Pound-Drever-Hall locking scheme free from Trojan operating points Manuel Zeyen1Lukas Aﬀolter1Marwan Abdou Ahmed2Thomas Graf2Oguzhan Kara1Klaus Kirch1 3 Miroslaw Marszalek1 3François Nez4Ahmed Ouf5Randolf Pohl5Siddharth Rajamohanan5Pauline

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Pound-Drever-Hall locking scheme free from Trojan operating points

Manuel Zeyen,1Lukas Aﬀolter,1Marwan Abdou Ahmed,2Thomas Graf,2Oguzhan Kara,1Klaus Kirch,1, 3

Miroslaw Marszalek,1, 3 François Nez,4Ahmed Ouf,5Randolf Pohl,5Siddharth Rajamohanan,5Pauline

Yzombard,4Aldo Antognini,1, 3 and Karsten Schuhmann1

1)Institute for Particle Physics and Astrophysics, ETH, 8093 Zurich, Switzerland.

2)Institut für Strahlwerkzeuge, Universität Stuttgart, Pfaffenwaldring 43, 70569 Stuttgart,

Deutschland.

3)Paul Scherrer Institute, 5232 Villigen, Switzerland.

4)Laboratoire Kastler Brossel, Sorbonne Université, CNRS, ENS-Université PSL, Collège de France, 75252 Paris Cedex 05,

France.

5)Johannes Gutenberg-Universität Mainz, QUANTUM, Institut für Physik & Exzellenzcluster PRISMA, 55128 Mainz,

Germany.

(*Authors to whom correspondence should be addressed: zeyenm@phys.ethz.ch; aldo.antognini@psi.ch)

(Dated: 12 October 2022)

The Pound-Drever-Hall (PDH) technique is a popular method for stabilizing the frequency of a laser to a stable optical

resonator or, vice versa, the length of a resonator to the frequency of a stable laser. We propose a reﬁnement of the

technique yielding an "inﬁnite" dynamic (capture) range so that a resonator is correctly locked to the seed frequency,

even after large perturbations. The stable but off-resonant lock points (also called Trojan operating points), present

in conventional PDH error signals, are removed by phase modulating the seed laser at a frequency corresponding to

half the free spectral range of the resonator. We verify the robustness of our scheme experimentally by realizing an

injection-seeded Yb:YAG thin-disk laser. We also give an analytical formulation of the PDH error signal for arbitrary

modulation frequencies and discuss the parameter range for which our PDH locking scheme guarantees correct locking.

Our scheme is simple as it does not require additional electronics apart from the standard PDH setup and is particularly

suited to realize injection-seeded lasers and injection-seeded optical parametric oscillators.

I. INTRODUCTION

The Pound-Drever-Hall (PDH) locking technique1is a pop-

ular method to stabilize the frequency of a laser to a stable

optical resonator. It has been used to achieve lasers with sub-

Hertz linewidth2,3and is applied in a wide range of ﬁelds,

such as gravitational wave detection4, atomic physics5,6and

metrology,7just to name a few. The PDH method can also

be used in the opposite way to stabilize the length of an op-

tical resonator to a stable single-frequency laser with equally

numerous applications8–11.

Despite its widespread and long-standing application, this

technique is continuously reﬁned and adapted to speciﬁc

applications12–16. The dynamic range of the standard PDH

technique is limited by the additional zero crossings of the er-

ror signal at off-resonant frequencies. This typically limits the

dynamic range (or capture range) of the lock to a fraction of

the free spectral range (FSR) of the resonator given by

∆f=c

2L,(1)

where cis the speed of light in the resonator and Lis the length

of the (linear) resonator. The PDH error signal has a zero

crossing right in the middle between two adjacent resonator

modes at frequency detuning ν=∆f/2 from the resonator

mode. This zero crossing represents a stable lock point, where

laser and resonator are stabilized in a totally off-resonant state.

Such undesired stable operating points are also called Tro-

jan operating points17,18. Since the modulation frequency is

typically much smaller than the FSR of the resonator, a large

disturbance (causing large laser frequency or resonator length

variations) may lead to an erroneous stabilization on the Tro-

jan operating point. When this occurs, the correct lock point

must be restored either manually or via an automated process,

which requires dedicated electronics and can take up to sev-

eral ms19–21.

In this paper, we demonstrate a simple way to avoid off-

resonant Trojan operating points in a PDH error signal, by

modulating the seed laser at νM=∆f/2, i.e. at half the FSR

of the resonator. In doing so, the off-resonant lock point be-

tween two resonances is made unstable, resulting in an “in-

ﬁnite” capture range so that re-locking always occurs on a

resonance independently of the size of the perturbation. Our

scheme is particularly well suited for injection-seeding lasers

or optical parametric oscillators.

The paper is organized as follows: In section II we present

the theory of the PDH error signal, highlighting the peculiar-

ities related to the use of νM=∆f/2. We also emphasize the

parameter range in which the locking scheme works best and

we link our scheme to recent ideas which extend the linear

range of the PDH error signal22–24. In section III we present

an implementation of our scheme in an injection-seeded thin-

disk laser (TDL).

II. PDH SCHEME WITHOUT TROJAN OPERATING

POINTS

A. Analytical expression for the classic PDH error signal

In general, a resonator, where the losses mainly occur at

the end-mirrors, can be simpliﬁed to a two-mirror resonator,

where the mirrors have power reﬂectivities R1and R2. With-

out loss of generality, possible intra-resonator gain can be in-

arXiv:2210.05501v1 [physics.optics] 11 Oct 2022

cluded in R2so that R2>1 is possible. For such a general

resonator we deﬁne its ﬁnesse25 as

F=2π

−ln(R1R2).(2)

Fis a measure of the sharpness of the resonances and can be

approximated as the ratio

F ≈ ∆f

δf,(3)

where δfis the FWHM linewidth of the resonances. In this

study νdenotes the relative detuning between the seed laser

frequency and the nearest resonance frequency of the res-

onator TEM00 modes.

Depending on ν, part of the phase modulated seed light is

reﬂected from the resonator. Its electric ﬁeld amplitude and

phase relative to the input light is given by the complex reﬂec-

tion coefﬁcient26

F(ν) = √R1−

(1−R1)√R2exph2i(πν

∆f+φ)i

1−√R1R2exph2i(πν

∆f+φ)i,(4)

where φis an additional phase shift which the light might

acquire over one resonator round trip (e.g. by propagating

through a gain medium). For simplicity, in the following we

set φ=0.

From this reﬂection coefﬁcient, the well known PDH error

signal is obtained as27

ε(ν,νM) =−2√P

cPs Im[F(ν)F∗(ν+νM)

−F∗(ν)F(ν−νM)],(5)

where νMis the modulation frequency, P

cand P

sare the power

in the carrier and the sidebands of the seed laser respectively,

Im[...] takes the imaginary part and * denotes complex con-

jugation. Note that this is the error signal only for the case of

demodulation at νMand phase delay ∆ϕ=0 (see Subsec.II E).

The modulation frequency νMcan be expressed in terms of

the FSR as νM=ξ∆f, where 0 <ξ≤1 so that the PDH error

signal can be re-written as

ε(ν,ξ) =−2√P

cPs Im[F(ν)F∗(ν+ξ∆f)

−F∗(ν)F(ν−ξ∆f)].(6)

Inserting Eq. (4) into Eq. (6) we ﬁnd

ε(ν,ξ) =4√P

sin(ξ π)

G1(ν)G2(ν,ξ)

G1(ν+ξ∆f)

−G2(ν,−ξ)

G1(ν−ξ∆f),(7)

with

G1(ν) = 1+γ2−2γcos2πν

∆f(8)

and

G2(ν,ξ) =(γ2R1−R2)cos(ξ π)

+γ−γ3+γ(R2−R1)cos2πν

∆f+ξ π,(9)

Trojan

lock-point

FIG. 1. Top: Simulation of a typical open-loop PDH error signal

obtained by scanning the resonator length. The shaded area indi-

cates the capture range of the feedback loop, i.e. the detuning region

between resonator mode and seed laser frequencies that can be cor-

rected. The arrows indicate the direction of the correction of the feed-

back loop. In this example the laser was modulated at νM=0.2∆f

so that the red and blue sideband (RSB and BSB, respectively) lie at

±0.2∆ffrom the carrier frequency. Bottom: Corresponding simu-

lated intra-resonator intensity.

where

γ=√R1R2.(10)

Instead of using the parameter γ, these equations can also be

expressed in terms of the ﬁnesse by performing the substitu-

tion

γ=exp−π

F.(11)

Figure 1shows an example of an error signal for ξ=0.2 and

F=40 (R1=0.6 and R2=1.42). The capture range of the

error signal is indicated as gray shaded area. The black full

arrows indicate the direction of the feedback loop correction

moving the system to the correct lock point (TEM00 reso-

nance), whereas the open arrows indicate the region where

the feedback loop steers the system erroneously towards the

Trojan operating point. In the next subsection we present a

method to avoid this stable but unwanted lock point.

B. Modulating at half the free spectral range

If we modulate the seed laser at half the FSR of the res-

onator, i.e. νM=∆f/2 or ξ=1/2, Eq. (7) simpliﬁes to

ε(ν,1

2) = 8√P

sγ3−γ(1+R2−R1)sin2πν

∆f

1+γ4−2γ2cos4πν

∆f.(12)

In this case the blue and the red sidebands from neighboring

resonances overlap and the error signal is free from Trojan

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Pound-Drever-HalllockingschemefreefromTrojanoperatingpointsManuelZeyen,1LukasAolter,1MarwanAbdouAhmed,2ThomasGraf,2OguzhanKara,1KlausKirch,1,3MiroslawMarszalek,1,3FrançoisNez,4AhmedOuf,5RandolfPohl,5SiddharthRajamohanan,5PaulineYzombard,4AldoAntognini,1,3andKarstenSchuhmann11)InstituteforParticlePh...

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Pound-Drever-Hall locking scheme free from Trojan operating points Manuel Zeyen1Lukas Aﬀolter1Marwan Abdou Ahmed2Thomas Graf2Oguzhan Kara1Klaus Kirch1 3 Miroslaw Marszalek1 3François Nez4Ahmed Ouf5Randolf Pohl5Siddharth Rajamohanan5Pauline.pdf

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Pound-Drever-Hall locking scheme free from Trojan operating points Manuel Zeyen1Lukas Aﬀolter1Marwan Abdou Ahmed2Thomas Graf2Oguzhan Kara1Klaus Kirch1 3 Miroslaw Marszalek1 3François Nez4Ahmed Ouf5Randolf Pohl5Siddharth Rajamohanan5Pauline

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