GENERAL THEORY FOR

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GENERAL THEORY FOR DISTRIBUTED
REFLECTORS J. BUUS Gayton Photonics Ltd 6
Baker Street Gayton Northants NN7 3EZ UK Tel
44 (0) 1604 859253 Fax [phone]
859256 Email buus_at_compuserve.com
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OVERVIEW

• 1. ADDITION RULES FOR MULTIPLE REFLECTORS
• 2. PERIODIC STRUCTURES, COUPLED MODE THEORY
• 3. MORE GENERAL STRUCTURES
• 4. MODULATED PERIODIC STRUCTURES
• APPROXIMATIONS
• CONCLUSIONS AND OUTLOOK

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ADDITION OF REFLECTIONS
For more on this tanh substitution and its
applications see 1
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SMALL REFLECTIONS
If the reflection per unit length is k and all
reflections are in phase
This gives a differential equation which leads to
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TRANSFER MATRICES
Er1
Er2
Er3
Es1
Es2
Es3
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PERIODIC INDEX VARIATION
Assume that the solution is of the form
R(z) and S(z) are the amplitudes of a forward and
a backward wave, and they vary slowly with z
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COUPLED MODE EQUATIONS
Direct substitution gives 20 terms! 4 combine to
2, leaving 18 2 contain second derivatives of
R(z) and S(z) 6 contain Dn2 4 are not phase
matched The remaining 6 are in 2 sets with the
same exp factor
For an alternative derivation of the C.M.E. see
2
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SOLUTION
Note that this can be written as a transfer matrix
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GENERAL REFLECTION EQUATION
Phase factor Depletion term Differential
Fresnel coefficient
Ordinary, first order, nonlinear differential
equation No general analytical solution
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APPLICATION TO PERIODIC STRUCTURES
For a modulated periodic structure
higher terms
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LINEARIZATION
Neglecting the higher order terms, replacing j by
-1
dividing by (1-r2) and using ratanh(r)
and replacing r/(1-r2) by r gives a linear
equation for r
This substitution was proposed in ref 3
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GRATING REFLECTION
The phase factor in front of tanh depends on the
reference plane used and has no significance The
integral of rapidly varying terms will average to
0 Introduce window function w(z) and use db-b0
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FOURIER TRANSFORM
for rectangular window
If g(z) or f(z) are periodic, the FT becomes a
Fourier series, with each term corresponding to
a reflection peak. Each of these peaks will have
sidelobes due to the FT of the window function w.
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UNIFORM GRATING
For high values of kL it is more accurate to use
a modified detuning
The modified detuning was derived in ref 4
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REFLECTION AS FUNCTION OF DETUNING
0.7000
0.6000
0.5000
0.4000
0.3000
0.2000
0.1000
0.0000
0.00
2.00
4.00
6.00
8.00
10.00
Uniform grating with kL1 (left) and 3 (right)
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SAMPLED GRATING
1
g(z)
0
0
Lg
Ls
Ideal comb for M specific
values of N
Consider phase modulated gratings
For g(z)1, Parcevals theorem
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PHASE MODULATED GRATING
Grating design from ref 5
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CONCLUSIONS AND OUTLOOK

• Several powerful tools are available
• Analytical solutions or good approximations
  can be
• found for many special cases
• Good test cases for numerical methods
• Basis for synthesis of reflection functions
• Challenges
• Find better modified detuning
• Find an expression for the reflection phase

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REFERENCES
1 S.W. Corzine et al, IEEE J. Quantum
Electron., Vol. 27, 2086-2090, 1991. 2 D.G.
Hall, Optics Comm., Vol. 82, 453-455,
1991. 3 D.L. Jaggard and Y. Kim, J.O.S.A. A,
Vol. 2, 1922-1930, 1985. 4 M.C. Parker et al,
J. Optics A, Vol. 3, 171-183, 2001. 5 H. Ishii
et al, IEEE J.S.T.Q.E. Vol. 1, 401-407,
1995. 6 Some of the material from this
presentation is discussed in more detail in J.
Buus et al, Tunable laser diodes and related
optical sources, Wiley, 2005.
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