ECE 536 – Integrated Optics and Optoelectronics Lecture 7 – February 16, 2021

Spring 2021

Tu-Th 11:00am-12:20pm

Prof. Umberto Ravaioli ECE Department, University of Illinois

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Lecture 7 Outline

• Recap of slab optical wave guide

• Refractive index of III-V material. Example of AlGaAs

• Confinement factor (optical power) in slab waveguide

• Electron-Photon interaction

• Optical transitions using Fermi Golden Rule

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Summary of symmetric slab wave guide

0

𝑑

2

−𝑑

2

𝑧

𝑥

continuity of 𝐸𝑦

continuity of 𝐻𝑧

𝐻𝑧 = −𝑖1

𝜔𝜇 𝜕𝐸𝑦

𝜕𝑥

from Faraday’s law even modes

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Summary of symmetric slab wave guide

0

𝑑

2

−𝑑

2

𝑧

𝑥

continuity of 𝐸𝑦

continuity of 𝐻𝑧

𝐻𝑧 = −𝑖1

𝜔𝜇 𝜕𝐸𝑦

𝜕𝑥

from Faraday’s law odd modes

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Summary of symmetric slab wave guide

Coordinate transformation

𝑅 = 𝜔𝑑

2 𝜇1𝜀1 − 𝜇𝜀 = 𝑘0

𝑑

2 𝑛1

2 − 𝑛2

𝑋2 + 𝑌2 = 𝑅2

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Graphical solution

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Normalization constant for the optical mode

Assume total guided power to be unity

(for 𝜇 = 𝜇1)

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Confinement factor

Fraction of optical power guided in the core of the waveguide

In the limit of thin waveguide (very small 𝑑) and 𝜇 = 𝜇1 it can be shown that the TE0 confinement factor is

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Confinement factor

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From TE to TM solutions – Duality Principle

Equivalent solutions can be obtained by making the following replacements:

TM)

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Asymmetric wave guide

0

𝑑

2

−𝑑

2

𝑧

𝑥

𝑛2 = 𝜇2𝜀2

𝑛1 = 𝜇1𝜀1

𝑛3 = 𝜇3𝜀3

In an asymmetric wave guide, usually 𝑛1 > 𝑛3 > 𝑛2 The top layer of the guide (material 2) is often air, but in some cases it can also be replaced by a metallization

𝑥

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Various possible geometries

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Guiding structures may be realized as part optoelectronic devices realized with III-V semiconductors.

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Refractive index of 𝐀𝐥𝐱𝐆𝐚𝟏−𝐱𝐀𝐬

Refractive index “below the band gap” for many III-V compound materials is with good approximation (imaginary part is considered to be negligible below the bandgap)

Real part of material permittivity

The most popular models are based on semi-empirical fitting of measurements (Adachi, 1985). More complete model based on the Kramers-Kronig relations (Adachi, 1988) addresses the full dielectric function for the complete energy range.

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Refractive index of 𝐀𝐥𝐱𝐆𝐚𝟏−𝐱𝐀𝐬

The approximate model for refractive index below the band gap is mainly related to properties of the band structure

∆

𝐸

𝑘

Parameters used in Chuang’s book

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𝜀1

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Another parameter set – Adachi (1985)

𝐴 𝑥 = 6.3 + 19.0𝑥 𝐵 𝑥 = 9.4 − 10.2𝑥

𝐸𝑔 𝑥 = 1.425 + 1.155 𝑥 + 0.37 𝑥2 [eV]

∆ 𝑥 = 0.34 [eV]

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Another parameter set – Adachi (1985)

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Example of complete analysis – Adachi (1988)

measurements

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Experimental measurements

S. Gehrsitz, F. K. Reinhart, C. Gourgon, N. Herres, A. Vonlanthen and H. Sigg, “The refractive index of AlxGa1-xAs below the band gap: Accurate determination and empirical modeling,” Journal of Applied Physics, vol. 87, p. 7825, 2000.

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Experimental measurements

S. Gehrsitz et al. (2000)

The Electron-Photon Interaction Hamiltonian

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Representative crystal momentum for electron in a crystal

𝐩 = 𝑘 ≈𝜋

𝑎. With 𝑎𝑜 ≈ 5.5 Å,

𝐩 ≈𝜋 × 1.054 × 10−34

5.5 × 10−10= 6.02 × 10−25 J s/m

𝑘𝑜𝑝𝑡 ≈2𝜋

≈

2𝜋 × 1.054 × 10−34

10−6= 6.6 × 10−28 J s/m

Momentum of a photon at 1m wavelength

In free space (Compton effect)

incoming photon

scattered photon with lower energy and momentum

accelerated particle

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Energy in light wave

𝐔𝐰𝐚𝐯𝐞 = 𝜀𝐸2 There is no obvious dependence on frequency

Number of photons/m3

𝐔𝐩𝐡𝐨𝐭𝐨𝐧𝐬 = 𝑁 ∙ 𝜔

There is an obvious dependence on frequency

Since we must have

𝐔𝐰𝐚𝐯𝐞 = 𝐔𝐩𝐡𝐨𝐭𝐨𝐧𝐬

the number of photons per unit volume 𝑁 must be proportional to 𝐸2 with constant of proportionality which depends on the wave frequency

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We have seen earlier the Fermi Golden Rule

Given a time-harmonic optical perturbation applied at t=0

Transition rate of a particle from state “𝑖” to state “𝑓”

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Optical Transitions Using Fermi Golden Rule

Hamiltonian describing the electron-photon interaction in a semiconductor

momentum variable

electron charge

EM vector potential periodic crystal potential

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Expansion of the Hamiltonian

Hamiltonian for interaction with light:

Expand the squared term

𝐻0 𝐻′ Unperturbed Hamiltonian

Perturbed Hamiltonian

Simplifications:

• In practical cases 𝑒𝑨 ≪ 𝐩 so the term 𝑒2𝐴2

2𝑚0 is neglected

• Choice of Coulomb gauge 𝛻 ∙ 𝐀 = 0 such that 𝐩 ∙ 𝐀 = 𝐀 ∙ 𝐩

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Expansion of the Hamiltonian

Hamiltonian for interaction with light:

Expand the squared term

𝐻0 𝐻′ Unperturbed Hamiltonian

Perturbed Hamiltonian

Simplifications:

• In practical cases 𝑒𝑨 ≪ 𝐩 so the term 𝑒2𝐴2

2𝑚0 is neglected

• Choice of Coulomb gauge 𝛻 ∙ 𝐀 = 0 such that 𝐩 ∙ 𝐀 = 𝐀 ∙ 𝐩

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Coulomb Gauge

Lorenz Gauge

The number of possible gauges is infinite. We are free to some extent to choose the gauge which simplifies solution of a problem, as long as the underlining physics is not affected. Coulomb gauge is often used when no sources are present. The Lorenz gauge is necessary when the action of “retarded potentials” has to be considered.

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Optical Fields

unit vector (direction of Electric Field with assumption below)

We define the perturbing field through the magnetic vector potential

vanishes for optical field

Assumption

Electric Field (V/m)

Magnetic Field (A/m)

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Poynting Vector

direction of power flow

We have used

Time Average Poynting Flux

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Interaction Hamiltonian

Substitute the assumed magnetic vector potential into the expanded Hamiltonian

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Interaction Hamiltonian

𝐻′(𝐫, 𝑡) = 𝐻′ 𝐫 𝑒−𝑖𝜔𝑡 + 𝐻′+ 𝐫 𝑒+𝑖𝜔𝑡

Hermitian adjoint (conjugate) of 𝐻′(𝐫)

Transition Rate due to

Electron-Photon Interaction

Emission Absorption

𝐸𝑎 𝐸𝑎

𝐸𝑏 𝐸𝑏

initial state

initial state

final state

final state

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Transition Rate: Absorption

Transition Rate for Photon Absorption (single electron)

Total Upward Transition Rate per unit volume (s-1 cm-3)

probability of occupation

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Transition Rate: Emission

Transition Rate for Photon Emission (single electron)

Total Downward Transition Rate per unit volume (s-1 cm-3)

probability of occupation

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Net Absorption Rate

Net Upward Transition

Using

Optical Absorption Coefficient

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Absorption Coefficient

Absorption Coefficient [cm-1] = fraction of photons absorbed per unit distance

We have already derived what we need for the absorption coefficient.

𝛼 =Number of photons absorbed per second per unit volume

Number of photons injected per second per unit area

1.0 cm

1.0 cm2

optical intensity

energy per photon=

𝑃

𝜔

𝑅

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Absorption Coefficient

We found earlier

optical intensity

energy per photon=

𝑃

𝜔=

𝑛𝑟𝑐𝜀0𝜔2𝐴02

2 𝜔

Long wavelength (dipole) approximation: wavelength of light is large compared to the spatial distance between energy levels involved on the transition

substitute into 𝜶

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Absorption Coefficient

with

The intensity factors 𝐴02 cancel out. The absorption

coefficient is independent of optical intensity in linear regime.

Recall also:

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We had

The Hamiltonian can also be written in terms of the electric dipole moment which has form

Using the property 𝐻0 = unperturbed Hamiltonian

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The absorption coefficient becomes

With

In practice, scattering causes linewidth broadening. The delta function can be replaced by a Lorentzian function

Real and Imaginary Parts of the Permittivity Function

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The absorption coefficient and the permittivity are related by

Absorption Coefficient and Permittivity

(Factor of 2 because 𝛼 refers to absorption of optical intensity)

Rearranging:

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The real part of the permittivity can be obtained from the imaginary part using Kramers-Kronig relation (in Appendix 5A)

Absorption Coefficient and Permittivity

Note: for a semiconductor, 𝐸𝑎 and 𝐸𝑏 are obtained from the band structure

denotes the Principal Value of the integral

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Reading Assignments:

Chapter 7 of Chuang’s book Section 9.1 of Chuang’s book

Chapter 7 of Coldren & Corzine’s book (supplemental) Appendix 3 (optical waveguides) Appendix 9 (Fermi Golden rule)

References downloadable from the digital library:

S. Adachi, “Properties of Semiconductor Alloys: Group IV, III-V and II-VI Semiconductors,” Wiley, 2009

S. Adachi, “Physical Properties of III-V Semiconductor Compounds – InP, InAs, GaAs, GaP, InGaAs, and InGaAsP,” Wiley, 1992