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Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 11, Iss. 9 — May. 5, 2003
  • pp: 1106–1113
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Design criteria for Herriott-type multi-pass cavities for ultrashort pulse lasers

Alphan Sennaroglu and James G. Fujimoto  »View Author Affiliations


Optics Express, Vol. 11, Issue 9, pp. 1106-1113 (2003)
http://dx.doi.org/10.1364/OE.11.001106


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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

Herriott-type multi-pass cavities (MPC) can be used in a wide range of femtosecond laser designs. MPCs enable the laser pulse energy to be increased by extending the cavity path length and decreasing the repetition rate. MPCs also enable the development of very compact lasers while preserving standard repetition rates and pulse energies. In its simplest form, an MPC consists of a stable two-mirror resonator and a mechanism for injecting and extracting light beams. When the MPC parameters are properly adjusted, the incident beam injected with the correct offset and tilt, undergoes multiple bounces before exiting. The successive bounces of the beam viewed in a given reference plane (for example on one of the end mirrors) form an elliptical or circular spot pattern. Using appropriate design conditions, the MPC can leave the Gaussian beam q parameter invariant. This means that diffractive beam spreading effects are exactly cancelled as a result of the periodic focusing inside the cavity. First introduced by Herriott et al [1

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

,2

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

], MPCs have long been used in many applications such as accurate optical loss measurements [2

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

], stimulated Raman scattering [3

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

,4

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]

], long-path absorption spectroscopy [5

5. J. B. McManus, P. L. Kebabia, and M. S. Zahniser, “Astigmatic mirror multipass absorption cells for long-path-length spectroscopy,” Opt. Lett. 34, 3336–3348 (1995).

], and high-speed path-length scanning [6

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]

].

Fig. 1. A schematic of a general multi-pass cavity. One round trip is represented by the ABCD matrix.

Several issues need to be addressed in the practical design of MPCs for high-energy ultrafast lasers. In general, the cavity operating point is very important for Kerr lens modelocked operation and modelocking is only possible in a small subset of the cavity stability region. If the cavity repetition rate is increased by extending the length of a standard four-mirror cavity, the cavity operating point, including the stability region and spot sizes change dramatically. This makes re-establishing KLM operation very difficult. These problems are solved by keeping the q parameter invariant. In addition, even if the pulse energy changes as a result of using an MPC, keeping the same q parameter operating point is important if the laser is modelocked in the soliton pulse regime. In this case, the cavity dispersion is proportionally scaled up as the pulse energy is increased to enable the generation of pulses with approximately the same duration. For these reasons, maintaining the same q parameter is desirable. This is guaranteed if the MPC is designed in such a way that the exiting beam has the same q parameter as the beam incident on the MPC. In what follows, we will refer to this as the “q preserving” configuration of the MPC. For the design of MPC femtosecond lasers, it is helpful to have general guidelines on how to construct q preserving MPCs. In addition, it is also important to know how the optical path length and the repetition rate vary for different “q preserving” configurations in order to maximize the output energy of the oscillator.

Let us begin by considering the general multi-pass cavity (MPC) shown in Fig. 1. The ray transfer matrix MT, which represents one round trip, is given by

MT=[ABCD].
(1)

Because the ray necessarily comes back to the same region in the cavity after each round trip, the determinant of MT is unity. Let the initial ray be represented by a vector ri given by

ri=[r0r0].
(2)

In the above, r0 and r0’ 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 rn becomes

rn=MTnri.
(3)

The stability of the cavity, in other words, the condition that the rays remain confined to the optical axis after an arbitrary number of round trips, is guaranteed when A and D satisfy the condition

A+D21.
(4)

In this case, the eigenvalues λ1,2 and the associated eigenvectors ν1,2 of MT can be expressed as

λ1=eiθ,v1=1B(BC)[B(Aeiθ)]
λ2=eiθ,v2=1B(BC)[B(Aeiθ)].
(5)

In Eq. (5), cosθ=A+D2 . The stability condition can be understood by noting that if Eq. (4) is not satisfied, then the magnitudes of the eigenvalues in Eq. (5) can be larger than 1, and the ray no longer remains confined to the optical cavity after multiple bounces. By using the well-known techniques of matrix algebra, it can be shown that MTn becomes

MTn=[AD2sinnθsinθ+cosnθBsinnθsinθCsinnθsinθDA2sinnθsinθ+cosnθ].
(6)

We can give a physical interpretation for the angle θ appearing in Eqs. (5) and (6) by considering the effect of the MPC on an incident ray with non-zero off-set and inclination along the x- and y- directions. By using Eq. (6), the transverse displacements xn and yn of the ray after n round trips can be expressed as

xn=x0cosnθ+(xo(AD)+2Bx0'2sinθ)sinnθ
yn=y0cosnθ+(yo(AD)+2By0'2sinθ)sinnθ,
(7)
Fig. 2. The spot pattern formed by the beams upon successive transits will in general be elliptical at a given reference plane in the cavity. A circular spot pattern is obtained at the input reference plane when the position and tilt of the incident ray is adjusted according to Eq. 8.

where ere (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

y0=0
y0'=x0sinθB,
x0'=x02B(DA)
(8)

then the ray describes a circle of radius x0. 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.

nθ=mπ.
(9)

Here, n and m are any two integers. Equation (9) summarizes the most important design rule for the construction of q preserving MPCs. Stated in words, whenever the angle, θ between successive bounces of the beam is π times a rational number, the ratio of two integers (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]

].

For example, Fig 3(a) with m=1 and n=9, corresponds to a spot pattern where the 9th 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 9th 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 9th bounce overlaps with the 3rd 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 9th 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.

Finally, Fig 3(e), with m=5 and n=9, shows a case where the 9th 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.

Fig. 3. Spot patterns corresponding some possible q preserving configurations for the case where m=1,2,3,4,5 and n=9.

The general results derived above can also be examined for the specific case of an MPC consisting of a curved (M1, Radius of curvature=R) and a flat (M2) high reflector as schematically shown in Fig. 4. The separation between the MPC mirrors is L0. 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 zR1 of the MPC to be located at the position of the flat mirror M2 (See Fig. 4).

The ray transfer matrix MT, representing one round trip starting at the input reference plane, becomes

MT=[12L0R2L0(1L0R)2R12LR],
(10)

with

cosθ=12L0R.
(11)

By varying L0, we can find the possible values of θ that satisfy the q-preserving condition in Eq. (9).

Fig. 4. A sketch of the MPC consisting of a flat and a curved mirror with radius R or focal length f=R/2, separated by a distance L0. This cavity can be analyzed to calculate the optimum configuration which minimizes the repetition rate or maximizes the beam path length for a given cavity size.

This simple MPC design permits the derivation of closed form expressions for the mirror separation and the pulse repetition rate. By using these results, optimum spot patterns that minimize the pulse repetition rate can be calculated. Note that the optical path length introduced by the MPC is simply 4nL0, giving a repetition frequency frep of

frep=c2nR(1cos(mπn)),
(12)
Fig. 5. Calculated variation of (a) the pulse repetition rate and (b) the corresponding mirror separation for the q preserving configurations of the multi-pass cavity for different values of m for the flat - curved MPC case.

for the MPC when additional arm lengths of the resonator are neglected. In Eq. (12), c is the speed of light. The q preserving condition in Eq. (9) was used to obtain Eq. (12). In practical femtosecond lasers, the overall repetition rate will be smaller than frep given in Eq. (12) due to the additional length of the KLM resonator. Figures 5(a) and (b) show the variation of frep and the corresponding mirror separation L0 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 nopt of n, independent of R, which gives the largest possible optical length and the lowest repetition rate.

For example, in the case of m=2, nopt=3 and the lowest possible repetition rate that can be obtained is c/9R. In the general case, nopt is the integer closest to the exact solution of the transcendental equation

1=cos(mπnopt)+mπnoptsin(mπnopt).
(13)

Table 1 gives the values of nopt, the lowest possible repetition rate fmin, and the corresponding MPC separation Lopt 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.

Table 1. Calculated values of nopt, the achievable minimum repetition rate fmin, and the corresponding MPC separation Lopt for different values of m for the flat-curved MPC.

table-icon
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In conclusion, we have presented a generalized analysis of multi-pass cavities that can be used for high pulse energy or compact femtosecond laser oscillators. Analytical criteria for designing q preserving cavities are derived. An analysis of a specific MPC consisting of a flat and a curved mirror showed that there is an optimum spot pattern, characterized by the optimum value nopt 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

The authors would like to thank Andrew Kowalevicz and Aurea Zare, for helpful discussions. This work is supported in part by contracts NSF ECS-0119452, AFOSR F49620-01-1-0084, and AFOSR F49620-01-1-0186. This work has also been supported in part by the Turkish Academy of Sciences, in the framework of the Young Scientist Award Program (AS/TUBAGEBIP/2001-1-11).

References and links

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]

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]

5.

J. B. McManus, P. L. Kebabia, and M. S. Zahniser, “Astigmatic mirror multipass absorption cells for long-path-length spectroscopy,” Opt. Lett. 34, 3336–3348 (1995).

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]

7.

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.

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:Al2O3 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:Al2O3 laser operating with net positive and negative intracavity dispersion,” Opt. Lett. 26, 560–562 (2001). [CrossRef]

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

  1. D. R. Herriot, H. Kogelnik, and R. Kompfner, �??Off-axis paths in spherical mirror interfometers,�?? Appl. Opt. 3, 523-526 (1964). [CrossRef]
  2. D. R. Herriot 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]
  5. 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).
  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]
  7. 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.
  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:Al2O3 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:Al2O3 laser operating with net positive and negative intracavity dispersion,�?? Opt. Lett. 26, 560-562 (2001). [CrossRef]

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