ece 695 - lecture 11pbermel/ece695/lectures/ece695-lecture11-s17.pdflecture 11: beam propagation...
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![Page 1: ECE 695 - Lecture 11pbermel/ece695/Lectures/ECE695-Lecture11-S17.pdfLecture 11: Beam Propagation Method Prof. Peter Bermel February 3, 2017. Outline •Vectorial BPM Solver •Tunable](https://reader035.vdocuments.us/reader035/viewer/2022071216/6047e65fff084a19b1171c43/html5/thumbnails/1.jpg)
ECE 695Numerical Simulations
Lecture 11: Beam Propagation Method
Prof. Peter Bermel
February 3, 2017
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Outline
• Vectorial BPM Solver
• Tunable Photonic Crystal Fibers
• Electro-Optic Modulator
• Electro-Optic Switch
2/3/2017 ECE 695, Prof. Bermel 2
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VBPM on a Waveguide: Problem Description
• Cross section defined above; 𝜆 = 1.3 𝜇m• Propagation along z is semi-infinite• Must grid space with first-order triangular
elements in cross-sectional plane; choose PML to reduce reflections to 10-100
• Will vary Dz for maximum effectiveness
2/3/2017 ECE 695, Prof. Bermel
=3.20
=3.26
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VBPM on a Waveguide
• Fundamental mode is calculated accurately with 12,800 first-order triangular elements
2/3/2017 ECE 695, Prof. Bermel 4
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VBPM on a Waveguide
• Propagation step size in Z, known as DZ, should equal transverse dimensions for best accuracy
2/3/2017 ECE 695, Prof. Bermel 5
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VBPM on a Waveguide: Longitudinal Imaginary Propagation
• With optimal step size, can solve the fundamental mode of both polarizations in a pretty modest number of steps!
2/3/2017 ECE 695, Prof. Bermel 6
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VBPM on a Waveguide: Accuracy
• Accuracy of calculation of waveguide coupling length as a function of mesh divisions N
2/3/2017 ECE 695, Prof. Bermel 7
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VBPM on a Waveguide
• Accuracy of coupling length as a function of DZ saturates below one wavelength
2/3/2017 ECE 695, Prof. Bermel 8
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VBPM on a Photonic Crystal Fiber
• Originally conceived of by P.J. Russell
• Confines light to core without total internal reflection!
2/3/2017 ECE 695, Prof. Bermel 9
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VBPM on a PhC Fiber
• Effective index vs. PhC period
2/3/2017 ECE 695, Prof. Bermel 10
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VBPM on a PhC Fiber
• Hy field distributions for the fundamental TE modes
2/3/2017 ECE 695, Prof. Bermel 11
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VBPM on a PhC Fiber
• Confinement loss decreases sharply as period Lincreases
2/3/2017 ECE 695, Prof. Bermel 12
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VBPM on a PhC Fiber
• Variation of the effective mode area with PhCperiod L
2/3/2017 ECE 695, Prof. Bermel 13
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VBPM on a PhC Fiber
• Effective index increases modestly with increasing period L, indicating increased mode confinement
2/3/2017 ECE 695, Prof. Bermel 14
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VBPM on a PhC Fiber
2/3/2017 ECE 695, Prof. Bermel
• Calculated dispersion relation (effective index versus wavelength) for a PhC Fiber
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VBPM on a PhC Fiber
2/3/2017 ECE 695, Prof. Bermel
• Obtained dispersion 𝐷 = 𝑑2𝑘/𝑑𝜔2 from earlier data
• Note modest changes in parameters flip sign of D
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Tunable PhC Fiber
• Cross-section of a PhC fiber filled with electrostatically tunable liquid crystals
2/3/2017 ECE 695, Prof. Bermel
S. Obayya, “Computational Photonics” (Wiley, 2010)
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Liquid Crystals
• Liquid crystals consist of many stiff molecules
• LC order in between that of liquids and crystals
• LCs have a uniaxial dielectric function:
𝜖𝑖𝑗 = 𝜖𝑜 + 𝛿𝜖 𝑛𝑖 𝑛𝑗
• The director is oriented along applied electrostatic fields
2/3/2017 ECE 695, Prof. Bermel 18
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Tunable PhC Fiber
• Variation of LC refractive indices both on and off-axis, consistent with normal dispersion
2/3/2017 ECE 695, Prof. Bermel
S. Obayya, “Computational Photonics” (Wiley, 2010)
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Tunable PhC Fiber
• Dominant and non-dominant HE (quasi-TE) modes for tunable PhC fiber
2/3/2017 ECE 695, Prof. Bermel
S. Obayya, “Computational Photonics” (Wiley, 2010)
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Tunable PhC Fiber
• Wavelength dependence of the effective index (left) and dispersion (right)
2/3/2017 ECE 695, Prof. Bermel
S. Obayya, “Computational Photonics” (Wiley, 2010)
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Tunable PhC Fiber
• Polarization conversion versus propagation distance Z
2/3/2017 ECE 695, Prof. Bermel
S. Obayya, “Computational Photonics” (Wiley, 2010)
22
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Electro-Optic Modulation
• The refractive index matrix for a Pockels medium subject to an external electric field in the xy-plane can be written as follows:
𝑛 =
𝑛𝑜 + 𝛿𝑛𝑥𝑥 𝛿𝑛𝑥𝑦 0
𝛿𝑛𝑦𝑥 𝑛𝑜 0
0 0 𝑛𝑜 − 𝛿𝑛𝑧𝑧
• Where:
𝛿𝑛𝑥𝑥 = 𝛿𝑛𝑧𝑧 =1
2𝑛𝑜
3 𝑟41𝐸𝑦
𝛿𝑛𝑥𝑦 = 𝛿𝑛𝑦𝑥 =1
2𝑛𝑜
3 𝑟41𝐸𝑥
2/3/2017 ECE 695, Prof. Bermel 23
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Electro-Optic Modulator
• Schematic diagram of the electro-optic modulator, made from epitaxial GaAs/AlGaAslayers
2/3/2017 ECE 695, Prof. Bermel
S. Obayya, “Computational Photonics” (Wiley, 2010)
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Electro-Optic Modulator
• Electric modulation field distributions for Ex (left-hand side) and Ey (right-hand side)
2/3/2017 ECE 695, Prof. Bermel
S. Obayya, “Computational Photonics” (Wiley, 2010)
25
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Electro-Optic Modulator
• Key quantity 𝑉𝜋𝐿, product of voltage and electrode separation necessary to create a p phase shift, is measured as a function of core height for a few designs
2/3/2017 ECE 695, Prof. Bermel
S. Obayya, “Computational Photonics” (Wiley, 2010)
26
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Electro-Optic Modulator
• Here, 𝑉𝜋𝐿 is measured as a function of core width for several designs – greater widths are more sensitive to voltage
2/3/2017 ECE 695, Prof. Bermel
S. Obayya, “Computational Photonics” (Wiley, 2010)
27
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Electro-Optic Modulator
• Here, 𝑉𝜋𝐿 increases with buffer thickness, caused by diminishing field strength in the core region
2/3/2017 ECE 695, Prof. Bermel
S. Obayya, “Computational Photonics” (Wiley, 2010)
28
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Electro-Optic Modulator
• On the other hand, optical loss decreases with buffer thickness increases for similar reasons
2/3/2017 ECE 695, Prof. Bermel
S. Obayya, “Computational Photonics” (Wiley, 2010)
29
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Electro-Optic Modulator
• Effective impedance of microwaves and refractive index of IR signals cross over only at selected buffer thicknesses that vary greatly with core height
2/3/2017 ECE 695, Prof. Bermel
S. Obayya, “Computational Photonics” (Wiley, 2010)
30
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Electro-Optic Modulator
• Here, the buffer thickness needed to achieve a given level of loss is calculated as a function of Al doping concentration Xf
2/3/2017 ECE 695, Prof. Bermel
S. Obayya, “Computational Photonics” (Wiley, 2010)
31
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Electro-Optic Switch
• Coupling length required for power transfer decreases as a function of EO index tuning
2/3/2017 ECE 695, Prof. Bermel
S. Obayya, “Computational Photonics” (Wiley, 2010)
32
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Electro-Optic Switch
• Power transferred as a function of position for waveguides both with and without EO tuning
2/3/2017 ECE 695, Prof. Bermel
S. Obayya, “Computational Photonics” (Wiley, 2010)
33
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Electro-Optic Switch
• Variation of output and maximum power transfer as a function of EO index tuning
2/3/2017 ECE 695, Prof. Bermel
S. Obayya, “Computational Photonics” (Wiley, 2010)
34
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Next Class
• We will cover other FEM applications in heat transfer and electronic transport
2/3/2017 ECE 695, Prof. Bermel 35