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**Optical Waveguide and Resonator**

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**Free Space and Plane Wave**

Propagating wave in free space – plane wave Free space without dispersion – modulated plane wave can still propagate without any distortion, since the slow time-varying of the EM wave on its envelope, frequency, or phase won’t bring any effect to the propagation Free space with dispersion – modulated plane wave has its envelope gradually expanded in time-domain in propagation, eventually converts itself to harmonic plane wave How can we force the EM wave to propagate along a specific direction in the 3D world?

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**Concept of Waveguide The wave has to localized in certain directions**

How to localize the wave? – Convert the traveling wave into the standing wave Introduce the transverse resonance z x s-wave reflection at the boundary: standing wave is formed underneath the boundary

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**Concept of Waveguide x The resonance condition for standing**

wave in transverse direction (x): z A necessary condition is: How to make it possible? TIR – dielectric waveguide Conductor reflection – metallic waveguide Photonic crystal – Bragg waveguide Plasma reflection – plasmonic polariton waveguide

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**Dielectric Waveguide If becomes purely imaginary, or:**

The resonance condition becomes: or: Dispersion relation for the dielectric slab waveguide Even mode Odd mode Obviously, we have: With definition we find: - waveguide effective index

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**Dielectric Waveguide Dispersion relation E-field (y-component)**

Symmetric (even mode): Anti-symmetric (odd mode): A, B, C – given by the tangential boundary condition H-field is given by the Faraday’s law, with x and z components only – that’s the TE wave

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Dielectric Waveguide Similarly, the guided TM wave solution can be derived from the reflection of the p-wave at the boundary The E-field of the TM wave has abrupt change at the boundary! Hence, the effective index of the TM wave is smaller than that of the TE wave. Application examples: Single mode waveguide – higher order mode cut-off 2D waveguide – no analytical solution Slot waveguide – utilizing the abrupt change of the E-field normal to the boundary, for TM wave guidance only

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**Metallic Waveguide By letting**

in previous derivations, we will be able to obtain the EM wave solution in metallic waveguide. 1D (slab) or 2D dielectric waveguide – support TE and TM waves, not TEM wave 1D (slab) metallic waveguide – support all TEM, TE, and TM waves 2D (hollow) metallic waveguide – support TE and TM waves, not TEM wave

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**Hollow Metallic Waveguide – How to Treat Wave Equations**

Once the propagation along z is identified b Substitute it back into the wave equation a y x where For any component, we have Variable separation where By imposing boundary conditions

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**Hollow Metallic Waveguide – Characteristics**

We find: where and (from the divergence-free condition) Once the E-field is obtained, the H-field can be found by Faraday’s law. For a given set of (m, n) there are two independent modes. For the 3 coefficients, only 2 of them are independent. For a fixed set of (m, n), if one independent mode (A) is chosen as the transverse E-field, the other independent mode (B) must have the E-field with a non-zero z component, and mode A must have its H-field with a non-zero z component. Therefore: Similar to the dielectric waveguide, a hollow metallic waveguide doesn’t support the TEM wave. All its supported modes can be classified as the TE wave (mode A) and the TM wave (mode B).

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**Hollow Metallic Waveguide – Characteristics**

Cut-off frequency Applications: - HPF - Single mode waveguide

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**Transmission Line – TEM Wave**

For TEM wave: Hence: TEM solution is the same as the static electric and magnetic field solution. I E H E H parallel lines or twisted pair coaxial cable

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**Transmission Line – TEM Wave**

Reason: in the source-less cross-sectional region perpendicular to the propagation direction, both E- and H- fields have to be curl-free and divergence-free – the same as the static E- and M- fields, all field components are described by the Laplace equation. Therefore, only for a cross-sectional structure that supports both static E- and H- fields, it will be able to support the TEM wave. A hollow metallic waveguide doesn’t support a static H-field, hence it cannot support the TEM wave. A necessary condition for a TEM waveguide: at least two pieces of disconnected conductors in the cross-sectional region for supporting the static H-field. Why we emphasize this point? – pre-knowledge on field types will help us to construct the solution: identify non-zero components and the function form, by substituting such constructed solution back into the wave equation, we will get the problem solved as the wave equation will usually be greatly simplified.

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**TEM Wave - Characteristics**

TEM wave – “localized” plane wave with k, E, H mutually orthogonal: k is along the direction in which the waveguide (transmission line) is extended; E- and H- fields are restricted in the 2D cross-section, with their longitudinal dependence identical to the plane wave, and transverse dependence identical to the static E- and H- fields with the same boundary condition. Propagation of the TEM wave relies on the free charge and conduction current on the metal (conductor) – dielectric surface. Namely, the TEM wave is a resonance between the EM fields and the free charge distribution. For the TEM wave, we can readily introduce the voltage and current concept to turn a field problem into a circuit problem. The TEM wave can be supported by the dual conductor transmission line: Parallel lines or twisted pair Coaxial cable Printed metal stripe lines (on PCB or other substrates) The TEM wave has no cut-off frequency, it can send DC power through.

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Resonator In its propagation along the waveguide, the EM wave oscillates in the cross-section to form a standing wave. For a given cross-sectional area and material refractive index distribution, there is a cut-off frequency associated with. A waveguide cannot support a EM wave with its frequency below the cut-off frequency, simply because the phase matching condition cannot possibly be satisfied. For the EM wave with its frequency higher than the cut-off frequency, there exists a real propagation constant given by the dispersion relation. Hence the wave can propagate along the waveguide. Simply because the waveguide is open in its propagation direction, there is no constraint imposed on the phase of the guided EM wave. Hence once the frequency is beyond the cut-off, there is a solution for the guided EM wave with a real propagation constant. In this sense, the waveguide is actually a high-pass filter.

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Resonator Once the waveguide is even “closed” along the propagation direction, extra boundary matching conditions are imposed on both ends. Since the only degree of freedom is the phase of the guided EM wave, the phase will therefore be fixed by the “closing” of the waveguide. As such, the associated frequency will be fixed. In conclusion, for a resonator formed by a “closed” waveguide, it only supports EM waves with a discrete set of specific frequencies. That’s why a closed cavity (with constraints imposed on every dimension in the 3D space) is essential to form a resonator that supports discrete frequencies. Examples

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Resonator More specifically, once we force the guided EM wave to reflect at the two ends of a waveguide, the EM wave will form a standing wave in all dimensions in the closed waveguide (or cavity), a EM wave resonator is therefore formed for the creation or selection of a single or a discrete set of frequencies.

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**Resonator as an Oscillator**

E-field inside the cavity Boundary condition at the two ends For cavity with zero-gain Meaning less since no energy can be collected from the outside. When gain is available the resonance condition becomes: Amplitude Phase

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**Resonator as a Filter L R R If there is wave incidence (from left)**

At the right hand side E-field transmissivity At the left hand side E-field reflectivity

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