## Design criteria for Herriott-type multi-pass cavities for ultrashort pulse lasers

Optics Express, Vol. 11, Issue 9, pp. 1106-1113 (2003)

http://dx.doi.org/10.1364/OE.11.001106

Acrobat PDF (140 KB)

### Abstract

We investigate the general characteristics of Herriott-type multipass cavities (MPC) for femtosecond lasers. MPCs can be used to increase the laser pulse energy by extending the laser cavity path length and decreasing the repetition rate, as well as to make standard repetition rate lasers more compact. We present an analytical design condition for MPCs which preserve the Gaussian beam q parameter, enabling the laser path length to be extended while leaving the Kerr-lens modelocking operating point of the cavity invariant. As a specific example, we analyze q preserving MPCs consisting of a flat and curved mirror to obtain analytical expressions for the cavity length. This predicts the optimum MPC designs that minimize the pulse repetition rate for given specifications. These design conditions should prove useful for designing a wide range of high pulse energy or compact femtosecond lasers.

© 2003 Optical Society of America

1. D. R. Herriott, H. Kogelnik, and R. Kompfner, “Off-axis paths in spherical mirror interfometers,” Appl. Opt. **3**, 523–526 (1964). [CrossRef]

2. D. R. Herriott and H. J. Schulte, “Folded optical delay lines,” Appl. Opt. **4**, 883–889 (1965). [CrossRef]

2. D. R. Herriott and H. J. Schulte, “Folded optical delay lines,” Appl. Opt. **4**, 883–889 (1965). [CrossRef]

3. W. R. Trutna and R. L. Byer, “Multiple-pass Raman gain cell,” Appl. Opt. **19**, 301–312 (1980). [CrossRef] [PubMed]

4. B. Perry, R. O. Brickman, A. Stein, E. B Treacy, and P. Rabinowitz, “Controllable pulse compression in a multiple-pass-cell Raman laser,” Opt. Lett. **5**, 288–290 (1980). [CrossRef] [PubMed]

6. P-L Hsiung, X. Li, C. Chuboda, I. Hartl, T. H. Ko, and J. G. Fujimoto, “High-speed path-length scanning with a multiple-pass cavity delay line,” Appl. Opt. **42**, 640–648 (2003). [CrossRef] [PubMed]

8. S. H. Cho, B. E. Bouma, E.P. Ippen, and J. G. Fujimoto, “Low-repetition-rate high-peak-power Kerr-lens mode-locked Ti:Al_{2}O_{3} laser with multi-pass cavity,” Opt. Lett. **24**, 417–419 (1999). [CrossRef]

9. S. H. Cho, F. X. Kartner, U. Morgner, E. P. Ippen, J. G. Fujimoto, J. E. Cunningham, and W. H. Knox, “Generation of 90-nJ pulses with a 4-MHz repetition-rate Kerr-lens mode-locked Ti:Al_{2}O_{3} laser operating with net positive and negative intracavity dispersion,” Opt. Lett. **26**, 560–562 (2001). [CrossRef]

_{T}, which represents one round trip, is given by

_{T}is unity. Let the initial ray be represented by a vector

_{0}and r

_{0}’ give the initial ray displacement from the optical axis and the initial inclination of the ray, respectively. We also note in passing that in typical MPC designs, the angles of inclination are small and radii of curvature of the mirrors very large. Therefore, astigmatic effects arising from tilted beam incidence can be neglected. After n round trips, the ray vector

_{1,2}and the associated eigenvectors

_{T}can be expressed as

_{n}and y

_{n}of the ray after n round trips can be expressed as

*x*

_{0},

*x*

_{0}') and (

*y*

_{0},

*y*

_{0}') are the initial off-set and inclination in the x- and y- directions, respectively, at the input reference plane. If the initial ray parameters are adjusted so that

_{0}. Furthermore, θ corresponds to the change in angular position of the spots around the circle formed by successive bounces of the beam after each round trip. The resulting circular spot pattern at the location of the reference plane is schematically shown in Fig. 2.

*I*, where

*I*is the unity matrix. In the general case, we note from Eq. (6) that

^{m}

*I*whenever

*m/n*), the Gaussian q parameter remains invariant after n round trips. Note that m gives the number of semicircular arcs that the bouncing beam traverses on one of the mirrors before the q parameter is transformed back to its initial value. For each value of m, every integer value of n corresponds to a q preserving configuration of the MPC. When m is even, the bouncing beam traverses an integral number of full circular trajectories and comes back to the initial entry position before exiting the MPC. This q-preserving case is identical with Herriott’s “reentrant” condition discussed in Ref. [1

1. D. R. Herriott, H. Kogelnik, and R. Kompfner, “Off-axis paths in spherical mirror interfometers,” Appl. Opt. **3**, 523–526 (1964). [CrossRef]

^{th}bounce is on the opposite side of the mirror, or at an angle of π away, from the initial beam. In Fig 3(b) with m=2 and n=9, the 9

^{th}bounce in the spot pattern is at the same position as the initial beam. The beam has bounced in a circular pattern with a net angular sweep of 2π around the circle. In Fig 3(c) with m=3 and n=9, the 9

^{th}bounce overlaps with the 3

^{rd}bounce. This design cannot be used in practice because it is impossible to extract the beam after 9 bounces without blocking it during earlier bounces. Figure 3(d) with m=4 and n=9 is analogous to the case when m=2, where the 9

^{th}bounce is at the same position as the initial beam, except that the beam has bounced with a net angular sweep of 4π around the circle.

^{th}bounce is again on the opposite side of the mirror from the initial beam. In this case, the beam has bounced in a circular pattern with a net angular sweep of 5π. Additional cases for other values of m are not shown, but can easily be constructed by extending these results.

_{0}. A small flat mirror (M3) injects the incident beam into the MPC. A total of n round trips are completed when the beam is extracted from the MPC following a reflection from a curved pick-up mirror (M4) with the same radius of curvature as that of M1. For this particular case, we investigate the effect of the spot pattern on the repetition rate of the MPC. We choose the input reference plane z

_{R1}of the MPC to be located at the position of the flat mirror M2 (See Fig. 4).

_{T}, representing one round trip starting at the input reference plane, becomes

_{0}, we can find the possible values of θ that satisfy the q-preserving condition in Eq. (9).

_{0}, giving a repetition frequency

*f*

_{rep}of

*f*

_{rep}given in Eq. (12) due to the additional length of the KLM resonator. Figures 5(a) and (b) show the variation of f

_{rep}and the corresponding mirror separation L

_{0}as a function of n for the case where R=2 meters. As noted earlier, each integer value of n (shown by markers of different shapes in Figs. 5 (a) and (b)) gives a q preserving configuration. Note that for each value of m, there is an optimum value n

_{opt}of n, independent of R, which gives the largest possible optical length and the lowest repetition rate.

_{opt}=3 and the lowest possible repetition rate that can be obtained is c/9R. In the general case, n

_{opt}is the integer closest to the exact solution of the transcendental equation

_{opt}, the lowest possible repetition rate f

_{min}, and the corresponding MPC separation L

_{opt}for different values of m. These equations are helpful for general laser design because they give the lowest possible repetition rate or longest path length that can be built with a given q preserving MPC configuration.

_{opt}of n, for which the pulse repetition frequency is minimum. The generalized analysis presented here can be readily applied to MPCs with more complicated mirror geometries. The closed form solutions presented here should enable a more efficient design and optimization of high pulse energy and compact femtosecond lasers.

## Acknowledgments

## References and links

1. | D. R. Herriott, H. Kogelnik, and R. Kompfner, “Off-axis paths in spherical mirror interfometers,” Appl. Opt. |

2. | D. R. Herriott and H. J. Schulte, “Folded optical delay lines,” Appl. Opt. |

3. | W. R. Trutna and R. L. Byer, “Multiple-pass Raman gain cell,” Appl. Opt. |

4. | B. Perry, R. O. Brickman, A. Stein, E. B Treacy, and P. Rabinowitz, “Controllable pulse compression in a multiple-pass-cell Raman laser,” Opt. Lett. |

5. | J. B. McManus, P. L. Kebabia, and M. S. Zahniser, “Astigmatic mirror multipass absorption cells for long-path-length spectroscopy,” Opt. Lett. |

6. | P-L Hsiung, X. Li, C. Chuboda, I. Hartl, T. H. Ko, and J. G. Fujimoto, “High-speed path-length scanning with a multiple-pass cavity delay line,” Appl. Opt. |

7. | A.R. Libertun, R. Shelton, H.C. Kapteyn, and M. M. Murnane, “A 36nJ-15.5 MHz extended-cavity Ti:sapphire oscillator,” in |

8. | S. H. Cho, B. E. Bouma, E.P. Ippen, and J. G. Fujimoto, “Low-repetition-rate high-peak-power Kerr-lens mode-locked Ti:Al |

9. | S. H. Cho, F. X. Kartner, U. Morgner, E. P. Ippen, J. G. Fujimoto, J. E. Cunningham, and W. H. Knox, “Generation of 90-nJ pulses with a 4-MHz repetition-rate Kerr-lens mode-locked Ti:Al |

**OCIS Codes**

(080.0080) Geometric optics : Geometric optics

(140.0140) Lasers and laser optics : Lasers and laser optics

(320.0320) Ultrafast optics : Ultrafast optics

**ToC Category:**

Research Papers

**History**

Original Manuscript: March 11, 2003

Revised Manuscript: April 22, 2003

Published: May 5, 2003

**Citation**

Alphan Sennaroglu and James Fujimoto, "Design criteria for Herriott-type multi-pass cavities for ultrashort pulse lasers," Opt. Express **11**, 1106-1113 (2003)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-9-1106

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### References

- D. R. Herriot, H. Kogelnik, and R. Kompfner, �??Off-axis paths in spherical mirror interfometers,�?? Appl. Opt. 3, 523-526 (1964). [CrossRef]
- D. R. Herriot and H. J. Schulte, �??Folded optical delay lines,�?? Appl. Opt. 4, 883-889 (1965). [CrossRef]
- W. R. Trutna and R. L. Byer, �??Multiple-pass Raman gain cell,�?? Appl. Opt. 19, 301-312 (1980). [CrossRef] [PubMed]
- B. Perry, R. O. Brickman, A. Stein, E. B Treacy, and P. Rabinowitz, �??Controllable pulse compression in a multiple-pass-cell Raman laser,�?? Opt. Lett. 5, 288-290 (1980). [CrossRef] [PubMed]
- J. B. McManus, P. L. Kebabia, and M. S. Zahniser, �??Astigmatic mirror multipass absorption cells for longpath-length spectroscopy,�?? Opt. Lett. 34, 3336-3348 (1995).
- P-L Hsiung, X. Li, C. Chuboda, I. Hartl, T. H. Ko, and J. G. Fujimoto, �??High-speed path-length scanning with a multiple-pass cavity delay line,�?? Appl. Opt. 42, 640-648 (2003). [CrossRef] [PubMed]
- A.R. Libertun, R. Shelton H.C. Kapteyn, and M. M. Murnane, �??A 36nJ-15.5 MHz extended-cavity Ti:sapphire oscillator,�?? in CLEO�??99 Technical Digest, (Institute of Electrical and Electronics Engineers, New York, 1999), pp. 469-470.
- S. H. Cho, B. E. Bouma, E.P. Ippen, and J. G. Fujimoto, �??Low-repetition-rate high-peak-power Kerr-lens mode-locked Ti:Al2O3 laser with multi-pass cavity,�?? Opt. Lett. 24, 417-419 (1999). [CrossRef]
- S. H. Cho, F. X. Kartner, U. Morgner, E. P. Ippen, J. G. Fujimoto, J. E. Cunningham, and W. H. Knox, �??Generation of 90-nJ pulses with a 4-MHz repetition-rate Kerr-lens mode-locked Ti:Al2O3 laser operating with net positive and negative intracavity dispersion,�?? Opt. Lett. 26, 560-562 (2001). [CrossRef]

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