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• Experiment 4: The Fabry-Perot Interferometer

The FabryPerot interferometer, simply referred to as the Fabry-Perot, is an impor- tant application of multiple wave interference in optics. It consists of two partially reflecting surfaces aligned with each other in such a way that many waves of light derived from the same incident wave can interfere. The resulting interference patterns may be used to analyze the spectral character of the incident beam.

4.1 The Fabry-Perot Equation

Consider a Fabry-Perot consisting of two parallel reflecting surfaces, separated by a distance d as shown in Figure (4.1)1. Let n be the index of refraction of the medium between the mirrors.

A plane monochromatic wave is incident on the FabryPerot plate at an angle φ. Let E0A1 be the ray representing the direction of propagation of the incident wave. At the first surface, this wave is divided into two plane waves, one reflected in the direction of A1E ′1, and the other transmitted into the plate in the direction A1B1. This latter wave is incident on the second surface at angle θ and is there divided into two plane waves, one transmitted in the direction B1E1, the other reflected back at the Fabry-Perot in the direction B1A2. This process of division of the wave remaining inside the plate continues as shown in Figure (4.1). The total transmitted field can be calculated by adding the contributions from each of the transmitted waves. To carry out this sum we need to know their relative phases and amplitudes. These can be calculated as follows.

We note that the phase of each transmitted wave differs from that of the preceding

1In its simplest form a Fabry-Perot could be a glass plate of fixed thickness with parallel polished surfaces. This form of the Fabry-Perot is referred to as an etalon.

cv Text Box

• 2 Fabry-Perot Interferometer

W θ φ

A1

B1

A2

E1 C1

A3

A4 E2

E3

E4

E1

E2

E3

E4ʹ′

ʹ′

ʹ′

ʹ′

Eo d

Transmitted waves

Incident wave

φ

nn1=1

B2

B3

B4

n2=1

Reflected waves

Single surface reflectivity R

Figure 4.1: A Fabry-Perot gives rise to multiple beam interference both in transmis- sion and reflection.

wave by an amount corresponding to the difference in their paths. For example, the optical paths of E2 and E1 differ by

nB1A2B2 −B1C1 = 2nd

cosφ − 2d

cosφ sinφ sin θ =

2nd

cosφ − 2d

cosφ sinφ(n sinφ)

= 2nd

cosφ

( 1− sin2 φ

) = 2ndcosφ . (4.1)

This path difference corresponds to a phase difference

2δ = 2π × 2nd cos θ λ

= 4πnd cos θ

λ = ω2nd cos θ

c . (4.2)

We can now write down the relative amplitudes of transmitted waves. We shall assume that both reflecting surfaces are identical and each surface, when it alone is present, reflects a fraction R of the intensity of the light

I R

Iinc = R . (4.3)

• Fabry-Perot Interferometer 3

Then the ratios of reflected-to-incident and transmitted-to-incident electric field am- plitudes at each interface are 2

Eref Einc

= − √ R , (4.4)

Etrans Einc

= √

1−R . (4.5)

We are now ready to calculate the transmission of the Fabry-Perot. We will write down all the fields at the same instant of time t.

The field of a plane wave is of the form

E(t) = E0e −iωt+ikz (4.6)

where k = 2πn/λ and z is the distance of propagation. Let us set z = 0 at point A1, at the input face so that the incident wave can be written as

Einc = E0e −iωt . (4.7)

The transmitted field amplitude across the first surface, according to Eq.(4.5), is √

1−R E0 and after crossing the second surface, it is √

1−R · √

1−RE0 = (1−R) E0. The transmitted field E1 at B1 is then

E1 = (1−R)E0e−i(ωt−k A1B1) . (4.8)

where the distance A1B1 = d/ cosφ. Writing the constant phase k A1B1 = kd/ cosφ = δ0 we can write E1 as

E1 = (1−R)E0e−i(ωt−δ0) . (4.9)

To write down E2 we note that the wave transmitted across the first surface is reflected twice inside the plate before being transmitted. Its amplitude will thus gather amplitude factors

√ 1−R (

√ R) ( √ R) √

1−R = (1−R)R and an additional phase factor ei2δ relative to E1,

E2 = (1−R)RE0e−i(ωt−δ0)ei2δ (4.10)

2The minus sign takes into account the phase change at reflection from a denser medium.

• 4 Fabry-Perot Interferometer

Similar considerations for other waves lead to the following expressions for the trans- mitted waves

E1 = E0(1−R)e−i(ωt−δ0)

E2 = E0(1−R)Re−i(ωt−δ0)ei2δ

E3 = E0(1−R)R2e−i(ωt−δ0)ei4δ

E4 = E0(1−R)R3e−i(ωt−δ0)ei6δ

· · · EN = E0(1−R)RN−1e−i(ωt−δ0)ei(N−1)2δ

(4.11)

If the incident wave and the plates are wide enough and reflectivity is high there will be a large number of contributions. For all practical purposes we can take the number of transmitted waves to be infinitely large. The total transmitted field is then obtained by summing the infinite geometric series 3

ET = E1 + E2 + E3 + · · · = E0(1−R)e−i(ωt−δ0)

( 1 +Rei2δ +R2ei4δ + · · ·

) = E0(1−R)e−i(ωt−δ0)

1−Rei2δ . (4.12)

Since the (time averaged) intensity is proportional to the modulus squared of the field amplitude, the transmitted intensity IT is given in terms of the incident wave intensity I0 as

IT = I0(1−R)2

1 +R2 − 2R cos 2δ =

I0(1−R)2

1 +R2 − 2R+ 2R(1− cos 2δ) (4.13)

= I0(1−R)2

(1−R)2 + 4R sin2 δ (4.14)

This can be written in the form

IT = I0

1 + F sin2 δ , (4.15)

where F is the coefficient of finesse

F = 4R

(1−R)2 , (4.16)

3An infinite geometric series 1 + x + x2 + x3 + · · · with x < 1 has the sum 1/(1− x).

• Fabry-Perot Interferometer 5

and the phase δ from Eq. (4.2) is given by

δ = 2πnd cos θ

λ . (4.17)

From Eq. (4.15) we see that the transmitted intensity is a periodic function of δ that varies between a maximum and a minimum as δ changes

[IT ]max = I0 , δ = pπ , p an integer (4.18)

[IT ]min = I0

1 + F , δ =

( p+

1

2

) π (4.19)

0.5

0.9

R= 0.1

δ

pπ (p+1)π

Io

1 2 Io

IT

0

Figure 4.2: Fabry-Perot transmission as a function of δ.

Figure (4.2) shows the transmitted intensity I T

as a function of δ. Note that the peaks get narrower as the mirror reflectivity (and therefore the coefficient of finesse F ) increases. When peaks are very narrow, light can be transmitted only if the plate separation d, refractive index n, and the wavelength λ satisfy the precise relation

δ = 2πnd cos θ

λ = integer× π ≡ pπ , (4.20)

otherwise no light is transmitted. It is this property that permits the Fabry-Perot to act as very narrow band-pass filter for fixed d.

• 6 Fabry-Perot Interferometer

If the incident light contains many wavelengths of varying intensities, we can analyze its spectrum (wavelength/frequency and intensity) by scanning the length d of the Fabry-Perot because for a given separation d, the Fabry-Perot transmits only the wavelength that satisfies Eq. (4.20). In this mode the Fabry-Perot is referred to as a spectrum analyzer. Also note that as we scan the Fabry-Perot the transmission pattern will repeat when δ increases by π.

Integer p in Eq. (4.20) is referred to as the order of the transmission peak (or the fringe). Note that p has its maximum value for θ = 0. If the width of the pump beam was very large, and all the rays of light were incident at the same angle, we would not receive any light at angles other than those satisfying Eq. (4.20). In practice, the incident light often has some divergence so that the relation (4.20) is satisfied for other angles as well [see Figure (4.3)]. Consequently, in transmitted light we will see concentric rings corresponding to rays entering the FabryPerot at angles θ1 and θ2, if Eq. (4.20) is satisfied.

θ1 θ2

Side view Front view

Figure 4.3: The output of a Fabry-Perot illuminated by a diverging set of rays consists of concentric rings.

4.2 FabryPerot Finesse

The peaks in Figure (4.2) are not infinitely sharp because the surfaces cannot be made perfectly reflecting. This limits the instrument’s (spectral) resolution [Its ability to tell two closely spaced wavelengths or frequencies apart]. A number F called “finesse” describes the resolution of the instrument. Its significance is shown in Figure (4.4). Note that successive transmission maxima are separated by ∆δ = π and the peak width δc is defined be the full width at half maximum (FWHM). The finesse for

• Fabry-Perot Interferometer 7

the instrument is defined as the maximum number of resolvable peaks that can be inserted in the interval π,

F = π δc

(4.21)

δ

pπ (p+1)π

Io

1 2 Io

IT

0

δc

π

Figure 4.4: Finesse is a measure of the sharpness of transmission peaks.

To determine δc, we look for the values of δ, for which the transmitted intensity is reduced to half o

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