EE530 Lecture 4

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EE530 Lecture 4

• Please add your own figures

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Dielectric function

• Describes the polarization of the medium
• De E
• The dielectric response describes propagation
  (real e) and absorption (imaginary part of e)
• e e1ie2 n n1in2
• e n2
• Propagation of plane wave is the amplitude

propagation absorption The absorption describes
attenuation of the amplitude of the wave as it
traverses medium to a depth z Intensity IE2 I
I0 exp(- g z) where g2n2(w/c)z2kn2k0z is the
attenuation factor
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Photon density of states equation 1-3-1.4 Sakoda

• Let us derive the important equation 1.4 (Sakoda)
  for density of photon modes
• Consider a box of side L and volume VL3
• Electromagnetic waves reside in the box
• Integral number of waves need to fit inside box
  nl L, or lL/n for each direction
• Since k2p/l,
• kx 2pn1/L ky 2pn2/L, kz 2pn3/L
• Each unit cube of k-space has the volume
  W(2p/L)3 (2p)3/V and contains two modes
  because of the 2 polarizations
• Let us find the number of modes with frequencies
  between w and wdw or equivalently magnitude of
  wavevector between k and kdk
• Volume of shell in the wave-vector space is
  dV4pk2dk
• This volume contains the number of modes
• N(k)dk 2 4pk2dk /W
• 2 4pk2dk V/(2p)3
• N(k)dk Vk2dk/p2
• Convert from wave vector to frequency w then
  wck dkdw/c
• D(w) dw N(k)dk

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Modes in dielectric media

• Replace c by c/n the slower velocity in the
  dielectric

• Density of modes are increased by the factor of
  n3
• More wavelengths fit inside the dielectric and
  modes are compressed

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Photonic band gap crystal

• Photonic band gap crystal changes the density of
  photon states
• DOS is zero over some frequency range photonic
  band gap
• Photonic crystal can also increase the DOS over
  some frequency range
• Pictures of photonic crystals (Sakoda)

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Optical properties of PBG

• RTA 1
• Absorption A0
• In band gap no transmission T0, R1 all waves
  are reflected
• Outside band gap R, T non-zero
• See figures of R vs frequency, T vs frequency

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Spontaneous emission

• Transition from ground state to excited state
• hn E2-E1
• If DOS D(n) 0 then the transition is forbidden
• Excited state can have very long lifetime
• Ca still excite from ground state to higher
  levels E3, E4 decay from E2 to E1 is affected
• Change properties of radiation field increase or
  decrease spontaneous emission rate
• Why is spontaneous emission rate so important?
• Spontaneous emission governs the properties of
• Lasers (light emission limited by spontaneous
  emission)
• Chemical reactions, catalysis (depends on excited
  state), change catalysis by changing excited
  state lifetime
• Solar cells (lifetime of photoexcited carrier)
• Light emitting diode (LED)
• Many of these suggestions are very novel but have
  not been demonstrated yet

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Light emitting diode (LED)

• Basic structure is forward biased p-n junction
• Holes from p and electrons from n side recombine
  in junction regaio to give off light
• Hetero-junction LED
• Lens on top of LED often focuses light into fiber
• Single mode LED (Yablonovitch 1993, J. Opt Soc
  Am)
• LED surrounded by optical cavity
• Optical cavity allows only a single light mode to
  propagate e.g. l1 d
• Single mode LED similar to laser but not coherent
• Can make optical cavity out of low loss PBG
  crystal
• High fraction of spontaneous emission funnelled
  into desired mode (b factor large)
• LED require a minimum injected current for light
  emission (Threshold current)
• PBG crystal increase b, lower threshold current
• New application cavities at optical frequencies

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One-dimensional PBG DBR

• Vertical cavity surface emitting laser (VCSEL)
• Active layer separated by two distributed Bragg
  reflectors DBR (alternating stack of dielectric
  layers that act as mirrors) Bragg grating
• Laser principle When enough intensity builds up
  there is population inversion and stimulated
  emission dominates over
• m l m 2 neffd
• This wavelength is perfectly reflected from the
  DBR
• DBR s act as 1-dimensional photonic band gap
  crystals