Download 4.6.2 Reflection, Refraction, Diffraction

4.6.2 Reflection, Refraction, Diffraction
• So far we have mainly studied how to confine light using
PhCs.
• There are also interesting effects associated with free
propagation of waves in and around PhCs.
• Consider the case when an incident plane wave strikes
an interface of a PhC
• Some light will be reflected at the interface
• Some light (at least outside of a photonic bandgap) can
be transmitted or refracted, propagating at some angle
within the PhC.
• In addition, depending on frequency, the interface
periodicity and the band structure there may also be a
finite number of additional reflected or refracted waves.
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• We use our knowledge that in a linear system with
discrete translational symmetry is conserved up to the
addition of reciprocal lattice vectors.
• As translational symmetry at in interface is only conserved
along directions parallel to the interface, only the wave
vector ‖ is conserved
• Suppose that the periodicity along the interface is Л and
the incident plane wave has a wave vector ( ‖ , ) and
frequency .
• Then, any reflected or refracted wave must also have
frequency and a wave vector
, ′ for any integer .
‖+
Л
• Note that the periodicity Л of the interface need not be
the same as the lattice constant of the PhC.
• Indeed, the interface need not be periodic at all, but we
restrict ourselves to this case.
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• Illustration of reflection and refraction of a plane wave
incident on a square lattice of dielectric rods
• Interface with period Л =
direction
2 in the diagonal (1,1,0)
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Reflection at a PhC interface
• The familiar specular reflected wave corresponds to = 0,
that is = ‖ ,
• If the incident medium has a refractive index , then
=
‖
+
=
‖
+(
)
• It follows
=±
minus sign relevant for propagation
away from the interface.
• The familiar law of reflection of equal incident and reflected
angles is fulfilled.
• The existence of diffractive reflections with ≠ 0 depends on
the frequency, periodicity and incident angle.
• We can still apply conservation of frequency:
=−
2
−( ‖+
)
Л
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• For too small , too large and/or too small Л,
becomes
imaginary evanescent field that decays away exponentially
from the interface
• Non-evanescent diffractive reflections occur for
2
>c ‖+
/
Л
• For an incident angle θ > 0 we can write
sin %
.
‖ =
c
• The first diffractive reflection will occur at = −1 for
1
Л Л
= >
λ
(1 + sin % )
2
• For air ( = 1) no diffractive reflections occur for
Л
≤ 0.5
λ
• When diffractive reflections occur, each diffractive order starts at
glancing angle
Л
• Also minimum condition for metasurfaces ≤ 1 for % = 1 ! 5
*
• For refraction the analysis is similar as for reflection, but
we have to consider the band structure of the PhC
• There are different ways to visualize a band structure
• Example: dispersion relation of homogeneous space in 2D
light cone
The isofrequency
diagram is a
useful tool to
analyse refracted
waves in the PhC 6
• Example: constant frequency contours for the first TM
band of our square lattice of dielectric rods.
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• Discussion:
• At low frequencies the constant frequency diagram
approaches a circular shape since the photonic crystal
behaves as a uniform dielectric with properties
determined by the averaging of the different material
domains.
• With increasing frequencies the constant frequency
contour starts deviating from the circular shape.
• For periods approaching half a wavelength strong
effects on propagation leading to effects like superprism,
slow light, negative refraction/diffraction, and selfcollimation
• For a period equal to half the wavelength no light
propagation / gap.
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• We can use the isofrequency diagram to determine the
number of refracted waves (if any) and in what
states/directions they propagate.
• The amplitudes of the refracted and reflected waves,
however, depend on the surface termination and require
a more detailed solution of Maxwell’s equations.
• The group velocity +, is perpendicular to the
contours and points in the direction of increasing .
• We consider our example of a square lattice of dielectric
rods
-.
= 0.276 and draw the
• We choose a frequency
/
isofrequency contours for air (black circle, =
) and
for the first TM band of a square lattice of dielectric rods
(red contours) in -space at. The Brillouin zone is shown
in grey.
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Procedure:
• We draw a black dot for the
wave vector corresponding
to the incident angle %
• The black arrow is the
incident group velocity
• To select modes with the
same ‖ , we draw a dashed
line through the incident
perpendicular to the interface
direction
• The place(s) where the
dashed line intersects with
the PhC contours determines
the refracted wave(s)
• Not all intersection points
correspond to distinct
refracted waves.
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• We can eliminate intersections
corresponding to group
velocities pointing towards the
interface (pink dots)
• We need only keep one of the
intersection points differing by
a reciprocal lattice vector
• Therefore, in our example we
only find a single refracted
wave
• The gradient determines the
group velocity direction
(shown at various -points as
arrows.)
• In our example the refracted
wave lies at the same side of
the normal as the incident
wave negative refraction
• Note: negative refraction ≠
negative refractive index 11
• There are two ways to get multiple refracted waves:
• On the one hand, we could intersect multiple bands of the
band structure. For this we would have to superimpose
multiple contours on the isofrequency plot
• On the other hand, we could cut our interface at a different
angle. Then the fixed ‖ -line can intersect a given contour at
inequivalent points in different periodic unit cells.
• Conversely, there are some cases for which we will get no
refracted waves:
• Inside a band gap
• Inside a stop band: in our example, for a larger incident angle
% we would not intersect the PhC any contours, leading to
total reflection
• Comment: we don’t need to draw dashed lines with ‖ +
Л
for all because the periodic replication of the PhC contour is
equivalent to doing this.
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• The higher complexity of the band structure and isofrequency
diagram of a PhC compared to that of free space gives rise to
a range of unusual refraction and diffraction effects
• For example, if a contour line becomes flat, it is possible to
negate the effect of diffractive spreading of a narrow beam in
a uniform medium.
3• Suppose we have a finite-width beam propagating in 2
direction. Uncertainty principle: as the beam becomes narrow
we will have more nonnegligible 4 -components,
corresponding to propagation at different angles the beam
spreads
3, the
• If the contour line is flat and nearly perpendicular to 2
group velocities of many different 4 values will point
3 direction “supercollimation” 13
approximatelt in the 2
• Conversely, if a contour line exhibits sharp corners, for
refracted waves near these corners, we can have
dramatic changes in the group velocity direction for only
slight changes of the incident angle or the frequency.
• In analogy to a normal prism, which splits different
wavelengths into different angles, this effect was termed
“superprism effect”, because the PhC can split a small
range of wavelengths over a wide range of angles.
PRB 58, R10096 (1998)
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• Construction of the diffracted waves
PRB 58, R10096 (1998)
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• Measurement at 956 nm wavelength. Photographs
showing light-beam swing inside the photonic crystal.
PRB 58, R10096 (1998)
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4.7. PhC applications & devices
We already discussed a few possibilities:
• Frequency dependent mirrors
• Waveguides for transporting electromagnetic
energy from one place to another
• Slow light waveguides
• Photonic crystal particle accelerator waveguides
• Passband filters based on defect cavities
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A waveguide integrated filter
• A cavity formed by removing a single rod of a square lattice of
dielectric rods is adjajent to two waveguides, formed by removing
a row of rods
• Near the cavity resonance frequency light from the input
waveguide can couple to the cavity, and the cavity can couple to
the output waveguide high transmission (100%)
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• Can be described by coupled-mode theory
A channel drop filter
• Two waveguides coupled via a pair of resonant modes
• Can redirect light from the input to the output port, but
only around the cavity resonance frequency
• Other frequencies propagate along the waveguide
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unimpeded
Can we make optical isolators?
Optics Express Vol. 21, Issue 1, pp. 220-228 (2013)
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Optics Express Vol.
21, Issue 1, pp. 220228 (2013)
• No, just a mode converter: an optical isolator needs to
suppress back-reflection irrespective of modal content.
• A simple way to suppress back-reflection is to attenuate it,
but in reciprocal structures this implies, unfortunately, that
the forward light is also attenuated.
• More generally, one cannot construct an optical isolator with
any structure having a symmetric scattering matrix.
• Instead, one needs a non-reciprocal system, e.g. including
magneto-optical materials, or nonlinear or time dependent
structures (see e.g. Science 335, 38 (2012))
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• Connected to time-reversal invariance:
• If we take the complex conjugate of the Master equation and
use the fact that the eigenvalues are real for lossless materials,
we obtain:
( ) ∗
∗
8
67 8 =
Θ
7 8
6 7∗ 8 =
Θ
8(
)
7∗ 8
• 7∗ 8 satisfies the same eigenvalue equation as 7 8 with the
same eigenvalue
• Since 7 : = ; : <, : , 7∗ 8 is just the Bloch state at −
• It follows: 8
= 8 −
• Taking the complex conjugate of quivalent to 7 8 is equivalent
to reversing the sign of the time in Maxwell’s equations
• Band structures are symmetric even if the PhC structure is not
• Exception: magneto-optic materials
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A PhC microcavity laser
Painter et al., Science 284, 1819-1821 (1999)
• Light is confined to a single defect of a nanofabricated
two-dimensional (2D) photonic crystal
• Active (gain) region: InGaAsP quantum wells
• Flexibility in geometry allow fine-tuning of the defect
mode radiation pattern as well as the emission
wavelength.
• The compact size and high spontaneous emission
coupling factor of the defect microcavity also make it
interesting as a low-noise low-threshold light source.
Painter et al., Science 284, 1819-1821 (1999)
• Spectrum of the laser line just • “L-L-curve” (light out
above threshold
vs light in) shows the
lasing threshold
• Inset: spontaneous emission
well below threshold
All-optical transistor
• A cavity similar to the one for the waveguide-integrated
filter but with Kerr-type nonlinearity
• Suppose we have a cavity with resonance frequency = ,
and we have input power at a frequency below = .
• As we increase the power, >
will increase due to the
nonlinearity. This will shift =
to lower frequencies and pull
the resonance through and
coupling/transmission are
enhanced
• Output power depends on
past values hysteresis
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