Fermat&acirc;€™s Principle &acirc;€“ Concave Mirror

# Fermat&acirc;€™s Principle &acirc;€“...

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Fermat’s Principle – Concave Mirror z 0 represents an arbitrary reference plane. () 0 22 az z by f z =− + =++ With the sign conventions for “b”; b is negative as shown: 2 2 f z =− + + Determine the two optical path lengths. Remember that the index changes sign after a reflection: (n = 1 in air) ( ) 0 1 a b OPL na z z OPL n b y f z y f z == + ⎛⎞ =− − + + = ⎜⎟ ⎝⎠ Fermat’s principle requires that the total OPL be constant: ab OPL OPL Constant += z a b y -f z 0 z = f(y)

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This sum can be evaluated exactly along the optical axis (z = 0 at y = 0): 0 ab OPL OPL z f += + Equate this value with the total OPL for an arbitrary y: () 2 2 00 22 2 2 2 2 4 4 zf zz y f z fz y ff z z y z z yf z y z f −+=
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Fermat&acirc;€™s Principle &acirc;€“...

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