The Fabry-Perot Cavity

The Fabry-Perot cavity is just about the most fundamental building block of the Michelson-based gravitational wave interferometer. This Appendix will derive some useful formulas.

Figure A.1 shows the model of the Fabry-Perot cavity used in this Appendix, and the fields which will be of interest: the reflected, pickoff, and transmitted fields. The

I I

. c . c

Reflected PD Pickoff PD

Figure A.1: The model of a Fabry-Perot cavity. The three fields of interest are marked by pickoffs and respective photodiodes. These are the reflected, pickoff, and transmitted fields. The mirror reciprocity convention used is the real, anti-symmetric convention, and the signs for the mirror reflectivities are indicated. The phase of the cavity, ¢, is taken to be round trip.

fields at the reflected and pickoff photodiodes both have some method of guiding the light to the photodiode, such as a low reflectance pickoff optic placed in the beam.

The efficiency of this cavity field-to-photodiode mechanism is usually less than one (especially for the pickoff field). In this Appendix the efficiency will be assumed to be unity, that is the fields discussed are the direct, inline fields of the cavity.

Probably the most cited characteristic of a cavity is the finesse, F. This is given in any standard optics text,[65] although the typical assumption is that the reflectivity of the front and back mirrors are equal. Removing this assumption, the finesse, which

characterizes the fringe width relative to the spacing of the fringes, is

(A.l)

The behavior of the cavity is a function of the mirrors and the round trip phase of the light. The cavity is periodic in its behavior as the phase increments by 21r radians.

This defines a quantity known as the free spectral range, which is the frequency at which the phase changes by 21r for a given length.

c fJsr

=

2[ (A.2)

where lis the one way length of the cavity, and the phase is given by

cp =

2?r2lf jc.

A.l The Reflected Field

The reflectivity, as a function of the mirror parameters and the round trip phase is given by

Tfrbe-i4>

Tc

=

^{T j - .}

1 - r frbe-•4>

r f - A frbe-i4>

1-rfrbe-i<l>

(A.3)

(A.4)

Lower case r's and t's will be used for amplitude reflectivity and transmissivity, while upper case R's and T's will be used for power reflectance and transmittance. The quantity A ₁ is given by

(A.5)

and is associated with the losses of the front mirror. Resonance is satisfied for an integer 21r propagation phase. The reflectivity of the cavity, which is a complex

number, can be represented in a magnitude and phase form.

(A.6) (A.7) (A.8)

Examples of the cavity reflectivity are shown in Figure A.2. The two curves in the

.: . ....... ; ..

..... ..... ·······••#•'•

_ R

1=0.8, f\ ^=0.8

__ R₁ = 0.8, f\ = 0.9 .... · .. . . ·> ••...••••.. ·:· ...••. ···:

_ . R1 = 0.9, f\ = 0.8

-2 -1 0 2 3

. . . . . . . ·~ . :1 .. . :. ' \ .

-3 -2 -1 0 2 3

Round trip phase ¢

Figure A.2: Examples of cavity reflectivity for the cases where the mirrors are matched Tt

= n,

the front mirror has higher transmission than the back Tt >

n,

^{and when}

these mirrors are reversed.

magnitude plot which correspond to R1

=

0.8 and Rb

=

0.9 and the converse are identical. Clearly, however, their phases are not. It needs to be noted that, on resonance, the cavity reflectivity is less than either of the individual cavity mirrors, whereas the reflectivity is greater than either of the individual cavity mirrors for most of the phase range away from resonance. This is a general feature of all cavities.

A.l.l Cavity Coupling

One of the curves shows the reflectivity going to zero at the resonance point. From Eq. (A.3), this happens when

(A.9)

One way to view this is that the reflectivity of the front mirror is equal to a modified back mirror whose reflectivity is decreased by a factor due to losses. Another, perhaps more enlightening approach, is to first assume the losses are small. Then Eq. (A.3) can be solved for the transmittance of the front mirror which satisfies this condition.

(A.lO)

This condition is called "optimally coupled," or "impedance matched." Stated ver- bally, this condition says the transmittance of the front mirror is equal to the sum of the losses of the cavity, which includes the transmittance of the back mirror.

The optimally coupled point is a discontinuity in the reflectivity of a cavity, as a function of the front mirror parameters. For values _{T 1} > ^Tfovt, the cavity is referred to as "over-coupled." Contrariwise, a cavity with ^TJ < ^Tfovt is referred to as "under- coupled." The reflectivity of the cavity on resonance as a function of the front mirror parameters is shown in Figure A.3. As the front mirror departs from the optimally coupled point, the cavity reflectivity increases to either the reflectivity of the back or front mirror, whichever is highest. Perhaps the most interesting thing is the discon- tinuity in phase. The phase of the reflectivity of a resonant under-coupled cavity is positive, and the reflectivity phase of the over-coupled cavity is negative.

Figure A.2 can also be re-visited in this context. The two cases, R1

=

0.8%, Rb

=

0.9%, and R 1

=

0.9%, Rb

=

0.8% correspond to over- and under-coupled cavities, respectively. As noted earlier, the magnitude of the reflectivity is the same as a function of the cavity phase. However, the phase of the reflectivities are very different. For an under-coupled cavity, the phase of the reflectivity only wiggles a little around 0 radians as the cavity phase is swept through resonance. The over-coupled