# Composite acousto-optical modulation

Ruijuan Liu  Yudi Ma  Lingjing Ji  Liyang Qiu  Minbiao Ji  Zhensheng Tao
Saijun Wu  Department of Physics, State Key Laboratory of Surface Physics and
Key Laboratory of Micro and Nano Photonic Structures (Ministry of
Education),Fudan University, Shanghai 200433, China<EMAIL_ADDRESS>
###### Abstract

We propose a composite acousto-optical modulation (AOM) scheme for wide-band,
efficient modulation of CW and pulsed lasers. We show that by adjusting the
amplitudes and phases of weakly-driven daughter AOMs, diffraction beyond the
Bragg condition can be achieved with exceptional efficiencies. Furthermore, by
imaging pairs of AOMs with opposite directions of sound-wave propagation, high
contrast switching of output orders can be achieved at the driving radio
frequency (rf) limit, thereby enabling efficient bidirectional routing of a
synchronized mode-locked laser. Here we demonstrate a simplest example of such
scheme with a double-AOM setup for efficient diffraction across an octave of
rf bandwidth, and for routing a mode-locked pulse train with up to $f_{\rm
rep}=400$ MHz repetition rate. We discuss extension of the composite scheme
toward multi-path routing and time-domain multiplexing, so as to individually
shape each pulses of ultrafast lasers for novel quantum control applications.

††journal: oe††articletype: Research Article

##  1 Introduction

The developments of ultrafast technologies [1, 2, 3, 4] have dramatically
enhanced our ability to access physics at various time scales with finest
precision. To utilize the pulsed lasers for nonlinear optical control [5, 6,
7] and spectroscopic measurements [8, 9], pulse shaping techniques have been
developed in the frequency and time domain to produce optical waveforms with
optimally tailored spectra-phases [10, 11]. Notably, existing pulse shaping
techniques are designed to modulate a pulse train in a quasi-static manner. To
individually modulate each output pulses from a mode-locked laser is a largely
unexplored scenario, whose realization may greatly expand the applications of
ultrafast lasers for quantum control [6, 7, 12, 13], metrology [14, 15], and
imaging [16, 17, 18]. Unfortunately, the repetition rates of compact mode-
locked lasers are high. To individually control every pulses requires precise
modulation within nanoseconds, a demanding requirement in the presence of a
general technical gap today on wideband laser modulation at the $\sim$GHz-
level [19, 20]. To modulate individual pulses, the standard method for pulse
picking is through electro-optical modulation (EOM) with Pockels cells.
However, the Pockels cells can hardly operate beyond a 10 MHz rate since it is
difficult to generate the powerful high-voltage waveforms while managing the
dissipation [21]. Acousto-optical modulators (AOM) transduce low-voltage rf
signals to control the optical output with accurately programmable Bragg
diffraction [22, 23, 24, 25]. An excellent example to illustrate AOM pulse
modulation is to shift the phase of individual pulses so as to stabilize the
carrier-envelop phase (CEP) of a frequency comb [26, 27, 28]. However,
efficient AOM diffraction relies on phase-matching the light beams with the
sound waves. Deviation of the operation parameters from the pre-aligned Bragg
condition leads to reduced diffraction efficiency and distorted diffraction
phases. Therefore, in traditional AOM applications, the tuning range and the
control bandwidth for multi-color light are severely limited.

In this work we propose a composite AOM scheme for wide-band, efficient
modulation of continuous-wave (CW) and mode-locked lasers. Inspired by the
coherent control techniques for atom interferometry [29, 30, 31] and nuclear
magnetic resonances (NMR) [32, 33, 34], we identify a new class of techniques
to operate AOMs beyond the Bragg condition and for rapid modulation with
close-to-unity diffraction efficiency. The key idea is summarized in Fig. 1.
By optically linking a series of AOMs with 4-F imaging systems (Fig. 1b), the
Bragg diffraction by a single AOM is coherently split into a $n$-AOM process.
The $2n-1$ amplitude and relative phase degrees of freedom,
$\\{A_{j},\varphi_{j}\\}$, can be continuously adjusted to optimize for
specific light-control applications. In particular, when each “daughter” AOM
is only weakly driven, a 2-mode approximation maps the composite diffraction
dynamics to time-domain spin control, where composite pulse techniques are
developed to universally enhance the resilience to the control errors [32, 33,
34]. The key performance parameters such as the diffraction efficiency can
then be optimized for e.g., efficient diffraction of multi-color beams over a
broad range of sound wave frequency even when the Bragg condition is severely
violated.

We show that an application of the composite scheme with $n=2$ already enables
exceptionally high diffraction efficiency-bandwidth combinations. In
particular, in this demonstration a diffraction efficiency above $85\%$ is
maintained over nearly an octave of the driving rf frequency without optical
re-alignments. A peak diffraction efficiency of $\sim 95\%$ is accompanied by
the extinction of the undiffracted path at a $\sim 15$ dB level, allowing the
double-AOM system acting as a high contrast, bi-directional router for the
first time to our knowledge. When the sound waves in AOM2 is chosen to
propagate in the opposite direction as those in AOM1 (Fig. 1c, Fig. 2), the
interference of the diffraction amplitudes leads to rapid switching of the
output order at twice the rf driving frequency $\omega_{\rm S}$, which is
exploited for routing a pulse train with nanosecond switching time. We discuss
the possibility of constructing a high-contrast composite AOM network for
shaping individual pulses of a mode-locked laser with up to GHz repetition
rates.

##  2 Principles

Figure 1: Operational principle of the composite AOM scheme. (a) Light beams
with wavevector ${\bf k}_{a,b}$ and wavefronts $C_{a,b}$ are Bragg-coupled by
the sound wave with wavevector ${\bf k}_{\rm S}$. The blue arrow points to the
location of the rf-to-sound transducer represented by the horizontal blue line
in the schematic, which decides the direction of sound-wave propagation
(marked by the lower semi-transparent arrow). The approximate phase-matching
${\bf k}_{a}-{\bf k}_{b}\approx{\bf k}_{\rm S}$ is illustrated on the right.
When the AOM is weakly driven, the diffraction orders off the Bragg condition
(represented by dashed lines) can be ignored. The input-output relation for
the optical wavefront can then be represented by $|\mathcal{E}_{\rm
out}\rangle=U|\mathcal{E}_{\rm in}\rangle$, with evolution operator
$U(t,re^{i\varphi})=(t,re^{i\varphi};-r^{*}e^{-i\varphi},t^{*})$ controlled by
the rf signal to transform a $(C_{a},C_{b})$ spinor. This input-output
relation are applied iteratively in (b) for 4F-imaged identical AOMs with
$U_{j}=U(t_{j},r_{j}e^{i\varphi_{j}})$. With the output wavefront of each AOM,
$|\mathcal{E}_{j,\rm out}\rangle$ imaged to the exit window of the next AOM,
the wavefront at the entrance window of the next AOM is give by
$|\mathcal{E}_{j+1,\rm in}\rangle=U_{0}^{-1}|\mathcal{E}_{j,\rm out}\rangle$,
with $U_{0}^{-1}$ effectively evolving the wavefront backward over the
interaction length $L$ in absence of acousto-optical diffraction. (c) A simple
composite AOM network with $\omega_{\rm S}^{\rm A}=\pi(2N+1)f_{\rm rep}$/2 and
$\omega_{\rm S}^{\rm{B,C}}=\pi(2N^{\prime}+1)f_{\rm rep}/4$ to route four
temporally adjacent pulses in a $f_{\rm rep}$ pulse train from left into four
output paths on the right, with close to unity efficiency and controllable
relative phases. The network can be reversed for time-domain multiplexing,
i.e., to combine the relatively delayed ultrafast pulses into a single beam.

###  2.1 Coherent control of light with AOM

We consider composite AOM setup schematically illustrated in Fig. 1b where the
light diffracted by one AOM is imaged to another via a 4-F imaging system. For
each single AOM operation (Fig. 1a), the para-axial propagation of light in
the index field $n({\bf r},t)$ along ${\bf e}_{z}$, with
$E=\mathcal{E}e^{i\bar{n}k_{0}z}$ obeying $(\nabla^{2}+n^{2}k_{0}^{2})E=0$, is
characterized by a slowly varying amplitude $\mathcal{E}$ with

$i\partial_{z}\mathcal{E}=-\frac{1}{2\bar{n}k_{0}}\nabla_{\bot}^{2}\mathcal{E}-\delta
nk_{0}\mathcal{E}.$ (1)

Here $\bar{n}$ is the refractive index of the AOM crystal, for example,
$\bar{n}\approx 2.3$ for TeO2 at near infrared. An rf-driven traveling sound
wave with a strain amplitude $\eta$ leads to a rapidly varying refractive
index deviation $\delta n=n-\bar{n}$ as

$\delta n=\eta p\frac{1-\bar{n}^{2}}{2\bar{n}}\cos(k_{\rm S}x-\omega_{\rm
S}t+\varphi),$ (2)

where the constant $p$ is decided by the photo-elastic effect [35].

The slowly varying envelops of the density wave are described by the phase
$\varphi({\bf r}-{\bf v}_{\rm S}t)$ and the amplitude $\eta({\bf r}-{\bf
v}_{\rm S}t)$ functions, which are controlled by those of the driving rf-
fields through rf-to-sound transducers, and propagate with the group velocity
${\bf v}_{S}$ along ${\bf e}_{x}$, with $v_{\rm S}=\omega_{\rm S}/k_{\rm S}$
for the linearly dispersive crystal. For example, in TeO2 the longitudinal
sound velocity is $v_{\rm S}=4260~{}$m/s. The detailed spatial dependence is
determined by the crystal geometry and the location of the transducers. Here,
we simply assume a stationary and spatially uniform $\varphi$, and a
stationary amplitude following a step-wise distribution as constant $\eta$ for
$0<z<L$ and zero otherwise. The treatment can be generalized to account for
realistic sound wave distributions in the crystal.

The equivalence between Eq (1) and the two-dimensional Schrödinger equation
allows us to introduce Dirac notation $|\mathcal{E}(z)\rangle$, with
$\mathcal{E}({\bf r}_{\perp},z)=\langle{\bf r}_{\perp}|\mathcal{E}(z)\rangle$,
to conveniently describe the wavefront of the optical mode-function
propagating along $z$. The input-output relation for the wavefront through the
$0<z<L$ interaction can then be generally expressed as $|\mathcal{E}_{\rm
out}\rangle=U|\mathcal{E}_{\rm in}\rangle$ by an evolution operator
$U=e^{-iHL}$ that integrates the interaction by the Hamiltonian

$H=-\frac{1}{2\bar{n}k_{0}}\nabla_{\perp}^{2}+\eta
p\frac{\bar{n}^{2}-1}{2\bar{n}}k_{0}\cos(k_{\rm S}x+\varphi).$ (3)

Notice this $H$ governs the paraaxial wavefront propagation along $z$, with
$t$ merely as a parameter. We have thus absorbed the $-\omega_{\rm S}t$ term
in Eq. (2) into $\varphi(t)$ for conciseness in the following. Equation (3)
can be re-written in the $k$-space as $H=H_{0}+\int d^{2}{\bf k}_{\perp}V({\bf
k}_{\perp})$, with $H_{0}=\frac{1}{2\bar{n}k_{0}}{\bf k}_{\perp}^{2}$ and

$V({\bf k}_{\perp})=\frac{K}{2}e^{i\varphi}|{\bf k}_{\perp}+{\bf k}_{\rm
S}\rangle\langle{\bf k}_{\perp}|+h.c..$ (4)

Physically, $V$ couples an “infinite” set of wavefronts shifted in ${\bf
k}_{\perp}$-space by multiple ${\bf k}_{\rm S}=k_{\rm S}{\bf e}_{x}$. The
coupling constant $K=\eta pk_{0}(\bar{n}^{2}-1)/2\bar{n}$ can be referred to
as a “spatial Rabi frequency” [30]. We restrict ourselves to the input
wavefronts near a lowest order Bragg-resonance, $\langle{\bf
r}_{\perp}|\mathcal{E}(z=0)\rangle=C_{a}({\bf r}_{\perp},0)e^{-ik_{\rm
S}x/2}+C_{b}({\bf r}_{\perp},0)e^{ik_{\rm S}x/2}$ with $C_{a,b}$ confined to
$|k_{x}|<k_{\rm S}/2$ in the $k$-space. Considering a diffraction “pulse area”
$\Theta=KL$, a weak driving condition of

$\Theta\ll\frac{k_{\rm S}^{2}}{\bar{n}k_{0}}L$ (5)

allows us to truncate the wave propagation by Eq. (1) in $k$ to obtain
$\langle{\bf r}_{\perp}|\mathcal{E}(z)\rangle\approx C_{a}({\bf
r}_{\perp},z)e^{-ik_{\rm S}x/2}+C_{b}({\bf r}_{\perp},z)e^{ik_{\rm S}x/2}$.
The pair of Bragg-coupled wavefronts obey

$\begin{array}[]{l}i\partial_{z}C_{a}=\frac{({\bf k}_{\perp}-{\bf k}_{\rm
S}/2)^{2}}{2\bar{n}k_{0}}C_{a}+\frac{\Theta}{2L}e^{i\varphi}C_{b},\\\
i\partial_{z}C_{b}=\frac{({\bf k}_{\perp}+{\bf k}_{\rm
S}/2)^{2}}{2\bar{n}k_{0}}C_{b}+\frac{\Theta}{2L}e^{-i\varphi}C_{a}.\end{array}$
(6)

in the $k-$space, i.e. with $C_{a,b}({\bf k}_{\perp},z)$ to be the 2D Fourier
transform of $C_{a,b}({\bf r}_{\perp},z)$. By parametrizing
$(C_{a},C_{b})=(\cos(\frac{\theta}{2})e^{-i\phi/2},\sin(\frac{\theta}{2})e^{i\phi/2})$
and introducing a wavevector mismatch $\delta k=k_{x}k_{\rm S}/\bar{n}k_{0}$,
the dynamics of the spinor is mapped to that for a unit vector ${\bf
r}=(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)$ on a Bloch sphere (Fig.
2b):

$\frac{d}{dz}{\bf r}={\bf K}\times{\bf r}.$ (7)

Here ${\bf K}=(K\cos\varphi,K\sin\varphi,\delta k)$ is a “spatial Rabi
vector”. Clearly, for incident beam with ${\bf k}_{\perp}=\\{k_{x},k_{y}\\}$,
$|k_{x}|\ll\frac{\bar{n}k_{0}}{k_{\rm S}L}$ is required to support a ${\bf
k}_{\perp}$-insensitive $\Theta$-operation near the Bragg condition. One might
try to improve the broadband performance simply by reducing the interaction
length $L$. However, for a fixed $\Theta$-operation such as to Bragg-deflect
the incident beam with $\Theta=\pi$, a reduced $L$ typically leads to
increased AOM strength $K=\Theta/L$ to compromise the weak drive condition by
Eq. (5), resulting in high-order diffraction losses. Indeed, commercially
available AOMs are usually optimized near $\Theta=\pi$ with an interaction
length $L$ that balances the broadband operation with the weak drive
condition. The compromise is partially responsible for the never perfect
Bragg-diffraction using commercially available AOMs.

So far, we merely reformulate the standard AOM Bragg-diffraction theory with
notations of quantum mechanics. Nevertheless, they allow us to confirm the
validity of the 2-mode approximation and to apply composite control theory
[30, 33, 34] for the $C_{a,b}$ control next.

###  2.2 Composite AOM for error-resilient modulation

We consider the setup in Fig. 1b where multiple AOMs are optically linked by
the 4-F optics. Within the 2-mode approximation, each “daughter-AOM” $j$
transforms $\\{C_{a},C_{b}\\}$ with a unitary matrix
$U(t_{j},re^{i\varphi_{j}})=(t_{j},r_{j}e^{i\varphi_{j}};-r^{*}_{j}e^{-i\varphi_{j}},t_{j}^{*})$,
which can be obtained by integrating Eq. (6) and being visualized on the Bloch
sphere (Fig. 2b). Furthermore, before the second diffraction, since the 4-F
system effectively images the output at $z=L$ to the exit port of the next
AOM, the wavefront should first evolve backward along $z$ with
$U_{0}^{-1}=e^{iH_{0}L}$. Here $U_{0}^{-1}=e^{iH_{0}L}$ can also be obtained
from Eq. (6) with $\Theta=0$ and visualized on the Bloch sphere as precession
of ${\bf r}$ around ${\bf K}_{0}=(0,0,-\delta k)$. More generally, the
transformation of the wavefront by the composite AOM is described as
$|\mathcal{E}_{\rm out}\rangle=\bar{U}|\mathcal{E}_{\rm in}\rangle$, with
(with $\prod_{j=1}^{n}$ for multiplying from left)

$\bar{U}=U_{0}\prod_{j=1}^{n}\left(U_{0}^{-1}U_{j}\right).$ (8)

One should thus be able to adjust $\\{A_{j},\varphi_{j}\\}$ of each daughter
AOM to optimize $\bar{U}$ and the output wavefront $|\mathcal{E}_{\rm
out}\rangle$. Specifically, within the 2-mode approximation, the matrix
$\bar{U}=(\bar{t},\bar{r};-\bar{r}^{*},t^{*})$ is constructed in essentially
the same way as those for achieving 2-level robust population inversion [30,
33] and universal qubit gates [34]. The wavevector mismatch $\delta k$ and
diffraction pulse area $\Theta_{j}$ are in direct analogy to the detuning and
the pulse area in 2-level spin-controls [33, 34]. The associated formula can
thus be applied here to improve the robustness of beam-splitting [34] or
deflection [30, 33] against deviation of these AOM operation parameters, which
may arise during e.g. broadband diffraction of ultrafast pulses [26, 27, 28].
In addition, high-order diffraction leakage which generally limits the
efficiency of single AOMs may be dynamically suppressed in the composite
scheme, similar to those in multi-level quantum control [32]. To this end, we
note similar topic to be investigated further for robust coherent control of
matterwave [30, 31].

Figure 2: The double-AOM scheme for efficiently routing a mode-locked pulse
train with AOM diffraction beyond the Bragg condition. In (a), AOM1,2 with
$L_{1}\approx L_{2}$ are imaged by a 4-F system, with directions of sound wave
propagation effectively to be opposite to each other. . A pulse train from a
picosecond oscillator with $T_{\rm rep}=1/f_{\rm rep}$ is subsequently
diffracted by AOM1,2. When the sound-wave frequency is set as $\omega_{\rm
S}=\pi f_{\rm rep}(2N+1)/2$ with $N$ as an integer,
$\Delta\varphi_{1,2}(t)=2\omega_{\rm S}t+\Delta\varphi_{1,2}(0)$ is
incremented by $\pi$ for any pulse pair separated by $T_{\rm rep}$. By
adjusting $\Delta\varphi_{1,2}(0)$ and AOM driving amplitudes $A_{1,2}$, full
switching from $\bar{R}=1$ to $\bar{T}$=1 can be achieved. The effective
diffraction mechanisms under the 2-mode approximation are illustrated in (b,i)
and (b,ii) on a Bloch sphere, for the case of a large wavevector mismatch
$\delta k\approx\pi/L\sqrt{2}$ (corresponding to the diffraction path marked
with dashed blue line in (a)). Typical experimental measurements for routing a
$f_{\rm rep}=80$ MHz picosecond pulses are given on (c,i) leading to 30:1 and
100:1 side-pulse suppression. The time-dependent diffraction $\bar{R}$ and
transmission $\bar{T}$ measured by a CW laser are given in (c,ii), with
$\\{(\bar{T})_{\rm max},(\bar{R})_{\rm max}\\}=\\{0.94,0.93\\}$,
$\\{(\bar{T})_{\rm min},(\bar{R})_{\rm min}\\}=\\{0.04,0.02\\}$. The slightly
decreased contrast in the CW case is likely related to the finite response
time of the photo-detector.

###  2.3 A double-AOM scheme for coherent optical routing

In the following we study a simplest example of a double-AOM scheme in Fig. 2,
which is a building block of the Fig. 1c network, to demonstrate the utility
of the composite diffraction for broadband, flexible control of CW and mode-
locked lasers.

With Eqs. (6)(7)(8), it is straightforward to understand the operation
principle of the double-AOM scheme in Fig. 2. As illustrated in Fig. 2a,
practically the 4-F optics to image the wavefronts through AOM1,2 are composed
of $l_{1,2}$ in the standard 4-F configuration to ensure the imaging
precision. The sound waves in AOM2 can be chosen to propagate in the same or
opposite direction as those in AOM1. As by Eqs. (2)(4), a flip of ${\bf
k}_{\rm S}$ sign effectively change the time-dependence of the sound-wave
phase $\varphi_{j}(t)=\varphi_{j}(0)\pm\omega_{\rm S}t$.

To understand the dynamics of double-AOM diffraction, it suffices to analyze
the composite $\bar{U}$ by Eq. (8) at a specific time $t$ for AOM1,2 with a
specific driven amplitude $A_{1,2}$ and phase $\varphi_{1,2}$. According to
Eq. (7), to achieve an optimal diffraction $\bar{R}\equiv|\bar{r}|^{2}$ (Fig.
2(b,i)), the coherent control strategy should be that AOM1 rotate the state
vector ${\bf r}$ to the equator of the Bloch sphere with ${\bf K}_{1}$, which,
after a free precession by ${\bf K}_{0}$, be rotated again by ${\bf K}_{2}$
with optimal $\Delta\varphi_{2,1}=\varphi_{2}-\varphi_{1}$ to complete the
spin inversion. Optimal diffraction with $\bar{R}\approx 1$ could thus be
obtained even at large wavevector mismatch $\delta k\leq\pi/\sqrt{2}L$, by
properly adjusting $\\{A_{1,2},\varphi_{1,2}\\}$. The strategy is in direct
analogy to the double-diffraction method in atom interferometry [29, 30, 31].
To minimize $\bar{R}$ so as to recover $\bar{T}\equiv|\bar{t}|^{2}$ to unity
(Fig. 2(b,ii)), an additional $\pi$ should be added to $\Delta\varphi_{2,1}$
so that AOM2 undoes the rotation of ${\bf r}$ by AOM1. On the other hand, the
optical phase of the deflected beam can be flexibly controlled via
$\bar{\varphi}_{1,2}=(\varphi_{1}+\varphi_{2})/2$, such as for CEP adjustments
[26, 27, 28] or coherent pulse stacking [36, 37].

Importantly, in contrast to single-AOM diffraction which requires
$\Theta\approx\pi$ to achieve the best efficiency, here for the double-AOM
scheme $\Theta_{1,2}=\pi/2$ are halved at $\delta k=0$. The weak drive
condition by Eq. (5) is accordingly better satisfied toward the 2-mode
approximation (Eq. (6)). In addition, the $U_{0}^{-1}$ reversal by the 4-F
imaging partially rephases the $\delta k$-induced broadening, much like a spin
echo in NMR, thereby improving the tolerance of efficient diffraction to the
$\delta k$-spreading. These advantages are expected to improve further with
larger $n$ and additional AOMs. In particular, we expect it becomes easier for
the double-AOM scheme to achieve a close-to-unity composite $\bar{R}$ than
those with single AOMs. Furthermore, when $\bar{R}$ is close enough to unity,
the undiffracted path (Channel A) is shut off. Therefore, by switching on/off
the composite modulation, the double-AOM system should be able to route the
incident light into the channel A/B, respectively. We refer this highly
efficient AOM function as bidirectional optical routing.

We now discuss the time-dependence of the double-AOM in the Fig. 2a setup.
Since the 4-F linked AOM1,2 have opposite directions of sound-wave
propagation, the relative phase $\Delta\varphi_{1,2}(t)=2\omega_{\rm
S}t+\Delta\varphi_{1,2}(0)$ scans rapidly at twice the rf frequency. For the
optimally adjusted $\\{A_{1},A_{2}\\}$ as discussed above, an $\omega_{\rm
S}=(2N+1)\pi f_{\rm rep}/2$ with integer $N$ should result in a $\pi$ phase
increment in $\Delta\varphi_{1,2}$ for two successive pulses separated by
$T_{\rm rep}=1/f_{\rm rep}$. By pre-adjusting $\varphi_{1,2}(0)$, rapid
switching of the composite AOM output between $\bar{R}=1$ and $\bar{T}=1$ can
be achieved (Fig. 2c) to route adjacent pulses from a synchronized mode-locked
laser into two directions. Meanwhile, the common phase
$\bar{\varphi}_{1,2}=(\varphi_{1}+\varphi_{2})/2$ remains as a rapidly
adjustable degree of freedom to shift the optical phase in the diffraction
path B.

##  3 Experimental demonstration

###  3.1 Alignment procedure

We demonstrate the double-AOM scheme for efficiently modulating pulsed lasers.
As being summarized by Fig. 2a, a pair of AOMs (AA optoelectronic, MT-110-A1
with an estimated $L\approx 7\sim 8$ mm) are aligned to operate at
$\omega_{\rm S}=2\pi\times 100$ MHz to Bragg-deflect the laser beam from a
single-mode optical fiber into port-$a$. Phase-stable rf signals from a
programmable multi-channel synthesizer (NovaTech 409B) are amplified to drive
the sound waves. AOM2 is imagined to AOM1 by a pair of achromatic lenses
$l_{1,2}$ with $F$=100 mm focal length. Here, the primary reason for choosing
the achromatic lenses is to ensure precise 4F-imaging with a spatial bandwidth
beyond ${\bf k}_{\rm S}$ and a field of view to cover the incident beam size.
The waist radius $w\approx 110~{}\mu$m for the incident Gaussian beam is large
enough to support $|r_{1,2}|^{2}\approx 80\%$ with each of the two AOMs (Fig.
4(a)). To achieve composite AOM, the relative alignment starts with switching
on a single AOMj only, with $A_{j}$ adjusted to achieve $|r_{j}|^{2}\approx
0.5$ half deflection. Nearly identical beam splitting by AOM1,2 into the
output Channels $A$ and $B$, with nearly identical output shapes in each case,
are confirmed by recording the outputs with a digital camera. Next, fine
alignments are achieved by switching on both AOM1,2 with reduced
$|r_{1,2}|^{2}\sim 0.3$ and by injecting the single-mode fiber with a
synchronized pulse train with $\tau=0.5$ ns pulse duration [12] at $f_{\rm
rep}=100$ MHz. By shifting the $l_{1,2}$ location and finely orienting AOM2,
the AOM1,2 diffractions are mode-matched, leading to uniformly varying output
beams on the camera when $\Delta\varphi_{1,2}(0)$ is slowly scanned. Further
improvements to the alignment is achieved by optimizing the contrast of the
time-dependent transmission $\bar{T}$ (from Channel A) and diffraction
$\bar{R}$ (from Channel B) at 200 MHz with a CW laser, using a multi-mode-
fiber coupled fast photodetector (Thorlabs DXM12DF), see Fig. 2c for example.
Finally, the rf amplitudes $A_{1,2}$ are scanned to finely locate
$|r_{1,2}|_{\rm opt}^{2}\approx 0.5$ that maximize the composite diffraction
efficiency $\bar{R}_{\rm max}$. Typical $|r_{1,2}|^{2}$-scan 2D data for the
CW laser modulation are shown in Fig. 3a. The optimal $|r_{1,2}|_{\rm
opt}^{2}$ values at $\omega_{\rm S}=2\pi\times$100 MHz are marked with white
dashed lines.

The moderate $\omega_{\rm S}=2\pi\times 100~{}$MHz in this report was chosen
to fit parallel experimental progresses in our lab. In separate setups we
confirm an increased $\omega_{\rm S}$ to 200 MHz does not lead to measurable
degradation of the double-AOM performance. In addition to the echo-type
rephasing as in Fig. 2b to suppress $\delta k$ dephasing (Eq. (6)), the
efficient operation of the double-AOM scheme at high $\omega_{\rm S}$ is also
likely facilitated by a weaker requirement on the driving rf power than in
traditional AOM applications.

###  3.2 An $f_{\rm R}$=400 MHz synchronized pulse router

From the CW laser measurements as in Fig. 2(c,ii), we see that the full
switching from $\bar{R}\approx 1$ to $\bar{T}\approx 1$ for the double-AOM
output is within a $\tau_{\rm R}=2.5$ ns transient time. Therefore, even at
this moderate $\omega_{\rm S}=2\pi\times 100$ MHz we are still able to realize
a pulse router with nanosecond switching speed with a switching rate of
$f_{\rm R}=400~{}$MHz. More generally, the double-AOM system can route a
synchronized mode-locked pulse train with $f_{\rm rep}=f_{\rm R}/(2N+1)$
repetition rate into two directions with integer $N$. In this work, we choose
$N=2$ to route a picosecond mode-locked laser (Spectra-Physics Tsunami system)
with $f_{\rm rep}$=80 MHz.

To achieve the pulse routing, the Novatech rf source is synchronized with the
laser pulses by phase-locking the Novatech clock with a 10 MHz laser
synchronization signal. By adjusting $\Delta\varphi_{1,2}(0)$, adjacent pulses
injected to port-$a$ with $T_{\rm rep}=12.5$ ns inter-pulse spacing are
alternating routed into Channel A and B, as demonstrated in Fig. (2c).
Excluding the $\sim 5\%$ overall insertion loss, the efficiency for pulse
picking in Channel A(B) reaches $96\%$ ($94\%$), with a contrast reaching 30:1
(100:1) respectively relative to the suppressed side pulses. Notice this
contrast is compromised slightly by an $\delta\tau\approx 100~{}$ps relative
jitter between the laser pulses and the rf signal, which is primarily limited
by the phase locking quality and should be improvable to 1 ps level via e.g.,
direct digital synthesizing of the rf signals.

We find $\Delta\varphi_{1,2}$ stays optimal in our temperature stabilized lab
for tens of hours at least without noticeable drifts. The excellent passive
phase-stability is due to common-mode rejection of drifting and vibrational
noises by the two diffraction orders (Fig. 2a), which share all the optics
during their propagation and are within one millimeter to each other
spatially.

We think the $\sim 95\%$ composite diffraction efficiency and the $15\sim 20$
dB side-pulse suppression are fairly close to the theoretical limit decided by
the incident beam size $w$ and the interaction length $L$ (Sec. 3.3). Our
numerical model suggests that both the efficiency and contrast can be improved
by increasing $w$ and $L$, i.e., optimally using the apertures of large AOMs.
For example, with $w=0.5~{}$mm and $L=15$ mm, composite efficiency $\sim 99\%$
appears achievable, a prediction subjected to experimental verification where
much finer alignments are likely required particularly by taking into account
variation of the sound fields experienced by the large wavefronts. A more
straightforward method to further improve the performance at the moderate $w$
and $L$ is to increase the number of AOMs involved in the composite
diffraction so as to further improve the small-$\Theta$ condition and to
reduce the $\delta k$-broadening (Eqs. (5)(6)). Furthermore, the side-pulse
suppression can be doubled to $30\sim 40$ dB level simply by applying the
double-AOM scheme again to a selected output channel, at a small cost of
overall efficiency.

###  3.3 Ultra-broadband diffraction

Figure 3: Fine tuning of $|r_{1,2}|^{2}$ to optimize $\bar{R}_{\max}$ for the
double-AOM scheme. The experimentally measured $\bar{T}_{\rm max,min}$ and
$\bar{R}_{\rm max,min}$ are given in column (a,c) (with raw data re-
interpolated to evenly spaced $|r_{1,2}|^{2}$ grids) to be compared with the
simulated data in column (b,d) based on the Eq. (1) model with $L=8~{}$mm. The
AOMs as by Fig. 2a are optimally aligned at $\omega_{\rm S}=2\pi\times 100$
MHz, with corresponding data displayed in column (a,b). Data for $\omega_{\rm
S}=2\pi\times 140$ MHz operation which substantially violates the Bragg
condition are shown in column (c,d). Figure 4: Optimal diffraction efficiency
for single AOM (a) and double AOMs (b), measured by monitoring the CW
transmission with a fast photo-detector. Simulated data based on the effective
Eq. (1) model parametrized by the interaction length parameter $L$ (mm) are
displayed for comparison. For the $\bar{R}_{\rm max}$ measurements in (b), the
efficiencies are likely to be slightly underestimated due to the finite
detector bandwidth.

We now demonstrate the efficient double-AOM deflection when the Bragg
condition is substantially violated by a large $\delta k$ mismatch. For the
purpose, we shift the sound-wave frequency $\omega_{\rm S}$ away from 100 MHz
and measure the maximum diffraction efficiency $\bar{R}_{\rm max}$ optimized
with $\\{A_{j},\varphi_{j}\\}$, at sound wave frequency as large as
$\omega_{\rm S}=2\pi\times 140$ MHz. The electronic optimization process to
obtain $\bar{R}_{\rm max}$ in Fig. 3c is identical to those for achieving
$|r_{1,2}|^{2}_{\rm opt}$ in Fig. 3a for the case of $\omega_{\rm
S}=2\pi\times 100$ MHz, by retrieving $\\{\bar{R}_{\rm min},\bar{R}_{\rm
max}\\}$,$\\{\bar{T}_{\rm max},\bar{T}_{\rm min}\\}$ from Channel A/B with CW
laser measurements (Fig. 2(c,ii)). With the Bragg condition aligned at
$\omega_{\rm S}=2\pi\times$100 MHz, a $40\%$ deviation of $\omega_{\rm S}$
leads to $k_{x}=0.4k_{\rm S}$ and $\delta k=0.4k_{\rm
S}^{2}/\bar{n}k_{0}\approx\pi/(\sqrt{2}\times 10~{}{\rm mm})$. According to
the analysis in Sec 3.1, if the sound field interaction length $L$ is within
10 mm, then the 2-mode theory predicts a maximum $|r_{1,2}|^{2}\geq 0.5$ and
nearly perfect $\bar{R}_{\rm max}\approx 1$. Practically, we find
$|r|^{2}_{1,2}$ saturate to 0.3 and 0.4 respectively (Fig. 4a), likely due to
a lack of rf power in our setup to fully drive the sound wave in presence of
significant impedance mismatch. Nevertheless, guided by the effective theory
we are able to locate $\bar{R}_{\rm max}=0.82$ at optimal $|r_{1,2}|^{2}$ and
$\Delta\varphi_{1,2}$, as in Fig. 3c and Fig. 4b. We further measure
$\bar{R}_{\rm max}$ for $\omega_{\rm S}$ between 70 MHz and 140 MHz, with
$\bar{R}_{\rm max}>90\%$ uniformly achieved between $80$ and $130$ MHz (Fig.
4b).

To further understand the double-AOM system beyond the 2-mode approximation,
we numerically integrate the 1D AOM model outlined in Sec. 2 according to the
original Eq. (1). As discussed there, in the simple model the sound fields
have a “step-function” distribution with an effective interaction length $L$.
We simulate the performance of single and double-AOM with the model with
various $L$ to match the experiment. By comparing the simulations with the
experimental measurements, as in Fig. 4, an effective $L=7\sim 8$ mm
interaction length can be inferred with which we further simulate the
$|r_{1,2}|^{2}$-tuning process as those in Fig. 3(b,d), with very good global
agreements. It should be noted that the actual sound field distribution is
more complicated and may vary among AOMs. We expect even better performance
once these details are included, followed by fine tuning of specific two-AOM
systems.

The broadband deflection supported by the double-AOM system allows us to
synchronize the composite diffraction with the mode-locked laser at
$\omega_{\rm S}=2\pi\times 80~{}$MHz, without realigning the optics, to
directly observe the channel A/B output intensity profiles with a slow camera.
Figure 5 gives such measurements when $\Delta\varphi_{1,2}(0)$ is adjusted to
give $\bar{R}\approx 0.5$, 0 and 1 respectively. As in Figs. 5b,5c, for the
case of $\bar{R}\approx 0,~{}1$, the residual output in the A/B ports are so
weak, that we have to over-expose the camera to see the residual features.
These residual features provide us with clue on what is limiting the pulse
routing efficiency and contrast for future improvements of the double-AOM
system.

We note that to achieve the ultra-broadband performance with the $n=2$
composite scheme, the composite parameter $\Delta\varphi_{1,2}$ and
$|r_{1,2}|^{2}$ controlled by each AOM needs to be adjusted for specific
$\omega_{\rm S}$ in a point-by-point basis. The control bandwidth for the
required adjustments of diffraction amplitudes $r_{1,2}$ by AOM1,2 is still
limited to a few MHz level, decided by the input beam size $w$ and the
associated $\delta k$ spreading [23]. As discussed in Sec. 3.2, the “time-
reversal” induced by the 4-F imaging enhances the $\delta k$-tolerance and
therefore supports a moderately enhanced control speed. The benefit is
expected to increase with $n$, a topic associated with robust AOM control in
general [33, 34] for future investigations. On the other hand, the double-AOM
scheme enables broadband coherent frequency sweeps by rapidly following
$r_{1,2}$ with $\omega_{\rm S}$ according to a pre-characterized map. Beyond
the synchronized pulse routing, the MHz-level control bandwidth is typically
large enough for basic and advanced CW and pulsed AOM applications [26, 38]
where the ability to keep nearly perfect diffraction quality is highly
preferred during long-range (coherent) frequency sweeps.

Figure 5: The output of the composite AOM recorded by a digital camera. The
alignment of the double-AOM is optimized at $\omega_{\rm S}=2\pi\times
100~{}$MHz to route a $f_{\rm rep}=80$ MHz picosecond laser alternating into
Channel A/B respectively. Here we instead set $\omega_{\rm S}=2\pi\times 80$
MHz to synchronize with $f_{\rm rep}$. By adjusting $\Delta\varphi_{1,2}(0)$,
$\bar{R}\approx 0.5,~{}0,~{}1$ are achieved for all the pulses for the slow
camera recording, as being displayed in (a),(b),(c) respectively. The re-
scaled colorbar on the right is for Figs. (b, c) to highlight the weak
residuals in the A/B channels.

##  4 Discussions

###  4.1 Multi-path routing and time-domain multiplexing

The high pulse picking efficiency toward $99\%$ level should support one to
iteratively apply the double-AOM system for multi-directional pulse routing
and time-domain multiplexing of a synchronized ultrafast laser. An example of
the network is illustrated in Fig. 1c, where the output path number is doubled
with another layer of double-AOM systems. In this simple network, when the
driving rf frequencies are set as $\omega_{\rm S}^{\rm A}=\pi(2N+1)f_{\rm
rep}/2$ and $\omega_{\rm S}^{{\rm B,C}}=\pi(2N^{\prime}+1)f_{\rm rep}/4$ and
by adjusting $\Delta\varphi_{1,2}^{\rm A,B,C}$ and optimizing $A_{1,2}^{\rm
A,B,C}$, an ultrafast pulse train with $f_{\rm rep}$ repetition rate from left
can be routed into four output paths on the right, with stable efficiencies
close to unity. The common phase $\bar{\varphi}_{1,2}^{\rm A,B,C}$ can be
adjusted in the feedback loops to stabilize the relative phases to the desired
values among the four output paths. Depending on the applications, the pulse
routing contrast can be enhanced by additional time-domain-filtering as
discussed in Sec. 3.2, or by expanding the double-AOM units themselves into
$n>2$ composite AOM sequences (Fig. 1b) to further enhance the performance.

With the pulse routing speed at the driving-rf frequency limit, the composite
network as in Fig. 1c can easily operate with ultrafast laser pulses at a GHz
repetition rate, for example by choosing $\omega_{\rm S}^{\rm A}=2\pi\times
250$ MHz and $\omega_{\rm S}^{\rm B,C}=2\pi\times 125$ MHz. After the spatial
separation of the temporally adjacent pulses, one shall individually shape
each pulses [10, 11] and then direct them to a target experiment [6, 12].
Alternatively, the network can be reversed to coherently combine the
individually shaped pulses back to a single spatial mode via time-domain
multiplexing, thereby realizing individual shaping of the adjacent ultrafast
pulses at the high repetition rate.

To reduce the insertion losses, we note the fact that the composite AOM only
needs to be weakly driven makes AOM based on crystalline quartz, which are
typically more difficult to be deeply driven but are more amenable to
ultrafast control with low insertion losses [39], a favorable choice for
constructing the network. In addition, since the rapid switching between
$\bar{R}=1$ and $\bar{T}=1$ as in Fig. 2 is a result of diffraction
interference and does not require tightly focusing the laser beams, the high-
speed pulse routing can operate with much higher power than those by
traditional AOM-based pulse pickers when subjecting to a same level of optical
intensity damage threshold.

###  4.2 Summary and outlook

Acousto-optical modulators are ubiquitously equipped in modern research labs
across fields for modulation of amplitude, phase, frequency and propagation
direction of light. In this work, we have proposed a novel approach to control
light with acousto-optical modulation by coherently splitting a single AOM
diffraction into an n-AOM process. We have shown that the diffraction dynamics
of the weakly driven n-AOM system can be mapped to time-domain spin control
where composite techniques are developed to universally enhance the resilience
to control errors [32, 33, 34]. We notice related schemes of composite-pulse-
inspired error-resilient 2-mode optical control have been derived [40, 41, 42]
in scenarios where a mode-truncation as in this work is not required. The
discussions in this work have been restricted to isotropic acousto-optical
interactions. With proper modification of the simple theory, the composite
diffraction technique may also be applied to acousto-optical devices based on
birefringent response of slow shear waves [43, 38].

Experimentally, we have demonstrated the utility of the composite scheme with
a simplest $n=2$ example for achieving exceptional diffraction efficiency,
ultra-broadband operation, and high-contrast, bidirectional routing of a mode-
locked laser. We note that when neither the wide-band diffraction nor the
$\bar{\varphi}_{1,2}$ diffraction phase adjustability is important, the
double-AOM scheme as in Fig. 2a can be replaced by double-passing a single AOM
through retro-reflection. This can be achieved by inserting a mirror in
between AOM1,2 so that AOM2 simply becomes the mirror-image of AOM1. Notably,
although the resulting composite AOM looks similar to traditional application
of double-pass AOMs [22], the modulation characteristics are substantially
different. The results will be given in a separate technical paper. The more
compact single-AOM realization of the double-AOM system can be installed, for
example, for better suppressing the side pulses during the pulse routing (Sec.
3). More generally, we expect extension of the kind to $n>2$ for further
enhancing the performance and functionality of the composite AOM.

We expect the composite technique to enable advanced AOM applications with
both CW [23, 38, 24] and pulsed lasers [26, 27, 36, 44, 45]. For example, the
exceptionally broad rf-tuning range may help the composite AOM scheme to
stabilize the carrier-envelop phase [26, 27, 28] of a few-cycle femtosecond
pulse, in combination with the efficient and stable pulse compression
technique recently developed [46]. Our method may also extend the modulation
schemes and pulse shaping techniques used for coherent Raman imaging, enabling
fast frequency, amplitude and polarization modulations, to achieve spectral
multiplexing and background suppression [16, 17, 18]. Finally, as outlined in
sec. 4.1, the high speed pulse routing by the double-AOM scheme may be
extended into a composite network for generating high speed, complex pulse
sequence with ultrafast lasers, for novel applications in quantum information
processing [6, 13] and nonlinear quantum optics [12]. In particular, the high
speed complex pulse sequence may help to extend the composite control
techniques [33, 47, 34] in NMR to optical domain for precise quantum control
of strong transitions in atoms and molecules [20, 7].

## Funding information

National Key Research Program of China (2017YFA0304204); National Natural
Science Foundation of China (12074083,61975033,1874121); The Shanghai
Municipal Science and Technology Basic Research Project (19JC1410900).

## Acknowledgements

We thank Ruochen Gao, Yuxiang Zhao, Jiangyong Hu and Xing Huang for
experimental assistance and discussions. We thank Prof. Yanting Zhao and Prof.
Lin Zhou for helpful discussions during the development of this project.

## 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.

## References

  * [1] H. A. Haus, “Mode-locking of lasers,” IEEE Journal on Selected Topics in Quantum Electronics 6, 1173 (2000).
  * [2] S. T. Cundiff and J. Ye, “Colloquium: Femtosecond optical frequency combs,” Reviews of Modern Physics 75, 325 (2003).
  * [3] F. Krausz and M. Ivanov, “Attosecond physics,” Reviews of Modern Physics 81, 163 (2009).
  * [4] S. A. Diddams, K. Vahala, and T. Udem, “Optical frequency combs: Coherently uniting the electromagnetic spectrum,” Science 369, 267 (2020).
  * [5] T. Laarmann, I. Shchatsinin, R. Wu, A. Pechen, D. I. Bondar, and A. N. Pechen, “Control of quantum phenomena: past, present and future,” New Journal of Physics 12, 075008 (2010).
  * [6] J. Mizrahi, C. Senko, B. Neyenhuis, K. G. Johnson, W. C. Campbell, C. W. S. Conover, and C. Monroe, “Ultrafast Spin-Motion Entanglement and Interferometry with a Single Atom,” Phys. Rev. Lett. 110, 203001 (2013).
  * [7] C. P. Koch, “Quantum control of molecular rotation,” Reviews of Modern Physics 91, 35005 (2019).
  * [8] M. Maiuri, M. Garavelli, G. Cerullo, M. Maiuri, M. Garavelli, and G. Cerullo, “Ultrafast spectroscopy : state of the art and open challenges,” Journal of the American Chemical Society 9, 10533 (2019).
  * [9] N. Picqué and T. W. Hänsch, “Frequency comb spectroscopy,” Nature Photonics 13 (2019).
  * [10] A. M. Weiner, “Femtosecond pulse shaping using spatial light modulators,” Review of Scientific Instruments 71, 1929–1960 (2000).
  * [11] A. Monmayrant, “A newcomer ’ s guide to ultrashort pulse shaping and characterization,” J. Phys. B: At. Mol. Opt. Phys 43, 103001 (2010).
  * [12] Y. He, L. Ji, Y. Wang, L. Qiu, J. Zhao, Y. Ma, X. Huang, S. Wu, and D. E. Chang, “Geometric control of collective spontaneous emission,” Physical Review Letters 125, 213602 (2020).
  * [13] E. Torrontegui, D. Heinrich, M. I. Hussain, R. Blatt, and J. J. García-Ripoll, “Ultra-fast two-qubit ion gate using sequences of resonant pulses,” New J. Phys. 22 (2020).
  * [14] C. Sanner, N. Huntemann, R. Lange, C. Tamm, and E. Peik, “Autobalanced Ramsey Spectroscopy,” Phys. Rev. Lett. 120, 53602 (2018).
  * [15] V. I. Yudin, M. Y. Basalaev, A. V. Taichenachev, J. W. Pollock, Z. L. Newman, M. Shuker, A. Hansen, M. T. Hummon, R. Boudot, E. A. Donley, and J. Kitching, “General Methods for Suppressing the Light Shift in Atomic Clocks Using Power Modulation,” Phys. Rev. Appl. 14, 2015–2017 (2020).
  * [16] D. Zhang, M. N. Slipchenko, D. E. Leaird, A. M. Weiner, and J.-X. Cheng, “Spectrally modulated stimulated Raman scattering imaging with an angle-to-wavelength pulse shaper,” Optics Express 21, 13864 (2013).
  * [17] M. Andreana, M.-A. Houle, D. J. Moffatt, A. Ridsdale, E. Buettner, F. Légaré, and A. Stolow, “Amplitude and polarization modulated hyperspectral Stimulated Raman Scattering Microscopy,” Optics Express 23, 28119 (2015).
  * [18] R. He, L. Zhang, X. Miao, and M. Ji, “Dual-phase stimulated Raman scattering microscopy for rapid, label-free histology,” Optica 4, 44 (2017).
  * [19] C. E. Rogers III and P. L. Gould, “Nanosecond Pulse Shaping with Fiber-Based Electro-Optical Modulators and a Double-Pass Tapered Amplifier,” Optics Express 24, 2596 (2015).
  * [20] Y. Ma, X. Huang, X. WANG, L. Ji, Y. He, L. Qiu, J. ZHAO, Y. WANG1, S. Wu, Y. Wang, and S. Wu, “Precise pulse shaping for quantum control of strong optical transitions,” Optics Express 28, 17171 (2020).
  * [21] S.-w. Chiow, S. Herrmann, and S. Chu, “Nanosecond electro-optical switching with a repetition rate above 20 MHz.” Rev. Sci. Instrum. 78, 124702 (2007).
  * [22] E. A. Donley, T. P. Heavner, F. Levi, M. O. Tataw, and S. R. Jefferts, “Double-pass acousto-optic modulator system,” Rev. Sci. Instrum. 76, 063112 (2005).
  * [23] J. Thom, G. Wilpers, E. Riis, and A. G. Sinclair, “Accurate and agile digital control of optical phase , amplitude and frequency for coherent atomic manipulation of atomic systems,” Opt. Express 21, 18712 (2013).
  * [24] C. Zhou, C. He, S. T. Yan, Y. H. Ji, L. Zhou, J. Wang, and M. S. Zhan, “Laser frequency shift up to 5 GHz with a high-efficiency 12-pass 350-MHz acousto-optic modulator,” Rev. Sci. Instrum. 91, 033201–1 (2020).
  * [25] V. Stummer, T. Flöry, G. Krizsán, G. Polónyi, E. Kaksis, A. Pugžlys, J. Hebling, J. A. Fülöp, and A. Baltuška, “Programmable generation of terahertz bursts in chirped-pulse laser amplification,” Optica 7, 1758 (2020).
  * [26] S. Koke, C. Grebing, H. Frei, A. Anderson, A. Assion, and G. Steinmeyer, “Direct frequency comb synthesis with arbitrary offset and shot-noise-limited phase noise,” Nat. Photonics 4, 462–465 (2010).
  * [27] Z. Chen, M. Yan, T. W. Hänsch, and N. Picqué, “A phase-stable dual-comb interferometer,” Nat. Commun. 9, 3035 (2018).
  * [28] J. Hirschman, R. Lemons, E. Chansky, G. Steinmeyer, and S. Carbajo, “Long-term hybrid stabilization of the carrier-envelope phase,” Opt. Express 28, 34093 (2020).
  * [29] Y.-J. Wang, D. Anderson, V. Bright, E. Cornell, Q. Diot, T. Kishimoto, M. Prentiss, R. Saravanan, S. Segal, and S. Wu, “Atom michelson interferometer on a chip using a Bose-Einstein condensate,” Physical Review Letters 94, 090405 (2005).
  * [30] S. Wu, Y.-J. Wang, Q. Diot, and M. Prentiss, “Splitting matter waves using an optimized standing-wave light-pulse sequence,” Phys. Rev. A 71, 043602 (2005).
  * [31] K. J. Hughes, B. Deissler, J. H. T. Burke, and C. A. Sackett, “High-fidelity manipulation of a Bose-Einstein condensate using an optical standing wave,” Physical Review A 76, 035601 (2007).
  * [32] G. T. Genov and N. V. Vitanov, “Dynamical Suppression of Unwanted Transitions in Multistate Quantum Systems,” Phys. Rev. Lett. 110, 133002 (2013).
  * [33] G. T. Genov, D. Schraft, T. Halfmann, and N. V. Vitanov, “Correction of Arbitrary Field Errors in Population Inversion of Quantum Systems by Universal Composite Pulses,” Phys. Rev. Lett. 113, 043001 (2014).
  * [34] G. H. Low, T. J. Yoder, and I. L. Chuang, “Methodology of resonant equiangular composite quantum gates,” Phys. Rev. X 6, 041067 (2016).
  * [35] A. Feldman, “Measurement of the Photoelastic Contents of Optical Materials,” Opt. Eng. 17, 453 (1978).
  * [36] H. Tunnermann and A. Shirakawa, “Delay line coherent pulse stacking of a symmetric mode-locked pulse train,” Opt. Lett. 42, 4829 (2017).
  * [37] R. Liu, Y. Zhao, Y. Ma, and S. Wu, under preparations .
  * [38] M. Endres, H. Bernien, A. Keesling, H. Levine, E. R. Anschuetz, A. Krajenbrink, C. Senko, V. Vuletic, M. Greiner, and M. D. Lukin, “Atom-by-atom assembly of defect-free one-dimensional cold atom arrays,” Science 354, 1024 (2016).
  * [39] Y. Vidne, M. Rosenbluh, and T. W. Hansch, “Pulse picking by phase-coherent additive pulse generation in an external cavity,” Optics Letters 28, 2396 (2003).
  * [40] G. T. Genov, A. A. Rangelov, and N. V. Vitanov, “Efficient broadband frequency generation in composite crystals,” J. Opt. 16, 062001 (2014).
  * [41] A. B. M. Essaadi, M. A. M. S. Á. Ópez, P. A. G. A. Artínez, and I. G. M. Oreno, “Achromatic linear retarder with tunable retardance,” Opt. Lett. 43, 3277 (2018).
  * [42] J. F. F. Bulmer, J. A. Jones, and I. A. Walmsley, “Drive-noise tolerant optical switching inspired by composite pulses,” Opt. Express 28, 8646 (2020).
  * [43] P. Tournois, “Compensation of group delay time dispersion in laser systems,” Opt. Commun. 4018, 245 (1997).
  * [44] A. Klenke, M. Muller, H. Stark, M. Kienel, C. Jauregui, A. Tunnermann, and J. Limpert, “Coherent Beam Combination of Ultrafast Fiber Lasers,” IEEE J. Sel. Top. Quantum Electron. 24, 0902709 (2018).
  * [45] T. S. Khwaja and T. H. Yoon, “Phase-stable and selectable repetition-rate division of an optical frequency comb,” Phys. Rev. A 102, 043515 (2020).
  * [46] S. Zhang, Z. Fu, B. Zhu, G. Fan, Y. Chen, S. Wang, Y. Liu, A. Baltuska, C. Jin, C. Tian, and Z. Tao, “Solitary beam propagation in periodic layered Kerr media enables high-efficiency pulse compression and mode self-cleaning,” Light: Science and Applications 10, 53 (2021).
  * [47] X. Rong, J. Geng, F. Shi, Y. Liu, K. Xu, W. Ma, F. Kong, Z. Jiang, Y. Wu, and J. Du, “Experimental fault-tolerant universal quantum gates with solid-state spins under ambient conditions,” Nat. Commun. 6, 8748 (2015).