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Class6OK - Read Hecht, from Chapter 6: 6.1 to 6.3 Matrix...

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Matrix method in paraxial optics When a ray passes through a optical system (no matter how complicated) there are three types of processes: 1) translation (i.e. the ray continues in a straight line); 2) refraction ; 3) reflection . The ray transfer matrix has a special form for each one of these three processes. Once we derive these 3 special forms for the ray transfer matrix we can calculate the ray propagation through any optical system (in the paraxial approximation).
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Translation matrix (n=const) α 2 1 Input plane x=x Optical system Output plane Y 2 n d 1 = Τ 1 0 1 n d T: Translation matrix = 1 1 2 2 1 0 1 Y n n d Y n
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Refraction matrix 2 1 Y Y = ) sin( ) sin( 2 1 r i n n θ = R y 1 ) sin( = ≈ϕ ϕ + = 1 i R y n n n i 1 1 1 1 1 + = S n 1 A n 2 Y 1 C θ i θ r θ 1 R θ 2 2 r i n n = 2 1 | | 2 + = r R y n n n r 2 2 2 2 2 + = 2 + = r 2 2 | | = ( θ 2 < 0, by the sign convention)
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Refraction matrix 2 1 Y Y = r i n n θ = 2 1 1 2 1 1 1 2 2 ) ( Y R n n n n + = 2 1 Y Y = = 1 1 1 1 2 2 2 2 1 0 1 Y n R n n Y n = 1 0 1 D R R Y n n n i 1 1 1 1 1 + = R Y n n n r 2 2 2 2 2 + = R : refraction matrix D : refracting power R n n D ) ( 1 2 =
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Matrix method: summary = 1 1 2 2 1 0 1 Y n n d Y n α Translation: (1) = Τ 1 0 1 n d = 1 1 1 1 2 2 2 2 1 0 1 Y n R n n Y n θ = 1 0 1 D R R n n D ) ( 1 2 = Refraction: (2) = 1 1 1 2 2 2 1 0 2 1 Y n R n Y n = 1 0 1 D M R n D 2 = Reflection: (3) (n=n 2 =-n 1 ) Note : det(T)=det(R)=det(M)=1
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Example I: Single lens α 2 1 Y 1 2 d n 3 Surface 1, R 1 Surface 2, R 2 = 1 1 1 1 2 1 2 2 3 2 2 2 3 1 0 1 1 0 1 1 0 1 Y n R n n n d R n n Y n Translation Refraction at surface 2 Refraction at surface 1 (4)
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Example I: Single lens = 1 0 1 1 0 1 1 0 1 1
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This note was uploaded on 10/19/2009 for the course PHY 31 taught by Professor Cebe during the Fall '08 term at Tufts.

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Class6OK - Read Hecht, from Chapter 6: 6.1 to 6.3 Matrix...

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