††thanks: A footnote to the article title

Also at ]Hubei Key Laboratory of Modern Manufacturing Quantity Engineering,
School of Mechanical Engineering, Hubei University of Technology, Wuhan, Hubei
430068, China.

# Cavity locking with spatial modulation of optical phase front for laser
stabilization

Sheng Feng<EMAIL_ADDRESS>Songqing You  [  Peng Yang  Fenglei Zhang
Yunlong Sun  School of Electrical and Electronic Infomation Engineering, Hubei
Polytechnic University, Huangshi, Hubei 435003, P.R. China  Boya Xie  Hubei
Key Laboratory of Modern Manufacturing Quantity Engineering, School of
Mechanical Engineering, Hubei University of Technology, Wuhan, Hubei 430068,
China.

###### Abstract

We study optical cavity locking for laser stabilization through spatial
modulation of the phase front of a light beam. A theoretical description of
the underlying principle is developed for this method and special attention is
paid to residual amplitude modulation (RAM) caused by experimental
imperfections, especially the manufacture errors of the spatial phase
modulator. The studied locking method owns the common advantages of the Pound-
Drever-Hall method and the tilt-locking one, and it can provide a more artful
way to eliminate RAM noise in phase modulation for the ultimate stability of
lasers. In situations where cost and portability are a practical issue, the
studied method allows one to realize compact laser stabilization systems
locked to Fabry-P$\acute{\mbox{e}}$rot cavities without use of expensive bulky
devices, such as signal generators and electro-optic modulators.

††preprint: APS/123-QED

Ultra-stable lasers with high-spectral purity play an essential role in a
variety of advanced research fields, such as optical atomic clocks [1, 2, 3],
tests of relativity [4, 5], gravitational wave detectors [6], and photonic
microwave synthesizers [7]. State-of-the-art stable lasers locked to high-
finesse Fabry P$\acute{\mbox{e}}$rot (FP) cavities delivering single-mode
electromagnetic waves have been realized with integral linewidths below 1 Hz
[8, 9]. Another research direction of relevance is to construct portable
stable laser systems [10] towards applications wherein robustness and
integration are primary concerns, with a 20-ms stability of $10^{-13}$ already
reported [11]. Despite several locking schemes of potential interest [12, 13,
14, 15, 16], most of these performance demonstrations have been accomplished
by use of the standard Pound-Drever-Hall (PDH) one [17] in which the phase
front of the optical carrier undergoes temporal modulation that generates
radio frequency (r.f.) sidebands.

A well-known issue with the PDH modulation technique is residual amplitude
modulation (RAM) that arises when the r.f. modulation sidebands are unequal in
magnitude, not exactly out of phase, or both. RAM couples frequency offset
noise to the servo error signal and thereby aggravates the laser frequency
instability. A few RAM reduction schemes [18, 19, 20, 21] have been
implemented with the most remarkable one demonstrated recently [22] that
utilized an active servo loop involving both DC electric field and temperature
corrections applied to the phase modulator. RAM control on the $10^{-6}$ level
was reported with a RAM-induced frequency instability comparable to or lower
than thermal noise limit [22].

Instead of using temporal modulation as the PDH method does [17], we explore
in this work spatial modulation of the optical phase front of the carrier for
FP cavity locking. Aside from its own merit of fundamental interest, the
studied cavity locking method provides handy ways for RAM reduction; moreover,
it has the common advantages of the PDH method and the tilt-locking one the
later of which nonetheless utilizes travelling-wave cavities for laser
stabilization [23]. In comparison with the PDH one, the studied method can
provide larger error signals for the servo loop and may eliminate the need of
electro-optic modulators (EOM’s) and signal generators that are expensive,
bulky, and otherwise necessary for temporal phase modulation, which is
beneficiary to low-cost portable laser stabilization systems locked to FP
cavities [10, 11].

In what follows, let describe the principle underlying the studied cavity
locking method through spatial modulation of optical phase front. For
simplicity without loss of generality, let consider a collimated laser beam in
a Gaussian mode, $E_{\mbox{i}}(\vec{r},t)$, that is coupled into an FP cavity
through a focal lens (Fig. 1). A transparent optical element, namely a spatio-
optic modulator (SOM), is inserted into the incident light beam before the
lens such that the phase front of the beam undergoes spatial modulation. The
spatially modulated light field $E_{\mbox{m}}(\vec{r},t)$ is described as,

$E_{\mbox{m}}(\vec{r},t)=E_{\mbox{i}}(\vec{r},t)U(\vec{r})=E_{\mbox{i}}(\vec{r},t)e^{i\varphi(\vec{r})}\
,$ (1)

wherein $U(\vec{r})$ is the SOM modulation function, and the vector $\vec{r}$
represents the two-dimensional coordinates in the plane perpendicular to the
propagation axis of the light beam. Unity transmittance is assumed for the SOM
and $\varphi(\vec{r})$ is the spatial distribution of the change to the light
field phase front caused by the SOM.

To get an insight into the cavity locking method, let assume
$|\varphi(\vec{r})|<<1$ for the moment. Under this assumption, the modulated
light field $E_{\mbox{m}}(\vec{r},t)$ reads

$\displaystyle E_{\mbox{m}}(\vec{r},t)$ $\displaystyle=$ $\displaystyle
E_{\mbox{i}}(\vec{r},t)+i\varphi(\vec{r})E_{\mbox{i}}(\vec{r},t)\ ,$ (2)

from which it follows that the modulated light field becomes a superposition
of the incident light field, $E_{\mbox{i}}(\vec{r},t)$, and another effective
field, $E_{\mbox{e}}(\vec{r},t)\equiv
i\varphi(\vec{r})E_{\mbox{i}}(\vec{r},t)$, as a consequence of spatial
modulation. For the purpose of this work, the SOM may be designed for
$\varphi(\vec{r})$ to satisfy

$\iint d\vec{r}E_{\mbox{i}}^{*}(\vec{r},t)E_{\mbox{e}}(\vec{r},t)=\iint
d\vec{r}\varphi(\vec{r})|E_{\mbox{i}}(\vec{r},t)|^{2}=0\ ,$ (3)

in which the integral area covers the whole cross section of the incident
beam. In other words, the spatial mode of the effective field,
$E_{\mbox{e}}(\vec{r},t)$, is orthogonal to that of the incident field,
$E_{\mbox{i}}(\vec{r},t)$, and hence the two fields may not simultaneously
resonate inside a narrow-linewidth optical cavity due to their spatial mode
orthogonality [24].

Figure 1: (color online) Schematics for cavity locking by spatial modulation
of optical phase front. SOM: Spatio-optic modulator. PBS: Polarizing beam-
splitter. $\lambda/4$: Quarter wave plate. L1: Focal lens coupling the
incident light into the cavity. $M_{1,2}$: Cavity mirrors. PZT: Piezo-
electrical transducer. L2,3: Imaging lenses used to map the SOM images to the
corresponding photo-detectors. PD1,2: Dual-quadrant photo-detectors. F: An
optional optical element used to tailor the spatial mode amplitudes of the
light field received by PD1 for RAM cancellation.

Now let suppose that the incident field $E_{\mbox{i}}(\vec{r},t)$ is near
resonance in the FP cavity of Fig. 1, and that the cavity linewidth is narrow
enough so that the effective field $E_{\mbox{e}}(\vec{r},t)$ is far-off
resonance. Then after reflection from the cavity, the modulated light field
becomes [12]

$\displaystyle E_{\mbox{r}}(\vec{r},t)$ $\displaystyle=$
$\displaystyle\left\\{\sqrt{R_{1}}\left[1+i\varphi(\vec{r})\right]-\frac{T_{1}}{\sqrt{R_{1}}}\frac{e^{i\delta}\sqrt{R}}{1-e^{i\delta}\sqrt{R}}\right\\}E_{\mbox{i}}(\vec{r},t)$
(4) $\displaystyle=$ $\displaystyle\left[\chi
e^{i\theta}+i\varphi(\vec{r})\right]\sqrt{R_{1}}\ E_{\mbox{i}}(\vec{r},t)\ ,$

in which

$\chi e^{i\theta}\equiv
1-\frac{T_{1}}{R_{1}}\frac{e^{i\delta}\sqrt{R}}{1-e^{i\delta}\sqrt{R}}\ ,$ (5)

and $R_{1},T_{1}$ are respectively the reflectivity and transmissivity of the
cavity entrance mirror M1, $R\equiv R_{1}R_{2}$ ($R_{2}$ the reflectivity of
the other cavity mirror M2), and $\delta$ stands for the round trip phase
delay of the light field inside the cavity.

In what follows, we will show how to exploit an optical element of spatial
modulation, i.e. an SOM, to implement cavity locking based on Eq. (4). Given
that the incident light field $E_{\mbox{i}}(\vec{r},t)$ is in a Gaussian mode
which is symmetric with respect to the light propagation axis,

$E_{\mbox{i}}(\vec{r},t)=E_{\mbox{i}}(-\vec{r},t)\ ,$ (6)

one may utilize an SOM with a spatial modulation function that introduces an
anti-symmetric phase change to the incident beam around its axis,

$\varphi(\vec{r})=-\varphi(-\vec{r})\ .$ (7)

From Eqs. (6) and (7) it immediately follows that Eq. (3) holds true and,
hence, so does Eq. (4). An example of a designed SOM satisfying Eq. (7) is
illustrated in Fig. 2 and the phase front change $\varphi(\vec{r})$ of the
light field due to the spatial modulation reads

$\varphi(\vec{r})=\left\\{\begin{array}[]{ll}\pm\varphi_{0}&|\vec{r}\mp
r_{0}|<r_{0}\\\ 0&\textrm{otherwise}\end{array}\right.\ ,$ (8)

in which $\pm\varphi_{0}$ are constant modulation-induced phase changes in the
corresponding areas, $S_{1,2}$, which are circular here centered at the points
of ($\pm r_{0},0$) respectively with the same radius of $r_{0}$.

Figure 2: (color online) An example of a designed SOM for cavity locking.
$\Delta\equiv(\varphi_{0}/2\pi)\cdot\lambda/(n-1)$ with $\lambda$ being the
wavelength of the light beam and $n$ the refractive index of the SOM
material). (Top): Front view of the SOM. (Bottom): The SOM surface profile
along the dash line as indicated on the top.

Next we will show how an error signal may be obtained in the cavity locking
method. To this end, one may use a dual-quadrant photo-detector to receive the
light beam reflected from the cavity. An optical imaging system is utilized to
project the image of the SOM onto the photo-detector, with each quadrant
aligned with respect to each of the circles, $S_{1,2}$, on the SOM. The output
photo-electric currents from the two quadrants are subtracted to produce an
error signal $\varepsilon(\delta)$ as described by,

$\varepsilon(\delta)=\iint_{S_{1}}d\vec{r}|E_{\mbox{r}}(\vec{r},t)|^{2}-\iint_{S_{2}}d\vec{r}|E_{\mbox{r}}(\vec{r},t)|^{2}\
.$ (9)

To make it clearer, one may plug Eqs. (4) and (8) into Eq. (9), leading to

$\displaystyle\varepsilon(\delta)$ $\displaystyle=$
$\displaystyle\iint_{S_{1}}d\vec{r}R_{1}|(\chi
e^{i\theta}+i\varphi_{0})E_{\mbox{i}}(\vec{r},t)|^{2}$ (10)
$\displaystyle-\iint_{S_{2}}d\vec{r}R_{1}|(\chi
e^{i\theta}-i\varphi_{0})E_{\mbox{i}}(\vec{r},t)|^{2}$ $\displaystyle=$
$\displaystyle 4R_{1}I_{0}\ \varphi_{0}\ \chi\sin\theta$ $\displaystyle=$
$\displaystyle 4R_{1}I_{0}\ \varphi_{0}\
\mbox{Im}\left(1-\frac{T_{1}}{R_{1}}\frac{e^{i\delta}\sqrt{R}}{1-e^{i\delta}\sqrt{R}}\right)\
,$

in which Im($\cdot$) stands for the imaginary part of a complex number and one
has invoked Eq. (6) from which it follows that
$\iint_{S_{1}}d\vec{r}|E_{\mbox{i}}(\vec{r},t)|^{2}=\iint_{S_{2}}d\vec{r}|E_{\mbox{i}}(\vec{r},t)|^{2}\equiv
I_{0}$. The error signal for cavity locking varies as a function of cavity
detuning $\delta$ from its peak resonance according to Eq. (10).

In what follows, let turn to the RAM problem associated with the studied
cavity locking method due to SOM manufacture errors. Specifically, the anti-
symmetry given by Eq. (7) may be broken to some extent in practice resulting
from the difference between the areas, optical transmissivities, modulation
depths of $S_{1}$ and $S_{2}$; the SOM symmetry breaking, together with
unexpected input beam drifting, unbalanced quantum efficiencies and unequal
gains of the two quadrants of the photo-detector, will invalidate the
assumptions of both Eq. (8) and
$\iint_{S_{1}}d\vec{r}|E_{\mbox{i}}(\vec{r},t)|^{2}=\iint_{S_{2}}d\vec{r}|E_{\mbox{i}}(\vec{r},t)|^{2}$,
causing RAM noises in the error signal. One should note that both the real RAM
noises and RAM-like noises are referred to as “RAM noises” in this work when
they cannot be distinguished in experiment.

To account for these effects in theory, one may define different phase
modulations $\varphi_{1,2}$ for each of the two circular areas $S_{1,2}$
respectively with radii $r_{1,2}<r_{0}$,

$\varphi(\vec{r})=\left\\{\begin{array}[]{ll}\varphi_{1}&|\vec{r}-r_{0}|<r_{1}\\\
0&\textrm{otherwise}\\\ \varphi_{2}&|\vec{r}+r_{0}|<r_{2}\end{array}\right.\
,$ (11)

and photo-currents $I_{1,2}$,

$I_{1}\equiv G_{1}\iint_{S_{1}}d\vec{r}|E_{\mbox{i}}(\vec{r},t)|^{2},\ \
I_{2}\equiv G_{2}\iint_{S_{2}}d\vec{r}|E_{\mbox{i}}(\vec{r},t)|^{2}\ .$ (12)

Here the normalized coefficients $G_{1,2}$ account for all the effects
resulting from the differences in the optical transmissivities of the areas
$S_{1,2}$, beam drifting, the gains and quantum efficiencies of the two
detector quadrants.

With Eqs. (4), (11), and (12), the error signal becomes

$\displaystyle\varepsilon(\delta)$ $\displaystyle=$ $\displaystyle
G_{1}\iint_{S_{1}}d\vec{r}R_{1}|(\chi
e^{i\theta}+i\varphi_{1})E_{\mbox{i}}(\vec{r},t)|^{2}$ (13) $\displaystyle-
G_{2}\iint_{S_{2}}d\vec{r}R_{1}|(\chi
e^{i\theta}-i\varphi_{2})E_{\mbox{i}}(\vec{r},t)|^{2}$ $\displaystyle=$
$\displaystyle 4R_{1}(\bar{I}\ \bar{\varphi}+\Delta
I\Delta\varphi)\mbox{Im}\left(1-\frac{T_{1}}{R_{1}}\frac{e^{i\delta}\sqrt{R}}{1-e^{i\delta}\sqrt{R}}\right)$
$\displaystyle+2R_{1}\left[2\bar{I}\ \bar{\varphi}\Delta\varphi+\Delta
I(\chi^{2}+\bar{\varphi}^{2}+(\Delta\varphi)^{2})\right]\ ,$

in which $\bar{I}=(I_{1}+I_{2})/2$, $\Delta I=(I_{1}-I_{2})/2$,
$\bar{\varphi}=(\varphi_{1}+\varphi_{2})/2$, and
$\Delta\varphi=(\varphi_{1}-\varphi_{2})/2$. The first term on the right hand
side of Eq. (13) is the error signal required for cavity locking, whereas the
second term describes RAM in the studied scheme that introduces frequency
offset noise into the servo error signal leading to degradation of the laser
frequency stability.

To get rid of RAM noises in the scheme, one may adjust the coefficients
$G_{1,2}$ to change the value of $\Delta I=(I_{1}-I_{2})/2$ according to Eq.
(12) such that

$\Delta
I=-\frac{2\bar{I}\bar{\varphi}\Delta\varphi}{\chi^{2}+\bar{\varphi}^{2}+(\Delta\varphi)^{2}}\
,$ (14)

which guarantees zero RAM, i.e., the second term in Eq. (13) becomes null.
Equality (14) can be realized without much difficulty in practice by varying
the optical transmissivities of the areas $S_{1,2}$ on the SOM and/or the
gains of the two quadrants of the detector. Moreover, unexpected displacement
or tilting of the phase-modulated beam can be monitored with another dual-
quadrant photo-detector (PD1 in Fig. 1) and corrections may be applied to the
input beam or the error signal accordingly for further RAM reduction. As for
the PDH scheme, to suppress RAM is not a trivial work [22]; therefore, our
proposed scheme should be a promising alternative to the PDH one in
applications where RAM noise suppression to lower levels may be demanded [10,
11].

Figure 3: (color online) Error signals for cavity locking as a function of the
cavity detuning $\delta$ in comparison with that of the PDH scheme. From the
simulation results, it follows that the error signal is near its maximum when
$\varphi_{0}=\pi/3$ (blue curve) and its size is 1.9 times that of the PDH
error signal for the modulation depth of 1.08 (green curve). In the
simulation, $R_{1}=99.999\%$ and $R_{2}=1$ are assumed. Moreover, the total
area of $S_{1}$ and $S_{2}$ on the SOM covers half of the cross section of the
incident beam.

In the preceding analysis, it was assumed that $|\varphi(\vec{r})|<<1$, which
is nonetheless usually invalid in practice when large error signal sizes are
desired. In the following, let relax this assumption and discuss how to
implement cavity locking by spatial modulation of optical phase front. To that
end, Eq. (2) may be replaced by

$E_{\mbox{m}}(\vec{r},t)=AE_{\mbox{i}}(\vec{r},t)+E_{\mbox{e}}(\vec{r},t)\ ,$
(15)

wherein the effective field now becomes

$E_{\mbox{e}}(\vec{r},t)=\left[e^{i\varphi(\vec{r})}-A\right]E_{\mbox{i}}(\vec{r},t)\
.$ (16)

Here the constant coefficient $A$ is defined as

$A\equiv\frac{\iint
d\vec{r}\cos\varphi(\vec{r})|E_{\mbox{i}}(\vec{r},t)|^{2}}{\iint
d\vec{r}|E_{\mbox{i}}(\vec{r},t)|^{2}}\ ,$ (17)

which ensures the orthogonality between the effective field
$E_{\mbox{e}}(\vec{r},t)$ and the incident field $E_{\mbox{i}}(\vec{r},t)$

$\iint d\vec{r}E_{\mbox{i}}^{*}(\vec{r},t)E_{\mbox{e}}(\vec{r},t)=0\ ,$ (18)

as long as the anti-symmetry (7) of the phase modulation function holds valid.
Then after reflection from the cavity, the modulated light field becomes [12]

$\displaystyle E_{\mbox{r}}(\vec{r},t)$ $\displaystyle=$
$\displaystyle\left\\{\sqrt{R_{1}}e^{i\varphi(\vec{r})}-A\frac{T_{1}}{\sqrt{R_{1}}}\frac{e^{i\delta}\sqrt{R}}{1-e^{i\delta}\sqrt{R}}\right\\}E_{\mbox{i}}(\vec{r},t)$
(19) $\displaystyle\equiv$ $\displaystyle\left[\chi
e^{i\theta}+i\sin\varphi(\vec{r})\right]\sqrt{R_{1}}\ E_{\mbox{i}}(\vec{r},t)\
,$

in which

$\chi
e^{i\theta}\equiv\cos\varphi(\vec{r})-A\frac{T_{1}}{R_{1}}\frac{e^{i\delta}\sqrt{R}}{1-e^{i\delta}\sqrt{R}}\
.$ (20)

After substituting Eqs. (19) and (8) into Eq. (9), one arrives at

$\displaystyle\varepsilon(\delta)$ $\displaystyle=$
$\displaystyle\iint_{S_{1}}d\vec{r}R_{1}|(\chi
e^{i\theta}+i\sin\varphi_{0})E_{\mbox{i}}(\vec{r},t)|^{2}$ (21)
$\displaystyle-\iint_{S_{2}}d\vec{r}R_{1}|(\chi
e^{i\theta}-i\sin\varphi_{0})E_{\mbox{i}}(\vec{r},t)|^{2}$ $\displaystyle=$
$\displaystyle 4R_{1}I_{0}\ \sin\varphi_{0}\
\mbox{Im}\left(\cos\varphi_{0}-A\frac{T_{1}}{R_{1}}\frac{e^{i\delta}\sqrt{R}}{1-e^{i\delta}\sqrt{R}}\right)\
,$

from which it follows that the error signal for cavity locking varies as the
cavity detunes from its peak resonance (Fig. 3) when $\sin\varphi_{0}\neq 0$.

From Fig. 3 it follows that the size of the error signal can be almost twice
that of the PDH one with optimized modulation depth of $\beta=1.08$, for which
the PDH scheme needs to pay the price of high-order harmonic r.f. modulation
side-bands in the optical spectrum of the incident beam. We stress that the
error signal in this scheme is generated as the differential photo-current
from the two quadrants of the detector and, thereby, low-frequency optical and
detection noises can be substantially suppressed. Despite this, the error
signal may still be contaminated by residual low-frequency noises. For cases
where these low-frequency noises are detrimental to laser stability, a
possible solution could be to use a dual-frequency incident light field with
two equal-strength components such that the error signal is up-shifted to r.f.
bands. When both optical frequency components are chosen to simultaneously
resonate in the cavity, the r.f. error signal reads

$\displaystyle\varepsilon(\delta)$ $\displaystyle=$ $\displaystyle
4R_{1}I_{AC}\ \sin\varphi_{0}\ \chi\sin\theta$ (22) $\displaystyle=$
$\displaystyle 4R_{1}I_{AC}\ \sin\varphi_{0}\
\mbox{Im}\left(\cos\varphi_{0}-A\frac{T_{1}}{R_{1}}\frac{e^{i\delta}\sqrt{R}}{1-e^{i\delta}\sqrt{R}}\right)\
,$

wherein $I_{AC}=\bar{I}\cos\Omega t$ with $\bar{I}$ being the average power of
the incident beam and $\Omega$ the optical frequency span between the two
frequency components. A dual-frequency light beam may be created by a dual-
frequency laser [25] or by use of an acousto-optic modulator and a single-
frequency laser [26].

Last but not least, let discuss the differences between the studied cavity
locking method and the tilt-locking one, the latter of which is modulation
free [13]. By slightly tilting or laterally displacing the input beam of a
well-aligned and mode-matched cavity, a TEM10 mode may be generated to
interfere with the fundamental TEM00 mode to produce an error signal. This
particular way to create error signal makes the system very sensitive to light
beam drifting noises as evidenced by the fact that unexpected beam drifting is
indistinguishable from deliberate beam alignment for error signal generation;
therefore, double pass configuration with travelling-wave cavities must be
used to suppress these noises [23]. This leads to the major disadvantage of
the tilt-locking scheme, i.e., its incompatibility with the standing-wave
cavity configuration used by the PDH method. On the contrary, spatial phase
modulation in the studied method produces the error signal in a different way
and hence is not so sensitive to beam drifting as the tilt locking one; in
addition, residual beam drifting noises can be further corrected as discussed
in the previous text. Therefore, our proposed scheme needs not stick to
travelling-wave cavities and allows one to implement laser stabilization with
FP cavities, just as the PDH scheme does. In other words, both the techniques
of spatial modulation (our method) and temporal modulation (the PDH one) of
optical phase front can share the same optical platform for laser
stabilization, which will save much time and man power for cavity design.

To conclude, we have studied optical cavity locking for laser stabilization
through spatial modulation of the phase front of an optical carrier. A
theoretical description of the underlying principle has been developed for the
cavity locking method with a special attention paid to RAM noises caused by
experimental imperfections. While the studied method has the common advantages
of the PDH method and the tilt-locking one, it can provide a more artful way
to eliminate RAM noise in phase modulation. This method can implement laser
stabilization with FP cavities like the PDH one, but giving larger error
signals and cleaner spectra for the carriers than the later. In situations
where cost and portability are a practical issue, the studied scheme allows
one to realize laser stabilization in the PDH optical platform without use of
expensive and bulky devices such as signal generators and electro-optic
modulators.

Funding This work was supported by the National Natural Science Foundation of
China (12074110).

Disclosures The authors declare no conflicts of interest.

Data availability Data underlying the results presented in this paper are not
publicly available at this time but may be obtained from the authors upon
reasonable request.

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