Abstract

Dispersive thin-film stacks are interesting as compact, cost-effective devices for temporal dispersion compensation and wavelength multiplexing. Their performance depends on the total group delay or spatial shift that can be achieved. For general multilayer stacks, no analytic model exists relating the performance to the stack parameters such as the refractive indices and the number of layers. We develop an empirical model by designing and analyzing 623 thin-film stacks with constant dispersion. From this analysis we conclude that, for given stack parameters, the maximum constant dispersion value is inversely proportional to the wavelength range over which the dispersion is achieved. This is equivalent to saying that, for constant dispersion, there is a maximum possible spatial shift (or group delay) that can be achieved for a given material system and number of layers. This empirical model is useful to judge the feasibility of dispersive photonic nanostructures and photonic crystal superprism devices and serves as a first step in the search for an analytic performance model. We predict that an 8-channel wavelength multiplexer can be realized with a single 21-µm-thick SiO2-Ta2O5 thin-film stack.

© 2005 Optical Society of America

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References

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    [Crossref]
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    [Crossref] [PubMed]
  3. P. Tournois, P. Hartemann, “Bulk chirped Bragg reflectors for light pulse compression and expansion,” Opt. Commun. 119, 569–575 (1995).
    [Crossref]
  4. N. Matuschek, F. X. Kärtner, U. Keller, “Analytical design of double-chirped mirrors with custom-tailored dispersion characteristics,” IEEE J. Quantum Electron. 35, 129–137 (1999).
    [Crossref]
  5. G. Lenz, C. K. Madsen, “General optical all-pass filter structures for dispersion control in WDM systems,” J. Lightwave Technol. 17, 1248–1254 (1999).
    [Crossref]
  6. C. K. Madsen, J. H. Zhao, Optical Filter Design and Analysis—A Signal Processing Approach (Wiley, New York, 1999).
    [Crossref]
  7. M. Jablonski, Y. Takushima, K. Kikuchi, “The realization of all-pass filters for third-order dispersion compensation in ultrafast optical fiber transmission systems,” J. Lightwave Technol. 19, 1194–1205 (2001).
    [Crossref]
  8. M. Gerken, D. A. B. Miller, “Multilayer thin-film structures with high spatial dispersion,” Appl. Opt. 42, 1330–1345 (2003).
    [Crossref] [PubMed]
  9. M. Gerken, D. A. B. Miller, “Wavelength demultiplexer using the spatial dispersion of multilayer thin-film structures,” IEEE Photon. Technol. Lett. 15, 1097–1099 (2003).
    [Crossref]
  10. M. Gerken, “Wavelength multiplexing by spatial beam shifting in multilayer thin-film structures,” Electrical Engineering Ph.D. dissertation (Stanford University, Stanford, Calif., 2003).
  11. M. Gerken, D. A. B. Miller, “The relationship between the superprism effect, group delay, and stored energy in 1-D photonic crystals and photonic nanostructures,” in MRS Spring Meeting (Materials Research Society, Warrendale, Pa., 2003), paper J2.7.
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    [Crossref] [PubMed]
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  16. G. Steimeyer, “Dispersion oscillations in ultrafast phase-correction devices,” IEEE J. Quantum Electron. 39, 1027–1034 (2003).
    [Crossref]
  17. T. Baba, T. Matsumoto, “Resolution of photonic crystal superprism,” Appl. Phys. Lett. 81, 2325–2327 (2002).
    [Crossref]
  18. R. Shankar, Principles of Quantum Mechanics (Plenum, New York, 1994).
    [Crossref]
  19. A. B. Migdal, Qualitative Methods in Quantum Theory (Addison-Wesley, Redwood City, Calif., 1989).
  20. N. Matuschek, F. X. Kärtner, U. Keller, “Theory of double-chirped mirrors,” IEEE J. Sel. Top. Quantum Electron. 4, 197–208 (1998).
    [Crossref]
  21. I. N. Bronstein, K. A. Semendjajew, G. Musiol, H. Muehlig, Taschenbuch der Mathematik (Verlag Harri Deutsch, Frankfurt am Main, Germany, 1993), pp. 232–237.

2003 (3)

M. Gerken, D. A. B. Miller, “Wavelength demultiplexer using the spatial dispersion of multilayer thin-film structures,” IEEE Photon. Technol. Lett. 15, 1097–1099 (2003).
[Crossref]

G. Steimeyer, “Dispersion oscillations in ultrafast phase-correction devices,” IEEE J. Quantum Electron. 39, 1027–1034 (2003).
[Crossref]

M. Gerken, D. A. B. Miller, “Multilayer thin-film structures with high spatial dispersion,” Appl. Opt. 42, 1330–1345 (2003).
[Crossref] [PubMed]

2002 (1)

T. Baba, T. Matsumoto, “Resolution of photonic crystal superprism,” Appl. Phys. Lett. 81, 2325–2327 (2002).
[Crossref]

2001 (1)

1999 (2)

N. Matuschek, F. X. Kärtner, U. Keller, “Analytical design of double-chirped mirrors with custom-tailored dispersion characteristics,” IEEE J. Quantum Electron. 35, 129–137 (1999).
[Crossref]

G. Lenz, C. K. Madsen, “General optical all-pass filter structures for dispersion control in WDM systems,” J. Lightwave Technol. 17, 1248–1254 (1999).
[Crossref]

1998 (1)

N. Matuschek, F. X. Kärtner, U. Keller, “Theory of double-chirped mirrors,” IEEE J. Sel. Top. Quantum Electron. 4, 197–208 (1998).
[Crossref]

1995 (1)

P. Tournois, P. Hartemann, “Bulk chirped Bragg reflectors for light pulse compression and expansion,” Opt. Commun. 119, 569–575 (1995).
[Crossref]

1994 (1)

1990 (1)

1989 (1)

Baba, T.

T. Baba, T. Matsumoto, “Resolution of photonic crystal superprism,” Appl. Phys. Lett. 81, 2325–2327 (2002).
[Crossref]

Belkind, A.

Bronstein, I. N.

I. N. Bronstein, K. A. Semendjajew, G. Musiol, H. Muehlig, Taschenbuch der Mathematik (Verlag Harri Deutsch, Frankfurt am Main, Germany, 1993), pp. 232–237.

Chong, E. K. P.

E. K. P. Chong, S. H. Zak, An Introduction to Optimization (Wiley, New York, 1996).

Dobrowolski, J. A.

Ferencz, K.

Gerken, M.

M. Gerken, D. A. B. Miller, “Wavelength demultiplexer using the spatial dispersion of multilayer thin-film structures,” IEEE Photon. Technol. Lett. 15, 1097–1099 (2003).
[Crossref]

M. Gerken, D. A. B. Miller, “Multilayer thin-film structures with high spatial dispersion,” Appl. Opt. 42, 1330–1345 (2003).
[Crossref] [PubMed]

M. Gerken, “Wavelength multiplexing by spatial beam shifting in multilayer thin-film structures,” Electrical Engineering Ph.D. dissertation (Stanford University, Stanford, Calif., 2003).

M. Gerken, D. A. B. Miller, “The relationship between the superprism effect, group delay, and stored energy in 1-D photonic crystals and photonic nanostructures,” in MRS Spring Meeting (Materials Research Society, Warrendale, Pa., 2003), paper J2.7.

Hartemann, P.

P. Tournois, P. Hartemann, “Bulk chirped Bragg reflectors for light pulse compression and expansion,” Opt. Commun. 119, 569–575 (1995).
[Crossref]

Ho, F. C.

Jablonski, M.

Kärtner, F. X.

N. Matuschek, F. X. Kärtner, U. Keller, “Analytical design of double-chirped mirrors with custom-tailored dispersion characteristics,” IEEE J. Quantum Electron. 35, 129–137 (1999).
[Crossref]

N. Matuschek, F. X. Kärtner, U. Keller, “Theory of double-chirped mirrors,” IEEE J. Sel. Top. Quantum Electron. 4, 197–208 (1998).
[Crossref]

Keller, U.

N. Matuschek, F. X. Kärtner, U. Keller, “Analytical design of double-chirped mirrors with custom-tailored dispersion characteristics,” IEEE J. Quantum Electron. 35, 129–137 (1999).
[Crossref]

N. Matuschek, F. X. Kärtner, U. Keller, “Theory of double-chirped mirrors,” IEEE J. Sel. Top. Quantum Electron. 4, 197–208 (1998).
[Crossref]

Kemp, R. A.

Kikuchi, K.

Koss, V. A.

Krausz, F.

Lenz, G.

MacLeod, H. A.

H. A. MacLeod, Thin-Film Optical Filters (Institute of Physics, Philadelphia, Pa., 2001).
[Crossref]

Madsen, C. K.

G. Lenz, C. K. Madsen, “General optical all-pass filter structures for dispersion control in WDM systems,” J. Lightwave Technol. 17, 1248–1254 (1999).
[Crossref]

C. K. Madsen, J. H. Zhao, Optical Filter Design and Analysis—A Signal Processing Approach (Wiley, New York, 1999).
[Crossref]

Matsumoto, T.

T. Baba, T. Matsumoto, “Resolution of photonic crystal superprism,” Appl. Phys. Lett. 81, 2325–2327 (2002).
[Crossref]

Matuschek, N.

N. Matuschek, F. X. Kärtner, U. Keller, “Analytical design of double-chirped mirrors with custom-tailored dispersion characteristics,” IEEE J. Quantum Electron. 35, 129–137 (1999).
[Crossref]

N. Matuschek, F. X. Kärtner, U. Keller, “Theory of double-chirped mirrors,” IEEE J. Sel. Top. Quantum Electron. 4, 197–208 (1998).
[Crossref]

Migdal, A. B.

A. B. Migdal, Qualitative Methods in Quantum Theory (Addison-Wesley, Redwood City, Calif., 1989).

Miller, D. A. B.

M. Gerken, D. A. B. Miller, “Wavelength demultiplexer using the spatial dispersion of multilayer thin-film structures,” IEEE Photon. Technol. Lett. 15, 1097–1099 (2003).
[Crossref]

M. Gerken, D. A. B. Miller, “Multilayer thin-film structures with high spatial dispersion,” Appl. Opt. 42, 1330–1345 (2003).
[Crossref] [PubMed]

M. Gerken, D. A. B. Miller, “The relationship between the superprism effect, group delay, and stored energy in 1-D photonic crystals and photonic nanostructures,” in MRS Spring Meeting (Materials Research Society, Warrendale, Pa., 2003), paper J2.7.

Mistree, F.

T. E. Shoup, F. Mistree, Optimization Methods with Applications for Personal Computers (Prentice-Hall, Upper Saddle River, N.J., 1987).

Muehlig, H.

I. N. Bronstein, K. A. Semendjajew, G. Musiol, H. Muehlig, Taschenbuch der Mathematik (Verlag Harri Deutsch, Frankfurt am Main, Germany, 1993), pp. 232–237.

Musiol, G.

I. N. Bronstein, K. A. Semendjajew, G. Musiol, H. Muehlig, Taschenbuch der Mathematik (Verlag Harri Deutsch, Frankfurt am Main, Germany, 1993), pp. 232–237.

Semendjajew, K. A.

I. N. Bronstein, K. A. Semendjajew, G. Musiol, H. Muehlig, Taschenbuch der Mathematik (Verlag Harri Deutsch, Frankfurt am Main, Germany, 1993), pp. 232–237.

Shankar, R.

R. Shankar, Principles of Quantum Mechanics (Plenum, New York, 1994).
[Crossref]

Shoup, T. E.

T. E. Shoup, F. Mistree, Optimization Methods with Applications for Personal Computers (Prentice-Hall, Upper Saddle River, N.J., 1987).

Spielmann, C.

Steimeyer, G.

G. Steimeyer, “Dispersion oscillations in ultrafast phase-correction devices,” IEEE J. Quantum Electron. 39, 1027–1034 (2003).
[Crossref]

Szipöcs, R.

Takushima, Y.

Tournois, P.

P. Tournois, P. Hartemann, “Bulk chirped Bragg reflectors for light pulse compression and expansion,” Opt. Commun. 119, 569–575 (1995).
[Crossref]

Zak, S. H.

E. K. P. Chong, S. H. Zak, An Introduction to Optimization (Wiley, New York, 1996).

Zhao, J. H.

C. K. Madsen, J. H. Zhao, Optical Filter Design and Analysis—A Signal Processing Approach (Wiley, New York, 1999).
[Crossref]

Appl. Opt. (3)

Appl. Phys. Lett. (1)

T. Baba, T. Matsumoto, “Resolution of photonic crystal superprism,” Appl. Phys. Lett. 81, 2325–2327 (2002).
[Crossref]

IEEE J. Quantum Electron. (2)

N. Matuschek, F. X. Kärtner, U. Keller, “Analytical design of double-chirped mirrors with custom-tailored dispersion characteristics,” IEEE J. Quantum Electron. 35, 129–137 (1999).
[Crossref]

G. Steimeyer, “Dispersion oscillations in ultrafast phase-correction devices,” IEEE J. Quantum Electron. 39, 1027–1034 (2003).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (1)

N. Matuschek, F. X. Kärtner, U. Keller, “Theory of double-chirped mirrors,” IEEE J. Sel. Top. Quantum Electron. 4, 197–208 (1998).
[Crossref]

IEEE Photon. Technol. Lett. (1)

M. Gerken, D. A. B. Miller, “Wavelength demultiplexer using the spatial dispersion of multilayer thin-film structures,” IEEE Photon. Technol. Lett. 15, 1097–1099 (2003).
[Crossref]

J. Lightwave Technol. (2)

Opt. Commun. (1)

P. Tournois, P. Hartemann, “Bulk chirped Bragg reflectors for light pulse compression and expansion,” Opt. Commun. 119, 569–575 (1995).
[Crossref]

Opt. Lett. (1)

Other (9)

H. A. MacLeod, Thin-Film Optical Filters (Institute of Physics, Philadelphia, Pa., 2001).
[Crossref]

C. K. Madsen, J. H. Zhao, Optical Filter Design and Analysis—A Signal Processing Approach (Wiley, New York, 1999).
[Crossref]

M. Gerken, “Wavelength multiplexing by spatial beam shifting in multilayer thin-film structures,” Electrical Engineering Ph.D. dissertation (Stanford University, Stanford, Calif., 2003).

M. Gerken, D. A. B. Miller, “The relationship between the superprism effect, group delay, and stored energy in 1-D photonic crystals and photonic nanostructures,” in MRS Spring Meeting (Materials Research Society, Warrendale, Pa., 2003), paper J2.7.

R. Shankar, Principles of Quantum Mechanics (Plenum, New York, 1994).
[Crossref]

A. B. Migdal, Qualitative Methods in Quantum Theory (Addison-Wesley, Redwood City, Calif., 1989).

E. K. P. Chong, S. H. Zak, An Introduction to Optimization (Wiley, New York, 1996).

T. E. Shoup, F. Mistree, Optimization Methods with Applications for Personal Computers (Prentice-Hall, Upper Saddle River, N.J., 1987).

I. N. Bronstein, K. A. Semendjajew, G. Musiol, H. Muehlig, Taschenbuch der Mathematik (Verlag Harri Deutsch, Frankfurt am Main, Germany, 1993), pp. 232–237.

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Figures (5)

Fig. 1
Fig. 1

Wavelength demultiplexer for a multilayer thin-film stack with high spatial dispersion.

Fig. 2
Fig. 2

Forty-layer SiO2-Ta2O6 design for a 40° incidence angle. The dispersion was specified as 0.571 µm/nm.

Fig. 3
Fig. 3

Dependency of the shift on the average refractive index (crosses are for 40- and 60-layer desings, plusses are for 100 and 120 layers, and diamonds are for 200 layers).

Fig. 4
Fig. 4

Two poor models for comparison. (a) Assuming the shift to be inversely proportional to the average refractive index, (b) assuming the shift to be inversely proportional to the refractive index cubed.

Fig. 5
Fig. 5

Wavelength range divided by the shift model of Eq. (5) as a function of the specified dispersion. The thick solid line represents the expected value from the model of Eq. (5). Designs below the line are worse than the model and designs above the line are better than the model. The diamonds represent the fraction of the 623 designed stacks with less than 10% relative standard deviation and the plusses represent the stacks with 10–25% relative standard deviation. The large circles are the chirped designs discussed in Ref. 8, the X is the numerically optimized design in Ref. 8, and the square is the coupled-cavity design in Ref. 8.

Tables (2)

Tables Icon

Table 1 Sequence of Algorithms for Numerical Optimization

Tables Icon

Table 2 Distribution of the Design Parameters for the 623 Analyzed Designs

Equations (18)

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τ group = s x ν g x .
1 ν g x , appr 1 c sin ( θ ) ( n eff 2 ¯ ) ,
MF = ( 1 N i = 1 N | Q i T Q i Δ Q i | p ) 1 / p ,
( Δ λ ) max = 1 c disp 16 L Δ n n avg 2 sin ( θ ) ,
Δ s = c disp ( Δ λ ) max = 16 L Δ n n avg 2 sin ( θ ) .
Δ τ = Δ s ν g x 16 L Δ n n avg 2 sin ( θ ) 1 c sin ( θ ) ( n eff 2 ¯ ) 16 Δ n L c ,
N channels 1 + Δ s λ c π sin ( θ ) [ 1 sin ( θ ) 2 / n s 2 ] 2 c cross talk ,
N channels 1 + L λ c 8 c cross talk Δ n n avg 2 π sin ( θ ) 2 [ 1 sin ( θ ) 2 n s 2 ] .
N SiO 2 Ta 2 O 5 , 45 ° 1 + 0 . 53 L λ c .
τ group = s x ν g x = s x β ω | K = const = 2 L ν g z = 2 L K ω | β = const = ϕ refl ω | β = const ,
ϕ refl = 2 L K .
ϕ appr ( β , ω ) 2 i { [ ( ω c n i ) 2 β 2 ] 1 / 2 d i } .
K appr ( β , ω ) = 1 L i { [ ( ω c n i ) 2 β 2 ] 1 / 2 d i } .
K appr ( β , ω ) ω | β = const = 1 L i { n i 2 c d i [ n i 2 ( β c ω ) 2 ] 1 / 2 } ,
K appr ( β , ω ) β | ω = const = 1 L i { β c ω d i [ n i 2 ( β c ω ) 2 ] 1 / 2 } .
1 ν g x , appr = ( β ω ) appr | K appr = const = K appr / ω K appr / β ω β c 2 i { n i 2 d i / [ n i 2 ( β c ω ) 2 ] 1 / 2 } i { d i / [ n i 2 ( β c ω ) 2 ] 1 / 2 } .
1 ν g x , appr = ( β ω ) appr | K = const = 1 c sin ( θ ) ( n eff 2 ¯ ) ,
( n eff 2 ¯ ) = i { n i 2 d i / [ n i 2 sin ( θ ) 2 ] 1 / 2 } i { d i / [ n i 2 sin ( θ ) 2 ] 1 / 2 } .

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