Page 440 - Engineering Electromagnetics, 8th Edition

422 ENGINEERING ELECTROMAGNETICS

material of index n, whose thickness, l,is set to transmit wavelengths which satisfy
the condition λ = λ 0 /n = 2l/m. Often we want to transmit only one wavelength,
not several, as (37) would allow. We would therefore like to assure that adjacent
wavelengths that are passed through the device are separated as far as possible, so
that only one will lie within the input power spectrum. In terms of wavelength as
measured in the material, this separation is in general given by
2l 2l 2l . 2l
λ m−1 − λ m = λ f = − = =
m − 1 m m(m − 1) m 2
Note that m is the number of half-wavelengths in region 2, or m = 2l/λ = 2nl/λ 0 ,
where λ 0 is the desired free-space wavelength for transmission. Thus

. λ 2 2
λ f = (43a)
2l
In terms of wavelength measured in free space, this becomes
. λ 2 0
λ f 0 = n λ f = (43b)
2nl
λ f 0 is known as the free spectral range of the Fabry-Perot interferometer in terms
of free-space wavelength separation. The interferometer can be used as a narrow-
band filter (transmitting a desired wavelength and a narrow spectrum around this
wavelength) if the spectrum to be filtered is narrower than the free spectral range.

EXAMPLE 12.4
Suppose we wish to filter an optical spectrum of full width λ s0 = 50 nm (measured
in free space), whose center wavelength, λ 0 ,isin the red part of the visible spectrum
at 600 nm, where one nm (nanometer) is 10 −9 m. A Fabry-Perot filter is to be used,
consisting of a lossless glass plate in air, having refractive index n = 1.45. We need
to find the required range of glass thicknesses such that multiple wavelength orders
will not be transmitted.
Solution. We require that the free spectral range be greater than the optical spectral
width, or λ f 0 > λ s . Using (43b)

λ 2 0
l <
2n λ s0
So
600 2 3
l < = 2.5 × 10 nm = 2.5 µm
2(1.45)(50)
where 1µm (micrometer) = 10 −6 m. Fabricating a glass plate of this thickness or less
is somewhat ridiculous to contemplate. Instead, what is often used is an airspace of
thickness on this order, between two thick plates whose surfaces on the sides opposite
the airspace are antireflection coated. This is in fact a more versatile configuration
because the wavelength to be transmitted (and the free spectral range) can be adjusted
by varying the plate separation.