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MASSACHUSETTS INSTITUTE OF TECHNOLOGY2.710 Optics Fall ’05Problem Set #5 Posted Wednesday, Oct. 19, 2005 — Due Wednesday, Oct. 26, 20051. Frequencies and wavelengths1.a) MIT’s WMBR radio station transmits at frequency 88.1MHz. What isWMBR’s wavelength in air?1.b) Microwave antennas, especially those of radio stations, are often of the“λ/2” type, i.e. the height of the antenna equals one half of the wavelength.If an antenna is 10m tall, what is the frequency that it is transmitting?1.c) An obsessive flutist keeps playing the A tone (440Hz) on WMBR. What isthe wavelength of that sound?1.d) Soft x–rays have a wavelength of 1nm. What is the frequency?1.e) According to quantum mechanics, particles can also be thought of as waves.The “dispersion relation” for these waves, is different than that for lightwaves that we saw in class; instead, particles obey the “de Broglie” disper-sion relationship, which isλ =hmv,where λ is the wavelength associated with the particle, m is the particle’srest mass, v is the (non-relativistic) velocity, and h is Planck’s constant.Based on this equation, what is the electron’s wavelength?1.f) Water waves also obey a different dispersion relation, given byω2= gk tanh (kd) ,where, as usual, ω = 2πν, k = 2π/λ; g is the acceleration of gravity, and dis the depth of the water. If the depth of a pool is 3m, and the wavelengthis 10cm, what is the frequency of the water waves?2. Plane waves and phasor representations Throughout this problem, by “com-plete expression” of a wave we mean the space-time representation, e.g. f(x, y, z, t) =A cos(kx−ωt) is a plane wave of wave-vector magnitude k and frequency ω prop-agating in the direction of the ˆx coordinate axis. By “phasor expression” wemean the complex representation of the wave, e.g. Aeikxfor the same wave.2.a) Write the complete and phasor expressions for a plane wave f1(x, y, z, t)propagating at an angle 30◦relative to the ˆz axis on the xz-plane (i.e.,the plane y = 0). The wavelength is λ = 1µm, and the wave speed isc = 3 × 108m · sec−1.2.b) Write the complete and phasor expressions for a plane wave f2(x, y, z, t)of the same wavelength and wave speed as f1but propagating at angle 60◦relative to the ˆz axis on the yz-plane.2.c) Use the complete expression to plot f1(x, y, z = 0, t = 0) and f2(x, y, z =0, t = 0) using Matlab. (Note: you will probably need to use surf or anequivalent command.)2.d) The plane z = 0 is illuminated by the superposition of the two waves f1and f2. Plot the waveform received at points A, B, C, D, E with Cartesiancoordinates, respectively,(0, 0, 0),µ14, −14√3, 0¶,µ12, −12√3, 0¶,µ34, −34√3, 0¶,µ1, −12√3, 0¶(all units in microns) as function of time. What do you observe?3. Superposition of scalar waves Given two wavesE1(z, t) =E0cos (k1z − ω1t) ,E2(z, t) =E0cos (k2z − ω2t) ,where ω1= 100Hz, ω2= 101Hz and the wavenumbers k1, k2obey the usualdispersion relation for light waves.3.a) Show that the superposition can be expressed as a product of two tones,one at frequency lower than both the waves, and one at higher frequency.3.b) Plot your result. What do you observe?4. Superposition of polarized waves Given two wavesE1(z, t) =ˆxE0cos (kz − ωt) ,E2(z, t) =ˆyE0cos (kz − ωt) ,where E0is a constant electric field magnitude, and k, ω are the wavenumberand angular frequency of the waves.4.a) What is the superposition of the two waves E1, E2? What is the type ofits polarization?4.b) Now suppose the second wave E2is delayed by π/ 2. What is the polariza-tion of the superposition wave?4.c) Now suppose that, in addition, the second wave E2is attenuated to 0.717E0.If you were to trace the tip of the superposition electric field vector, as inslide #21 of lecture 7–a, what would the locus inscribed by the tip look


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MIT 2 710 - Problem Set 5

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