GENERAL THEORY FOR DISTRIBUTED REFLECTORS J. BUUS Gayton Photonics Ltd 6 Baker Street Gayton

1. GENERAL THEORY FOR DISTRIBUTED REFLECTORS J. BUUS Gayton Photonics Ltd 6 Baker Street Gayton Northants NN7 3EZ UK Tel +44 (0) 1604 859253 Fax +44 (0) 1604 859256 Email buus@compuserve.com

2. 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

3. ADDITION OF REFLECTIONS For more on this tanh substitution and its applications see [1]

4. SMALL REFLECTIONS If the reflection per unit length is k and all reflections are in phase This gives a differential equation which leads to

5. L r TRANSFER MATRICES Er1 Er2 Er3 Es1 Es2 Es3

6. 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

7. 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]

8. SOLUTION Note that this can be written as a transfer matrix

9. GENERAL REFLECTION EQUATION Phase factor Depletion term Differential Fresnel coefficient Ordinary, first order, nonlinear differential equation No general analytical solution

10. APPLICATION TO PERIODIC STRUCTURES For a modulated periodic structure + higher terms

11. LINEARIZATION Neglecting the higher order terms, replacing j by -1 dividing by (1-r2) and using r=atanh(r) and replacing r/(1-r2) by r gives a linear equation for r This substitution was proposed in ref [3]

12. 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 d=b-b0

13. 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.

14. UNIFORM GRATING For high values of kL it is more accurate to use a modified detuning The modified detuning was derived in ref [4]

15. Exact Modified Simple 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 kL=1 (left) and 3 (right)

16. 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, Parceval’s theorem:

17. PHASE MODULATED GRATING Grating design from ref [5]

18. 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

19. 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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