Response Function of the Pound-Drever-Hall (PDH) Measurement

The Pound-Drever-Hall (PDH) method is a “self-heterodyne” technique used to measure the phase of the field reflected from the cavity [82, 54]. In PDH, a local oscillator (LO) is generated by phase modulating the input field at a frequency Ω₀ much larger than the cavity linewidth; the resulting

“PDH” sidebands are directly reflected from the cavity. The remaining carrier portion of the input field is tuned into resonance with the cavity (∆ = 0). A fraction gets into the cavity and leaks back out with a phase shift that is proportional to δωc(t). When combined on a photodetector, the beat between the reflection of far-detuned sidebands and the near-resonant carrier produces a rapidly fluctuating photocurrent fluctuationδi(t) =(t) sin(Ω0t+φ) whose slowly varying envelope (t) is proportional toδωc(t). The PDH “error signal”,(t), is extracted using a mixer. Important advantages of the PDH technique are (1) Ω0 is typically an RF frequency, near which technical noise (e.g., noise produced by photodetector electronics) can be significantly reduced, and (2) the photosignal is produced without modulating — to first order — the intensity of the intracavity field, hence significantly reducing radiation pressure effects.

8.4.1 Fast Modulation

We here carry out a derivation of the response function for the PDH method that is appropriate for modulation frequencies much smaller than the PDH sideband frequency, Ω₀ which is much larger

MIM cavity PBS

EOM PBS

PD

Spectrum Analyzer

Figure 8.2: Block diagram of the Pound-Drever-Hall measurement

than the cavity linewidth Ω0 >> κ. We first expand the slowly varying envelope of the phase modulated input field using the Jacobi-Anger Expansion:

Ein(t) =E0e^{iαsin(Ω0t)}=E0

X

n

Jn(α)e^inΩ0t, (8.17)

whereαis the phase modulation depth,J_nis the n^th-order Bessel function of the first kind, and the sum is understood to extend over all integers from−∞to∞.

Letting a(t) =hai+δa(t), we can express the equation of motion for the field to first order in

δa/haias:

˙

a(t) =−(κ+i∆)a(t) +iδωc(t)hai+√

2κ1Ein(t) (8.18a) Ein(t) =E0

X

n

Jn(α)e^inΩ0t (8.18b)

E_out(t) =√

2κ₂a(t) (8.18c)

Eref(t) =−Ein(t) +√

2κ1a(t), (8.18d)

wherehai=√

2κ1E0J0(α)/(κ+i∆).

Inserting the expression forhaiinto (8.18) and applying the Fourier transform to both sides gives

a(Ω) =

√2κ1

κ+i(∆ + Ω) X

n

Jn(α)δ(Ω−nΩ0)−iδωc(Ω) κ+i∆

!

E0 (8.19a)

E_ref(Ω) =√

2κ₁a(Ω)−E_in(Ω). (8.19b)

For PDH, the input field carrier frequency is usually on resonance (∆ = 0) and the phase modulation frequency is large Ω₀>> κ. In this case we can simplify (8.19c) to include only the directly reflected field and the sidebands generated on the carrier inside the cavity:

Eref(Ω)≈ 2κ1

κ+iΩ−1

E0

X

n

Jn(α)δ(Ω−nΩ0)− i κ+iΩ

2κ1

κ δωc(Ω)J0(α)E0. (8.20) It’s instructive at this point to move back into the time domain and consider a sinusoidal frequency modulation,δωc(t) =δωc,0cos(Ωmt)<< κ. In this case:

E_ref(t) =X

n

2κ₁ κ+inΩ0

−1

E₀J_n(α)e^inΩ0t−i2κ₁

κ δω_c,0J₀(α)1 2

e^iΩmt κ+iΩm

+c.c.

E₀. (8.21)

The average power in the reflected beam is approximately

h|Eref(t)|²i ≈ |E0|² 1−(4κ1κ2/κ²)J₀²(α))

. (8.22)

The relevant fluctuating part of the reflected power is contributed by terms which are oscillating at frequencies±Ω0±Ωm. Using Ω0>> κ,J1(α) =−J₋₁(α), keeping only terms linear inδωc/κ <<

1, and ignoring terms oscillating at 2Ω₀, we obtain

δ|Eref(t)|²≡ |Eref(t)|²− h|Eref(t)|²i (8.23a)

=

−J1(α)e^iΩ0t−J₋₁(α)e^−iΩ0t−iδω_c,0J₀(α)2κ₁ κ

1 2

e^iΩmt κ+iΩm

+c.c.

2

|E0|² (8.23b)

=

2J₁(α) sin(Ωt) +δω_c,0J₀(α)2κ₁ κ Re

e^iΩmt κ+iΩm

²

|E₀|² (8.23c)

≈4J0(α)J1(α)|E0|²δωc,0

κ 2κ1

κ s

1

1 + (Ωm/κ)²cos

Ωmt−tan⁻¹ Ωm

κ

sin(Ω0t).

(8.23d) The PDH error signal is obtained by passing i(t) = R|Eref(t)|² through a mixer to extract the slowly varying envelope of the sin(Ω₀t) quadrature. For an ideal mixer with no conversion loss, the resulting photocurrent is:

(t) = 4J₀(α)J₁(α)R|E₀|²δω_c,0 κ

2κ₁ κ

s 1

1 + (Ωm/κ)²cos Ω_mt−tan⁻¹(Ω_m/κ)

(8.24a)

= 4J0(α)J1(α) hii

1−(4κ1κ2/κ²)J₀²(α) δωc,0

κ 2κ1

κ s

1

1 + (Ωm/κ)²cos

Ωmt−tan⁻¹ Ωm

κ

.

(8.24b)

The relationship between the error signal and the cavity resonance frequency fluctuations is given by the response functionG,ω_c(Ω):

G_,ωc(Ω)≡ (Ω)

δωc(Ω) = 8(κ₁/κ)J₀(α)J₁(α) 1−(4κ1κ2/κ²)J0(α)²hii1

κ

s 1

1 + (Ω/κ)²e^{−itan−1(Ω/κ)}. (8.25) In the limit that the reflectivity of the membrane is small, our symmetric (r1=r2) MIM cavity is described byκ1≈κ2≈κ/2 andα≈1, which gives the transfer function:

|G,ωc(Ω)|²≈10× 1 κ²

1 1−Ω²/κ²

hii². (8.26)

8.4.2 The Effect of Mode-Mismatch

In general the input field is not perfectly mode-matched to the cavity, and we must account for this effect on the measurement response function. Mode-mismatch may be due to the polarization or spatial profile of the input beam. To obtain a simple model, we ignore the former by assuming that the input field is linearly polarized along one of the birefringent axes of the cavity. We then divide the input beam into two non-interfering spatial modes, one which interacts with the cavity,E~_in^a, and

one which is directly reflected,E~^b_in:

E~in(t) =E~^a_in(t) +E~_in^b (t). (8.27)

Vector notation is here used to represent the fact that modesaandb are spatially orthogonal over the transverse plane, and as such do not interfere. The orthogonality rule is E~_in^a (t)·E~^a∗_in(t) =

|E_in^a (t)|², ~E_in^a(t)·E~^b∗_in(t) = 0. The power in the beam is then expressed as:

Pin(t) =E~in(t)·E~_in^∗ (t) =|E_in^a(t)|²+|E_in^b (t)|². (8.28) Field E~_in^a couples to the cavity according to Eq. 8.4, but field E~_in^b is directly reflected. The reflected and transmitted fields can be written

E~ref(t) =E~_ref^a (t) +E~_ref^b (t) (8.29a)

=E~_ref^a (t)−E~_in^b (t) (8.29b)

E~_out(t) =E~_out^a (t). (8.29c)

Since the two spatial modes don’t interfere, the total transmitted and reflected power is

Pref(t) =|E_ref^a (t)|²+|E_in^b (t)|² (8.30a)

P_out(t) =|E_out^a (t)|². (8.30b)

We now define the mode-matching efficiencyξ² as the ratio of the average power in field “a” to the total average power in the field:

ξ²≡ h|E_in^a(t)|²i

h|E_in^a(t)|²i+h|E_in^b (t)|²i= hP_in^ai

hPini. (8.31)

For the detuned probe measurement, only light that is coupled to the cavity passes through to the transmission photodetector. Importantly, this means that the transfer function|Gi,ω_c(Ω)|²does not depend on the mode matching efficiency.

More care must be taken with the PDH measurement (in reflection). In this case the input field can be modeled

E~in(t) =E~₀^ae^{iαasin(Ω0t)}+E~₀^be^{iαbsin(Ω0t)} (8.32) where the subscripts on the modulation depth indicate a possibly different phase-modulation depth for the two spatial modes.

The mean photocurrent and the error signal for the PDH measurement become:

hii=R|E₀^a|² 1−(4κ1κ2/κ²)J0(α)²

+R|E₀^b|² (8.33a)

(t)≈4J0(α)J1(α)R|E₀^a|²δωc,0

κ 2κ1

κ

s 1

1 + (Ω_m/κ)²cos

Ωmt−tan⁻¹ Ωm

κ

(8.33b)

which gives the following modified PDH response function:

G_,ωc(Ω) = 8(κ₁/κ)J₀(α)J₁(α)

1−(4κ1κ2/κ²)J0(α)²+ (1−ξ²)/ξ²hii1 κ

s 1

1 + (Ω/κ)²e^{−itan−1(Ω/κ)}. (8.34)