## High-visibility interference fringes with femtosecond laser radiation

Optics Express, Vol. 17, Issue 25, pp. 23016-23024 (2009)

http://dx.doi.org/10.1364/OE.17.023016

Acrobat PDF (312 KB)

### Abstract

We propose and experimentally demonstrate an interferometer for femtosecond pulses with spectral bandwidth about 100 nm. The scheme is based on a Michelson interferometer with a dispersion compensating module. A diffractive lens serves the purpose of equalizing the optical-path-length difference for a wide range of frequencies. In this way, it is possible to register high-contrast interference fringes with micrometric resolution over the whole area of a commercial CCD sensor for broadband femtosecond pulses.

© 2009 OSA

## 1. Introduction

1. A. F. Fercher, W. Drexler, C. K. Hitzenberger, and T. Lasser, “Optical coherence tomography-principles and applications,” Rep. Prog. Phys. **66**(2), 239–303 (
2003). [CrossRef]

_{o}and

*Δ*λ the center wavelength and the bandwidth of the source respectively. Optical coherence tomography (OCT) takes advantage of low-coherence interferometry to provide cross-section images of biological tissue. OCT images can be recorded either point by point through a fast two-dimensional scanning system or using wide-field techniques, which provide en-face

*(XY)*tomographic images without transverse scanning [2

2. L. Vabre, A. Dubois, and A. C. Boccara, “Thermal-light full-field optical coherence tomography,” Opt. Lett. **27**(7), 530–532 (
2002). [CrossRef] [PubMed]

3. E. Cuche, P. Poscio, and C. Depeursinge, “Optical tomography by means of a numerical low-coherence holographic technique,” J. Opt. **28**(6), 260–264 (
1997). [CrossRef]

4. L. Martínez-León, G. Pedrini, and W. Osten, “Applications of short-coherence digital holography in microscopy,” Appl. Opt. **44**(19), 3977–3984 (
2005). [CrossRef] [PubMed]

9. E. N. Leith and G. J. Swanson, “Achromatic interferometers for white-light optical processing and holography,” Appl. Opt. **19**, 638–644 (
1980). [CrossRef] [PubMed]

6. A. A. Maznev, T. F. Crimmins, and K. A. Nelson, “How to make femtosecond pulses overlap,” Opt. Lett. **23**(17), 1378–1380 (
1998). [CrossRef] [PubMed]

10. J. Hebling, I. Z. Kozma, and J. Kuhl, “Compact high-aperture optical setup for excitation of dynamic gratings by ultrashort light pulses,” J. Opt. Soc. Am. B **17**(10), 1803–1805 (
2000). [CrossRef]

7. Z. Ansari, Y. Gu, M. Tziraki, R. Jones, P. M. W. French, D. D. Nolte, and M. R. Melloch, “Elimination of beam walk-off in low-coherence off-axis photorefractive holography,” Opt. Lett. **26**(6), 334–336 (
2001). [CrossRef] [PubMed]

## 2. Frequency-domain analysis of interference of short pulses traveling through dispersive media

_{o},

*a*(

*t*). The superposition of the pulse with a time-delayed and a spatially tilted replica of itself can be implemented with the help of a Michelson interferometer. When the pulses overlap, for slow detection compared with the pulse duration, the time-integrated intensity of their sum iswhere Re designs the real part and

*∆Φ(*ω + ω

_{o}

*)*is the phase difference between the object and the reference pulse for the spectral component with frequency

*(*ω + ω

_{o}

*)*at the superposition point.

*∆Φ(*ω + ω

_{o}

*)*is expanded around the carrier frequency up to second order aswhere

*τ*and

*τ*´ are the first-order and second-order derivatives of the phase difference with respect to frequency and evaluated at the carrier frequency. The above quantities are identified as the differential group delay (DGD) and the differential group delay dispersion (DGDD) between the object and the reference waves for ω

_{o}. We assume the slowly varying envelope approximation and the Taylor series expansion of the phase to be valid. The above assumptions set the ultimate limit for the maximum bandwidth of

*S*(ω) which, in turn, leads to pulse duration of a few cycles of the optical carrier [11

11. T. Brabec and F. Krausz, “Nonlinear optical pulse propagation in the single-cycle regime,” Phys. Rev. Lett. **78**(17), 3282–3285 (
1997). [CrossRef]

12. H. Xiao and K. E. Oughstun, “Failure of the group-velocity description for ultrawideband pulse propagation in causally dispersive, absorptive dielectric,” J. Opt. Soc. Am. B **16**(10), 1773–1785 (
1999). [CrossRef]

*∆*e) and the mirror tilts

*θ*

_{1}and θ

_{2}through the expression

*∆Φ(*ω

*) = [2∆*e

*+*x

_{S}(

*sinθ*-

_{2}*sinθ*] ω/c. In the above expression

_{1})*c*denotes the speed of light and x

_{S}the transverse coordinate at the sensor plane. Note that

*∆Φ(*ω

*)*is linear with frequency so that

*τ*´ =

*0*. The two usual cases

*θ*and

_{1}= θ_{2}*∆*e

*= 0*can be obtained from the above result. For the interference of parallel plane waves, we recover the interferogram trace

*τ*

*= 2Δ*e/c. Also, the same trace holds when

*θ*≠

_{1}*θ*and

_{2}*∆*e

*= 0*but now the interference pattern is codified in the tranverse coordinate x

_{S}at the output plane through the relationship

*τ*

*=*x

_{S}

*(sinθ*-

_{2}*sinθ*/c.

_{1})*/*

_{o}*Δ*ω

*=*λ

*/*

_{o}*Δ*λ with Δω the spectral width [8]. Also the transverse position of the central fringe with the length mismatch is given by x

_{S}

*= 2Δ*e/

*(sinθ*-

_{2}*sinθ*which accounts for the walk-off effect.

_{1})12. H. Xiao and K. E. Oughstun, “Failure of the group-velocity description for ultrawideband pulse propagation in causally dispersive, absorptive dielectric,” J. Opt. Soc. Am. B **16**(10), 1773–1785 (
1999). [CrossRef]

6. A. A. Maznev, T. F. Crimmins, and K. A. Nelson, “How to make femtosecond pulses overlap,” Opt. Lett. **23**(17), 1378–1380 (
1998). [CrossRef] [PubMed]

*∆Φ(*ω

*x*

_{o}) = 4π_{S}/d, with d the grating period, is achieved. In this way,

*τ*

*=*

*0*and

*τ*´

*=*0. This results in a pure cosenoidal interferogram with no fringe modulation

*=*d/

*2*.

## 3. Optical setup design

*Δ*e

*= 0*. Here, we take advantage of the inherent dispersive nature of diffractive optical elements to perform frequency equalization of the optical path length difference (OPLD). Specifically we recall that the focal length

*Z(*ω

*)*of a diffractive lens is a linear function of the wave frequency; i.e.,

*Z(*ω

*)*=

*Z*ω/ω

_{o}*, where*

_{o}*Z*denotes the focal length for the carrier ω

_{o}*. For this case, the value of the phase difference is given bywhere the frequency-dependent OPLD, ΔL*

_{o}*(*ω

*)*is evaluated asIn the above equation the symbols A and B stand for first-row elements of the paraxial ABCD matrix between the source and the observation planes which, indeed, are wavelength dependent. Again, x

_{S}denotes transverse coordinate at the sensor and f is the focal length of the achromatic objective.

*τ*

*=*0 and

*τ*´

*= ΔΦ*

_{o}/*ΔΦ*is given by Eq. (6) for ω

_{o}*=*ω

*after the conditions derived in Eq. (8) are substituted into matrix elements A and B. Also, the value for*

_{o}*τ*´ is derived from the second derivative of Eq. (6) with achromatic compensation. Note that Eq. (5) indicates that the Fourier transform relation between the complex degree of coherence and the spectral energy is substituted by a Fresnel transformation. It is one of the main results of our work as it is the only way the optical design in Fig. 2 can be carried out. In Fig. 3 we plot the visibility function V

*(*x

_{S}

*)*for the interferograms at the sensor for the optical setups in Fig. 1 (solid curve) and 2 (dashed curves) as a function of the normalized sensor coordinate x

_{S}/p. We assume a Gaussian spectrum centered at ω

*and full-width-at-half-maximum (FWHM) width*

_{o}*Δ*ω. For the plot, we choose ω

*ω =*

_{o}= 2.40 10^{14}Hz and Δ*2.44 10*λ

^{13}Hz (i.e.,_{o}= 789 nm and

*Δ*λ =

*81.6 nm)*. It appears that the diffractive lens enhances the sensor region over which interference fringes are observable by at least an order of magnitude. In fact, it can be demonstrated that the gain factor, defined as the ratio between the FWHM width of the function V

*(*x

_{S}

*)*for the interference fringes obtained with the Michelson interferometer with and without DCM, respectively, is given by

*∆Φ(*ω + ω

_{o}

*) and the presence of spatial variations in the input femtosecond beam*. To take into account this effect, we test the goodness of the quadratic approximation in Eq. (3) in the same plot. To this end, visibility is evaluated directly through Eq. (2). Now, the full ω–dependence of the OPLD is retained for the calculation. Results are plotted in short-dashed line. Only a slight deviation is observed for a sensor area covering a huge amount of fringe periods.

## 4. Experimental results

*and*

_{o}*Δ*ω are those used for the plot in Fig. 3. Normally, the pulses are transform-limited to roughly 10 fs. However, the pulse can be pre-chirped so that at the sensor results in a transform-limited signal to increase temporal resolution of the interference as explained in Section 2. Fringes were captured by a CCD camera with a pixel size of 6 μm and a sensor area of 7.68 × 6.14 mm

^{2}. The focal lengths for the refractive achromatic objective

*f*and the diffractive lens

*Z*were chosen as 200 mm and 143 mm, respectively. The diffractive lens was located at the image focal plane of the objective and the sensor was positioned 71.5 mm from the dispersive component. All the above parameters were chosen just to meet achromatic requirements specified through Eq. (8).

_{o}*∆Φ(*ω + ω

_{o}

*) and the presence of spatial variations in the input femtosecond beam*.

## 5. Conclusions

_{S}/p is increased by a factor

## Acknowledgements

## References and Links

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

2. | L. Vabre, A. Dubois, and A. C. Boccara, “Thermal-light full-field optical coherence tomography,” Opt. Lett. |

3. | E. Cuche, P. Poscio, and C. Depeursinge, “Optical tomography by means of a numerical low-coherence holographic technique,” J. Opt. |

4. | L. Martínez-León, G. Pedrini, and W. Osten, “Applications of short-coherence digital holography in microscopy,” Appl. Opt. |

5. | P. Massatsch, F. Charrière, E. Cuche, P. Marquet, and C. D. Depeursinge, “Time-domain optical coherence tomography with digital holographic microscopy,” Appl. Opt. |

6. | A. A. Maznev, T. F. Crimmins, and K. A. Nelson, “How to make femtosecond pulses overlap,” Opt. Lett. |

7. | Z. Ansari, Y. Gu, M. Tziraki, R. Jones, P. M. W. French, D. D. Nolte, and M. R. Melloch, “Elimination of beam walk-off in low-coherence off-axis photorefractive holography,” Opt. Lett. |

8. | B. E. A. Saleh, and M. C. Teich, |

9. | E. N. Leith and G. J. Swanson, “Achromatic interferometers for white-light optical processing and holography,” Appl. Opt. |

10. | J. Hebling, I. Z. Kozma, and J. Kuhl, “Compact high-aperture optical setup for excitation of dynamic gratings by ultrashort light pulses,” J. Opt. Soc. Am. B |

11. | T. Brabec and F. Krausz, “Nonlinear optical pulse propagation in the single-cycle regime,” Phys. Rev. Lett. |

12. | H. Xiao and K. E. Oughstun, “Failure of the group-velocity description for ultrawideband pulse propagation in causally dispersive, absorptive dielectric,” J. Opt. Soc. Am. B |

13. | P. Andrés, J. Lancis, E. E. Sicre, and E. Bonet, “Achromatic Fresnel diffraction patterns,” Opt. Commun. |

14. | J. Lancis, E. E. Sicre, A. Pons, and G. Saavedra, “Achromatic white-light self-imaging phenomenon-an approach using the Wigner distribution function,” J. Mod. Opt. |

15. | G. Mínguez-Vega, O. Mendoza-Yero, M. Fernández-Alonso, P. Andrés, V. Climent, and J. Lancis, “Experimental generation of high-contrast Talbot images with an ultrashort laser pulse,” Opt. Commun. |

**OCIS Codes**

(030.1640) Coherence and statistical optics : Coherence

(260.3160) Physical optics : Interference

(320.5550) Ultrafast optics : Pulses

**ToC Category:**

Ultrafast Optics

**History**

Original Manuscript: June 3, 2009

Revised Manuscript: September 23, 2009

Manuscript Accepted: October 27, 2009

Published: December 2, 2009

**Citation**

Raúl Martínez-Cuenca, Lluís Martínez-León, Jesús Lancis, Gladys Mínguez-Vega, Omel Mendoza-Yero, Enrique Tajahuerce, Pere Clemente, and Pedro Andrés, "High-visibility interference fringes with
femtosecond laser radiation," Opt. Express **17**, 23016-23024 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-25-23016

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### References

- A. F. Fercher, W. Drexler, C. K. Hitzenberger, and T. Lasser, “Optical coherence tomography-principles and applications,” Rep. Prog. Phys. 66(2), 239–303 (2003). [CrossRef]
- L. Vabre, A. Dubois, and A. C. Boccara, “Thermal-light full-field optical coherence tomography,” Opt. Lett. 27(7), 530–532 (2002). [CrossRef] [PubMed]
- E. Cuche, P. Poscio, and C. Depeursinge, “Optical tomography by means of a numerical low-coherence holographic technique,” J. Opt. 28(6), 260–264 (1997). [CrossRef]
- L. Martínez-León, G. Pedrini, and W. Osten, “Applications of short-coherence digital holography in microscopy,” Appl. Opt. 44(19), 3977–3984 (2005). [CrossRef] [PubMed]
- P. Massatsch, F. Charrière, E. Cuche, P. Marquet, and C. D. Depeursinge, “Time-domain optical coherence tomography with digital holographic microscopy,” Appl. Opt. 44(10), 1806–1812 (2005). [CrossRef] [PubMed]
- A. A. Maznev, T. F. Crimmins, and K. A. Nelson, “How to make femtosecond pulses overlap,” Opt. Lett. 23(17), 1378–1380 (1998). [CrossRef] [PubMed]
- Z. Ansari, Y. Gu, M. Tziraki, R. Jones, P. M. W. French, D. D. Nolte, and M. R. Melloch, “Elimination of beam walk-off in low-coherence off-axis photorefractive holography,” Opt. Lett. 26(6), 334–336 (2001). [CrossRef] [PubMed]
- B. E. A. Saleh, and M. C. Teich, Fundamentals of Photonics, 2nd ed. (Wiley, 2007), Chap. 11.
- E. N. Leith and G. J. Swanson, “Achromatic interferometers for white-light optical processing and holography,” Appl. Opt. 19, 638–644 (1980). [CrossRef] [PubMed]
- J. Hebling, I. Z. Kozma, and J. Kuhl, “Compact high-aperture optical setup for excitation of dynamic gratings by ultrashort light pulses,” J. Opt. Soc. Am. B 17(10), 1803–1805 (2000). [CrossRef]
- T. Brabec and F. Krausz, “Nonlinear optical pulse propagation in the single-cycle regime,” Phys. Rev. Lett. 78(17), 3282–3285 (1997). [CrossRef]
- H. Xiao and K. E. Oughstun, “Failure of the group-velocity description for ultrawideband pulse propagation in causally dispersive, absorptive dielectric,” J. Opt. Soc. Am. B 16(10), 1773–1785 (1999). [CrossRef]
- P. Andrés, J. Lancis, E. E. Sicre, and E. Bonet, “Achromatic Fresnel diffraction patterns,” Opt. Commun. 104(1-3), 39–45 (1993). [CrossRef]
- J. Lancis, E. E. Sicre, A. Pons, and G. Saavedra, “Achromatic white-light self-imaging phenomenon-an approach using the Wigner distribution function,” J. Mod. Opt. 42(2), 425–434 (1995). [CrossRef]
- G. Mínguez-Vega, O. Mendoza-Yero, M. Fernández-Alonso, P. Andrés, V. Climent, and J. Lancis, “Experimental generation of high-contrast Talbot images with an ultrashort laser pulse,” Opt. Commun. 281, 374–379 (2008).

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