Download PT symmetry in optics

University of Ljubljana
Faculty of Mathematics and Physics
Seminar Ib
PT symmetry in optics
Author:
Jakob Frontini
Mentor:
doc. dr. Miha Ravnik
Abstract
In this seminar the notion of non-Hermitian parity-time (PT) symmetry is presented in the context of advanced optics and photonics. Recently, PT symmetry was
explored as exciting possible mechanism for designing and compensating losses in
optical metamaterials. The selected realisations of PT symmetric systems in optics
are presented, including two coupled PT symmetric waveguides, a PT symmetric
Bragg scatterer and a PT microring laser.
June 10, 2016
Contents
1 Introduction
1
2 Theoretical background
2
3 Coupled waveguides
3
4 Unidirectional invisibility
4
5 Single-mode laser
7
6 Conclusion
10
References
10
1
Introduction
The development of new artificial structures and materials is one of the major research
challenges in optics today. In most research so far, the design has been based on manipulating the refractive index profile. There is no doubt as to how useful this is, since this
has led to the design and fabrication of photonic crystals [1] and photonic crystal fibres
and to the exploration of nanoplasmonics [2] and negative-index metamaterials [3].
Recently, a new approach has been proposed: simultaneously using gain and loss
as a way of achieving optical behaviour that is at present unattainable with standard
arrangements. This is done by carefully designing the profile of not only the real, but
also the imaginary part of refractive index of the structure. Positive imaginary part
causes gain and negative part causes loss. Loss is abundant in physical systems and is
typically considered to be a problem. Gain, however, is important in optoelectronics
because it provides a mean to induce lasing or to overcome losses. The question that
arises is whether new artificially made optical structures with balanced gain and loss can
be produced.
Quantum mechanics is based on Hermitian operators that have real eigenvalues. But
a Hamiltonian does not need to be Hermitian in order to have real eigenvalues. There is
an alternative formulation of quantum mechanics, where the requirement of hermiticity
is replaced by a more general requirement of space-time reflection symmetry (PT symmetry). Certain optical systems are governed by equations that are similar to Schrödinger
equation, so the concept of PT symmetry can be extended to such systems. This means
that PT symmetry is important for developing new materials.
This seminar is organised as follows. In Chapter 2 the concept of parity-time (PT)
symmetric Hamiltonians in quantum mechanics is introduced and a corresponding concept in optics is established. Then a few realisations of PT symmetric systems in optics
are presented: in Chapter 3 a PT optical coupled system, in Chapter 4 a PT symmetric
Bragg scatterer and in Chapter 5 a PT microring laser. Conclusions are given in Chapter
6.
1
2
Theoretical background
New materials were recently developed by using the notion of “parity-time symmetry”
in optical systems. In 1998, Bender and Boettcher [4] showed that wide class of nonHermitian Hamiltonians can actually possess entirely real spectra as long as they respect
parity-time symmetry. In quantum mechanics, the action of parity operator P̂ is defined
by the relation (i, x, p) → (i, −x, −p), and that of time reversal operator T̂ by (i, x, p) →
(−i, x, −p), Where x and p represent position and momentum operators, respectively and
i is the imaginary unit. The P̂ T̂ symmetry of the Hamiltonian Ĥ means that it commutes
with the P̂ and T̂ operators:
Ĥ P̂ T̂ = P̂ T̂ Ĥ
(2.1)
Thus Ĥ and P̂ T̂ may share a common set of eigenfunctions. Let us consider a Hamiltonian
Ĥ = p2 /2 + V (x) with a complex potential V (x). It can be shown that the condition for
PT symmetry reduces to the requirement that the complex potential involved satisfies [5]
V (x) = V ∗ (−x)
(2.2)
In other words, the real part of the potential must be an even function of position and that
of the imaginary part must be odd. In this configuration, the eigenfunctions are no longer
orthogonal and hence the eigenmodes vector space is skewed. Even more fascinating is the
existence of a sharp, symmetry-breaking transition. In the broken parity-time symmetry
the eigenvalues of the system cease to be all real.
The notions of parity-time symmetry can be experimentally explored and used in
optics. Because such photonic systems are typically considered classical, they can be
realised without introducing any conflict with the hermiticity of quantum mechanics.
What allows this duality between quantum mechanics and optics is the formal equivalence
of the wave equation to the Schrödinger equation
∂ψ
h̄2 ∂ 2 ψ
=−
+ V (x)ψ.
ih̄
∂t
2m ∂x2
(2.3)
For example, under paraxial propagation conditions the electric field envelope E of the
optical beam is governed by the equation:
i
∂E
1 ∂ 2E
+
+ k0 [nR (x) + inI (x)]E = 0,
∂z
2k ∂x2
(2.4)
where k0 = 2π/λ, k = k0 n0 , λ is the wavelength of light in vacuum and n0 represents
the refractive index of the substrate. nR (x) is the real part of the index modulation, and
nI (x) denotes the imaginary component leading to gain or loss. We can see the similarities
between 2.3 and 2.4. The complex refractive index plays the role of the complex potential
and PT symmetry in the context of optics demands:
n(x) = n∗ (−x).
(2.5)
By solving Schrödinger equation we get energy eigenvalues, which are propagation constants in the case of paraxial optics.
2
3
Coupled waveguides
In integrated optics, a single PT element can be realised in the form of a coupled optical
system. Rüter et al. [6] studied two PT symmetric waveguides fabricated from iron-doped
LiNbO3 . If the distance between the two waveguides is small enough that the evanescent
tails of the guided modes overlap, then they become coupled. One of the waveguides
provides gain for the guided light while the other experiences the equal amount of loss.
Under these conditions, the optical field dynamics in the two coupled waveguides are
described by:
dE1
γ
i
− i E1 + κE2 = 0,
(3.1)
dz
2
dE2
γ
i
+ i E2 + κE1 = 0,
(3.2)
dz
2
where E1,2 represent field amplitudes in channels 1 and 2, κ is the coupling constant and
γ is the effective gain coefficient. We can rewrite the system of equations into a matrix
differential equation
!
!
!
d E1
i γ2 −κ
E1
·
=i
.
(3.3)
−κ −i γ2
E2
dz E2
q
The eigenvalues of the matrix in 3.3 are λ± = ±κ 1 − (γ/2κ)2 . The general solution can
be written as (E1 , E2 ) = c1 exp(iλ+ z)u+ + c2 exp(iλ− z)u− , where u± are the eigenvectors
and c1 and c2 are constants, which can be calculated from initial conditions.
The behaviour of this non-Hermitian system can be explained by considering the
structure of its eigenvectors, above and below the point γ/(2κ) = 1. Below this threshold
the eigenvectors are given by u± = (1, ± exp(±iθ)), with corresponding eigenvalues being
± cos θ, where sin θ = γ/2κ. At phase transition, the modes coalesce to u± = (1, i),
where amplitudes in both channels have the same magnitude. Above threshold (γ > 2κ),
u± = (1, i exp(∓θ)), where in this range cosh θ = γ/2κ and the eigenvalues (propagation
constants in the language of optics) are ∓ sinh θ. Unlike Hermitian systems, these eigenmodes are no longer orthogonal. This leads to important implications for beam dynamics
including a non-reciprocal response and power oscillations.
For a conventional Hermitian system (γ = 0), any superposition of the eigenmodes
leads to reciprocal wave propagation: the light obeys left-right symmetry. The situation
changes when gain and loss are involved in the coupled system (Fig. 1). If gain is below
threshold, the relative phase difference between the two fields increases with increasing
gain from their initial values at 0 and π, and finally, at threshold they coalesce at θ = π/2.
Remarkably, light propagation is now non-reciprocal: the pattern of beam propagation
differs depending on whether the initial excitation is on the left or right waveguide. This
behaviour is even more drastic above threshold. In this regime, light always leaves the
sample from channel 1, regardless of the input. This can be explained by noting that the
two complex eigenvalues occur in complex conjugate pairs, with the corresponding amplitudes either exponentially increasing or decaying. Thus only one eigenmode effectively
survives.
The experimental set-up is shown in Fig. 2. The coupled waveguide system in the
experiment [6] was based on Fe-doped LiNbO3 . Optical gain was provided through twowave mixing using the material’s photorefractive nonlinearity. A mask on top of the
3
Figure 1: Light propagation in a PT material. Left (right) panels correspond to initial
excitation in the left (right) waveguide in each case. Waveguides are coloured according
to the gain/loss parameter (γ): gray for passive (γ = 0), red for gain and green for loss.
(a) In a passive system, the total power (orange line) remains constant. (b) System below
the threshold γ < 2κ. Variation of total power can be seen. (c) γ > 2κ. Beam power
grows exponentially and the beam propagation is again non-reciprocal with respect to
the mirror axis of the two waveguides. [7]
sample was used to partially block the pump beam, to provide input in only one channel.
The output intensity and the phase relation between the channels (using interference with
a reference plane wave) were monitored using a CCD camera. Losses in the system arise
from the optical excitations of electrons from Fe2+ centres to the conduction band. The
optical two-wave mixing gain has a finite response time, so the evolution of the intensity
at the output was monitored as a function of time t.
4
Unidirectional invisibility
Lin et al [8] studied the possibility of synthesizing PT symmetric objects which can act
as unidirectional invisibility media near the spontaneous PT symmetry breaking points
(exceptional points). The subject of cloaking physics has attracted considerable interest
in the recent years. Lin proposed a new approach to achieve invisibility. Rather than
surrounding a scatterer with a cloak medium, in their case the invisibility arises because
of spontaneous PT symmetry breaking. This can be accomplished via designing a PT
synthetic Bragg structure. They considered scattering from such a structure (Fig. 3) and
investigated the consequences of PT symmetry in the scattering process.
4
Figure 2: Experimental set-up. An argon-ion laser beam (wavelength 514.5 nm) is coupled
into the arms of the structure fabricated on a photorefractive LiNbO3 substrate. An
amplitude mask blocks the pump beam from entering channel 2, thus enabling two-wave
mixing gain only in channel 1. The output intensity and phases were monitored with a
CCD camera.
Figure 3: PT symmetric Bragg scatterer. The wave entering from the left (upper figure)
goes through the structure unaffected, while a wave entering the same structure from the
right (lower figure) experiences enhanced reflection.
Passive gratings (involving no gain or loss) can act as high efficiency reflectors around
the Bragg wavelength. Instead, a PT symmetric object at the PT symmetric breaking
point is reflectionless over all frequencies around the Bragg resonance when light is incident from one side of the structure while from the other side its reflectivity is enhanced.
Additionally, in this same regime the transmission phase vanished, which is necessary condition for avoiding detectability. Surprisingly, these effects persist even in the presence
of Kerr nonlinearities.
These effects were demonstrated on a PT symmetric periodic structure or grating
with a refractive index profile n(z) = n0 + n1 cos(2βz) + in2 sin(2βz) for |z| < L/2.
This grating is surrounded by a homogeneous medium with a refractive index n0 for
|z| > L/2. In practice, refractive index modulations are small, e.g. n1 , n2 n0 . The
grating wave number β is related to its spacial periodicity Λ via β = π/Λ and in the
absence of gain (n2 = 0) the periodic index modulation leads to a Bragg reflection close
to the Bragg angular frequency ωB = cβ/n0 where c is the speed of light in vacuum. In
this arrangement, a time-harmonic electric field obeys the Helmholtz equation:
∂ 2 E(z) ω 2 2
+ 2 n (z)E(z) = 0,
∂z 2
c
5
(4.1)
which formally coincides with stationary 1D Schrödinger equation:
∂2
2m(V (z) − Ek )
ψk (z) = 0.
ψk (z) −
2
∂z
h̄
(4.2)
We would like to obtain the transmission T and reflection R coefficients for left (L) and
right (R) incidence waves. It turns out that for k ≈ β close to the Bragg point, they can
be written as
|λ|2
T =
(4.3)
|λ|2 cos2 (λL) + δ 2 sin2 (λL)
RL =
(n1 − n2 )2 k 2 /4n20
;
δ 2 + |λ cot(λL)|2
RR =
(n1 + n2 )2 k 2 /4n20
,
δ 2 + |λ cot(λL)|2
(4.4)
q
where λ = δ 2 − (n1 − n2 )2 k 2 /4n20 and δ = β − k is the detuning. The transmission is
the same for left and right incidence.
For n2 = 0 one gets the standard scattering features of periodic Bragg structures
(RL = RR ). Close to the Bragg point δ = 0 the reflection in the large L limit becomes
unity (transmission becomes zero), see Fig 4a. Instead, if n2 6= 0, an asymmetry between
left and right reflection starts to develop.
Figure 4: (a) Numerical evaluation from Eq. 4.1 of transmission T and reflection R for
a Bragg grating. The following parameters were used: n0 = 1, n1 = 10−3 , L = 12.5π and
β = 100. In case of a PT grating, the system is at the exceptional point when n2 = n1 .
In this case, RL is diminished for a broad frequency range, while RR is enhanced, in
agreement with the theoretical predictions. (b) Transmission phase φt as a function of
the detuning δ for the PT symmetric system of Fig. 3. The results of the exceptional
point (n1 = n2 ) are compared with the passive structure n2 = 0. (c) The corresponding
transmission delay times τt as a function of δ.
This asymmetry becomes most pronounced at n2 = n1 . At the Bragg point δ = 0. the
transmission is identical to unity, T = 1, while the reflection for the left incident waves is
RL = 0 (see Fig. 4a). This is a manifestation of the PT nature of this periodic structure.
At the same time, the reflection for right incident grows with the size L of the grating as
2
RR = L
k
2 n1
n0
2 sin(Lδ)
Lδ
6
2
δ→0
2
−−→ L
k
2 n1
n0
2
.
(4.5)
This was confirmed by using numerical simulations. This phenomenon is referred to
as unidirectional reflectivity. Furthermore, equations 4.4 indicate that a transformation
n2 → −n2 reverts the reflectivity of the system, allowing for reflectionless behaviour for
the right incident waves, i.e. RR = 0, while left incident waves now follow Eq. 4.5.
In other words, the phase delay between the real and imaginary part refractive index
dictates the unidirectional reflectivity of the system.
Reflectionless potentials in one-dimensional scattering configuration are in general not
invisible. This because the transmitted wave might depend on energy, thus leading to
distortion after passing through the sample. It is therefore necessary to examine the
phase φt of the transmission t = |t| exp(iφt ) and compare it to the phase acquired by a
wave propagating in a grating-free environment (φt = 0). It turns out that the phase φt
close to the Bragg point is
φt = arctan
λ
− tan(λL) + Lδ.
δ
(4.6)
At n1 = n2 , which means that λ = δ, the resulting transmission phase is φt = 0. Thus
interference measurements will fail to detect this periodic structure. Although the above
analytical result is performed near the Bragg point δ ≈ 0, numerical results shown in Fig.
4b indicate that these effects are valid over a wide range of frequencies. The results are
compared to a passive (n2 = 0) Bragg grating in Fig. 4b.
The dependence of the transmission delay time τt ≡ dφt /dk on the detuning δ was
also analysed. This quantity provides information about the time shift experienced by a
transmitted wave packet when its average position is compared to a corresponding one in
the absence of the scattering medium. Using Eq. 4.6, it shown that at the PT symmetry
breaking point the transmission delay time is τt = 0. The results for a structure at
n1 = n2 are shown in Fig. 4c together with those for a passive case.
5
Single-mode laser
Laser resonators support large number of closely spaced modes. As a result, the outputs
from such laser are subject to fluctuations and instabilities. During recent decades,
different strategies have been explored to achieve single mode operation, which is needed
for enhanced laser performance with higher monochromaticity, less mode competition and
better beam quality. Obtaining single mode operation depends on sufficient gain and
loss modulations, but factors such as inhomogeneous gain saturation can impede such
modulation. A general design concept with flexible design of cavity modes is desired.
Feng et al. [10] used the PT symmetry breaking concept to delicately manipulate
gain and loss of a microring resonator and observed single-mode laser oscillation of a
whispering-gallery mode. Whispering-gallery modes (WGMs) are a type of wave that can
travel around a concave surface. In a ring cavity, two counter-propagating modes interfere
with each other and form two orthogonal standing waves (WGMs) in the ring resonator.
These two WGMs are the eigenmodes of the microring resonator. They take place no
matter whether or not there is additional modulation in the ring resonator. Without
the coupling between the clockwise and counter-clockwise modes by the additional gain
and loss modulation, two WGMs would share the same eigen wavenumber; while with
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modulation, the coupling is introduced and the wavenumbers of two WGMs become
different.
In a ring cavity without any modulation, the clockwise mode and counter-clockwise
mode do not cross-talk. The mode equations can be written as
dA
= iβ0 A,
R dφ
dB
= iβ0 B,
R dφ
(5.1)
where A and B are the clockwise and the counter-clockwise travelling waves in a harmonic
form of A = A0 exp(iβRφ) and B = B0 exp(iβRφ), R is the radius of the resonator and
φ is the azimuthal angle. The wavenumbers associated with two standing modes are the
same as the original wavenumbers of the clockwise and counter-clockwise modes: β = β0 .
If the ring cavity is pumped, the created optical amplification makes β0 complex. If gain
is sufficient to compensate for the loss, lasing occurs. Since there is no limitation to the
azimuthal order of the WGMs, multi WGMs corresponding to different azimuthal orders
typically lase simultaneously, showing the multi-mode lasing spectrum similar to Fig. 7A.
When the PT modulation is added, the equations 5.1 have to be modified. Assuming
a small variation of index change and negligible scattering loss in the PT microring
resonator, the coupled mode equations can be written as
dA
= iβ0 A + in00 κB,
R dφ
dB
= iβ0 B − in00 κA,
R dφ
(5.2)
where n00 denotes the index modulation in only the imaginary part. The wave numbers
of the WGMs can be obtained, β = β0 ± iκn00 . manifesting non-threshold PT symmetry
breaking in PT microring lasers (Fig. 5B,C). This thresholdless condition is robust against
the nonlinearity during the pump process. For example, the real part index change by the
optical nonlinear effects only affects β0 in Eq. 5.2. However, with any arbitrary values for
β0 , the bifurcation in the imaginary part of β takes place immediately and the resulted
PT symmetry breaking is thresholdless with respect to n00 .
The PT microring laser can in principle support all WGM orders. However, the
continuous rotational symmetry is only valid for the desired WGM order, because the
introduced PT gain-loss modulation was specifically designed for the desired WGM order
and it does not affect the non-desired ones. Consequently, the coupled mode equations
for the non-desired WGMs can be written as in Eq. 5.1, meaning that they experience
balanced gain/loss modulation and thus remain below the lasing threshold.
The PT-synthetic microring resonator was designed with 500 nm thick InGaAsP multiple quantum wells, which have a high gain coefficient, on an InP substrate (Fig. 5A).
Periodically formed Cr-Ge structures on top of the InGaAsP introduce loss in such fashion that PT symmetry is satisfied for the desired WGM order. In the lasing mode, electric
fields are confined mainly in the amplification sections, exhibiting a net gain coefficient
6. For other WGMs, electric fields in both energy-degenerate WGMs are uniformly distributed in gain and loss regimes and so gain and loss average out.
8
Figure 5: (A) Schematic of the PT microring laser which, consists of Cr/Ge structures
on top of InGaAsP microring to mimic a gain/loss modulation. The diameter and width
of the microring resonator are 8.9 µm and 900 nm, respectively. Here, the designed
azimuthal order is m = 53. (B and C) The same eigenfrequency (197.6 THz) and complex
conjugate imaginary eigenspectra for the two modes at m = 53.
Figure 6: WGMs of different azimuthal orders in the PT microring laser. (A and B)
Electric field intensity distributions for lasing and absorption modes at m = 53. The
two modes have conjugate gain/loss coefficients. (C and D) Electric field intensity distributions at m = 54. The two modes share the same eigenfrequency and a similar loss
coefficient.
The PT microring laser with Cr/Ge modulations was fabricated using overlay electron
beam lithography and plasma etching. It was optically pumped with a femtosecond laser.
At pumping intensities above the lasing threshold, the single mode lasing peak is seen at
the wavelength of 1513 nm, confirming the theoretical predictions. The lasing linewidth
is about 1.7 nm, corresponding to a quality factor of about 890, that is limited by the
surface roughness of the sample. The lasing threshold at peak pump power density is
about 600 MW cm−2 .
For comparison, a control sample of a same-sized microring resonator without the
additional index modulation was fabricated. As expected, a typical multimode lasing
spectrum with different azimuthal orders m was observed (Fig. 7). Relative to the PT
microring laser, it can be seen that the resonance peaks for the same azimuthal order of
9
m = 53 are well matched. The power efficiency and the lasing threshold are also similar,
because the introduced loss minimally affects the desired lasing mode. An additional PT
microring laser with azimuthal order m = 55 was also fabricated. Its lasing emission also
agrees with the conventional WGM laser. By changing the desired azimuthal order of the
structured PT modulation, the single mode lasing frequency can be efficiently selected.
Figure 7: Comparison between PT laser and typical microring laser. (A) Multimode
lasing spectrum from a typical microring WGM laser, showing a series of lasing modes
corresponding to different azimuthal orders m. (B) Single-mode lasing spectra of the
PT microring lasers operating at the m = 53 and m = 55, consistent with the lasing
wavelength for the same m in (A). This shows that PT microring lasers efficiently select
the desired lasing mode and do not alter the original WGMs.
6
Conclusion
It was shown that the initial proposal of PT symmetric systems in quantum mechanics can
be extended to optics. Therefore optics has become an interesting platform for studying
the fundamentals of PT symmetry. The strategic modulation of gain and loss in the
PT symmetry breaking condition can broaden optical science at both semiclassical and
quantum levels.
Throughout the seminar we demonstrated some intriguing features of PT synthetic
materials, such as PT phase transition, unidirectional invisibility and single-mode lasing.
This could lead to a new generation of optical devices, materials and networks, including,
for example, unidirectional on-chip devices and high power laser system. Finally, ideas
from PT symmetry may provide a viable route to overcoming losses that have so far
hindered progress in other areas of applied physics such as plasmonics and metamaterials.
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