10.5 Dielectric Thin-Film Filter (TFF)
419
To see how it works, consider a light signal that is incident on the left surface S 1
of the etalon. After the light passes through the cavity and hits the inside surface S 2
on the right, some of the light leaves the cavity and some is reflected. The amount of
light that is reflected depends on the reflectivity R of the surface S 2 . If the roundtrip
distance between the two mirrors is an integral multiple of a wavelength λ (i.e.,
λ, 2λ, 3λ, etc.), then all the light at those wavelengths that pass through the right
facet add in phase. This means that these wavelengths interfere constructively in
the device output beam so they add in intensity. These wavelengths are called the
resonant wavelengths of the cavity. The etalon rejects all other wavelengths.
10.5.1 Applications of Etalon Theory
The transmission T of an ideal etalon in which there is no light absorption by the
dielectric mirrors is an Airy function given by
T =
1 +
4R
(1 − R)
2
sin
2
ϕ
2
−1
(10.53)
where R is the reflectivity of the mirrors (the fraction of light reflected by the mirror)
and ϕ is the roundtrip phase change of the light beam. If one ignores any phase
change at the mirror surface, then the phase change for a wavelength λ is
ϕ =
2π
λ
2n D cos θ
(10.54)
where n is the refractive index of the dielectric layer that forms the mirror, D is the
distance between the mirrors, and θ is the angle to the normal of the incoming light
beam.
Figure 10.24 gives a generalized plot of Eq. (10.53) over the range − 3π ≤ ϕ ≤ 3π
for three different reflectivities (R = 0.4, 0.7, and 0.9). Because ϕ is proportional to
the optical frequency f = 2π/λ, Fig. 10.24 shows that the power transfer function T
is periodic in f (or λ). The wavelength range that can pass through the filter is called
the passband. The peaks of the spacings occur at those wavelengths that satisfy the
condition Nλ = 2nD, where N is an integer. Thus in order for a single wavelength
or a narrow spectral segment to be selected by the etalon filter from a wider spectral
range, all the wanted wavelengths must lie within a passband of the filter transfer
function. The distance between adjacent peaks (shown in Fig. 10.24) is called the free
spectral range or FSR. The FSR can be expressed in terms of either the frequency
or the wavelength. Respectively, these expressions (and two common notations) are
FSR ν = ν F S R =
c
2n D
(10.55)
419
To see how it works, consider a light signal that is incident on the left surface S 1
of the etalon. After the light passes through the cavity and hits the inside surface S 2
on the right, some of the light leaves the cavity and some is reflected. The amount of
light that is reflected depends on the reflectivity R of the surface S 2 . If the roundtrip
distance between the two mirrors is an integral multiple of a wavelength λ (i.e.,
λ, 2λ, 3λ, etc.), then all the light at those wavelengths that pass through the right
facet add in phase. This means that these wavelengths interfere constructively in
the device output beam so they add in intensity. These wavelengths are called the
resonant wavelengths of the cavity. The etalon rejects all other wavelengths.
10.5.1 Applications of Etalon Theory
The transmission T of an ideal etalon in which there is no light absorption by the
dielectric mirrors is an Airy function given by
T =
1 +
4R
(1 − R)
2
sin
2
ϕ
2
−1
(10.53)
where R is the reflectivity of the mirrors (the fraction of light reflected by the mirror)
and ϕ is the roundtrip phase change of the light beam. If one ignores any phase
change at the mirror surface, then the phase change for a wavelength λ is
ϕ =
2π
λ
2n D cos θ
(10.54)
where n is the refractive index of the dielectric layer that forms the mirror, D is the
distance between the mirrors, and θ is the angle to the normal of the incoming light
beam.
Figure 10.24 gives a generalized plot of Eq. (10.53) over the range − 3π ≤ ϕ ≤ 3π
for three different reflectivities (R = 0.4, 0.7, and 0.9). Because ϕ is proportional to
the optical frequency f = 2π/λ, Fig. 10.24 shows that the power transfer function T
is periodic in f (or λ). The wavelength range that can pass through the filter is called
the passband. The peaks of the spacings occur at those wavelengths that satisfy the
condition Nλ = 2nD, where N is an integer. Thus in order for a single wavelength
or a narrow spectral segment to be selected by the etalon filter from a wider spectral
range, all the wanted wavelengths must lie within a passband of the filter transfer
function. The distance between adjacent peaks (shown in Fig. 10.24) is called the free
spectral range or FSR. The FSR can be expressed in terms of either the frequency
or the wavelength. Respectively, these expressions (and two common notations) are
FSR ν = ν F S R =
c
2n D
(10.55)