Page 156 - Principles and Applications of NanoMEMS Physics

144 Chapter 3

εµω 2 εµω 2
' k = i + k + k − k , 2 + k > . (196)
2
2
2
z x y 2 x y 2
c c
Then, using the formula for the transmission coefficient of a slab of width d,
i.e.,

T = ' tt exp ( 'ik z ) d , (197)
S 2
1− ' r exp ( ik2 ' ) d
z
where,

t and t’ are the vacuum/medium and medium/vacuum transmission
coefficients and r and r’ the corresponding reflection coefficients, given by,

2µ k µ k − k '
t = z r , = z z , (198)
µ k + k ' µ k + k '
z z z z

and

' t = 2 ' k z ' r , = ' k − µ k ' z . (199)
z
' k + µ k ' k + µ k
z z z z
If both the permeability and permittivity approach negative unity, then the
transmission coefficient becomes,
)
( 'ik
exp
d
' tt
lim T = 2 z ) = exp ( ik− ' z d ) = exp ( ik− z d ) . (200)
S lim
µ → − 1 µ → − 1 1 − ' r exp ( ik2 ' z d
ε → − 1 ε → − 1
Since k is imaginary, see (196), (200) is a growing exponential and the
z
wave is amplified.
By contrast, in a normal lens the large transverse wave vector of
propagating waves are evanescent and decay prior to reaching the focus, thus
the incomplete spectral contents makes it impossible to identically
reconstruct the image.