HW3_soln - ECE 455: Optical Electronics Homework 3...

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Unformatted text preview: ECE 455: Optical Electronics Homework 3 Solutions 1 Optical System Design First, we need to determine the beam waist at the output mirror of the laser cavity, M 2 , so that we can propagate that beam through the distance d 2 , through the lens f 1 and through the distance d 3 to the component to be etched. We will do this by noticing that the radius of curvature of mirror M 1 is R 1 = 300 cm = 3 m. We also know that d 1 = 0 . 8 m. R z = z " 1 + z z 2 # R 1 = d 1 " 1 + z d 1 2 # z = d 1 s R 1 d 1- 1 z = (0 . 6 m) v u u t (3 . 00 m) (0 . 6 m)- 1 = 1 . 2 m The minimum beam waist, w 2 , is related to z via: z = πnw 2 2 λ w 2 = s z λ πn w 2 = v u u t (1 . 2 m)(488 . 0 nm) π (1) = 431 . 7 μ m Now, switching to the q-method, we have that R 2 = ∞ m, so we know that: 1 q 2 = 1 R 2- j λ πnw 2 2 = 1 ∞- j 1 z =- j 5 6 1 m Now, we determine the ABCD matrix to be: T = " 1 d 3 1 #" 1- 1 f 1 1 #" 1 d 2 1 # = " 1- d 3 f 1 d 2 + d 3- d 2 d 3 f 1- 1 f 1 1- d 2 f 1 # Now, we propagate the Gaussian beam through this ABCD matrix, obtaining an expression for q component ≡ q 3 : 1 q 3 = C + D 1 q 2 A + B 1 q 2 = C + D- j 1 z A + B- j 1 z A + B j 1 z A + B j 1 z = AC + BD z 2- j 1 z ( AD- BC ) A 2 + B 2 z 2 1 Ignoring the radius of curvature and picking out only the beam waist portion, we have: λ πnw 3 2 = z ( AD- BC ) A 2 z 2 + B 2 We require that 2 w 3 < . 03 cm. Being careful not to mix units, we have: λ πnw 3 2 > 6 . 904 1 m 6 . 904 1 m < z ( AD- BC ) A 2 z 2 + B 2 If we leave z in units of m, and ensure that B is also in m, then we dispense with units and have: 5 . 753 < AD- BC A 2 z 2 + B 2 We have a great deal of freedom in being able to choose d 2 , d 3 and f 1 . For this reason, I will make a few choices that reduce A , B , C , and D nicely. If I select d 3 = f 1 , this results in A = 0 and: 5 . 753 <- C B = 1 f 1 d 3 = 1 f 1 2 . 1738 > f 1 2 f 1 < . 4169 m Theoretically, any value of d 2 will result in a spot size that fulfills the requirements. However, interestingly enough, making these choices for d 3 and f 1 simplifies the expression for the radius of...
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This note was uploaded on 02/28/2012 for the course ECE 455 taught by Professor Eden,j during the Fall '08 term at University of Illinois, Urbana Champaign.

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HW3_soln - ECE 455: Optical Electronics Homework 3...

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