Handout 29 - Handout 29 Optical Transitions in Solids Optical Gain and Semiconductor Lasers In this lecture you will learn Electron-photon

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1 ECE 407 – Spring 2009 – Farhan Rana – Cornell University Handout 29 Optical Transitions in Solids, Optical Gain, and Semiconductor Lasers In this lecture you will learn: • Electron-photon Hamiltonian in solids • Optical transition matrix elements • Optical absorption coefficients • Stimulated absorption and stimulated emission • Optical gain in semiconductors • Semiconductor heterostructure lasers ECE 407 – Spring 2009 – Farhan Rana – Cornell University Interactions Between Light and Solids The basic interactions between light and solids cover a wide variety of topics that can include: • Interband electronic transitions in solids • Intraband electronic transitions and intersubband electronic transitions • Plasmons and plasmon-polaritons • Surface plasmons • Excitons and exciton-polaritons • Phonon and phonon-polaritons • Nonlinear optics • Quantum optics • Optical spintronics
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2 ECE 407 – Spring 2009 – Farhan Rana – Cornell University Fermi’s Golden Rule: A Review Now suppose a time dependent externally applied potential is added to the Hamiltonian: t i t i o e V e V H H ω + + = ˆ ˆ ˆ ˆ Consider a Hamiltonian with the following eigenstates and eigenenergies: { integer ˆ = = m E H m m m o ψ Suppose at time t = 0 an electron was in some initial state k : ( ) p t = = 0 Fermi’s golden rule tells that the rate at which the electron absorbs energy from the time-dependent potential and makes a transition to some higher energy level is given by: h () ( ) δ π h h = p m p m m E E V p W 2 ˆ 2 The rate at which the electron gives away energy to the time-dependent potential and makes a transition to some lower energy level is given by: h ( ) h h + = p m p m m E E V p W 2 ˆ 2 ECE 407 – Spring 2009 – Farhan Rana – Cornell University Optical Transitions in Solids: Energy and Momentum Conservation k r h For an electron to absorb energy from a photon energy conservation implies: ( ) ( ) h r r + = i n f m k E k E Final energy Initial energy Photon energy Momentum conservation implies: q k k i f r h r h r h + = Initial momentum Final momentum Photon momentum Intraband photonic transitions are not possible: For parabolic bands, it can be shown that intraband optical transitions cannot satisfy both energy and momentum conservation and are therefore not possible Intraband Intraband Interband Note that the momentum conservation principle is stated in terms of the crystal momentum of the electrons. This principle will be derived later. E
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3 ECE 407 – Spring 2009 – Farhan Rana – Cornell University Electromagnetic Wave Basics Consider an electromagnetic wave passing through a solid with electric field given by: () ( ) t r q E n t r E o ω = r r r r . sin ˆ , The vector potential associated with the field is: t t r A t r E = , , r r r r t r q A n t r q E n t r A o o = = r r r r r r . cos ˆ . cos ˆ , The power per unit area carried by the field is given by the Poynting vector: () () η 2 ˆ 2 ˆ , , , 2 2 2 o o A q E q t r H t r E t r S P = = × = = r r r r r r r n o o
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This note was uploaded on 10/21/2009 for the course ECE 4070 taught by Professor Rana during the Spring '08 term at Cornell University (Engineering School).

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Handout 29 - Handout 29 Optical Transitions in Solids Optical Gain and Semiconductor Lasers In this lecture you will learn Electron-photon

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