## Fundamental limit for optical components

JOSA B, Vol. 24, Issue 10, pp. A1-A18 (2007)

http://dx.doi.org/10.1364/JOSAB.24.0000A1

Acrobat PDF (390 KB)

### Abstract

We show that there is a general limit to the performance of linear optical components, based only on their size, shape, and dielectric constants. The limit is otherwise independent of the design. The mathematics involved applies generally to linear systems with arbitrarily strong multiple scattering. Relevant optical structures include dielectric stacks, photonic crystals, nanometallics, metamaterials, and slow-light structures. The limit also covers acoustic and quantum-mechanical waves, and electromagnetic waves of any frequency. In an example, a one-dimensional glass/air structure, a thickness of at least

© 2007 Optical Society of America

## 1. INTRODUCTION

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

2. M. Sumetsky and B. Eggleton, “Modeling and optimization of complex photonic resonant cavity circuits,” Opt. Express **11**, 381–391 (2003). [CrossRef] [PubMed]

3. O. Schwelb, “Transmission, group delay, and dispersion in single-ring optical resonators and add/drop filters-a tutorial overview,” J. Lightwave Technol. **22**, 1380–1394 (2004). [CrossRef]

4. K. Yu and O. Solgaard, “Tunable optical transversal filters based on a Gires-Tournois interferometer with MEMS phase shifters,” IEEE J. Sel. Top. Quantum Electron. **10**, 588–597 (2004). [CrossRef]

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

6. B. Momeni and A. Abidi, “Optimization of photonic crystal demultiplexers based on the superprism effect,” Appl. Phys. B **77**, 555–560 (2003). [CrossRef]

7. C. Y. Luo, M. Soljacic, and J. D. Joannopoulos, “Superprism effect based on phase velocities,” Opt. Lett. **29**, 745–747 (2004). [CrossRef] [PubMed]

8. L. Wu, M. Mazilu, J.-F. Gallet, and T. F. Krauss, “Dual lattice photonic-crystal beam splitters,” Appl. Phys. Lett. **86**, 211106 (2005). [CrossRef]

9. B. Momeni and A. Abidi, “Systematic design of superprism-based photonic crystal demultiplexers,” IEEE J. Sel. Areas Commun. **23**, 1355–1364 (2005). [CrossRef]

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

11. M. Gerken and D. A. B. Miller, “Photonic nanostructures for wavelength division multiplexing,” Proc. SPIE **5597**, 82–96 (2004). [CrossRef]

12. M. Gerken and D. A. B. Miller, “Limits on the performance of dispersive thin-film stacks,” Appl. Opt. **44**, 3349–3357 (2005). [CrossRef] [PubMed]

13. M. Gerken and D. A. B. Miller, “The relationship between the superprism effect in one-dimensional photonic crystals and spatial dispersion in nonperiodic thin-film stacks,” Opt. Lett. **30**, 2475–2477 (2005). [CrossRef] [PubMed]

14. R. E. Klinger, C. A. Hulse, C. K. Carniglia, and R. B. Sargent, “Beam displacement and distortion effects in narrowband optical thin-film filters,” Appl. Opt. **45**, 3237–3242 (2006). [CrossRef] [PubMed]

15. Y. Jiao, S. H. Fan, and D. A. B. Miller, “Designing for beam propagation in periodic and nonperiodic photonic nanostructures: Extended Hamiltonian method,” Phys. Rev. E **70**, 036612-1–036612-9 (2004). [CrossRef]

16. Y. Jiao, S. H. Fan, and D. A. B. Miller, “Demonstration of systematic photonic crystal device design and optimization by low rank adjustments: an extremely compact mode separator,” Opt. Lett. **30**, 141–143 (2005). [CrossRef] [PubMed]

17. Y. Jiao, S. H. Fan, and D. A. B. Miller, “Systematic photonic crystal device design: global and local optimization and sensitivity analysis,” IEEE J. Quantum Electron. **42**, 266–279 (2006). [CrossRef]

18. P. J. van Heerden, “Theory of optical information storage in solids,” Appl. Opt. **2**, 393–400 (1963). [CrossRef]

20. H. Lee, X. G. Gu, and D. Psaltis, “Volume holographic interconnections with maximal capacity and minimal cross talk,” J. Appl. Phys. **65**, 2191–2194 (1989). [CrossRef]

21. X. M. Yi, P. Yeh, C. Gu, and S. Campbell, “Crosstalk in volume holographic memory,” Proc. IEEE **87**, 1912–1930 (1999). [CrossRef]

22. K. Tian and G. Barbastathis, “Cross talk in resonant holographic memories,” J. Opt. Soc. Am. A **21**, 751–756 (2004). [CrossRef]

23. J. Shamir, “Analysis of volume holographic storage allowing large-angle illumination,” J. Opt. Soc. Am. B **22**, 975–986 (2005). [CrossRef]

24. A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett. **24**, 711–713 (1999). [CrossRef] [CrossRef]

25. G. Lenz, B. J. Eggleton, C. K. Madsen, and R. E. Slusher, “Optical delay lines based on optical fibers,” IEEE J. Quantum Electron. **37**, 525–532 (2001). [CrossRef]

26. R. S. Tucker, P.-C. Ku, and C. J. Chang-Hasnain, “Slow-light optical buffers: capabilities and fundamental limitations,” J. Lightwave Technol. **23**, 4046–4066 (2005). [CrossRef]

27. M. D. Stenner, M. A. Neifeld, Z. Zhu, A. M. C. Dawes, and D. J. Gauthier, “Distortion management in slow-light pulse delay,” Opt. Express **13**, 9995–10002 (2005). [CrossRef] [PubMed]

28. M. Povinelli, S. Johnson, and J. Joannopoulos, “Slow-light, band-edge waveguides for tunable time delays,” Opt. Express **13**, 7145–7159 (2005). [CrossRef] [PubMed]

29. Z. S. Yang, N. H. Kwong, R. Binder, and A. L. Smirl, “Distortionless light pulse delay in quantum-well Bragg structures,” Opt. Lett. **30**, 2790–2792 (2005). [CrossRef] [PubMed]

30. M. R. Fisher and S.-L. Chuang, “Variable group delay and pulse reshaping of high bandwidth optical signals,” IEEE J. Quantum Electron. **41**, 885–891 (2005). [CrossRef]

31. J. Sharping, Y. Okawachi, J. van Howe, C. Xu, Y. Wang, A. Willner, and A. Gaeta, “All-optical, wavelength and bandwidth preserving, pulse delay based on parametric wavelength conversion and dispersion,” Opt. Express **13**, 7872–7877 (2005). [CrossRef] [PubMed]

32. M. S. Bigelow, N. N. Lepeshkin, H. Shin, and R. W. Boyd, “Propagation of smooth and discontinuous pulses through materials with very large or very small group velocities,” J. Phys.: Condens. Matter **18**, 3117–3126 (2006). [CrossRef]

33. A. V. Uskov, F. G. Sedgwick, and C. J. Chang-Hasnain, “Delay limit of slow light in semiconductor optical amplifiers,” IEEE Photon. Technol. Lett. **18**, 731–733 (2006). [CrossRef]

34. R. W. Boyd, D. J. Gauthier, A. L. Gaeta, and A. E. Willner, “Maximum time delay achievable on propagation through a slow-light medium,” Phys. Rev. A **71**, 023801 (2005). [CrossRef]

35. J. B. Khurgin, “Performance limits of delay lines based on optical amplifiers,” Opt. Lett. **31**, 948–950 (2006). [CrossRef] [PubMed]

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

2. M. Sumetsky and B. Eggleton, “Modeling and optimization of complex photonic resonant cavity circuits,” Opt. Express **11**, 381–391 (2003). [CrossRef] [PubMed]

3. O. Schwelb, “Transmission, group delay, and dispersion in single-ring optical resonators and add/drop filters-a tutorial overview,” J. Lightwave Technol. **22**, 1380–1394 (2004). [CrossRef]

4. K. Yu and O. Solgaard, “Tunable optical transversal filters based on a Gires-Tournois interferometer with MEMS phase shifters,” IEEE J. Sel. Top. Quantum Electron. **10**, 588–597 (2004). [CrossRef]

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

6. B. Momeni and A. Abidi, “Optimization of photonic crystal demultiplexers based on the superprism effect,” Appl. Phys. B **77**, 555–560 (2003). [CrossRef]

7. C. Y. Luo, M. Soljacic, and J. D. Joannopoulos, “Superprism effect based on phase velocities,” Opt. Lett. **29**, 745–747 (2004). [CrossRef] [PubMed]

8. L. Wu, M. Mazilu, J.-F. Gallet, and T. F. Krauss, “Dual lattice photonic-crystal beam splitters,” Appl. Phys. Lett. **86**, 211106 (2005). [CrossRef]

9. B. Momeni and A. Abidi, “Systematic design of superprism-based photonic crystal demultiplexers,” IEEE J. Sel. Areas Commun. **23**, 1355–1364 (2005). [CrossRef]

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

11. M. Gerken and D. A. B. Miller, “Photonic nanostructures for wavelength division multiplexing,” Proc. SPIE **5597**, 82–96 (2004). [CrossRef]

12. M. Gerken and D. A. B. Miller, “Limits on the performance of dispersive thin-film stacks,” Appl. Opt. **44**, 3349–3357 (2005). [CrossRef] [PubMed]

13. M. Gerken and D. A. B. Miller, “The relationship between the superprism effect in one-dimensional photonic crystals and spatial dispersion in nonperiodic thin-film stacks,” Opt. Lett. **30**, 2475–2477 (2005). [CrossRef] [PubMed]

14. R. E. Klinger, C. A. Hulse, C. K. Carniglia, and R. B. Sargent, “Beam displacement and distortion effects in narrowband optical thin-film filters,” Appl. Opt. **45**, 3237–3242 (2006). [CrossRef] [PubMed]

15. Y. Jiao, S. H. Fan, and D. A. B. Miller, “Designing for beam propagation in periodic and nonperiodic photonic nanostructures: Extended Hamiltonian method,” Phys. Rev. E **70**, 036612-1–036612-9 (2004). [CrossRef]

16. Y. Jiao, S. H. Fan, and D. A. B. Miller, “Demonstration of systematic photonic crystal device design and optimization by low rank adjustments: an extremely compact mode separator,” Opt. Lett. **30**, 141–143 (2005). [CrossRef] [PubMed]

17. Y. Jiao, S. H. Fan, and D. A. B. Miller, “Systematic photonic crystal device design: global and local optimization and sensitivity analysis,” IEEE J. Quantum Electron. **42**, 266–279 (2006). [CrossRef]

16. Y. Jiao, S. H. Fan, and D. A. B. Miller, “Demonstration of systematic photonic crystal device design and optimization by low rank adjustments: an extremely compact mode separator,” Opt. Lett. **30**, 141–143 (2005). [CrossRef] [PubMed]

17. Y. Jiao, S. H. Fan, and D. A. B. Miller, “Systematic photonic crystal device design: global and local optimization and sensitivity analysis,” IEEE J. Quantum Electron. **42**, 266–279 (2006). [CrossRef]

18. P. J. van Heerden, “Theory of optical information storage in solids,” Appl. Opt. **2**, 393–400 (1963). [CrossRef]

20. H. Lee, X. G. Gu, and D. Psaltis, “Volume holographic interconnections with maximal capacity and minimal cross talk,” J. Appl. Phys. **65**, 2191–2194 (1989). [CrossRef]

21. X. M. Yi, P. Yeh, C. Gu, and S. Campbell, “Crosstalk in volume holographic memory,” Proc. IEEE **87**, 1912–1930 (1999). [CrossRef]

22. K. Tian and G. Barbastathis, “Cross talk in resonant holographic memories,” J. Opt. Soc. Am. A **21**, 751–756 (2004). [CrossRef]

23. J. Shamir, “Analysis of volume holographic storage allowing large-angle illumination,” J. Opt. Soc. Am. B **22**, 975–986 (2005). [CrossRef]

24. A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett. **24**, 711–713 (1999). [CrossRef] [CrossRef]

25. G. Lenz, B. J. Eggleton, C. K. Madsen, and R. E. Slusher, “Optical delay lines based on optical fibers,” IEEE J. Quantum Electron. **37**, 525–532 (2001). [CrossRef]

26. R. S. Tucker, P.-C. Ku, and C. J. Chang-Hasnain, “Slow-light optical buffers: capabilities and fundamental limitations,” J. Lightwave Technol. **23**, 4046–4066 (2005). [CrossRef]

27. M. D. Stenner, M. A. Neifeld, Z. Zhu, A. M. C. Dawes, and D. J. Gauthier, “Distortion management in slow-light pulse delay,” Opt. Express **13**, 9995–10002 (2005). [CrossRef] [PubMed]

28. M. Povinelli, S. Johnson, and J. Joannopoulos, “Slow-light, band-edge waveguides for tunable time delays,” Opt. Express **13**, 7145–7159 (2005). [CrossRef] [PubMed]

29. Z. S. Yang, N. H. Kwong, R. Binder, and A. L. Smirl, “Distortionless light pulse delay in quantum-well Bragg structures,” Opt. Lett. **30**, 2790–2792 (2005). [CrossRef] [PubMed]

30. M. R. Fisher and S.-L. Chuang, “Variable group delay and pulse reshaping of high bandwidth optical signals,” IEEE J. Quantum Electron. **41**, 885–891 (2005). [CrossRef]

31. J. Sharping, Y. Okawachi, J. van Howe, C. Xu, Y. Wang, A. Willner, and A. Gaeta, “All-optical, wavelength and bandwidth preserving, pulse delay based on parametric wavelength conversion and dispersion,” Opt. Express **13**, 7872–7877 (2005). [CrossRef] [PubMed]

32. M. S. Bigelow, N. N. Lepeshkin, H. Shin, and R. W. Boyd, “Propagation of smooth and discontinuous pulses through materials with very large or very small group velocities,” J. Phys.: Condens. Matter **18**, 3117–3126 (2006). [CrossRef]

33. A. V. Uskov, F. G. Sedgwick, and C. J. Chang-Hasnain, “Delay limit of slow light in semiconductor optical amplifiers,” IEEE Photon. Technol. Lett. **18**, 731–733 (2006). [CrossRef]

34. R. W. Boyd, D. J. Gauthier, A. L. Gaeta, and A. E. Willner, “Maximum time delay achievable on propagation through a slow-light medium,” Phys. Rev. A **71**, 023801 (2005). [CrossRef]

35. J. B. Khurgin, “Performance limits of delay lines based on optical amplifiers,” Opt. Lett. **31**, 948–950 (2006). [CrossRef] [PubMed]

12. M. Gerken and D. A. B. Miller, “Limits on the performance of dispersive thin-film stacks,” Appl. Opt. **44**, 3349–3357 (2005). [CrossRef] [PubMed]

## 2. DERIVATION OF THE CENTRAL RESULT

*M*to the number of orthogonal functions that can be generated in a receiving volume as a result of scattering of a wave from a scattering volume. A key result of this paper is that an upper limit on

*M*can be deduced based on limits to quantities like the range of dielectric constant variation in the volume, irrespective of the details of the design of the structure. Knowing that limit, we will be able to use it to deduce limits for specific optical functions, such as dispersive elements.

### 2A. Mathematical Notation

*H*, then we can write the identity matrix or operator for that space

### 2B. Formulation of the Problem

*all*sources in the volume; it includes any source that we might regard as being generated as a result of the interaction between, e.g., waves and dielectrics, such as induced polarizations or currents. Hence

36. D. A. B. Miller, “Spatial channels for communicating with waves between volumes,” Opt. Lett. **23**, 1645–1647 (1998). [CrossRef]

37. D. A. B. Miller, “Communicating with waves between volumes – evaluating orthogonal spatial channels and limits on coupling strengths,” Appl. Opt. **39**, 1681–1699 (2000). [CrossRef]

### 2C. Proof of Core Sum Rule

*m*for which

*M*of possible orthogonal waves that could be generated in the receiving space by our strong scattering isi.e., we would have established a maximum dimensionality of the space of functions that can be generated in the receiving space by (strong) scattering.

### 2D. Interpretation of Bound

*M*is the maximum number of members of the orthogonal set

*M*is the maximum number of degrees of freedom we have in designing the form of the waves generated in the receiving volume. It is the maximum number of coefficients we can choose for linearly independent quantities in the generated wave amplitudes in the receiving volume. We will return below to consider such arguments explicitly in example one-dimensional cases. There we will also consider the straight-through and single-scattered waves.

### 2E. Evaluation of Bound

*m*for which the

*M*from Eq. (17).

*M*of orthogonal

*M*of such possible functions, to make this number as large as possible, we presume that all of the available resource for the

*M*possibilities, i.e., we now restrict the sums in Eqs. (27, 28) to be from 1 to

*M*. Given that, we know therefore that the average values,

*P*falls below 1, then we know there is at least one product

*m*that meet our criteria,

*M*of orthogonal waves that can be generated in

## 3. ONE-DIMENSIONAL EXAMPLE

### 3A. Choice of Spaces

*z*(i.e., we are working in transmission), though the final results for the

*z*(i.e., working in reflection).

*η*gives the fractional variation in the relative dielectric constant in the scatterer, i.e.,where

*c*is the velocity of light.

*η*(i.e.,

*η*can take on completely different spatial forms at different frequencies, or different spectral forms at different positions. Hence, for example, it is not restricted to structures such as dielectric stacks, which will have the same form of spatial distribution at all frequencies, nor to spatial patterns of one kind of atom. The sum does include the possibility therefore of two different spatial patterns of two (or more) different kinds of atoms or other interlaced materials.

### 3B. Limit for *M*

*z*, and would have chosen backwards-propagating Fourier source basis functions. The derivation would otherwise proceed identically, and the result for such a backscattered case would be identical to Eq. (37) and the equations following.

### 3C. Final Limit

### 3D. Interpretation of Limit for Separating Pulses

#### 3D1. Scaling the Number of Degrees of Freedom

*Informal argument*. We presume that these pulses of interest have some characteristic time duration

*Rigorous argument*. We can take a more rigorous approach to this argument. Suppose now that we say we are interested in a set of frequency bands, each of bandwidth

*η*over position and frequency in the same way as the original set. At least: (i) if the scattering space has a similar range of variation of

*η*over the entire scattering volume; and (ii) if there are no resonances in the dielectric response of the materials that are sharp compared to our frequency band of interest, then we can simply scale down the

#### 3D2. Representation of Pulses

*a*of each sine function and equal amounts

*b*of each cosine function, though

*a*and

*b*may be different numbers, with the ratio of

*a*and

*b*determining the phase of the resulting pulse within a given half-cycle of the carrier.) Different precise forms of pulse could have slightly different coefficients, but any pulse of approximately this minimum duration will essentially involve similar amplitudes from all the basis functions within the bandwidth (subject to the discussion of

*a*and

*b*above).

*a*and

*b*coefficient distinction) will form an exactly orthogonal set, with pulse centers spaced by integer multiples of a time

*a*finite,

*b*finite,

#### 3D3. Number of Required Basis Functions

37. D. A. B. Miller, “Communicating with waves between volumes – evaluating orthogonal spatial channels and limits on coupling strengths,” Appl. Opt. **39**, 1681–1699 (2000). [CrossRef]

### 3E. Limit to Separating Pulses

*Explicit calculation example*. Suppose then that we wish to separate pulses of 32 different equally spaced frequencies, all within a relatively narrow band (e.g., the telecommunications C-band) about a free-space wavelength of

## 4. CONCLUSIONS

## APPENDIX A: RELAXATION OF STRICT ORTHOGONALITY REQUIREMENT

*m*such that Eq. (13) is obeyed, instead of Eq. (14) we would havewhere

*m*for whichfor some (real and positive)

## APPENDIX B: DETAILED EVALUATION OF SUMS FOR ONE-DIMENSIONAL CASE

## Time Ranges

## Green’s Functions

*z*generated by a source amplitude

*z*in space at any time

*t*, either inside or outside the scattering space.

## Choosing the Source Function Set for the Scattering Space

### Form of the Operators Mapping from Source to Scattering Spaces

*z*. Using this form, we can evaluate

### Basis Set

37. D. A. B. Miller, “Communicating with waves between volumes – evaluating orthogonal spatial channels and limits on coupling strengths,” Appl. Opt. **39**, 1681–1699 (2000). [CrossRef]

*Condition for completeness of a basis set*. We want to work with some convenient basis set

*Evaluation of*

*Proposal of Fourier basis set*. The convenient set we propose to work with is the Fourier basis for

*n*can take on any of the

*p*. There will altogether be

*z*. Specifically, these are forward propagating source functions that we expect would lead to waves at larger

## Evaluation of N G S

*z*from sources at points of smaller or larger

*k*integral, so it can be dropped. We can now split the first term to eliminate the modulus. We obtain

*τ*variables, we have

*z*and

*τ*variables)To evaluate

*τ*, givingwhere

*τ*variables, we obtainHence,We will obtain an identical result if we consider the sine basis functions. Hence, with a total of

## Evaluation of N C

### Inequality for N C

*n*to include both the cosine and sine functions). We know, because of the sum rule Eq. (B20), that

### Evaluation of Bound on N C

*z*inside the source volume is dependent only on the wave at that precise point

*z*. Hence we have

*η*other than that

## APPENDIX C: CRITERION FOR COMPLETENESS FOR A BASIS SET IN H S

*reductio ad absurdum*that a set of functions

## APPENDIX D: SCALING OF N G S SUM INDEPENDENT OF BASIS

## ACKNOWLEDGMENTS

1. | G. Lenz and C. K. Madsen, “General optical all-pass filter structures for dispersion control in WDM systems,” J. Lightwave Technol. |

2. | M. Sumetsky and B. Eggleton, “Modeling and optimization of complex photonic resonant cavity circuits,” Opt. Express |

3. | O. Schwelb, “Transmission, group delay, and dispersion in single-ring optical resonators and add/drop filters-a tutorial overview,” J. Lightwave Technol. |

4. | K. Yu and O. Solgaard, “Tunable optical transversal filters based on a Gires-Tournois interferometer with MEMS phase shifters,” IEEE J. Sel. Top. Quantum Electron. |

5. | T. Baba and T. Matsumoto, “Resolution of photonic crystal superprism,” Appl. Phys. Lett. |

6. | B. Momeni and A. Abidi, “Optimization of photonic crystal demultiplexers based on the superprism effect,” Appl. Phys. B |

7. | C. Y. Luo, M. Soljacic, and J. D. Joannopoulos, “Superprism effect based on phase velocities,” Opt. Lett. |

8. | L. Wu, M. Mazilu, J.-F. Gallet, and T. F. Krauss, “Dual lattice photonic-crystal beam splitters,” Appl. Phys. Lett. |

9. | B. Momeni and A. Abidi, “Systematic design of superprism-based photonic crystal demultiplexers,” IEEE J. Sel. Areas Commun. |

10. | M. Gerken and D. A. B. Miller, “Multilayer thin-film structures with high spatial dispersion,” Appl. Opt. |

11. | M. Gerken and D. A. B. Miller, “Photonic nanostructures for wavelength division multiplexing,” Proc. SPIE |

12. | M. Gerken and D. A. B. Miller, “Limits on the performance of dispersive thin-film stacks,” Appl. Opt. |

13. | M. Gerken and D. A. B. Miller, “The relationship between the superprism effect in one-dimensional photonic crystals and spatial dispersion in nonperiodic thin-film stacks,” Opt. Lett. |

14. | R. E. Klinger, C. A. Hulse, C. K. Carniglia, and R. B. Sargent, “Beam displacement and distortion effects in narrowband optical thin-film filters,” Appl. Opt. |

15. | Y. Jiao, S. H. Fan, and D. A. B. Miller, “Designing for beam propagation in periodic and nonperiodic photonic nanostructures: Extended Hamiltonian method,” Phys. Rev. E |

16. | Y. Jiao, S. H. Fan, and D. A. B. Miller, “Demonstration of systematic photonic crystal device design and optimization by low rank adjustments: an extremely compact mode separator,” Opt. Lett. |

17. | Y. Jiao, S. H. Fan, and D. A. B. Miller, “Systematic photonic crystal device design: global and local optimization and sensitivity analysis,” IEEE J. Quantum Electron. |

18. | P. J. van Heerden, “Theory of optical information storage in solids,” Appl. Opt. |

19. | H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. |

20. | H. Lee, X. G. Gu, and D. Psaltis, “Volume holographic interconnections with maximal capacity and minimal cross talk,” J. Appl. Phys. |

21. | X. M. Yi, P. Yeh, C. Gu, and S. Campbell, “Crosstalk in volume holographic memory,” Proc. IEEE |

22. | K. Tian and G. Barbastathis, “Cross talk in resonant holographic memories,” J. Opt. Soc. Am. A |

23. | J. Shamir, “Analysis of volume holographic storage allowing large-angle illumination,” J. Opt. Soc. Am. B |

24. | A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett. |

25. | G. Lenz, B. J. Eggleton, C. K. Madsen, and R. E. Slusher, “Optical delay lines based on optical fibers,” IEEE J. Quantum Electron. |

26. | R. S. Tucker, P.-C. Ku, and C. J. Chang-Hasnain, “Slow-light optical buffers: capabilities and fundamental limitations,” J. Lightwave Technol. |

27. | M. D. Stenner, M. A. Neifeld, Z. Zhu, A. M. C. Dawes, and D. J. Gauthier, “Distortion management in slow-light pulse delay,” Opt. Express |

28. | M. Povinelli, S. Johnson, and J. Joannopoulos, “Slow-light, band-edge waveguides for tunable time delays,” Opt. Express |

29. | Z. S. Yang, N. H. Kwong, R. Binder, and A. L. Smirl, “Distortionless light pulse delay in quantum-well Bragg structures,” Opt. Lett. |

30. | M. R. Fisher and S.-L. Chuang, “Variable group delay and pulse reshaping of high bandwidth optical signals,” IEEE J. Quantum Electron. |

31. | J. Sharping, Y. Okawachi, J. van Howe, C. Xu, Y. Wang, A. Willner, and A. Gaeta, “All-optical, wavelength and bandwidth preserving, pulse delay based on parametric wavelength conversion and dispersion,” Opt. Express |

32. | M. S. Bigelow, N. N. Lepeshkin, H. Shin, and R. W. Boyd, “Propagation of smooth and discontinuous pulses through materials with very large or very small group velocities,” J. Phys.: Condens. Matter |

33. | A. V. Uskov, F. G. Sedgwick, and C. J. Chang-Hasnain, “Delay limit of slow light in semiconductor optical amplifiers,” IEEE Photon. Technol. Lett. |

34. | R. W. Boyd, D. J. Gauthier, A. L. Gaeta, and A. E. Willner, “Maximum time delay achievable on propagation through a slow-light medium,” Phys. Rev. A |

35. | J. B. Khurgin, “Performance limits of delay lines based on optical amplifiers,” Opt. Lett. |

36. | D. A. B. Miller, “Spatial channels for communicating with waves between volumes,” Opt. Lett. |

37. | D. A. B. Miller, “Communicating with waves between volumes – evaluating orthogonal spatial channels and limits on coupling strengths,” Appl. Opt. |

38. | R. Piestun and D. A. B. Miller, “Electromagnetic degrees of freedom of an optical system,” J. Opt. Soc. Am. A |

39. | G. W. Hanson and A. B. Yakovlev, |

40. | G. W. Hanson and A. B. Yakovlev, |

**OCIS Codes**

(230.4170) Optical devices : Multilayers

(260.2030) Physical optics : Dispersion

(290.4210) Scattering : Multiple scattering

(350.7420) Other areas of optics : Waves

**History**

Original Manuscript: November 6, 2006

Manuscript Accepted: December 17, 2006

Published: July 19, 2007

**Virtual Issues**

Photonic Metamaterials (2007) *JOSA A*

**Citation**

David A. B. Miller, "Fundamental limit for optical components," J. Opt. Soc. Am. B **24**, A1-A18 (2007)

http://www.opticsinfobase.org/josab/abstract.cfm?URI=josab-24-10-A1

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