 2: Sketch of the electromagnetic field propagation. ν CO2 = Hz. For the photon energy, we have another plug-and-chug relation: E γ = hν where h is Planck s constant. Using the value of this constant h = ev s we find E He:Ne = 2.28eV, E CO2 = 0.124eV. It is interesting to compare these energies to a thermal energy unit k b T eV at room temperature. Note, we thermally emit infrared light whose energy roughly, say within a factor of 2 or 3, corresponds approximately to k b T body eV giving λ 46µm. Calculating the wave vector is a simple task using the relation k = 2π/λ giving (of course after appropriate changes in units) k He:Ne = cm 1, k CO2 = cm 1. Finally we want to calculate the intensity in number of photons per second. We are given the beam power in the problem, which is the amount of energy crossing a plane perpendicular to the beam axis each second. Thus if we knew the energy for one photon, we could divide the power by this number to get the number of photons per second passing through a plane perpendicular to the beam. We have already calculated this number for both of our beams. Applying the result we find I He:Ne = photons, s 17 photons I CO2 = s 4

5 b) Assume you focus the light using appropriate optics down to a spot size of 10µm diameter. What is the irradiance (aka intensity) [most convenient in W/cm 2 ]? What is the electric field amplitude? What is the corresponding magnetic field amplitude? The spot radius is 5µm, thus the power per unit area or intensity I = P/A is I = W cm 2. The electric field amplitude can be calculated using I = cnϵ 0 2 E 0 2 = n 2Z 0 E 0 2. Inputing the approprate values for free space and sloving for E 0 we find E 0 = V m. And the corresponding magnetic field amplitude is given by B 0 = E0 c B 0 = T. Note that interatomic fields are roughly of the order V/m, so with a.1mw laser we won t be displacing the electrons far from their equilibrium position. As a result, the electron experiences a linear restoring force so that we are doing linear optics as in problem 5. With much stronger fields of the order of the interatomic field, we will begin to see the effects of nonlinearities in the restoring force which leads into topics of nonlinear optics where effects such as harmonic generation can occur. c) Now we use a 100 W light bulb converting say 2% of its electric power into light. Distribute the radiation evenly over a solid angle Ω = 1 sterad by a reflector. Calculate the light intensity and electric field of the light field at a distance of 1m from the bulb. Our light produces 2W of light all of which is collected into a solid angle of 1sterad. Solid angle is given by Ω = S/r 2. where S is the surface area of a spherical cap of radius r. So we wish to find S at r = 100cm which is just Ωr 2 thus I = W cm 2. E 0 = 38.8 V m. d) Discuss how intensity and field strength compare if you now were to use sunlight collected with a 5 cm diameter lens, still focused to a spot size of 10µm diameter (For this problem start from your knowledge of the luminosity of the sun of L = W, and the distance of the earth from the sun of km; neglect atmospheric effects). We can use proportional relation to derive luminosity of the sun in the lens : 4π( ) 2 = P : π(2.5) 2 5

6 Then we can simply claculate intensity P = 2.708W And we know I = cnϵ0 2 E 0 2 = n 2Z 0 E 0 2, thus I = P A = W π( = ) 2 cm 2 E 0 = V m. 6

Class XII Chapter 8 Electromagnetic Waves Physics

Class XII Chapter 8 Electromagnetic Waves Physics Question 8.1: Figure 8.6 shows a capacitor made of two circular plates each of radius 12 cm, and separated by 5.0 cm. The capacitor is being charged by an external source (not shown in the figure). The