What is chromatic resolving power of a fabre parot interferometer?
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2. Fabry–Perot resolution
2.1. How a Fabry–Perot interferometer works
The Fabry–Perot interferometer consists of two glass plates with parallel plane surfaces,
separated at a distance d. The media between the glass plates is air (n = 1). If a
monochromatic wave impinges upon the plate at angle , multiple reflections are generated.
If the inner surfaces are covered by a highly reflective film, the reflections in the glass plates
are negligible. Therefore, only the interferences produced by the multiple beams in the air
plate are observable. The intensity of the interference patterns is described by the following
expression:
I = a2
1 + 4r2
(1−r2)2 sin2 δ
2
-
, (1)
where a is the amplitude of the incident wave, r is the reflection coefficient of the coating film
and δ is the phase difference between two consecutive waves. Let -
be the optical length
difference. In this case, the optical length is -
= 2d cos . As usual, the optical length and
phase difference are related by δ = 2π
λ -
.
The intensity depends on the thickness d, the reflection coefficient r, the wavelength λ, the
intensity a2 of the incident plane wave and the incidence angle . If the light comes from an
extended source from all possible directions , and taking into account that the geometry of the
light distribution only depends on , the intensity pattern should exhibit rotational symmetry.The concept of resolving power in the Fabry–Perot interferometer 1113
The intensity maximum (Imax = a2) is obtained if the following condition is verified:
sin2 δ
2 = 0 thus δ = 2mπ and -
= 2d cos = mλ, (2)
where m = 0, ±1, ±2,.... The intensity profile therefore exhibits a sequence of maxima and
minima and, consequently, the interference pattern is characterized by a set of light circles.
The index m in the previous equation labels each circle. For instance, the maximum m value
is found when tends to zero. In particular, if a maximum is found when the incidence angle
is zero ( = 0), then m = 2d/λ.
2.2. The Rayleigh criterion
As we have explained before, the intensity pattern is a function of the wavelength. Each
wavelength coming from a light source generates a set of light circles. The resolving power
is a measure of the ability to discriminate between sets of circles generated by different
wavelengths. Moreover, it is also necessary to define mathematically a separation criterion
between two very close maxima. This criterion represents one’s visual ability to distinguish
two concentric circles.
Different resolution criteria are described in physical optics textbooks. Several authors
such as Born and Wolf [1] and Hecht [2] use Rayleigh’s criterion [6]. It was first introduced
by Lord Rayleigh in 1879 to determine whether two diffraction spots can be distinguished or
not. The criterion states that two intensity maxima are separated if the maximum value of the
first spot is superimposed on the first minimum of the second spot.
Rayleigh’s resolution limit seems to be rather arbitrary and is based on resolving
capabilities of the human eye. This limit was set to guarantee a clear distinction between
two close spots using the eye. When visual inspection is replaced by detectors, other less
restrictive criteria can be used. For instance, Sparrow’s criterion states that two diffraction
spots are just distinguished when the minimum between the two intensity maxima of the
composite intensity is undetectable [7].
Rayleigh’s criterion cannot be directly applied to the Fabry–Perot intensity profile because
of the slow decrease of these values. Consequently, the minimum is located far from
the maximum. To avoid this problem, an alternative definition is proposed: let Iλ(x) and
Iλ+-
λ(x) equal the intensity profiles along a diameter, generated by wavelengths λ and λ + -
λ
respectively [1]. Two maxima are resolved if the minimum value of Iλ(x) + Iλ+-
λ(x) verifies
the following condition:
min{Iλ(x) + Iλ+-
λ(x)} = 0.81 max{Iλ(x)}. (3)
When the Rayleigh criterion is applied to diffraction in a slit, we obtain the condition
shown in equation (3).
hope it is useful...!!
2.1. How a Fabry–Perot interferometer works
The Fabry–Perot interferometer consists of two glass plates with parallel plane surfaces,
separated at a distance d. The media between the glass plates is air (n = 1). If a
monochromatic wave impinges upon the plate at angle , multiple reflections are generated.
If the inner surfaces are covered by a highly reflective film, the reflections in the glass plates
are negligible. Therefore, only the interferences produced by the multiple beams in the air
plate are observable. The intensity of the interference patterns is described by the following
expression:
I = a2
1 + 4r2
(1−r2)2 sin2 δ
2
-
, (1)
where a is the amplitude of the incident wave, r is the reflection coefficient of the coating film
and δ is the phase difference between two consecutive waves. Let -
be the optical length
difference. In this case, the optical length is -
= 2d cos . As usual, the optical length and
phase difference are related by δ = 2π
λ -
.
The intensity depends on the thickness d, the reflection coefficient r, the wavelength λ, the
intensity a2 of the incident plane wave and the incidence angle . If the light comes from an
extended source from all possible directions , and taking into account that the geometry of the
light distribution only depends on , the intensity pattern should exhibit rotational symmetry.The concept of resolving power in the Fabry–Perot interferometer 1113
The intensity maximum (Imax = a2) is obtained if the following condition is verified:
sin2 δ
2 = 0 thus δ = 2mπ and -
= 2d cos = mλ, (2)
where m = 0, ±1, ±2,.... The intensity profile therefore exhibits a sequence of maxima and
minima and, consequently, the interference pattern is characterized by a set of light circles.
The index m in the previous equation labels each circle. For instance, the maximum m value
is found when tends to zero. In particular, if a maximum is found when the incidence angle
is zero ( = 0), then m = 2d/λ.
2.2. The Rayleigh criterion
As we have explained before, the intensity pattern is a function of the wavelength. Each
wavelength coming from a light source generates a set of light circles. The resolving power
is a measure of the ability to discriminate between sets of circles generated by different
wavelengths. Moreover, it is also necessary to define mathematically a separation criterion
between two very close maxima. This criterion represents one’s visual ability to distinguish
two concentric circles.
Different resolution criteria are described in physical optics textbooks. Several authors
such as Born and Wolf [1] and Hecht [2] use Rayleigh’s criterion [6]. It was first introduced
by Lord Rayleigh in 1879 to determine whether two diffraction spots can be distinguished or
not. The criterion states that two intensity maxima are separated if the maximum value of the
first spot is superimposed on the first minimum of the second spot.
Rayleigh’s resolution limit seems to be rather arbitrary and is based on resolving
capabilities of the human eye. This limit was set to guarantee a clear distinction between
two close spots using the eye. When visual inspection is replaced by detectors, other less
restrictive criteria can be used. For instance, Sparrow’s criterion states that two diffraction
spots are just distinguished when the minimum between the two intensity maxima of the
composite intensity is undetectable [7].
Rayleigh’s criterion cannot be directly applied to the Fabry–Perot intensity profile because
of the slow decrease of these values. Consequently, the minimum is located far from
the maximum. To avoid this problem, an alternative definition is proposed: let Iλ(x) and
Iλ+-
λ(x) equal the intensity profiles along a diameter, generated by wavelengths λ and λ + -
λ
respectively [1]. Two maxima are resolved if the minimum value of Iλ(x) + Iλ+-
λ(x) verifies
the following condition:
min{Iλ(x) + Iλ+-
λ(x)} = 0.81 max{Iλ(x)}. (3)
When the Rayleigh criterion is applied to diffraction in a slit, we obtain the condition
shown in equation (3).
hope it is useful...!!
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