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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

�‖ +��

Л, ��′ for any integer �.

• 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 = � 2 in the diagonal (1,1,0)direction

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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

> c �‖ +2��

Л/��

• For an incident angle θ > 0 we can write

�‖ = sin % ��

c.

• The first diffractive reflection will occur at � = −1 for

Л

2��=

Л

λ>

1

�� (1 + sin % )

• For air (�� = 1) no diffractive reflections occur for

Л

λ≤ 0.5

• When diffractive reflections occur, each diffractive order starts at glancing angle

• Also minimum condition for metasurfacesЛ

*≤ 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

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The isofrequency

diagram is a

useful tool to

analyse refracted

waves in the PhC

• 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 self-collimation

• 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

• We choose a frequency -.

��/= 0.276 and draw the

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. 10

• 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

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• 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 �‖ +��

Лfor 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 isofrequencydiagram 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.

• Suppose we have a finite-width beam propagating in 23-direction. Uncertainty principle: as the beam becomes narrow we will have more nonnegligible �4-components, corresponding to propagation at different angles � the beam spreads

• If the contour line is flat and nearly perpendicular to 23, the group velocities of many different �4 values will point approximatelt in the 23 direction � “supercollimation” 13

• 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.

14PRB 58, R10096 (1998)

• Construction of the diffracted waves

15PRB 58, R10096 (1998)

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PRB 58, R10096 (1998)

• Measurement at 956 nm wavelength. Photographs showing light-beam swing inside the photonic crystal.

4.7. PhC applications & devices

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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

A waveguide integrated filter

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• 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%)

• Can be described by coupled-mode theory

A channel drop filter

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• 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 unimpeded

Can we make optical isolators?

Optics Express Vol. 21, Issue 1, pp. 220-228 (2013) 20

• 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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Optics Express Vol. 21, Issue 1, pp. 220-228 (2013)

• 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:

Θ67�8∗

=8

�(�)

��7�8

∗

Θ67�8∗ =

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 22

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 above threshold

• Inset: spontaneous emission well below threshold

• “L-L-curve” (light out vs light in) shows the lasing 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 =.

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• 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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