## Theory of two beam interference with arbitrary spectra

Optics Express, Vol. 14, Issue 26, pp. 12751-12759 (2006)

http://dx.doi.org/10.1364/OE.14.012751

Acrobat PDF (256 KB)

### Abstract

A new formulation describing the interference term of a two beam interferometer with unequal Gaussian spectra propagating in different dispersive media is provided by defining a composite standard deviation and a composite center frequency of the interfering spectra. This formulation is generalized to arbitrary spectra by decomposing each spectrum into a linear composition of Gaussian distributions. The effective phase and group delays indicate the effect of the unequal spectral distributions and the dispersive media. An effective coherence length is derived, different than the coherence lengths of the interfering fields. The accuracy of the new formulation is proven experimentally by using optical coherence tomography systems.

© 2006 Optical Society of America

## 1. Introduction

2. R. S. Shankland, S. W. McCuskey, F. C. Leone, and G. Kuerti, “New analysis of the interferometer obervations of Dayton C. Miller,” Rev. Mod. Phys. **27**, 167–178 (1955). [CrossRef]

9. D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical Coherence Tomography,” Science **254**, 1178–1181 (1991). [CrossRef] [PubMed]

10. M. C. Booth, G. D. Giuseppe, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Polarization-sensitive quantum-optical coherence tomography,” Phys. Rev. A **69**, 043815- (2004). [CrossRef]

15. A. F. Fercher, W. Drexler, C. K. Hitzenberger, and T. Lasser, “Optical coherence tomography—principles and applications,” Rep. Prog. Phys. **66**, 239–303 (2003). [CrossRef]

16. W. Drexler, U. Morgner, R. K. Ghanta, F. X. Kartner, J. S. Schuman, and J. G. Fujimoto, “Ultrahigh-resolution ophthalmic optical coherence tomography,” Nat. Med. **7**, 502–507 (2001). [CrossRef] [PubMed]

15. A. F. Fercher, W. Drexler, C. K. Hitzenberger, and T. Lasser, “Optical coherence tomography—principles and applications,” Rep. Prog. Phys. **66**, 239–303 (2003). [CrossRef]

20. R. L. Forward, “Wideband laser-interferometer gravitational-radiation experiment,” Phys. Rev. D **17**, 379–390 (1978). [CrossRef]

## 2. Theory derivation

23. A. Puglisi, V. Loreto, U. M. B. Marconi, A. Petri, and A. Vulpiani, “Clustering and non-Gaussian behavior in granular matter,” Phys. Rev. Lett. **81**, 3848–3851 (1998). [CrossRef]

24. J. S. Olafsen and J. S. Urbach, “Velocity distributions and density fluctuations in a granular gas,” Phys. Rev. E **60**, 2468–2471 (1999). [CrossRef]

*a*,

_{i}*σ*,

_{i}*ω*and

_{i}*Δϕ*are respectively the amplitude ratio, standard deviation, the center frequency of

_{ij}*i*Gaussian distribution and the phase mismatch between the

^{th}*i*and

^{th}*j*components. The subscript

^{th}*j*represents the

*j*component of the complex conjugate wave. The first and second summations in Eq. (1.2) refer to the DC and interference parts respectively. The following analysis is derived using two Gaussian distributions, but can be generalized utilizing this assumption.

^{th}_{1}and ω

_{2}respectively. The standard deviations of the fields are σ

_{1}and σ

_{2}, respectively. Based on this asymmetrical Gaussian distribution assumption, the correlation function in the interference term should be a cross correlation function rather than an autocorrelation function. The interference term of the two beam interferometer can be written in the frequency domain as,

*ϕ*are the scale constant and the phase mismatch, while the

*S*(

*ω*,

*Δϕ*) is the cross spectral density.

*and the composite center frequency*σ ¯

*are respectively:*ω ¯

*only depends on the two initial standard deviations and is independent of the separation of the two spectra, while the composite center frequency*σ ¯

*depends not only on the separation of the two spectra, but also on their standard deviations. Importantly, the first exponential factor in Eq. (3) describes the attenuation of the correlation function due to difference between the two spectra.*ω ¯

*ϕ*

_{1}(

*ω*) and

*ϕ*

_{2}(

*ω*), accumulated in double-pass as a function of frequency for the reference and sample fields, and the phase mismatch Δ

*ϕ*(

*ω*) can be expressed as,

*l*

_{1}and

*l*

_{2}are the propagation distances in the reference and sample paths, respectively, and

*L*is the distance over which second order dispersion is encountered (assumed to be equal for both). For generality, we keep the dispersion coefficients β

_{1}, β

_{1}′ and β

_{1}″ for the first beam distinct from β

_{2}, β

_{2}′ and β

_{2}″ for the second beam. To derive group and phase delays, we substitute Eq. (4) and Eq. (5) into Eq. (7) and then rewrite it as,

*ω*=

*ω*

_{1}-

*ω*

_{2}, the phase delay Δ

*τ*and the group delay Δ

_{p}*τ*are defined as,

_{g}*β*

_{1}(

*ω*

_{1})=

*β*

_{2}(

*ω*

_{2}) and

*β*′

_{1}(

*ω*

_{1})=

*β*′

_{1}(

*ω*

_{2}) [26]. Through advantageous grouping and redefinition, Eq. (8) may be written in the more convenient form,

*β*″=

*β*″

_{1}(

*ω*

_{1})-

*β*″

_{2}(

*ω*

_{2}) is the mismatch of the group velocity dispersion (GVD), and the effective phase delay Δ

*τ*′

_{p}and the effective group delay Δ

*τ*′

_{g}are defined by,

*τ*′

_{p}=Δ

*τ*and Δ

_{p}*τ*′

_{g}=Δ

*τ*when

_{g}*ω*

_{1}=

*ω*

_{2}and Δ

*β*″=0. Since in general, the media in the two paths are different, Δ

*β*″ is not necessarily zero for

*ω*

_{1}=

*ω*

_{2}.

*A*=1, we have the final formula of the cross spectral density,

*I*should be the contributions of all the wavelengths under the interfering spectra. The integral is only performed over the last two factors in Eq. (14) because the first two factors are constants. After integrating for Eq. (14) over frequency, we obtain the modified expression for interferogram intensity:

*β*″ extends the envelope.

*β*″=0, we can significantly simplify Eq. (15) to:

*β*″=0, we obtain the effective coherence length

*l*′

*as a function of the two individual field distributions, or of the coherence lengths*

_{c}*l*

_{c1}and

*l*

_{c2}of the reference and sample fields,

*β*″≠0 according to Eq. (16).

## 3. Experiments

27. A. M. Rollins, M. D. Kulkarni, S. Yazdanfar, R. Un-Arunyawee, and J. A. Izatt, “In vivo video rate optical coherence tomography,” Opt. Express **3**, 219–229 (1998). [CrossRef] [PubMed]

28. Z. Hu and A. M. Rollins, “Quasi-telecentric optical design of a microscope-compatible OCT scanner,” Opt. Express **13**, 6407–6415 (2005). [CrossRef] [PubMed]

## 4. Conclusion

## Acknowledgments

## References and links

1. | A. A. Michelson, “Interferometer,” Am. J. Sci. |

2. | R. S. Shankland, S. W. McCuskey, F. C. Leone, and G. Kuerti, “New analysis of the interferometer obervations of Dayton C. Miller,” Rev. Mod. Phys. |

3. | R. P. Patten, “Michelson interferometer as a remote gauge,” Appl. Opt. |

4. | P. Becker, K. Dorenwendt, G. Ebeling, R. Lauer, W. Lucas, R. Probst, H.-J. Rademacher, G. Reim, P. Seyfried, and H. Siegert, “Absolute measurement of the (200) lattice plane spacing in a Silicon Crystal,” Phys. Rev. Lett. |

5. | M. S. Chapman, C. R. Ekstrom, T. D. Hammond, R. A. Rubenstein, J. Schmiedmayer, S. Wehinger, and D. E. Pritchard, “Optics and interferometry with Na |

6. | T. Fuji, M. Arakawa, T. Hattori, and H. Nakatsukaa, “A white-light Michelson interferometer in the visible and near infrared regions,” Rev. Sci. Instrum. |

7. | K. McKenzie, D. A. Shaddock, D. E. McClelland, B. C. Buchler, and P. K. Lam, “Experimental demonstration of a squeezing-enhanced power-recycled Michelson interferometer for gravitational wave detection,” Phys. Rev. Lett. |

8. | W. Marshall, C. Simon, R. Penrose, and D. Bouwmeester, “Towards Quantum superpositions of a mirror,” Phys. Rev. Lett. |

9. | D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical Coherence Tomography,” Science |

10. | M. C. Booth, G. D. Giuseppe, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Polarization-sensitive quantum-optical coherence tomography,” Phys. Rev. A |

11. | D. J. Faber, M. C. G. Aalders, E. G. Mik, B. A. Hooper, M. J. C. v. Gemert, and T. G. v. Leeuwen, “Oxygen saturation-dependent absorption and scattering of blood,” Phys. Rev. Lett. |

12. | D. L. Marks and S. A. Boppart, “Nonlinear interferometric vibrational imaging,” Phys. Rev. Lett. |

13. | R. A. Leitgeb, L. Schmetterer, C. K. Hitzenberger, A. F. Fercher, F. Berisha, M. Wojtkowski, and T. Bajraszewski, “Real-time measurement of in vitro flow by Fourier-domain color Doppler optical coherence tomography,” Opt. Lett. |

14. | J. Zhang, J. S. Nelson, and Z. Chen, “Removal of a mirror image and enhancement of the signal-to-noise ratio in Fourier-domain optical coherence tomography using an electro-optic phase modulator,” Opt. Lett. |

15. | A. F. Fercher, W. Drexler, C. K. Hitzenberger, and T. Lasser, “Optical coherence tomography—principles and applications,” Rep. Prog. Phys. |

16. | W. Drexler, U. Morgner, R. K. Ghanta, F. X. Kartner, J. S. Schuman, and J. G. Fujimoto, “Ultrahigh-resolution ophthalmic optical coherence tomography,” Nat. Med. |

17. | I. Hartl, X. Li, C. Chudoba, R. K. Ghanta, T. H. Ko, J. G. Fujimoto, J. K. Ranka, and R. S. Windeler, “Ultrahigh-resolution optical coherence tomography using continuum generation in an air-silica microstructure optical fiber,” Opt. Lett. |

18. | Y. Wang, Y. Zhao, J. S. Nelson, Z. Chen, and R. S. Windeler, “Ultrahigh-resolution optical coherence tomography by broadband continuum generation from a photonic crystal fiber,” Opt. Lett. |

19. | N. Nishizawa, Y. Chen, P. Hsiung, E. P. Ippen, and J. G. Fujimoto, “Real-time, ultrahigh-resolution, optical coherence tomography with an all-fiber, femtosecond fiber laser continuum at 1.5 mm,” Opt. Lett. |

20. | R. L. Forward, “Wideband laser-interferometer gravitational-radiation experiment,” Phys. Rev. D |

21. | W. Koechner, |

22. | K. J. Kuhn, |

23. | A. Puglisi, V. Loreto, U. M. B. Marconi, A. Petri, and A. Vulpiani, “Clustering and non-Gaussian behavior in granular matter,” Phys. Rev. Lett. |

24. | J. S. Olafsen and J. S. Urbach, “Velocity distributions and density fluctuations in a granular gas,” Phys. Rev. E |

25. | A. E. Siegman, |

26. | B. E. Bouma and G. J. Tearney, |

27. | A. M. Rollins, M. D. Kulkarni, S. Yazdanfar, R. Un-Arunyawee, and J. A. Izatt, “In vivo video rate optical coherence tomography,” Opt. Express |

28. | Z. Hu and A. M. Rollins, “Quasi-telecentric optical design of a microscope-compatible OCT scanner,” Opt. Express |

**OCIS Codes**

(030.1640) Coherence and statistical optics : Coherence

(110.4500) Imaging systems : Optical coherence tomography

(120.3180) Instrumentation, measurement, and metrology : Interferometry

(170.3890) Medical optics and biotechnology : Medical optics instrumentation

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: October 4, 2006

Revised Manuscript: December 14, 2006

Manuscript Accepted: December 14, 2006

Published: December 22, 2006

**Virtual Issues**

Vol. 2, Iss. 1 *Virtual Journal for Biomedical Optics*

**Citation**

Zhilin Hu and Andrew M. Rollins, "Theory of two beam interference with arbitrary spectra," Opt. Express **14**, 12751-12759 (2006)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-14-26-12751

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### References

- A. A. Michelson, "Interferometer," Am. J. Sci. 3, 120 (1881).
- R. S. Shankland, S. W. McCuskey, F. C. Leone, and G. Kuerti, "New analysis of the interferometer obervations of Dayton C. Miller," Rev. Mod. Phys. 27, 167-178 (1955). [CrossRef]
- R. P. Patten, "Michelson interferometer as a remote gauge," Appl. Opt. 10, 2717-2721 (1971). [CrossRef] [PubMed]
- P. Becker, K. Dorenwendt, G. Ebeling, R. Lauer, W. Lucas, R. Probst, H.-J. Rademacher, G. Reim, P. Seyfried, and H. Siegert, "Absolute measurement of the (200) lattice plane spacing in a Silicon Crystal," Phys. Rev. Lett. 46, 1540-1543 (1981). [CrossRef]
- M. S. Chapman, C. R. Ekstrom, T. D. Hammond, R. A. Rubenstein, J. Schmiedmayer, S. Wehinger, and D. E. Pritchard, "Optics and interferometry with Na2 Molecules," Phys. Rev. Lett. 74, 4783-4786 (1995). [CrossRef] [PubMed]
- T. Fuji, M. Arakawa, T. Hattori, and H. Nakatsukaa, "A white-light Michelson interferometer in the visible and near infrared regions," Rev. Sci. Instrum. 69, 2854-2858 (1998). [CrossRef]
- K. McKenzie, D. A. Shaddock, D. E. McClelland, B. C. Buchler, and P. K. Lam, "Experimental demonstration of a squeezing-enhanced power-recycled Michelson interferometer for gravitational wave detection," Phys. Rev. Lett. 88, 231102-231101 (2002). [CrossRef] [PubMed]
- W. Marshall, C. Simon, R. Penrose, and D. Bouwmeester, "Towards Quantum superpositions of a mirror," Phys. Rev. Lett. 91, 130401-130404 (2003). [CrossRef] [PubMed]
- D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, "Optical Coherence Tomography," Science 254, 1178-1181 (1991). [CrossRef] [PubMed]
- M. C. Booth, G. D. Giuseppe, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, "Polarization-sensitive quantum-optical coherence tomography," Phys. Rev. A 69, 043815- (2004). [CrossRef]
- D. J. Faber, M. C. G. Aalders, E. G. Mik, B. A. Hooper, M. J. C. v. Gemert, and T. G. v. Leeuwen, "Oxygen saturation-dependent absorption and scattering of blood," Phys. Rev. Lett. 93, 028102 (2004). [CrossRef] [PubMed]
- D. L. Marks, and S. A. Boppart, "Nonlinear interferometric vibrational imaging," Phys. Rev. Lett. 92, 123905 (2004). [CrossRef] [PubMed]
- R. A. Leitgeb, L. Schmetterer, C. K. Hitzenberger, A. F. Fercher, F. Berisha, M. Wojtkowski, and T. Bajraszewski, "Real-time measurement of in vitro flow by Fourier-domain color Doppler optical coherence tomography," Opt. Lett. 29, 171-173 (2004). [CrossRef] [PubMed]
- J. Zhang, J. S. Nelson, and Z. Chen, "Removal of a mirror image and enhancement of the signal-to-noise ratio in Fourier-domain optical coherence tomography using an electro-optic phase modulator," Opt. Lett. 30, 147-149 (2005). [CrossRef] [PubMed]
- A. F. Fercher, W. Drexler, C. K. Hitzenberger, and T. Lasser, "Optical coherence tomography—principles and applications," Rep. Prog. Phys. 66, 239-303 (2003). [CrossRef]
- W. Drexler, U. Morgner, R. K. Ghanta, F. X. Kartner, J. S. Schuman, and J. G. Fujimoto, "Ultrahigh-resolution ophthalmic optical coherence tomography," Nat. Med. 7, 502-507 (2001). [CrossRef] [PubMed]
- I. Hartl, X. Li, C. Chudoba, R. K. Ghanta, T. H. Ko, J. G. Fujimoto, J. K. Ranka, and R. S. Windeler, "Ultrahigh-resolution optical coherence tomography using continuum generation in an air-silica microstructure optical fiber," Opt. Lett. 26, 608-610 (2001). [CrossRef]
- Y. Wang, Y. Zhao, J. S. Nelson, Z. Chen, and R. S. Windeler, "Ultrahigh-resolution optical coherence tomography by broadband continuum generation from a photonic crystal fiber," Opt. Lett. 28, 182-184 (2003). [CrossRef] [PubMed]
- N. Nishizawa, Y. Chen, P. Hsiung, E. P. Ippen, and J. G. Fujimoto, "Real-time, ultrahigh-resolution, optical coherence tomography with an all-fiber, femtosecond fiber laser continuum at 1.5 mm," Opt. Lett. 29, 2846-2848 (2004). [CrossRef]
- R. L. Forward, "Wideband laser-interferometer gravitational-radiation experiment," Phys. Rev. D 17, 379-390 (1978). [CrossRef]
- W. Koechner, Solid-State Laser Engineering (Springer, New York, 1999).
- K. J. Kuhn, Laser Engeering (Prentice-Hall, Inc., Upper Saddle River, 1998).
- A. Puglisi, V. Loreto, U. M. B. Marconi, A. Petri, and A. Vulpiani, "Clustering and non-Gaussian behavior in granular matter," Phys. Rev. Lett. 81, 3848-3851 (1998). [CrossRef]
- J. S. Olafsen, and J. S. Urbach, "Velocity distributions and density fluctuations in a granular gas," Phys. Rev. E 60, 2468-2471 (1999). [CrossRef]
- A. E. Siegman, Laser (1986).
- B. E. Bouma, and G. J. Tearney, Handbook of optical coherence tomography (Marcel Dekker, New York, 2002).
- A. M. Rollins, M. D. Kulkarni, S. Yazdanfar, R. Un-Arunyawee, and J. A. Izatt, "In vivo video rate optical coherence tomography," Opt. Express 3, 219-229 (1998). [CrossRef] [PubMed]
- Z. Hu, and A. M. Rollins, "Quasi-telecentric optical design of a microscope-compatible OCT scanner," Opt. Express 13, 6407-6415 (2005). [CrossRef] [PubMed]

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