## Time-frequency distribution of interferograms from a frequency comb in dispersive media |

Optics Express, Vol. 19, Issue 4, pp. 3406-3417 (2011)

http://dx.doi.org/10.1364/OE.19.003406

Acrobat PDF (1482 KB)

### Abstract

We investigate general properties of the interferograms from a frequency comb laser in a non-linear dispersive medium. The focus is on interferograms at large delay distances and in particular on their broadening, the fringe formation and shape. It is observed that at large delay distances the interferograms spread linearly and that its shape is determined by the source spectral profile. It is also shown that each intensity point of the interferogram is formed by the contribution of one dominant stationary frequency. This stationary frequency is seen to vary as a function of the path length difference even within the interferogram. We also show that the contributing stationary frequency remains constant if the evolution of a particular fringe is followed in the successive interferograms found periodically at different path length differences. This can be used to measure very large distances in dispersive media.

© 2011 Optical Society of America

## 1. Introduction

1. D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff,“Carrier-Envelope Phase Control of Femtosecond Mode-Locked Lasers and Direct Optical Frequency Synthesis,” Science **288**, 635–639 (2000). [CrossRef] [PubMed]

7. J. Ye, H. Schnatz, and L. W. Hollberg, “Optical frequency combs: from frequency metrology to optical phase control,” IEEE J. Sel. Top. Quantum Electron. **9**, 1041–1058 (2003). [CrossRef]

8. J. Ye, “Absolute measurement of a long, arbitrary distance to less than an optical fringe,” Opt. Lett. **29**, 1153–1155 (2004). [CrossRef] [PubMed]

15. J. Lee, Y.-J. Kim, K. Lee, S. Lee, and S. Kim, “Time-of-flight measurement with femtosecond light pulses,” Nature Photon. **4**, 716–720 (2010). [CrossRef]

16. M. G. Zeitouny, M. Cui, N. Bhattacharya, S.A. van den Berg, A. J. E. M. Janssen, and H. P. Urbach, “From a discrete to a continuous model for interpulse interference with a frequency-comb laser,” Phys. Rev. A **82**. 023808 (2010). [CrossRef]

## 2. Cross-correlation in dispersive media

*x*-direction. A detailed description of our work can be found in [16

16. M. G. Zeitouny, M. Cui, N. Bhattacharya, S.A. van den Berg, A. J. E. M. Janssen, and H. P. Urbach, “From a discrete to a continuous model for interpulse interference with a frequency-comb laser,” Phys. Rev. A **82**. 023808 (2010). [CrossRef]

*ω*=

_{m}*mω*+

_{r}*ω*

_{0},

*m*= 0, 1, 2,... where

*ω*

_{0}is the common offset frequency and

*ω*is the repetition frequency

_{r}*f*expressed in angular notation

_{r}*l*is the cavity length,

_{pp}*c*is the velocity of light in the medium of the cavity and

*T*is the time distance between the pulses. The offset frequency

_{r}*ω*

_{0}is caused by the difference between the group velocity and the phase velocity inside the laser cavity. Both

*ω*

_{0}and

*ω*are stabilized to an atomic clock in most laboratories.

_{r}16. M. G. Zeitouny, M. Cui, N. Bhattacharya, S.A. van den Berg, A. J. E. M. Janssen, and H. P. Urbach, “From a discrete to a continuous model for interpulse interference with a frequency-comb laser,” Phys. Rev. A **82**. 023808 (2010). [CrossRef]

*X*) is the cross-correlation as a function of the delay distance (

*X*),

*S*(

*mω*+

_{r}*ω*

_{0}) ≡ Power Spectral Density (PSD) and

*n*() is the refractive index of the dispersive medium. Equation (1), which is a discrete representation of the correlation function, shows good agreement with the experimental results but is unable to provide a physical explanation of some properties of the cross-correlation function such as the the shift of the position of the maximum coherence, broadening effects and shape of the cross-correlation function. A continuous model was therefore developed for a better understanding of the problem.

**82**. 023808 (2010). [CrossRef]

*X*in

*h*is a parameter denoting the delay distance. This analysis is in the regime where inspite of the broadening the extent of the cross-correlation function is still smaller than the laser cavity length or the interpulse distance. This series expression, Eq. (2), for Γ(

_{X}*X*) reduces to at most a single term when

*h*(

_{X}*t*) has a support length ≤

*T*. The limitation posed by this for the case of propagation in air is discussed in a subsequent section. From a physical point of view, the integer

_{r}*ℓ*denotes the multiple of the laser cavity length at a given delay distance

*X*and

*ℓT*is the propagation time of a pulse in “vacuum”. In the case where

_{r}*X*> 0 the integer

*ℓ*must be negative. Defining the time variable (

*t*) as allows us to write the cross-correlation function as an integral given by, see [16

**82**. 023808 (2010). [CrossRef]

*X*, independent of the laser cavity length 2

*πc*/

*ω*, a cross-correlation pattern

_{r}*h*(

_{X}*t*) can be obtained by varying the time delay (

*t*) where −

*T*/2 ≤

_{r}*t*≤

*T*/2. In practice, this time (

_{r}*t*) is obtained by setting up a scanning short arm of one laser cavity length. For numerical comparison, both

*h*(

_{X}*t*) and Γ(

*X*), are normalised to unity.

*k*(

*ω*+

*ω*

_{0}) =

*αω*

^{2}+

*βω*+

*γ*for large delay distances, we can write Here,

*f*(

_{X}*t*) is defined as

*R*and

*θ*are radial and angular parameters which are functions of

*f*(

_{X}*t*) [16

**82**. 023808 (2010). [CrossRef]

*ζ*(

*X*) by Using this function we have defined the Non-linear Dispersion Depth (𝒟) of a given pulse in a particular dispersive medium as Equation (7) gives an indication of the effective distance of non-linear dispersion effects on cross-correlations obtained from a light source having a coherence time

*τ*and a carrier frequency

_{c}*ω*propagating in a refractive medium with a group delay dispersion

_{c}*α*at

*ω*. Short delay and large delay distances can be defined using 𝒟.

_{c}### 2.1. Typical characteristics of cross-correlations at large delay distances

**82**. 023808 (2010). [CrossRef]

13. M. Cui, M. G. Zeitouny, N. Bhattacharya, S. A. van den Berg, H. P. Urbach, and J. J. M. Braat, “High-accuracy long-distance measurements in air with a frequency comb laser,” Opt. Lett. **34**, 1982–1984 (2009). [CrossRef] [PubMed]

*ω*= 2.3254 × 10

_{c}^{15}rad/s, corresponding to a wavelength of 810 nm in vacuum, the bandwidth is typically Δ

*ω*≈ 5 × 10

^{14}rad/s, which correspond to a pulse width of Δ

*x*≈ 12

*μm*and a pulse duration of 40 fs. The frequency offset is typically

*ω*

_{0}≈ 113 × 10

^{7}rad/s and the repetition frequency is

*ω*= 6.28 × 10

_{r}^{9}rad/s, corresponding to a cavity length

*l*= 30 cm and period

_{pp}*T*≈ 1ns. The frequencies

_{r}*ω*

_{0}and

*ω*are synchronized to a cesium clock. The spectral content of the initial pulse is the main input for the numerical model. Using the corrected updated Edlén’s equation [17

_{r}17. K. P. Birch and M. J. Downs, “Correction to the Updated Edlén Equation for the Refractive Index of Air,” Metrologia **31**, 315–316 (1994). [CrossRef]

^{4}spectral lines fitted to the profile of the spectrum of the laser, are propagated and then recombined to form correlation patterns.

*X*has been taken to be long enough, 120 meters, so that the correlation patterns are observed to be linearly broadened. Three fringes are picked up from this correlation pattern and further analysed. The cross-correlation is shown in Fig. 1.a where the vertical lines indicate the positions of the fringes we investigate. The fringe pattern is obtained by varying the time delay

*t*around a delay distance

*X*= 120 m. At each small delay the frequency content of the interfering fields will be constantly in phase or out of phase depending on

*t*. We have chosen to analyze the cosine of the phase factor from Eq. (4). For each small delay scan one fringe the cosine of the phase function has been plotted. A 3-D plot can be obtained for each analyzed fringe showing the value of the cosine of the phase as a function of the time delay (

*t*) and the frequencies (or the wavelength) of the optical field. These plots are shown in Fig. 1.(b,c,d). The plots show that within one fringe the cosine of the phase function has either fast or slow oscillations as a function of the frequency. Using this oscillating function in Eq. (4), where

*S*(

*ω*+

*ω*

_{0}) is a slowly varying term, the integral transform is small. The integral will approach zero as the number of oscillations increases, Riemann-Lebesgue lemma. Thus this relatively high oscillatory part will have a minor contribution to the formation of the interference fringe. Only when the cosine of the phase function is slowly varying the contribution to the integral will be important. The plots analyzing the fringes show that the slowly varying part is always at a specific frequency for a particular fringe. In Fig. 1.(e,f,g) we plot the cosine of the phase function for each fringe as a function of the frequency. Five such samples at different intensity points of the fringe are plotted on top of each other. It is clearly seen that, for a given fringe, the position of the stationary frequency remains constant on the frequency-axis and changes only in value when

*t*changes. Between different fringes, the stationary frequency varies, depending on the fringe position in the correlation pattern.

### 2.2. Limit of the validity of the continuous model in standard air

*X*) contains one significant term only. This is the case when the length of the interval outside which

*h*is negligible does not exceed

_{X}*T*≈ 10

_{r}^{−9}s. We rewrite Eq. (4) as where represents the deviation of the refractive index

*n*from being constant. For

*X*= 0, we have and so, by Fourier inversion, Inserting Eq. (11) into Eq. (8), we get where

*ω*can be replaced by the essential integration range

*I*which is, in accordance with the values in Section 2.1 for the bandwidth Δ

*ω*and center frequency

*ω*, contained in the interval [2 × 10

_{c}^{15},2.6 × 10

^{15}]. Hence, instead of

*g*above, we consider in Eq. (12). Thus

_{X}*h*is obtained as the real part of the convolution of

_{X}*h*

_{X}_{=0}and

*h*

_{X}_{=0}is concentrated on a time interval of length, see Section 2.1, Δ

*x*/

*c*≈ 4 × 10

^{−14}s ≪ 10

^{−9}s = T

_{r}. So, the length of the time interval where

*h*is concentrated is chiefly determined by the set of points

_{X}*s*for which

*g*(

_{X,I}*s*) is non-negligible.

*s*where

*g*(

_{X,I}*s*) of Eq. 13 is non-negligible can be obtained by appealing to the stationary phase principle. Accordingly, the integral in Eq. (13) is non-negligible when the phase of the integrand has a stationary point

*ω*inside the integration range

*I*, and is expected to be small otherwise. The condition for

*ω*to be a stationary point is In order that Eq. (14) can hold for an

*ω*∈

*I*, we need that |

*s*| is at most of the order

*n*, see Eq. (9), given by Edlén’s equation [17

17. K. P. Birch and M. J. Downs, “Correction to the Updated Edlén Equation for the Refractive Index of Air,” Metrologia **31**, 315–316 (1994). [CrossRef]

*M*has order of magnitude 10

_{ξ},_{I}^{−7}. With

*n̄*= 1.00027,

*c*≈ 3 × 10

^{8}m/s, we see that the length of the interval of points

*s*where Eq. (14) holds is less than

*T*= 10

_{r}^{−9}s when

*X*≤ 1.5 × 10

^{6}m.

*g*is nonnegligible, but for the relevant cases that

_{X,I}*X*≤ 1000 km it seems that we can safely assume that the series in Eq. (2), that uses samples of

*h*at distances

_{X}*T*apart, has only one significant term. In any case, these distances are beyond the coherence length of present day laser systems but maybe are relevant in the future.

_{r}## 3. Stationary phase approximation of the cross-correlation function

**82**. 023808 (2010). [CrossRef]

*S*(

*ω*+

*ω*

_{0}), multiplied by the cosine of a phase factor. The stationary phase method is an appropriate mathematical tool to study the asymptotic behaviour of such integrals [18]. We rewrite Eq. (4) as where

*k*(

*ω*+

*ω*

_{0}) = (

*ω*+

*ω*

_{0})

*n*(

*ω*+

*ω*

_{0})/

*c*. The phase

*ϕ*(

*ω*) is stationary when

*ω*(

_{dom}*X,t*) for a given

*X*and

*t*, where

*k*

^{(1)}denotes the first derivative of the

*k*(

*ω*) vector. If we expand

*ϕ*(

*ω*) in a Taylor series about

*ω*and neglect terms of order higher than (

_{dom}*ω*−

*ω*)

_{dom}^{2}, we obtain

*C*

_{1},

*C*

_{2}are given below. With

*σ*= sign[

*ϕ*

^{(2)}(

*ω*)] and

_{dom}*ϕ*

^{(n)},

*S*

^{(n)}are the

*n*derivative of

^{th}*ϕ*,

*S*evaluated at the frequency

*ω*, the first two coefficients of the asymptotic series are given by [18] and

_{dom}*a*and

*b*are real, we can write

*a*cos

*x*−

*b*sin

*x*= ℛ cos(

*x*+ ϑ). In this case ℛ and ϑ are given by

19. K. E. Oughstun and N. A. Cartwright, “Physical significance of the group velocity in dispersive, ultrashort gaussian pulse dynamics,” J. Mod. Opt. **52**, 1089–1104 (2005 )and references therein. [CrossRef]

*S*(

*ω*+

*ω*

_{0}). This is in agreement with the previous experimental and numerical simulation [16

**82**. 023808 (2010). [CrossRef]

*ω*. Thus, with increasing delay distance the cross-correlation functions will spread linearly.

_{dom}^{5}frequencies. By using a scanning short arm with 10

^{4}steps, one has to compute a grid of 10

^{5}× 10

^{4}elements in order to obtain the final results. This can be simply reduced to a vector of 10

^{4}components by using the cross-correlation equations from the stationary phase method since we can associate one dominant frequency to each scanning step.

## 4. Stationary Phase Absolute Distance Metrology

*S*(

*ω*+

_{dom}*ω*

_{0}), at the dominant frequencies

*ω*. The envelope of the cross-correlation from Eq. (5) is given by the following integral

_{dom}*X*but at two different scanning positions of the short scanning arm,

*t*

_{1}and

*t*

_{2}, respectively. From the expression for

*ω*(

_{p}*X,t*) we obtain The time lag (Δ

*t*=

*t*

_{2}−

*t*

_{1}) can be measured by simply noting the piezo displacement (Δ

*x*), where

## 5. Conclusion

## Acknowledgments

## References and links

1. | D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff,“Carrier-Envelope Phase Control of Femtosecond Mode-Locked Lasers and Direct Optical Frequency Synthesis,” Science |

2. | R. Holzwarth, Th. Udem, T.W. Hänsch, J. C. Knight, W. J. Wadsworth, and P. St. J. Russell, “Optical Frequency Synthesizer for Precision Spectroscopy,” |

3. | S. T. Cundiff and J. Ye, “Colloquium: Femtosecond optical frequency combs,” Rev. Mod. Phys. |

4. | S. A. Diddams, J. C. Bergquist, S. R. Jefferts, and C. W. Oates, “Standards of Time and Frequency at the Outset of the 21st Century,” Science |

5. | Th. Udem, R. Holzwarth, and T. W. Hänsch, “Optical frequency metrology,” Nature |

6. | L. Hollberg, C. W. Oates, E. A. Curtis, E. N. Ivanov, S. A. Diddams, T. Udem, H. G. Robinson, J. C. Bergquist, R. J. Rafac, W. M. Itano, R. E. Drullinger, and D. J. Wineland, “Optical frequency standards and measurements,” IEEE J. Quantum Electron. |

7. | J. Ye, H. Schnatz, and L. W. Hollberg, “Optical frequency combs: from frequency metrology to optical phase control,” IEEE J. Sel. Top. Quantum Electron. |

8. | J. Ye, “Absolute measurement of a long, arbitrary distance to less than an optical fringe,” Opt. Lett. |

9. | M. Cui, R. N. Schouten, N. Bhattacharya, and S. A. van den Berg, “Experimental demonstration of distance measurement with a femtosecond frequency comb laser,” |

10. | Y. Salvade, N. Schuhler, S. Leveque, and S. Le Floch, “High-accuracy absolute distance measurement using frequency comb referenced multiwavelength source,” Appl. Opt. |

11. | K. -N. Joo, Y. Kim, and S. -W. Kim, “Distance measurements by combined method based on a femtosecond pulse laser,” Opt. Express |

12. | P. Balling, P. Křen, P. Mašika, and S. A. van den Berg, “Femtosecond frequency comb based distance measurement in air,” Opt. Express |

13. | M. Cui, M. G. Zeitouny, N. Bhattacharya, S. A. van den Berg, H. P. Urbach, and J. J. M. Braat, “High-accuracy long-distance measurements in air with a frequency comb laser,” Opt. Lett. |

14. | I. Coddington, W. C. Swann, L. Nenadovic, and N. R. Newbury, “Rapid and precise absolute distance measurements at long range,” Nature Photon. |

15. | J. Lee, Y.-J. Kim, K. Lee, S. Lee, and S. Kim, “Time-of-flight measurement with femtosecond light pulses,” Nature Photon. |

16. | M. G. Zeitouny, M. Cui, N. Bhattacharya, S.A. van den Berg, A. J. E. M. Janssen, and H. P. Urbach, “From a discrete to a continuous model for interpulse interference with a frequency-comb laser,” Phys. Rev. A |

17. | K. P. Birch and M. J. Downs, “Correction to the Updated Edlén Equation for the Refractive Index of Air,” Metrologia |

18. | V. A. Borovikov, |

19. | K. E. Oughstun and N. A. Cartwright, “Physical significance of the group velocity in dispersive, ultrashort gaussian pulse dynamics,” J. Mod. Opt. |

**OCIS Codes**

(070.4550) Fourier optics and signal processing : Correlators

(070.4790) Fourier optics and signal processing : Spectrum analysis

(120.3180) Instrumentation, measurement, and metrology : Interferometry

(320.1590) Ultrafast optics : Chirping

(320.7150) Ultrafast optics : Ultrafast spectroscopy

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: October 11, 2010

Revised Manuscript: December 11, 2010

Manuscript Accepted: December 13, 2010

Published: February 7, 2011

**Citation**

M. G. Zeitouny, M. Cui, A. J. Janssen, N. Bhattacharya, S. A. van den Berg, and H. P. Urbach, "Time-frequency distribution of interferograms from a frequency comb in dispersive media," Opt. Express **19**, 3406-3417 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-4-3406

Sort: Year | Journal | Reset

### References

- D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, “Carrier-Envelope Phase Control of Femtosecond Mode-Locked Lasers and Direct Optical Frequency Synthesis,” Science 288, 635–639 (2000). [CrossRef] [PubMed]
- R. Holzwarth, Th. Udem, T. W. H¨ansch, J. C. Knight, W. J. Wadsworth, and P. St. J. Russell, “Optical Frequency Synthesizer for Precision Spectroscopy,” Phys. Rev. Lett. 85, 2264–2267 (2000). [CrossRef] [PubMed]
- S. T. Cundiff, and J. Ye, “Colloquium: Femtosecond optical frequency combs,” Rev. Mod. Phys. 75, 325–342 (2003). [CrossRef]
- S. A. Diddams, J. C. Bergquist, S. R. Jefferts, and C. W. Oates, “Standards of Time and Frequency at the Outset of the 21st Century,” Science 306, 1318–1324 (2004). [CrossRef] [PubMed]
- Th. Udem, R. Holzwarth, and T. W. H¨ansch, “Optical frequency metrology,” Nature 416, 233–237 (2002). [CrossRef] [PubMed]
- L. Hollberg, C. W. Oates, E. A. Curtis, E. N. Ivanov, S. A. Diddams, T. Udem, H. G. Robinson, J. C. Bergquist, R. J. Rafac, W. M. Itano, R. E. Drullinger, and D. J. Wineland, “Optical frequency standards and measurements,” IEEE J. Quantum Electron. 37, 1502–1513 (2001). [CrossRef]
- J. Ye, H. Schnatz, and L. W. Hollberg, “Optical frequency combs: from frequency metrology to optical phase control,” IEEE J. Sel. Top. Quantum Electron. 9, 1041–1058 (2003). [CrossRef]
- J. Ye, “Absolute measurement of a long, arbitrary distance to less than an optical fringe,” Opt. Lett. 29, 1153–1155 (2004). [CrossRef] [PubMed]
- M. Cui, R. N. Schouten, N. Bhattacharya, and S. A. van den Berg, “Experimental demonstration of distance measurement with a femtosecond frequency comb laser,” J. Eur. Opt. Soc. Rapid Publ. 3, 08003 (2008). [CrossRef]
- Y. Salvade, N. Schuhler, S. Leveque, and S. Le Floch, “High-accuracy absolute distance measurement using frequency comb referenced multiwavelength source,” Appl. Opt. 47, 2715–2720 (2008) (and references therein). [CrossRef] [PubMed]
- K.-N. Joo, Y. Kim, and S.-W. Kim, “Distance measurements by combined method based on a femtosecond pulse laser,” Opt. Express 16, 19799–19806 (2008) (and references therein). [CrossRef] [PubMed]
- P. Balling, P. Křen, P. Mašika, and S. A. van den Berg, “Femtosecond frequency comb based distance measurement in air,” Opt. Express 17, 9300–9313 (2009). [CrossRef] [PubMed]
- M. Cui, M. G. Zeitouny, N. Bhattacharya, S. A. van den Berg, H. P. Urbach, and J. J. M. Braat, “High-accuracy long-distance measurements in air with a frequency comb laser,” Opt. Lett. 34, 1982–1984 (2009). [CrossRef] [PubMed]
- I. Coddington, W. C. Swann, L. Nenadovic, and N. R. Newbury, “Rapid and precise absolute distance measurements at long range,” Nat. Photonics 3, 351–356 (2009). [CrossRef]
- J. Lee, Y.-J. Kim, K. Lee, S. Lee, and S. Kim, “Time-of-flight measurement with femtosecond light pulses,” Nat. Photonics 4, 716–720 (2010). [CrossRef]
- M. G. Zeitouny, M. Cui, N. Bhattacharya, S. A. van den Berg, A. J. E. M. Janssen, and H. P. Urbach, “From a discrete to a continuous model for interpulse interference with a frequency-comb laser,” Phys. Rev. A 82, 023808 (2010). [CrossRef]
- K. P. Birch, and M. J. Downs, “Correction to the Updated Edl’en Equation for the Refractive Index of Air,” Metrologia 31, 315–316 (1994). [CrossRef]
- V. A. Borovikov, Uniform Stationary Phase Method, IEE Electromagnetic Wave Series (1994).
- K. E. Oughstun, and N. A. Cartwright, “Physical significance of the group velocity in dispersive, ultrashort Gaussian pulse dynamics,” J. Mod. Opt. 52, 1089–1104 (2005) (and references therein). [CrossRef]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.