Introduction to Optical Electronics

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Introduction to Optical Electronics
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Laser Amplilfiers
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Gain Phase CoefficientsLorentzian Lineshape
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Exercise 13.1-1Attenuation and Gain in a Ruby
Laser Amplifier

1. Consider a ruby crystal with two energy levels
   separated by an energy difference corresponding
   to a free-space wavelength ?0 694.3 nm, with a
   Lorentzian lineshape of width ?? 60 GHz. The
   spontaneous lifetime is tsp 3 ms and the
   refractive index of ruby is n 1.76. If N1 N2
   Na 1022 cm-3, determine the population
   difference N N2 N1 and the attenuation
   coefficient at the line center ?(?0) under
   conditions of thermal equilibrium (so that the
   Boltzmann distribution is obeyed) at T 300 K.
2. What value should the population difference N
   assume to achieve a gain coefficient ?(?0) 0.5
   cm-1 at the central frequency?
3. How long should the crystal be to provide an
   overall gain of 4 at the central frequency when
   ?(?0) 0.5 cm-1 ?

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Rate Equations

• Understanding Lifetimes
• ?1 and ?2 are overall lifetimes for atomic energy
  levels 1 and 2.
• Lifetime of level 2 has two contributions
• (where rates are inversely proportional to decay
  times)
• and
• Population densities N1 and N2 will vanish unless
  another mechanism is employed to increase
  occupation

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Rate Equations Absence of Amplifier Radiation

• Pumping Rates R1 R2 defined
• Rate Equations
• Steady-State Conditions

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Exercise 13.2-1Optical Pumping

• Assume that R1 0 and that R2 is realized by
  exciting atoms from the ground state E 0 to
  level 2 using photons of frequency E2 / h
  absorbed with a transition probability W. Assume
  that ?2 tsp, and ?1 ltlt tsp so that in steady
  state N1 0 and N0 R2 tsp. If Na is the total
  population of levels 0, 1, and 2, show that R2
  (Na 2N0)W, so that the population difference is
  N0 Na tsp W / (1 2 tsp W)

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Rate Equations Presence of Amplifier Radiation

• Pumping Rates
• Four Case Studies
• (Homogeneous Broadened Transitions)
• I? 0, R2(t) R20 u(t), R1(t) 0
• ?1 0, R2(t) R20 u(t)
• ?1 0, R2(t) R20, I? Pulse
• Steady State -

2
1
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Case 1I? 0, R1(t) 0, R2(t) R20 u(t)
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Solving Differential Equations

• Obtain Forms
• General Form
• Particular Form
• Homogeneous (Natural) and Particular (Forced)
  Response
• Particular Solution
• Note initial conditions not set
• Homogeneous
• Use initial conditions (removes the effect of the
  particular solutions i.c.)

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Case 1I? 0, R1(t) 0, R2(t) R20 u(t)
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Case 2?1 0, R2(t) R20 u(t)
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Case 2?1 0, R2(t) R20 u(t)
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Case 3?1 0, R2(t) R20, I? Pulse
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Case 3?1 0, R2(t) R20, I? Pulse
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Approach to Case 4Steady-State Rate Equations

• Rate Equations describe the rates of change of
  the population densities N1 and N2 as a result of
  pumping, radiative, and nonradiative transitions.
• Determine rate equations in the absence of
  Amplifier Radiation (i.e., no stimulated emission
  or absorption)
• Find steady-state population difference N0 N2
  N1
• Determine rate equations in the presence of
  Amplifier Radiation (non-linear interactions)
• Find steady-state population difference N f(N0)
• Determine the saturation time constant ?s

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Case 4 Steady State

• Pumping Rates
• Steady State

• Steady-state Population Differences
• N N2-N1
• N0 N2-N1 w/o amp. rad.
• ?s Saturation Time Constant

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Rate Equations in the Absence of Amplifier
Radiation

• Four-Level Pumping Schemes
• Three-Level Pumping Schemes

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Rate Equations in the Absence of Amplifier
Radiation

• Four-Level Pumping Schemes
• Three-Level Pumping Schemes

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Population Inversion
Population Difference Steady-State Difference Saturation Time Constant
Four-Level Laser
Three-Level Laser
What is the small-signal approximation?
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Exercise 13.2-3Pumping Powers in Three- and
Four-Level Systems

• Determine the pumping transition probability W
  required to achieve a zero population difference
  in a three- and a four-level laser amplifier
• If the pumping transition probability W 2 / tsp
  in the three-level system and W 1 / 2 tsp in
  the four-level system, show that N0 Na / 3.
  Compare the pumping powers required to achieve
  this population difference.

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Amplifier Nonlinearity Gain Coefficient
Note ?0(?) is called the small-signal gain
coefficient. Why?
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Amplifier Nonlinearity Gain
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Saturable Absorbers
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Gain CoefficientInhomogeneously Broadened Medium
Gain Coefficient
?
?