Reflected and Pickoff Signals

3.3 Signal Sensitivity

3.3.1 Reflected and Pickoff Signals

The simplified RSE interferometer of Figure 3.8 can be simplified even further to a single cavity, the power recycling cavity, when considering the fields actually incident on the various photodiodes. The front mirror is the power mirror, and the back mirror is the compound mirror formed by the signal cavity which has the Michelson mirror as its front mirror and the signal mirror as its back mirror. The fields then are the reflected, pickoff, and transmitted fields of a single cavity, with the appropriate substitutions. Sensitivity to a particular degree of freedom at a particular photodiode can then be formed from the simple relations found for a single cavity. For the reflected and pickoff signals, the derivatives with respect to the degrees of freedom take the

I

Laser

Compound signal cavity mirror

rs(ol>+, q>_, rp_, ¢.)

~-

_{1 I d}

I

0

^{I p t PD3}

I I

' - _ I

Figure 3.12: A single cavity representation of the RSE interferometer. The subscripts on the photodiode transmission functions r P• Pv, and dv indicate application of a cavity reflectivity, pickoff, and transmitted fields of this power recycling cavity. The subscripts on the end mirror indicates the reflectivity of a cavity applied to the signal cavity.

form of the following products.

orv or. orm orac or. orm orac o<I>+

orp o¢+

orv or. orm - - - - - or. orm o¢_

orv or.

- -or. o¢.

(3.44)

Specifically, these are for the reflected signals at the photodiode characterized by the transmission function r p· The subscript p is to indicate that the transmission function is for the effective single power recycling cavity. Similarly, the subscripts refers to the signal cavity, formed by the Michelson mirror and the signal mirror. The subscript m indicates the reflectivity and transmissivity of the Michelson mirror. The pickoff signals are formed by replacing rp subscript with Pv·

Individual Derivatives

The transmission functions to the reflected and pickoff photodiodes, relative to the input, are given by

{3.45) {3.46) These are explicitly functions of ¢+ only, so derivatives with respect to all other degrees of freedom require the functional dependence of the compound signal cavity mirror, as well as the derivative of rv with respect to r8 • All transmission functions and derivatives thereof are evaluated at the phases of the carrier in a broadband RSE.

The exponentials left in the formulas are set to 1 for the carrier, except for the signal cavity phase, which is replaced by the detuning phase. The phases need to be subsequently evaluated for the RF sidebands. The derivatives with respect to ¢+ are

8rv _{- ' l} T. prmrse -i¢+

(3.47) 8¢+ (1

+

r prmr se-icf>+ )2

8pv -itprmr se-i<l>+

{3.48) 8¢+ {1

+

rprmrse-i¢+ )²

and the derivative with respect to the end mirror is

{3.49) The signal cavity reflectivity is

(3.50) It can be noted that, for the carrier, r s reduces to r m regardless of the detuning. The only explicit dependence in r₈ is the signal cavity degree of freedom,

<Ps·

Both arm cavity degrees of freedom, <I>+ and <I>_, as well as

<P-,

are wrapped up in the Michelson

compound mirror. The derivatives of rs with respect to Tm and rl>s are

Finally, the arm cavities are expressed as sums and differences, Tacl

+

Tac2

Tac

=

2

~ Tacl- Tac2

Tac

=

2

(3.51)

(3.52) (3.53)

(3.54) (3.55)

In this analysis, the two arms are assumed to be equal. Hence, evaluated at DC, r ac is simply equal to a single arm cavity, while ~r ac is equal to 0. To express the derivative of the Michelson mirror with respect to the arm cavity degrees of freedom, the following relations are needed.

a a a

- - = - + -

a<I> + a¢3 a¢>4

a a a ^(3.56)

so that

arac =

_!!__(

Tacl

+

^{Tac2) =} arac(r/>ac)

a <I>+ a<I> + 2 a¢>ac

arac

=

_!!__(racl

+

^Tac2)

=

O a<I> _ a <I>_ 2

a~rac

=

_!!__(acl- ^Tac2)

=

O a<I>+ a<I>+ 2

(3.57)

a~rac

=

_!!__(acl-^Tac2)

=

arac(r/>ac)

a <I>- a<I>- 2 a¢>ac

r ac is the reflectivity of a single arm cavity, whose internal path phase is <Pac·

Brae

8</Jac ^(3.58)

It's been assumed here that the end mirror has very nearly unity reflection, retm ~ 1.

Writing the Michelson reflectivity and transmissivity in the following way indicates the functional dependence necessary for taking derivatives with respect to the arm cavity degrees of freedom. It's assumed for this purpose that the beamsplitter is matched, that is rbs

=

tbs.

r ^m

=

Abs(r ac( <I>+) cos( <P-) - i6.r ac( <I>-) sin( <P-)) tm

=

Abs(i rae( <I>+) sin(</J-) - 6.rac(<I>_) cos(</J-))

The losses of the Michelson are expressed in the term Am,

This is simply equal tor~ for the carrier.

(3.59) (3.60)

(3.61)

Every matrix element then will be a function of the derivatives of the fields at either the pickoff or the reflected port. These derivatives are found by applying the rules of Eq. (3.44). Frequently, a useful way to describe the sensitivity is to normalize the derivative of a field with the field itself as

_\t

^g~. This quantity gives the derivative of the phase of the complex quantity t with respect to some general parameter

e.

^When

e

^{is the cavity} phase, this is known as either the internal or external phase gain,⁶ depending on whether the field in question is the pickoff or reflected field. This notion is somewhat generalized here to include derivatives of the power recycling cavity transmission functions with respect to any of the degrees of

6In other literature, these have been referred to as the bounce number, and augmented bounce number, respectively.[32]

freedom.

N'

=

_1_8rc -ire 8¢

N

=

_1_8Pc -ipc 8cjJ

(3.62) (3.63) N' is the external phase gain, and N is the internal phase gain. Subscripts will be used in order to indicate which field (carrier or RF sideband) the derivative refers to, as well as the degree of freedom by which the derivative is taken.

Signal Sensitivity

The matrix elements, Eqs. (3.10) and (3.11) won't typically be as nice as the examples discussed in Section 3.1.1. The upper PM RF sideband and the SSB RF sideband tend to be somewhat off resonant. Their corresponding transmission functions and phase gains are complex. Simplification of the matrix elements as in Eq. (3.14) and Eq. (3.16) isn't possible, because the R {} argument needs to be kept, which makes it a little harder to understand the dependencies of the matrix elements.

The general matrix formula for the PM and SSB outputs can be re-arranged in the following fashion with a bit of algebra.

M_{r,j- -}^{p -}

IE

_{L JO}

12

^T J _1rp,o:^{10 {} _{:n. rp,-}* ( _Ho,j-^7\Tf N'* ) _{-,j e} -i[Jf.'

₊

_rp,+ * ( 1v^7\TI 0, j -N'* +,i ) e

i[Jf.'}

M_r,j-s

--IE

l

IJ

las::.rl ^{10 {} rp,-rp,s ^{* (N'} -,j ^{- N'*)} s,j e ifJ~}

(3.64) (3.65)

The power cavity reflectivity rp now have an additional subscript to indicate whether it refers to the carrier (0), the upper PM RF sideband (+),the lower PM RF sideband (-), or the SSB RF sideband ( s). The variable as is the amplitude of the SSB relative to the laser output amplitude, assumed to be real. These formulas apply to the matrix elements of the reflected photodiode, where the phase gain N' is evaluated for the various degrees offreedom. The substitutions N' ---+ Nand rp ---+ Pp supply the matrix elements for all signal ports of the pickoff photodiode, M:,i and M%,i.

The <I>+ degree of freedom is actually fairly simple, due to the fact that the RF

sidebands are very insensitive to the arm cavity modes, relative to the carrier sensitiv- ity. The N and N"s associated with the RF sidebands are 0, to good approximation.

(3.66) (3.67) (3.68) (3.69)

The constants

-1Ed

²Jo11 and

-1Ed

²J 1as have been dropped, since they're common to every element in the PM output and SSB output matrices, respectively.

The ¢+ degree of freedom affects every frequency involved in the interferometer, since all frequencies are meant to resonate in the power cavity. The matrix elements associated with this degree of freedom are

MP r,<l>+

=

r p,O ~ ^{r* p,-(N' 0,4>+ - N'* )e-if3[.' -,4>+

+

^r* p,+ (N' 0,4>+ - N'* +,4>+ )eif3[.'}

M:.<l>+

= Pp,o~

{P;,_(No,¢+- N:,<l>+)e-if3:

+

^P;,+(No,¢+-

N~,<t>Jeif3:}

Mr,</>+ ⁸

= ~

{r r* (N' p,- p,s -,4>+ - N'* s,</>+ )eif3;}

M!<t>+

= ~

{Pp,-P;,s(N-,4>+- N:,<t>Jeif3:}

(3.70) (3.71)

(3.72) (3. 73)

Both the reflected and pickoff photodiodes see the carrier on a bright fringe due to the reflectivity of the Michelson being ex: cos( cp_). The derivative of the carrier field with respect to the cjJ_ degree of freedom is zero. For the PM output signals, this allows some simplification of the formula.

(3.74) (3. 75) (3. 76) (3.77)

The <Ps degree of freedom is quite simple. First, the carrier is completely insensitive to this degree of freedom because of the dark fringe condition - no carrier light is in the signal cavity. Second, the upper RF sideband is largely non-resonant in the signal cavity. So both N and N' associated with the upper RF sideband as well as the single sideband can be set to 0.

Mt,q,. = -rp,orp,-N~,t/>. cos((J;) M:q,.

=

-pp,OPp,-N-,q,. cos((3:)

Mr,t/>. s _ - r p,-N' -,</>. ^{10 {} ;n r p,se * if3~ } lvf:.q,.

=

Pp,-N-,q,.R {P;,seif3%}