E o e associated with a light field (both the real part and the. ikr t. under the assumptions that J free

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1 Reiew Problems Chaters 1-5 True and False Questions E1. T or F: The otical index of any material aries with frequency. E2. T or F: The frequency of light can change as it enters a crystal. E3. T or F: The entire exression ikr -wt) E o e associated with a light field both the real art and the imaginary arts) is hysically releant. E4. T or F: The real art of the refractie index cannot be less than one. E5. T or F: s-olarized light and -olarized light exerience the same hase shift uon reflection from a material with comlex index. E6. T or F: When light is incident uon a material interface at Brewster's angle only one olarization can transmit. E7. T or F: When light is incident uon a material interface at Brewster's angle one of the olarizations stimulates dioles in the material to oscillate with orientation along the direction of the reflected k -ector. E8. T or F: The critical angle for total internal reflection exists on both sides of a material interface. E9. T or F: From aboe the surface of water, it is ossible to see objects ositioned anywhere under the water. E10. T or F: From beneath the surface of water, it is always ossible to see objects aboe the water. E11. T or F: An eanescent wae traels arallel to the surface interface on the transmitted side. E12. T or F: When -olarized light enters a material at Brewster s angle, the intensity of the transmitted beam is the same as the intensity of the incident beam. E13. T or F: For incident angles beyond the critical angle for total internal reflection, the Fresnel coefficients t s and t are both zero. E14. T or F: As light enters a crystal, the Poynting ector always obeys Snell's law. E15. T or F: As light enters a crystal, the k -ector does not obey Snell's for the extraordinary wae. Problems E16. a) Write down Maxwell's equations. b) Derie the wae equation for E under the assumtions that J free 0 and P eo c E. Note: 2 f ) f ) - f. c) Show by direct substitution that ikr t Ert, ) -w ) Ee o is a solution to the wae equation. Find the resulting connection between k and w. Gie aroriate definitions for c and n, assuming that c is real. d) If k kzˆ and E E xˆ, find the associated B -field. o o 95

2 e) The Poynting ector is S E B m o, where the fields are real. Derie an exression for I. S t E17. A horizontal and a ertical olarizer are laced in series, and horizontally olarized light with È Jones ector 1 Í 0 enters the system. Horizontal Vertical È1 Í0 È1 0 Í0 0 È0 0 Í0 1 Fast Axis Quarter- 45 o Wae 45 o Plate 1 È1 1 2 Í1 1 1 È 1 -i 2 Í -i 1 a) What is the Jones ector of the transmitted field? b) Now a olarizer at 45 o is inserted between the other two olarizers. What is the Jones ector of the transmitted field? How does the final intensity comare to initial intensity? c) Now a quarter wae late with a fast-axis angle of 45 o is inserted between the two olarizers instead of the olarizer of art b)). What is the Jones ector of the transmitted field? How does the final intensity comare to initial intensity? Fast axis x q slow axis z È1 Í 0 Half-wae late È1 0 Í 0 0 E18. a) Find the Jones matrix for half wae late with its fast axis making an arbitrary angle q with the x -axis. HINT: Project an arbitrary olarization with E x and E y onto the fast and slow axes of the wae late. Shift the slow axis hase by, and then roject the field comonents back onto the È 2 2 cos q - sin q 2sin qcosq horizontal and ertical axes. The answer is Í sin qcosq sin q - cos q 96

3 b) We desire to attenuate continuously a olarized laser beam using a half wae late and a olarizer aligned to the initial olarization of the beam see figure). The fast axis of the half wae late is initially aligned in the direction of olarization and then rotated through an angle q. What is the ratio of the intensity exiting the olarizer to the incoming intensity as a function of q? E19. Consider an interface between two isotroic media. The lane of incidence is shown below. y - axis k r E s E E E q r q t E s z - axis q i x - axis directed into age k t k i E s n i n t The incident field is defined by i E E ) y -z xe ) sin ˆ cosq ˆ sinq ˆ e + i i iki y qi zcosq-wit [ i + s ] [ ] a) By insection of the figure, write down similar exressions for the reflected and transmitted fields i.e. E r and E t ). b) Find an exression relating E i, E r, and E t using the boundary condition at the interface. From this exression obtain the law of reflection and Snell s law. c) The boundary condition requiring that the tangential comonent of B must be continuous leads to n E r - E ) n E - cosq cosq. Use this and the results from i ) t and ni Es Es ) i ntes E tan qi - qt ) - E tan qi + q t ). art b) to derie r You may use the identity E20. The Fresnel equations are Es sinqt cosqi - sinqicosqt rs E sinq s t cosqi + sinqicosqt Es 2sinqt cosqi ts E sinq s t cosqi + sinqicosqt E cosqtsinqt - cosqisinqi r E cosq sinq + cosq sinq sinqicosqi - sinqtcosqt sinq cosq + sinq cosq t t i i i i t t ) tan qi - qt tan q + q i t ). t 97

4 t E E 2cosqi sinqt cosq sinq + cosq sinq t t i i a) Find what each of these equations reduces to when q i 0. Gie your answer in terms of n i and n t. b) What ercent of light intensity) reflects from a glass surface n 15. ) when light enters from air n 1) at normal incidence? c) What ercent of light reflects from a glass surface when light exits into air at normal incidence? E21. Light goes through a glass rism with otical index n The light enters at Brewster's angle and exits at normal incidence. a) Derie and calculate Brewster's angle q B. You may use the results of E19 art c). b) Calculate f. c) What ercent of the light ower) goes all the way through the rism if it is -olarized? Ignore light that might make multile reflections within the rism and come out with directions other than that shown by the arrow. You may use the Fresnel coefficients gien in E20. d) What ercent for s-olarized light? q B n1.55 f o o o E22. A rism is a good deice for reflecting a beam of light arallel to the initial beam. The exiting beam will be arallel to the entering beam een when the incoming beam is not normal to the front surface although it needs to be in the lane of the drawing). n 15. q a) How large an angle q can be tolerated before there is no longer total internal reflection at both interior surfaces? Assume n 1 outside of the rism and n 15. inside. 98

5 b) If the light enters and leaes the rism at normal incidence, what will the difference in hase be between the s and -olarizations? You may use the Fresnel coefficients gien in E20. E23. Second harmonic generation the conersion of light with frequency w into light with frequency 2w ) can occur when ery intense laser light traels through a material. For good harmonic roduction, the laser light and the second harmonic light need to trael at the same seed in the material. In other words, both frequencies need to hae the same index of refraction so that harmonic light roduced down stream joins in hase with the harmonic light roduced u stream. This ensures a coherent building of the second harmonic field rather than destructie cancellations. The achieement of the same index of refraction for both w and 2w is called hase matching. Unfortunately, the index of refraction is almost neer the same for different frequencies in a gien material, owing to disersion. Howeer, we can achiee hase matching in some crystals where one frequency roagates as an ordinary wae and the other roagates as an extraordinary wae. We cause the two indices to be recisely the same by tuning the angle of the crystal. Consider a ruby laser l 694nm ) roagating in a uniaxial KDP crystal otassium dihydrogen hoshate). Remember that the indices of refraction are gien by n o and nn o e, where f is the angle made with the otical axis. At the frequency of a n 2 2 osin f + n 2 2 e cos f ruby laser, KDP has indices n o w ) and n e w ) At the frequency of the second harmonic, the indices are n o 2w ) and n e 2w ) Show that hase matching can be achieed if the laser is olarized so that it exeriences only the ordinary index and the second harmonic light is olarized erendicular to that. At what angle f does this occurs? Selected Answers E17. b) 14, c) 12. E20. b) 4% c) 4%. E21. b) 33 o, c) 95%, d) 79%. E22. a) 48. o, b) 74 o. E o. 99

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