# Phase Symmetry Breaking of Counterpropagating Light in Microresonators
for Switches and Logic Gates

Alekhya Ghosh 1,2,† Arghadeep Pal 1,2,† Shuangyou Zhang 1 Lewis Hill 1 Toby Bi
1,2 Pascal Del’Haye 1,2,∗ 1Max Planck Institute for the Science of Light,
Staudtstraße 2, D-91058 Erlangen, Germany
2Department of Physics, Friedrich Alexander University Erlangen-Nuremberg,
D-91058 Erlangen, Germany  †These authors contributed equally
<EMAIL_ADDRESS>
###### Abstract

The rapidly growing field of integrated photonics is enabling a large number
of novel devices for optical data processing, neuromorphic computing and
circuits for quantum photonics. While many photonic devices are based on
linear optics, nonlinear responses at low threshold power are of high interest
for optical switching and computing. In the case of counterpropagating light,
nonlinear interactions can be utilized for chip-based isolators and logic
gates. In our work we find a symmetry breaking of the phases of
counterpropagating light waves in high-Q ring resonators. This abrupt change
in the phases can be used for optical switches and logic gates. In addition to
our experimental results, we provide theoretical models that describe the
phase symmetry breaking of counterpropagating light in ring resonators.

Recent years have seen huge progress in the use of photonics for data
processing, neural networks and quantum circuits. However, it is still
challenging to add nonlinear responses to all-optical circuits. One solution
is the use of optical-to-electronic data conversion, with the drawback of
adding complexity and increases latency. In terms of all-optical systems,
there has been progress to realize optical switches and logic gates using
fibers [1, 2], photonic crystal waveguides [3, 4], plasmonic waveguides [5, 6,
7, 8], and microresonators [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19].
Compared to the rest, microresonators have gained much attention due to their
well-established and simple fabrication methods along with low threshold power
for driving optical nonlinearities.
In this work, we utilize the phase response of optical fields undergoing
spontaneous symmetry breaking (SSB) of intensities in microresonators to build
an all-optical switch. We further propose designs of all-optical XOR and
universal NAND gates realizable in integrated systems. SSB of counter-
propagating [20, 21, 22] or copropagating light fields with orthogonal
polarizations [23, 24, 25, 26] in microresonators can be used for applications
ranging from isolators to logic-gates and gyroscopes [13, 27, 28, 29, 30, 31,
32, 33]. Recent research has also predicted multi-level SSBs [34, 35, 36, 37].
All this previous research investigated the SSB of intensities of the optical
fields without taking into account the corresponding phase effects. Compared
to previous work, our model does not need any optical power bias in any
direction and is robust to large laser fluctuations and fabrication errors.

Figure 1: Spontaneous symmetry breaking (SSB) of intensities and phases. (a)
At low input power, when the resonator is pumped bi-directionally, the
intracavity power and phase profiles of the two circulating fields remain
degenerate as shown in panel (b). (c) Photograph of the microrod resonator
coupled to a tapered fiber that is used in the experiments. At high input
power, the system drives into one of two states: (d) or (g), with more power
circulating either in clockwise or counterclockwise direction. Panels (e) and
(h) present the corresponding Kerr-induced resonance shifts. Panels (f) and
(i) depict the SSB of both the propagating fields’ power and phase with
varying laser detuning corresponding to (d) and (g) respectively. The
clockwise (CW) and counter-clockwise (CCW) circulating directions are
indicated by red and black arrows (and lines). In panels (a, d, g) the widths
of the arrows correspond to intensities, solid and transparent arrows
represent the dominant and suppressed fields after SSB, respectively.

Figure 1 illustrates the symmetry breaking between two counter-propagating
fields in a bidirectionally pumped microresonator (Fig. 1(a)). At low input
powers, when two counterpropagating input lasers approach the cavity
resonances from the higher frequency sides, both directions see overlapping
Lorentzian power profiles, as shown in the upper panel of Fig. 1(b). The
corresponding phase profiles are depicted in the lower panel of Fig. 1(b).
Panels (d-g) present the SSB mechanism at high power. After a certain
threshold of input power [22, 21], when the input lasers from the two
directions approach the clockwise and counterclockwise resonances, an
infinitesimal power imbalance between the two directions (caused by noise
within the system), gets rapidly amplified, leading to SSB. This amplification
of imbalance is fueled by unequal self- and cross-phase modulation strengths
[38]. The direction that receives slightly higher power due to random
fluctuations pushes the counterpropagating resonance away from the input laser
due to strong cross-phase modulation.
In the upper panel of Fig. 1(e), the clockwise (CW) direction (black line) has
pushed the counter-clockwise (CCW) direction (red line) away, leading to a
reduction of light coupled into the CCW direction. More and more input light
couples into the CW direction as the laser approaches the CW resonance, with
the CCW resonance being pushed away further. Correspondingly, the phase
transmission profile of the CCW resonance is also pushed away from the input
laser (as shown in the lower panel of Fig. 1(e)). However, after a certain
detuning, self-phase modulation hinders the laser from approaching the CW
resonance anymore, and the cross-phase modulation from the CCW direction
becomes dominant, causing a reduction of the coupled power mismatch in the two
directions. The characteristic formation of symmetry breaking ”bubbles” can be
seen in Fig. 1(f) and Fig. 1(i) for both power and phase profiles of the
coupled fields as a function of detuning from the cold-resonances. Since SSB
is seeded by fluctuations of the input laser, the relative dominance of the
fields in the SSB region is random. Contrary to the case in Figs. 1(d-f), Fig.
1(g-i) represent the case where the CCW direction dominates.

Switching effects – The Kerr-shifts of resonances, that cause SSB of phases in
the case of identical input fields result in an enhancement of phase asymmetry
between counter-propagating fields when the inputs have non-zero relative
phase. This has been employed in this paper together with the SSB in
intensities, to demonstrate a novel all-optical switch, as shown in Fig. 2. In
simple words, occurrence of SSB in intensities of the circulating fields
triggers light transmission through the system, replicating the“ON” state of a
switch. Otherwise, the system does not transmit any light, corresponding to
the “OFF” state.

Figure 2: Schematics of the optical switch. (a) Symmetrical light intensities
of the counter-propagating fields leads to constructive interference in
interferometer at the output port, corresponding to the “OFF” state. (b)
Breaking this symmetry (in this case with suppressed transmission in
counterclockwise direction), prevents the destructive interference, leading to
the “ON” state of the switch. I: input port, O: output port.

The experimental setup for observing the switching effect is presented in Fig.
3(a). The amplified laser input at $1550$ nm is split into two halves and the
evanescent fields are coupled bidirectionally into a microresonator via a
tapered optical fiber. To access the light waves coming out of the resonator,
two optical circulators are used. Ten percent of the output from each arm goes
to a photodiode (PD1 or PD2) to observe the counter-propagating transmission
spectrum. The rest (90%) from both arms interferes at a 3-db-coupler, an
output from which is connected to a photodiode PD3 to examine the switching
effect. Here, the experiment is conducted using a $0.9$-mm-radius fused silica
rod resonator with a quality factor of ${10}^{8}$ fabricated via
$\text{CO}_{2}$ laser machining [39].

Figure 3: Demonstrations of switching. (a) Experimental setup for observing
the switching effect during spontaneous symmetry breaking (SSB) in a
microresonator. ECDL: External cavity diode laser, EDFA: Erbium-doped fiber
amplifier, VA: Variable attenuator, C: Circulator, PD: Photodiode. The top
panels of (b) and (c) show two experimentally observed transmission spectra of
the circulating fields with their respective bottom panels showing the
inteference signal from the two output directions recorded with PD3. (c) A
change of dominance of the fields within the intensity-SSB bubble (marked with
an arrow in the top panel) arises as the system jumps between the two stable
configurations, however, this does not affect the switching mechanism. (d)
Simulation of the transmitted power as a function of the laser detuning
frequency is shown in the top panel, with its switching action in the bottom
panel. Panel (e) depicts schematics of the switching setup used for analytical
modeling. Panel (f) shows theoretical predictions of the switching effect with
increasing input power imbalance. The power division factor (PDF) signifies
the fraction of the total input power going to one of the input ports. Thus,
the fraction of the power going to the other input port is $(1-\text{PDF})$.
(g) Varying input powers in two directions and their imbalance affect the
transmission difference (defined in text) of the switch.

In the upper panel of Fig. 3(b), the experiments exhibit SSB between the
coupled powers of the fields in the two directions below a certain detuning.
The upper panel of Fig. 3(c) shows an SSB bubble with interchanges between the
dominant and suppressed directions of circulating powers. The lower panels of
Fig. 3(b) and (c) show the interference of the two outputs of the two
directions. Below the SSB threshold, the output is close to zero. This can be
attributed to the destructive interference at the output (PD3), which comes
from the two successive $3$-dB couplers (one that splits the input into two
halves, one that interferes the two outputs from the resonators) each mixing
the respective input fields after introducing a relative quadrature phase
difference between them. Higher non-zero outputs are obtained during the
period of intensity-SSB (enclosed by dashed black boxes in Fig. 3) as a result
of a change in the relative phases of the two directions preventing a
destructive interference at the output $3$-dB coupler. Therefore, light
reaches PD3, only in the presence of intensity-SSB in the system, enabling the
use as an indicator for symmetry breaking and optical switch. The transmission
bandwidth of the switch is the same as the width of the intensity-SSB region.
We term the maximum (minimum) output at the interfering port to be the “high”
(“low”) of the switch and the difference between them as the transmission
difference of the switch.
Simulated results support the experiments, as shown in Fig. 3(d). The upper
panel shows the intensity profiles of the counter-propagating fields and the
lower panel depicts the interference pattern showing the switching effect. The
simplified model used for simulation is shown in Fig. 3(e).
Non-uniform power splitting at the two output ports of the first 3-dB coupler,
results in inequality in the input powers to the two counter-propagating
directions of the resonator. In this case, for lower detuning values, the
interference at the output exhibits a non-zero baseline in the absence of
symmetry breaking. However, the interference output increases with increasing
detuning, as seen in Fig. 3(f). Such a system can still be used as a switch,
where the non-zero baseline can be set as the “low” value of the switch.
However, Fig. 3(f) depicts that with increasing imbalance, the transmission
difference of the switch and the bandwidth decrease after an initial increase.
Figure 3(g) summarises the transmission difference of the switch as a function
of total input power and fraction of the total input power power pumping one
of the input directions.

SSB of phases – An initial phase difference of $\pi/2$ between the two input
lasers reaching the resonator gets enhanced during the SSB of circulating
intensities. A standing wave is generated in the coupling waveguide (tapered
fiber) as light fields are travelling in opposite directions through it. In
the presence of an anti-node of the standing wave at the coupling point, the
two input fields coupled into the resonator are considered to be in phase.
Under this condition, we observe SSB of the phases, which we explain
analytically in this section.
We first consider a high-$Q$ ring resonator made of a material having a
$\chi^{(3)}$ nonlinearity. Input lasers with normalized amplitude $f$ are
provided from both directions into the resonator at normalized detunings of
$\zeta_{0}$ through a coupling waveguide. As a result, CW (CCW) light
propagates through the resonator with normalized slowly-varying optical field
amplitude $E_{+}$ ($E_{-}$). The homogeneous solution states within the
resonator can be modeled using the normalized coupled Lugaito-Lefever
equations (LLE) [40]:

$\frac{\partial E_{\pm}}{\partial
t}=-E_{\pm}-i\zeta_{0}E_{\pm}+i|E_{\pm}|^{2}E_{\pm}+i2|E_{\mp}|^{2}E_{\pm}+f.$
(1)

The third and the fourth terms on the RHS describe the self- and cross-phase
modulation respectively.

Numerical simulations and analytical (see Appendices A & B) solutions of Eq.
(1) demonstrate occurrences of SSB bubbles both in phases and intensities
(Fig. 4(a,b)) of the circulating fields for an input power higher than the SSB
threshold. At higher input powers, we observe several different solutions (see
Appendices B and C).

Figure 4: Simulated solutions for symmetry breaking in phases. Panels (a,b)
show simulated results of circulating phases and intensities in the two
directions as a function of laser detuning within a microresonator
respectively, with each demonstrating spontaneous symmetry breaking (SSB). The
black dotted lines show analytical solutions, the red and blue solid lines
represent the circulating fields. The green dashed box signifies the
concurrence of the SSBs in phases and intensities.

Optical logic gates – Here, we propose a novel structure, as shown in Fig.
5(a), capable of performing logical operations on integrated platforms. The
operating principle is driven by the Kerr-enhancement of phase asymmetries
together with SSB of intensities of the circulating fields.
Figure 5(b) depicts the operating principles of XOR and NAND gates. The latter
is a universal gate and can be cascaded to generate any other logic circuit.
Figure 5(c) illustrates such a cascaded system. For the XOR gate, the input
powers just before ($P_{0}(\text{XOR})$) and after ($P_{1}(\text{XOR})$) the
SSB region are assigned as $0$ and $1$ logic inputs to the gate. An average of
the input powers reaches the microresonator in each direction and whenever it
lies in the intensity-SSB region (when either of the inputs is logic $1$), the
transmission is non-zero, corresponding to a logic output of $1$, similar to
the XOR gate truth table (Fig. 5(d)).
Similarly, for the NAND gate, as seen in Fig. 5(b), any input power value
within the SSB region ($P_{0}(\text{NAND})$) is chosen to be logic $0$,
whereas an input power beyond the SSB region ($P_{1}(\text{NAND})$) is
selected as logic $1$. These input power levels are chosen such that the mean
power reaching the resonator lies in the SSB region when either one or both
inputs correspond to a logic $0$, ($P_{0}(\text{NAND})$), resulting in non-
zero transmission. This leads to the NAND gate truth table in Fig. 5(e). This
proposed system can be easily integrated on chips to serve as a simple
platform for all-optical computing. In case of excess power loss, signals
could be re-amplified with integrated amplifiers [41].

Figure 5: Proposal for optical logic gates. Panel (a) depicts schematics of
the proposed optical logic gate setup. A and B represent the two inputs and Y
represents the output. (b) shows an input power scan of the circulating field
intensities. Different horizontal lines show different input power levels
required for $1$ or $0$ logic levels for XOR or NAND gate. (c) Schematic of a
photonic integrated circuit with multiple logic gates. Truth tables of XOR and
NAND gates are shown in panel (d) and (e).

Discussion and outlook – In summary, we investigate the evolution of phases of
the circulating fields in a bidirectionally pumped resonator during
spontaneous symmetry breaking of intensities. These phase responses can be
utlized for realizing components for photonic data processing. In a proof-of-
principle we demonstrate an all-optical switch that is “ON” during the
symmetry breaking of intensities. In addition, we propose designs for all-
optical logic gates. Moreover, in Appendix B we highlight that intensity
asymmetry in a coupled resonator system [35] is also associated with phase
asymmetry.
Recent advances in fabrication techniques [42, 43] made it feasible to realize
complex photonic devices on-chip. Phase effects in dispersion engineered
integrated systems [44, 45, 46, 47] with symmetry broken solitons [48, 49]
enables the exploration of novel dynamics. Moreover, SSB bifurcation points of
the phase and intensity of the circulating fields can be utilized for
microresonator based sensors [50]. Thus, the proposed devices could become
promising candidates for various all-optical systems including optical neural
networks, all-optical routers, and quantum information processors.
We would like to draw the reader’s attention to the following complementary
work that investigates phase effects of SSB [51].

Acknowledgements – This work was funded by the European Union’s H2020 ERC
Starting Grant “CounterLight” 756966, the Max Planck Society, and the Max
Planck School of Photonics.

AG did the theoretical calculations with help from AP. AP fabricated the
samples, performed the experiment with help from AG. All authors discussed the
results. AG, AP, and PD wrote the manuscript with input from all the authors.

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##  Appendix A Appendix A: SSB of phases - Analytical Solutions

The steady-state solutions of the system can be obtained by setting Eq. (1) to
$0$. Detailed calculations for obtaining the steady-state intensities have
already been performed in Ref. [21]. The circulating intensities in the two
directions follow the relation:

$\left(I_{+}-I_{-}\right)\left(I_{+}^{2}+I_{-}^{2}-2\zeta_{0}I_{+}-2\zeta_{0}I_{-}+I_{+}I_{-}+1+\zeta_{0}^{2}\right)=0,$
(2)

where, $I_{\pm}=|E_{\pm}|^{2}$. If we consider the circulating fields
$E_{\pm}$ with corresponding phases $\phi_{\pm}$, i.e.,
$E_{\pm}=\sqrt{I_{\pm}}e^{i\phi_{\pm}}$, the steady-state response for the
phases becomes,

$\left(x-y\right)\left(x^{2}-xy+y^{2}-\zeta_{0}x-\zeta_{0}y+3\right)=0,$ (3)

where $x=\tan(\phi_{+})$ and $y=\tan(\phi_{-})$. The expressions within the
first set of round brackets on the LHS of both Eqs. (2) and (3) represent the
symmetric lines, and the second set of round brackets represent the Kerr-
effect induced asymmetric ellipses, as seen in Fig. 6(a) where the two axes
are the phases of the fields in each direction. The inset in Fig. 6(a) shows
the intensity evolution of one circulating field with respect to its counter-
propagating field. The points where the symmetric lines meet the respective
asymmetric ellipses denote the endpoints of the SSB region and are given as:

$\displaystyle I_{\pm,0}=\frac{2\zeta_{0}\pm\sqrt{\zeta_{0}^{2}-3}}{3},$ (4)
$\displaystyle\phi_{\pm,0}=\tan^{-1}\left(\zeta_{0}\pm\sqrt{\zeta_{0}^{2}-3}\right).$
(5)

Figure 6: Analytical solutions for symmetry breaking in phase. Panel (a) shows
the relationships between the phases of the two counterpropagating fields. The
straight line represents the symmetric solutions, whereas the ellipses
correspond to the asymmetric solutions for different detunings. The inset
shows the corresponding relationships between the intensities of the two
counterpropagating fields ($I_{+}$ and $I_{-}$). Panel (b) shows the phases as
functions of input power with an inset illustrating the variation in the
corresponding field intensities as a function of input power for different
detuning values.

Figure 6(b) depicts the analytical solutions for the phases of the fields as a
function of input power. For low input power, the phases of the counter-
propagating fields remain symmetric. After a certain threshold of input power,
the symmetric line becomes unstable (similar to the case of SSB in intensities
[21]), and following a bifurcation point, the phases continue increasing but
along a symmetry broken solution set. At the beginning of the SSB region, the
phase of one field increases and the phase in the other direction decreases.
This dominant direction is randomly chosen. Phases of both fields reach
extrema, after which they approach each other again and finally converge in an
inverse bifurcation point. The following symmetric solution line regains its
stability after this inverse bifurcation point [21]. Similarly, intensities
also become asymmetric following the bifurcation, as shown in the inset of
Fig. 6(b). It should be noted that the SSB bifurcations occur at the same
input power for both the phases and intensities of the circulating fields.
With increasing detuning, the size of the bubble increases in Fig. 6(a,b).

##  Appendix B Appendix B: Simulations - phase oscillations and bistability
jumps

The upper (lower) panel of Fig. 7 shows detuning scans of phases (intensities)
of the circulating fields in the two directions of a microresonator.

For higher input power compared to the case of Fig. 7, we observe a horn-like
shape in the SSB bubble (i.e., the SSB bubble starts to fold on itself). This
happens both in phase (Fig. 7(a)) and intensity (Fig. 7(b)). Following the
horn-like region in the bubble, the asymmetric branches enter an oscillating
region. The oscillations in phases can are shown in Fig. 7(a) and happen
simultaneously with the oscillations in intensities shown in Fig. 7(b). We
report that oscillations in circulating field intensities in the SSB region
are accompanied by oscillations in the phases of the fields. The colored
dotted lines represent the maxima and minima of the oscillations for each
detuning. It can be observed that with increasing detuning, the amplitudes of
the phase oscillation increase, and after a certain detuning, the oscillations
of the phases of the two fields overlap. The average phases, depicted in solid
lines, regain symmetry in a small region within the oscillation region. The
amplitudes of the phase oscillations reach maxima and then reduce with further
increasing the detuning, causing another region of non-overapping oscillations
before they cease to exist. These different types of oscillations in phases
and the corresponding oscillations in intensities are depicted in the next
section. After the oscillatory region, the phases of the fields follow the
asymmetric branches and merge into a symmetric branch at the end of the SSB
bubble.

With further increasing the input power, the folds on the bubble turn sharper
causing bistability within the bubble, as shown in Fig. 7(c) and (d). The
bistability in intensity leads to a bistability jump inside the bubble, which
is large enough to cause the intermediate oscillatory stages and lead to the
fields jumping down to the stable symmetric lower branch. This bistability
jump can be observed in the phases of the circulating fields as well, as shown
in Fig. 7(c).

Figure 7: Simulated solutions for symmetry breaking in phases for various
input powers. (a,b) With increasing input power from the value used in Fig. 4,
both phases and intensities of the circulating fields show oscillations. The
colored dots represent the maxima and minima of the respective fields. (c,d)
For even higher input power, bistabilities appear within spontaneous symmetry
breaking bubbles. The black dotted lines show analytical solutions, the red
and blue solid lines represent the circulating fields. Used parameters:
$|f|^{2}=4$ [52]. More details on the oscillations in the intensities can be
found in Ref. [52].

##  Appendix C Appendix C: Examples of oscillations in phases

Figure 8 shows examples of oscillations in phases and intensities of
circulating counterpropagating fields in a microresonator that is pumped
symmetrically and bidirectionally.

Figure 8: Oscillations in phases and intensities under symmetry broken
conditions observed in simulations. The left (right) panel shows the temporal
oscillations in the phases (intensities) of the fields for input power,
$|f|^{2}=4$ and detuning values, $\zeta_{0}=~{}5.5652~{}(\text{panels
(a,b)}),~{}6.7826~{}(\text{panels (c,d)}),~{}7.0435~{}(\text{panels
(e,f)}),~{}8.8696~{}(\text{panels (g,h)}),~{}\text{and
}~{}10.087~{}(\text{panels (i,j)})$. The insets of the right panels show the
real-vs-imaginary phase space diagrams of the circulating fields. With
increasing detuning, the oscillations (in both phases and intensities) grow
initially, leading to overlapping oscillations (panels (c,d)) and perfect
periodic switchings (panels (e-h)). However, after a certain detuning, the
oscillations become non-overlapping (panels (i,j)).

##  Appendix D Appendix D: SSB of phases in coupled resonator systems

In this section, we demonstrate that SSB between the circulating field
intensities in the two coupled resonators are also accompanied by SSB in the
phases of the fields. We model the evaluations of the fields in the coupled
system depicted in Fig. 9(a) by a coupled LLE [40, 53]

$\frac{\partial E_{n}}{\partial
t}=-E_{n}-i\zeta_{0}E_{n}+ijE_{m}+i|E_{n}|^{2}E_{n}+f_{n},$ (6)

where $E_{n}$ is the normalized optical field envelope for the $n^{\text{th}}$
resonator, $\zeta_{0}$ is the normalized detuning, $j$ is the normalized
inter-resonator coupling rate and $f_{n}$ is the input to the $n^{\text{th}}$
resonator. Here, all the terms have been normalized with respect to half of
the total loss of each resonator. We consider here that the system is
perfectly symmetrical, therefore, $f_{n}=f$ is the same for both the
resonators.

From Fig. 9 (b) and (c), it can be stated that the phases of the circulating
fields in the two resonators undergo SSB as well when there is an SSB in their
corresponding intensities. Therefore, one can build all-optical switches or
optical digital logic gates using these resonators as well.

Figure 9: Simulated solutions for symmetry breaking in intensities and phases
in coupled resonator systems. Panel (a) shows the schematics of a coupled
resonator system, where both resonators are pumped in such a way that light
fields are propagating in one certain direction in both resonators (clockwise
in resonator 1 and counter-clockwise in resonator 2). Panels (b) and (c)
depict the spontaneous symmetry breakings in intensities and phases of the
circulating fields in the two resonators respectively. The dashed black line
presents the unstable analytical solutions for the field intensities within
the resonators.