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Experiment 4: The Fabry-PerotInterferometer

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 partiallyreflecting surfaces aligned with each other in such a way that many waves of lightderived from the same incident wave can interfere. The resulting interference patternsmay 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 adistance d as shown in Figure (4.1)1. Let n be the index of refraction of the mediumbetween the mirrors.

A plane monochromatic wave is incident on the FabryPerot plate at an angle .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 thedirection 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 and is there divided intotwo plane waves, one transmitted in the direction B1E1, the other reflected back atthe Fabry-Perot in the direction B1A2. This process of division of the wave remaininginside the plate continues as shown in Figure (4.1). The total transmitted field canbe calculated by adding the contributions from each of the transmitted waves. Tocarry out this sum we need to know their relative phases and amplitudes. These canbe 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 polishedsurfaces. This form of the Fabry-Perot is referred to as an etalon.

cvText Box

2 Fabry-Perot Interferometer

W

A1

B1

A2

E1C1

A3

A4 E2

E3

E4

E1

E2

E3

E4

Eod

Transmittedwaves

Incidentwave

nn1=1

B2

B3

B4

n2=1

Reflectedwaves

Single surfacereflectivity 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, theoptical paths of E2 and E1 differ by

nB1A2B2 B1C1 =2nd

cos 2d

cossin sin =

2nd

cos 2d

cossin(n sin)

=2nd

cos

(1 sin2

)= 2ndcos . (4.1)

This path difference corresponds to a phase difference

2 = 2 2nd cos

=4nd cos

=2nd cos

c. (4.2)

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

IR

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

ErefEinc

= R , (4.4)

EtransEinc

=

1R . (4.5)

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

The field of a plane wave is of the form

E(t) = E0eit+ikz (4.6)

where k = 2n/ 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 = E0eit . (4.7)

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

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

1R

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

E1 = (1R)E0ei(tk A1B1) . (4.8)

where the distance A1B1 = d/ cos. Writing the constant phase k A1B1 = kd/ cos =0 we can write E1 as

E1 = (1R)E0ei(t0) . (4.9)

To write down E2 we note that the wave transmitted across the first surface isreflected twice inside the plate before being transmitted. Its amplitude will thusgather amplitude factors

1R (

R) (R)

1R = (1R)R and an additionalphase factor ei2 relative to E1,

E2 = (1R)RE0ei(t0)ei2 (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(1R)ei(t0)

E2 = E0(1R)Rei(t0)ei2

E3 = E0(1R)R2ei(t0)ei4

E4 = E0(1R)R3ei(t0)ei6

EN = E0(1R)RN1ei(t0)ei(N1)2

(4.11)

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

ET = E1 + E2 + E3 + = E0(1R)ei(t0)

(1 +Rei2 +R2ei4 +

)=E0(1R)ei(t0)

1Rei2. (4.12)

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

IT =I0(1R)2

1 +R2 2R cos 2=

I0(1R)2

1 +R2 2R+ 2R(1 cos 2)(4.13)

=I0(1R)2

(1R)2 + 4R sin2 (4.14)

This can be written in the form

IT =I0

1 + F sin2 , (4.15)

where F is the coefficient of finesse

F =4R

(1R)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 from Eq. (4.2) is given by

=2nd cos

. (4.17)

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

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

[IT ]min =I0

1 + F, =

(p+

1

2

) (4.19)

0.5

0.9

R= 0.1

p (p+1)

Io

1 2 Io

IT

0

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

Figure (4.2) shows the transmitted intensity IT

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

=2nd cos

= integer p , (4.20)

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

6 Fabry-Perot Interferometer

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

Integer p in Eq. (4.20) is referred to as the order of the transmission peak (or thefringe). Note that p has its maximum value for = 0. If the width of the pump beamwas very large, and all the rays of light were incident at the same angle, we wouldnot 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 forother angles as well [see Figure (4.3)]. Consequently, in transmitted light we will seeconcentric rings corresponding to rays entering the FabryPerot at angles 1 and 2, ifEq. (4.20) is satisfied.

12

Side view Front view

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

4.2 FabryPerot Finesse

The peaks in Figure (4.2) are not infinitely sharp because the surfaces cannot be madeperfectly reflecting. This limits the instruments (spectral) resolution [Its ability totell two closely spaced wavelengths or frequencies apart]. A number F called finessedescribes the resolution of the instrument. Its significance is shown in Figure (4.4).Note that successive transmission maxima are separated by = and the peakwidth c 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 beinserted in the interval ,

F = c

(4.21)

p (p+1)

Io

1 2 Io

IT

0

c

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

To determine c, we look for the values of , for which the transmitted intensityis reduced to half of its peak value [See Figure (4.5)]. Near a transmission peak ofintegral order p, the points where the intensity falls to half of its maximum value areat = p 1

2c. At these points we obtain from Eq. (4.15)

1

2I0 =

I01 + F sin2(1

2c)

. (4.22)

This equation easily gives

sin

(1

2c

)=

1F

=1R2R

(4.23)

In most cases of practical interest, the width of the peak is small compared to thefree spectral range, c . This is the case when F is large. We can then use theapproximation sin(1

2c) 12c to obtain

c =2F

=(1R)R

. (4.24)

8 Fabry-Perot Interferometer

p

Io

1 2 Io

IT

0

c

pc/2 p +c/2

Figure 4.5: Peak width c is defined to be the full width at half maximum.

From our definition of finesse (4.20) then we find

F = c

=R

1R=

F

2. (4.25)

As an example, consider a Fabry-Perot consisting of two mirrors with

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