HW12 - ECE604 Homework 12 Out Tuesday(Session 27 Due...

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Unformatted text preview: ECE604 Homework 12 Out: Tuesday, April 20, 2004 (Session 27) Due: Tuesday, April 27 (Session 29) [Students with tape delays: due according to session number] Announcements: class evaluation forms will be filled out in class on Thursday, April 22 Final exam will be held on May 4, 2004, 1:00 pm — 3:00 pm, POTR 262 Problem numbers are from the text by Ramo, Whinnery and Van Duzer (3rd edition, 1994). 1. See attached 2) 3.19c 3) 12.3d 4) 12.3 g 5) 12.5c 6) 12.6a 7) 12.6b Problem #1 Consider a symmetrical optical resonantor (Fig. 1a) formed by two parallel mirrors with reflectivities R = r2, where R is the power reflectivity and r is the field reflectivity. For this problem you will model the resonator as a series of dielectrics (Fig. 1b) with impedances 711,712, and 111, respectively. Regions 1 and 3 are assumed to be made of the same material. The ratio 711/712 is chosen to be consistent with the field reflectivity r, i.e., we take r = n1 _n2 . T11+le ' F344)" ~zsrz’ f ' ‘2’} Wffiwfleflk ' ‘ awkgj a) For part (a) we will: assume the”r'esonance condition is satisfied, i.e., k2! = mn, where k2 is the propagation constant in region 2. As discussed in class, when k2! = mar, there is no reflected wave in region 1. On resonance we may write the E—field in the three regions as a function of 2 as follows: E1 = E1+e'}klz , z < 0 E2 = E2+C—jkzz +E2_C+jkzz , 0 < Z < Z E3 = E3+C~jk1(Z—[) , g < Z Similar expressions may be written for H. i) Use the boundary conditions at z = O to obtain expressions for E2+ and E2— in terms of E1+. Then use the boundary conditions at z = E to obtain an expression for E3+. ii) Use these results to obtain expressions for the output time—averaged intensity P3+, and the intensities of the forward travelling and backward travelling waves in region 2 (P2+ and P2- , respectively), all in terms of the input time—averaged intensity PH and the power reflectivity R. Also express P2- in terms of P2+. Then evaluate your expressions numerically for R = 99%. iii) Comment on the physical meaning of all the expressions from (ii). (b) Now we no longer assume the resonant condition, i.e., we do not assume k2! = mn. Instead we wish to calculate the fraction of the power transmitted as a function of frequency. Plot the fraction of the power transmitted assuming R = 99%, E = 150 um, and refractive index equal to one for optical frequencies in the range 2.975 x 1014 Hz to 3.025 x 1014 Hz. (Hint: there is no need to repeat the calculations of part (a). Rather, this problem can be solved easily by using the impedance transformation formula and writing a simple numerical code). Compare the width of your resonant peaks with the simple expression discussed in class. ...
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