- Email: [email protected]
Mutual Kerr-lens mode-locking
15 October 1999
Optics Communications 170 Ž1999. 85–92 www.elsevier.comrlocateroptcom
Mutual Kerr-lens mode-locking Matthew J. Bohn ) , R. Jason Jones, Jean-Claude Diels Department of Physics and Astronomy and Center for High Technology Materials, The UniÕersity of New Mexico, Albuquerque, NM 87131, USA Received 13 April 1999; received in revised form 5 August 1999; accepted 6 August 1999
Abstract Mutual Kerr-lens mode-locking is implemented to create synchronized, bidirectional pulse-trains in a mode-locked Ti:sapphire laser, in a linear as well as in a ring configuration. A simple ABCD matrix method is used to optimize the location of apertures in the cavity for preferential bidirectional mode-locked operation. q 1999 Elsevier Science B.V. All rights reserved.
1. Introduction When used for spectroscopy, ring cavities are preferred to linear operation, because of the absence of standing waves in the gain medium that cause gain competition between various modes. Bidirectional operation of a mode-locked laser is preferred for some specific applications such as rotation sensing, measurements of small displacements, fields, etc... w1x. Such an operation does not come easily in a Kerr-lens mode-locked Ti:sapphire ring laser, since the natural tendency of such a laser is to operate unidirectionally. We reported earlier bidirectional operation of a femtosecond Ti:sapphire laser through a combination of Kerr-lensing and saturable absorber w2x. For applications as a sensor, it is essential that there be no phase coupling between the oppositely directed waves, which precludes the use of the gain medium for mutual Kerr-lensing, as recently reported )
Corresponding author. DFMS Suite 6D2A, 2354 Fairchild Dr, USAFA, CO 80840. Tel.: q1-719-333-4470; fax: q1-719-3332114; e-mail: [email protected]
by Wang et al. w3x, as this would result in strong coupling of the oppositely travelling waves. In the work presented here, we analyze the operation of a bidirectional, mutually Kerr-lens mode-locked Ti:sapphire laser where we have separated the Kerr medium from the gain element. Because these elements are physically separated, the problems of the formation of a gain grating and subsequent gain competition are avoided. Three configurations are investigated: 1. a linear cavity with double pulses, which is a limiting case of an elongated ring laser; 2. a picosecond ring laser cavity without prisms for GVD compensation; and 3. a femtosecond ring laser with a sequence of 4 prisms for GVD compensation. 2. Theory The laser operating with counterpropagating pulses in the cavity is a particular case of a two-color laser w4–7x, since the central carrier frequency of each pulse is in general different Žthe purpose of the ring laser sensor being to observe a beat note between the two output pulse trains.. The important
0030-4018r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 Ž 9 9 . 0 0 4 4 1 - 1
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M.J. Bohn et al.r Optics Communications 170 (1999) 85–92
difference between our situation and that of Furst ¨ et al. w8x, is that, in the latter case, the modes of the laser outputs can overlap. In the ring laser sensor, where we are interested in measuring beat notes between the two pulse trains that are smaller than the mode spacing, there has to be two distinct combs of modes. To appreciate the difference, and to understand the role of mutual self-phase modulation versus mutual self-lensing, let us compare the output spectrum of a mode-locked laser with its longitudinal mode structure. The unequal spacing of the longitudinal mode structure can be measured by recording with a frequency counter the beating between adjacent longitudinal modes as a function of laser wavelength, while the laser is operating continuous-wave w9x. In the case of a continuous mode-locked train, the Fourier transform of a regularly spaced train of pulses of equal amplitude is a comb of equally spaced delta-functions, as we have verified experimentally with a 10 fs laser w10x. It is the phase modulation in the Kerr medium of the laser that is responsible for ‘shifting’ the modes into a regular comb. In the case of the two-color laser w8x, the cross phase modulation of co-propagating pulses, combined with dispersion, results in accelerating the pulse that is trailing, leading to perfect synchronization. In the frequency domain, this interpretation is equivalent to having the two combs of frequency components be part of a unique comb w11x. The situation is different in the case of the ring laser, where we want to minimize any frequency coupling between oppositely propagating pulses. On the other hand, the amplitude coupling has to be maximized. An ABCD matrix calculation of the cavity was made in order to optimize the mode-locked operation in bidirectional mode. The condition to be sought is that the loss be smaller for mode-locked bidirectional operation than for either continuous wave operation or mode-locked operation with two pulses in one direction. Because the lifetime of Ti:sapphire is much longer than the round-trip time, it is assumed that the total energy extracted from the cavity in one roundtrip is constant regardless of the mode of operation. Another approximation Žvalid within a factor of two. w2x is that the pulsewidths are equal. These combined assumptions require that the peak power in the bidirectional case be less than or equal to half the power in the single-pulse unidirectional case. The modifica-
tion to the nonlinear index of refraction due to two counter-propagating fields of direction i and j is: n i2 s n 2 Ž Ii q 2 Ij . .
Ž 1.
The factor two on the right hand side of Eq. Ž1. reflects the well known fact that cross-phase modulation is twice as effective as self-phase modulation for the same intensity w12x. Assuming equal intensity in the counter-propagating fields and similar waists for single pulse operation versus bidirectional operation yields the following approximate relationship for the nonlinear index of refraction in the ZnS crystal Žof length d . for single pulse Žunidirectional. versus double pulse Žbidirectional. operation: D n bidirectional ;
3 2
D n single
Leff d
Ž 2.
where Leff is the interaction length. Because the colliding pulses will only have an interaction length of approximately their pulsewidth, the mutual Kerrlens will be much smaller than the self Kerr-lens, if the length of the Kerr medium is much longer than the pulsewidth. Depending on the pulsewidth and crystal thickness, there may be enough nonlinearity to distinguish between single-pulse operation and bidirectional operation. For a laser gyro, we require that the pulses cross outside of the gain medium to prevent lock-in from scattering off a gain grating in the Ti:sapphire. To enhance the mutual Kerr-effect, we chose a high n 2 crystal, ZnS, which has a nonlinear index about 50 times larger than Ti:sapphire. We use the ABCD matrix method to determine where the slits should be located for bidirectional operation. We expect the simulation to yield only qualitative results because of the numerous approximations.
3. Numerical method for cavity analysis We use a simple ABCD matrix method to model the laser. The nonlinearity of the Ti:sapphire and the ZnS crystal is included by inserting a lens matrix of the appropriate focal length w13x at the location of the beam waist in the nonlinear medium: f nl s
1 n 0 p w 04 a 8 n 2 Pd
Ž 3.
M.J. Bohn et al.r Optics Communications 170 (1999) 85–92
where w 0 is the waist in the Ti:sapphire or ZnS, n 2 is the nonlinear index of refraction, P is the peak power, and d is the length of the crystal. The correction factor, a, ranging from 1 to 4, accounts for the use of a parabolic approximation for the beam profile. If a one-to-one correspondence is made in the Taylor series expansion, then the a-factor is 1. A recent empirical fit found the value of a to be 1.723 w14,15x for Gaussian beams. Eq. Ž3. is a good approximation for small nonlinearities Ž P - Pcr Žcritical power.. and thin Ž d - z 0 ŽRayleigh range.. samples. We are also assuming a TEM 00 mode, which was verified experimentally. For an infinitesimal slab, dz, a thin lens approximation can be used to determine the focal length of the nonlinear lens: d
1
ž /
s
f nl
a8 n 2 P n 0p w 4 Ž z .
dz
s
a8 n 2 P n 0p
f n l t o t al s
8 n2 P w 04
n 0p
Hz
z2
dz 4
1
Hz
w Ž z.
s
a8 n 2 P n 0p
f n l t o t al
w 04
z
z 2 2
ž ž // 1q
1
z
z0
2
ž
z0 z2 z 02 q z 22
yarctan
z1
q arctan
ž // z0
4
dz
4
dz
1 z 04
d eff
where z 0 is the confocal parameter and d eff is the effective length of the crystal defined by: z0
'
H wŽ z. H wŽ z.
n 0 pv 02rl,
d eff s
which means that the waist is at the left edge of the crystal. For lengths longer than the Rayleigh range, the nonlinearity approaches a constant. Typically the beam waist is near one edge of the crystal, such that small changes in the beam parameters can translate into large changes in the nonlinear lens. So far only the strength of the nonlinear lens has been discussed. The location, z, of the equivalent thin lens is an important parameter in determining the position and size of the beam waist in the cavity. It can be determined by calculating the centroid of the nonlinear lens:
dz
z2 1
Fig. 1. Plot of effective crystal length versus actual crystal length. Note for lengths longer than the Rayleigh range, the effective length is approximately a constant. a is an empirically determined constant w26x.
Ž 4.
The ABCD matrix method calculates the q beam parameter of the beam defined by 1rq Ž z . s 1rRŽ z . y i lrŽp w 2 Ž z ... From the beam parameter, we extract the radius of curvature of the beam, R 1 , and the beam waist, w1 Žthe 1re value of the field.. Since the crystals are not thin Žthe ZnS crystals used in these experiments were 3 mm and 1 mm., an integration is performed over the length of the crystal, d, from z 1 to z 2 : 1
87
z2
ž / z0
y
z 0 z1 z 02 q z 12
Ž 5.
It is instructive to plot d eff as a function of crystal length. Fig. 1 assumes w 0 s .001 cm and z 1 s 0,
z
'
2 Ž z 02 q z 12 .
y
z 04 2 Ž z 02 q z 22 .
d eff
Fig. 2 is a plot of Ž z y z . , the location of the nonlinear lens with respect to the center of the crystal, as a function of the location of the crystal with respect to the beam waist. Notice that the location of the nonlinear lens is always within the crystal. An iterative approach can be used where the waist is first calculated ignoring the nonlinearity and then the waist in the cavity is recalculated using the
88
M.J. Bohn et al.r Optics Communications 170 (1999) 85–92
Fig. 2. Location of the nonlinear lens plotted as a function of the location of the crystal with respect to the beam waist.
new waist. The waist can then be recalculated until the desired precision is reached. We have verified that, for our cavity, the process converges after a single iteration. The first iteration produces a correction of less than 0.5% from the waist calculated without the nonlinearity. After the first iteration, the change is less than the accuracy of the first order approximation used in Eq. Ž3.. Using this numerical method, we can calculate the waist in the cavity for continuous-wave operation, a single pulse in the cavity or double pulses in the cavity, by simply adjusting the strength of the nonlinearity in the ZnS crystal in accordance with Eq. Ž2. in the numerical calculation for the different regimes of operation. Fig. 3 is a plot of the beam waist as a function of position in the cavity. Notice that a slit at position B will increase the losses for continuous-wave operation Žpositive feedback. and slits at position A will limit single-pulse operation Žnegative feedback.. Judicious use of these two slits should yield bidirectional pulses. The distances from the Ti:sapphire crystal to the curved mirrors are included in Fig. 3 as d1 and d6. Likewise, d3 and d4 are the distances from the ZnS crystal to the curved mirrors. These distances are crucial to reproduce the results presented in this paper. 4. Experimental results ZnS is particularly attractive as a nonlinear crystal for the Ti:sapphire laser, because of its high nonlin-
earity Ž n 2 s 2.1 = 10y1 2 esu. combined with a low 2-photon absorption Ž b - .02 cmrGW. at l s 780 nm w16x. The nonlinear loss is low because the bandgap is at 3.6 eV which is beyond the 2-photon energy of the Ti:sapphire pulses Ž3 eV.. The close proximity of the two-photon energy to the bandgap enhances the effective x 3, although the interaction remains parametric Žno energy is transferred to the crystal. w17x. ZnS was first demonstrated as a Kerrlens mode-locker in the Ti:sapphire laser by Radwiecz et al. w18x. Because the ZnS is used to modelock the laser, the Ti:sapphire crystal can be tuned independently for maximum power w18x. Because the x 3 nonlinearity in ZnS is roughly 50 times larger than that of Ti:sapphire, the mode-locking threshold will be much lower. 4.1. Linear caÕity The gyroscopic response of a ring laser can be an undesirable noise factor in measuring a very small phase shift. Since the gyroscopic response is proportional to the ratio of area to perimeter, it can be eliminated by elongating the cavity. The limit of this elongation is a linear cavity, with two intracavity pulses per round-trip. Ultrasensitive phase measurements can be made by recording a beat note between the interwoven pulse trains of the laser output w19x.
Fig. 3. Beam radius as a function of position in the cavity. The calculation begins and ends in the Ti:sapphire crystal, TiS. Three different cases are plotted: continuous-wave, single-pulse and bidirectional pulses. d1 and d6 are the distances from the face of the Ti:sapphire to the curved mirrors. Likewise d3 and d4 are the distances from the ZnS crystal to the curved mirrors surrounding the ZnS crystal.
M.J. Bohn et al.r Optics Communications 170 (1999) 85–92
In the case of Kerr-lens mode-locking, a mechanism for creating multiple pulses w20x is enhanced Kerrlensing at the overlap of pulses in the Ti:sapphire crystal w21,22x. In the present case, mutual Kerr-lensing takes place in a ZnS crystal located at the center of the cavity, as shown in Fig. 4. For accurate centering of the ZnS crystal, one end mirror is positioned on a translation stage. Double pulses were observed over a broad range of end cavity placements Žup to 0.5 cm.. Although it is easier to achieve double pulses in this experimental setup Žin comparison to the dye jet w19x., they were less stable and the pulse train typically had approximately a 10–20% ripple. With careful alignment and the use of slits to control the spectrum, a very quiet mode of operation is achieved. The noise appears to be a result of the fluctuations of the Arq laser. The reflections off of an intracavity prism are used to extract the interwoven pulse trains, and, after appropriate delay line, make them interfere on a detector. A beat note of several kHz is observed, due to a small difference in intensity of the two intracavity pulses, resulting in a different nonlinear phase shift in the Ti:sapphire. There is significant second harmonic emission from the ZnS. Bidirectional operation is evident by the appearance of blue emission along the normal to the ZnS crystal. The various blue emissions will be discussed after the description of the ring laser configurations.
Fig. 4. Experimental setup for mutual Kerr-lens mode-locking in ZnS. The ZnS crystal is 3 mm long and at Brewster’s angle. The cavity fold mirrors all have a radius of curvature of 10 cm. The ZnS is equidistant from the end mirrors.
89
Fig. 5. Experimental arrangement for mutual Kerr-lens mode-locking. The bidirectional pulses meet in the ZnS crystal. The ZnS crystal is 3 mm in length.
4.2. Picosecond ring laser The bidirectional ring laser Žsee Fig. 5. is more difficult to align and typically requires a ‘kick’ to start the laser in the bidirectional mode, although self-starting operation was also observed. The laser is aligned by maximizing the blue light normal to the ZnS crystal. Once the bidirectional operation is achieved, slit A shown in Fig. 3 and in Fig. 5 is closed slightly to stabilize the bidirectional operation. According to the theory presented earlier Žsee Fig. 3., closing slit A should limit the peak power in the pulse and should therefore prevent single pulses. The two beams from the output coupler corresponding to the different directions of propagation in the cavity are made to interfere on a detector, after an appropriate delay. In Fig. 6 the top plot is the CCW intensity and the bottom is the beat signal. The beam intensity is modulated at the frequency of the beat signal, indicating that the beat frequency is close to the lock-in frequency. Auto- and cross-correlations performed on the laser indicate pulsewidths of 18.6 and 22.6 ps " 0.13 ps for the CCW and CW pulses, and 20 ps for the cross-correlation, which is close to the minimum expected for a Gaussian shape. Therefore the jitter of the crossing point is less than the precision of the autocorrelation measurement Ž0.13 ps.. Spectral measurements indicate that, even though the two pulses circulate in the same cavity, their average frequencies differ by as much as 2 nm. The widths of
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M.J. Bohn et al.r Optics Communications 170 (1999) 85–92
ity could be controlled by adjusting slits A and B as predicted by the theory presented in Fig. 3. The locations of slits A and B were qualitatively consistent with the theory presented earlier. From the two laser outputs a beat frequency of about 60 kHz is observed, which can be explained by the different powers in the CW and CCW directions at the ZnS. Assuming that the average power is different in the CW and CCW directions as a result of 50 mW of output coupled power, the beat frequency can be calculated to be: Fig. 6. CCW signal Žtop. and beat note Žbottom.. Notice that the beat frequency appears as an amplitude modulation on the CCW pulse train.
the spectra are: 2.65 and 2.5 nm " .04 nm for the CCW and CW pulse trains; however, the resolution of the monochromator was only 0.4 nm. The deconvolution of the spectra with the monochromator resolution yields a frequency bandwidth of: 1.24 and 1.17 THz for the CCW and CW pulses. The timebandwidth products are: 23.0 and 26.4 " .2 for the CCW and CW pulses. A possible explanation for the large time-bandwidth product is the large amount of SPM in the ZnS which would push the frequency away from line center.
Dn s
n 2 D Pave l
lp
w 02tp
F
2 n 2 D Pave
l2tp
f 285 kHz
Ž 6.
where l is the interaction length, n 2 is the nonlinear index of refraction, p w 02 is the spot size, D Pave is the average output coupled power and tp is the pulse width. Assuming that the ZnS crystal is centered in the beam waist leads to an upper bound of the beat frequency of 285 kHz. As the ZnS is translated beyond the beam waist, the beat frequency will decrease because the average spot size in the crystal will be greater. The Fourier transform of the beat note is plotted in Fig. 7, showing clearly the 2nd and 3rd harmonic in addition to the fundamental at 60 kHz. The lock-in frequency, l , can be calculated from the ratio, r, of the second harmonic amplitude to the fundamental
4.3. Femtosecond ring laser The introduction of a sequence of four dispersion compensating prisms into the ring laser cavity of Fig. 5 reduced the pulsewidth of the laser by roughly a factor of 100, down to 227 fs and 242 fs for the clockwise and counterclockwise pulses, respectively. Stable bidirectional pulse trains as short as 80 fs were also observed. Because the average power of the laser remained roughly constant, the peak power and thus the nonlinear drive are increased by a factor of 100. Several different operating regimes were observed including unidirectional operation, bidirectional unlocked operation, continuous wave operation and bidirectional operation that had asymmetrical powers and pulsewidths. Typically, the unidirectional operation did not favor any particular direction, but would alternate between CW and CCW directions after blocking the laser. The bidirectional-
Fig. 7. Fourier transform of the beat frequency. The geometric progression is predicted by the lock-in equation and yields information about the lock-in frequency.
M.J. Bohn et al.r Optics Communications 170 (1999) 85–92
Fig. 8. Wave vectors involved in SFG ŽSum Frequency Generation.. The beams are mixed in the intracavity ZnS crystal. The larger beams are the strong, intracavity fundamental beams. The weaker solid beams are the surface reflection of the fundamental. The dotted lines are the SFG beams.
amplitude of the beat signal, through the relation w23,24x:
ls
2 pr
Ž1yr 2 .
s 500 Hz
Ž 7.
Where p is the pulled Žor measured. frequency. From Eq. Ž7., the lock-in frequency is estimated to be of the order of 1.8 kHz. 4.4. Blue emission from the ZnS Blue radiation was observed emanating from the ZnS crystal in all configurations. Because ZnS has cubic symmetry, the index of refraction in the crystalline planes is nearly identical and therefore birefringent phase matching is not possible. We used two crystals for these experiments, the first is a 3 mm polycrystal and the second a 1 mm X-cut single crystal. For the picosecond laser without prisms, the 3 mm crystal is preferable because the nonlinear phase shifts accumulate over the full length of the crystal. The ZnS crystal is at Brewster’s angle inside the Ti:sapphire cavity. Collinear, non-phased matched second-harmonic is beyond the critical angle for total internal reflection and is trapped inside the crystal. A total of 8 beams of blue light are seen emerging from the ZnS, corresponding to reflected and transmitted surface second-harmonics w25x, and two beams normal to the ZnS surface, which appear only when the lasers are operating in bidirectional
91
mode. The latter are a result of the mixing of the fundamental in one direction with the Brewster’s reflection from the other direction, as illustrated in Fig. 8. The beams labeled SFG Žsum-frequency generation. are much brighter Žabout a factor of 100. in bidirectional mode-locked operation, as compared to continuous-wave lasing. Since the SFG beams completely disappear when the laser is running unidirectional, they provide a convenient means to monitor the bidirectionality of the laser. The SFG beams from the femtosecond laser were significantly weaker than the picosecond laser because of the reduced interaction length. To confirm that the mechanism is sum frequency generation, we compared the spectra of the surfacesecond harmonic to the SFG. Fig. 9 is a plot of the spectra of the surface second harmonic generation from the CW and CCW beams, and the spectrum of the SFG beam. Indeed, the SFG spectrum lies between the spectra of the two surface harmonic generation signals. This plot verifies that the beams are indeed a result of SFG. It is curious that the mean wavelength of the CW and CCW pulses can differ by several nanometers, yet the beat frequency differs by only a few kHz. The reason for the separation in wavelength is that the mechanism of sum frequency generation creates a loss that is frequency dependent. The frequency components that correspond to the shortest coherence length will be preferentially amplified in the cavity. Let v s s v 1 q v 2 be the sum frequency that is generated from the two counter propagating beams at v 1 and v 2 . Since the geome-
Fig. 9. Spectra of CW and CCW surface second-harmonic and the spectrum of sum frequency generated normal to the ZnS crystal.
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M.J. Bohn et al.r Optics Communications 170 (1999) 85–92
try is fixed, given an angle u between the two k vectors, the coherence length is given by l c s prw k v s y Ž k v 1 q k v 2 .cos u x or l c s p crw v s nŽ v s . y Ž v 1 nŽ v 1 . q v 2 n 2 Ž v 2 ..cos u x. If the phase mismatch is the smallest for v 1 s v 2 , one expects these two frequencies to drift away from each other, since this is a minimum loss configuration. As a precautionary measure, all possible fourwave mixing geometries were also considered. A total of 12 different possible Žnot probable. wavevectors were compared for energy and momentum conservation. k 4 s k1 q k 2 q k 3 Ž 8. The closest match had a wavevector mismatch of < D krk < ; 0.2, which is too large to be probable. 5. Conclusions We have demonstrated a new method of mutual Kerr-lens mode-locking which provides for a means to define the crossing point of counter-propagating waves in a ring or linear laser, without the use of a saturable absorber. Blue radiation emanating from the crystal has been identified as surface second harmonic and sum frequency generation. The latter, appearing only in condition of bidirectionality, provides a convenient means to verify that pulses are indeed crossing in the nonlinear crystal. This laser has promising applications for rotation sensing, motion sensing, electro-optic sampling and phase spectroscopy. This work was supported by the National Science Foundation under grant number ECS 9970082. References w1x S. Diddams, B. Atherton, J.-C. Diels, Appl. Phys. B 63 Ž1996. 473.
w2x M.J. Bohn, J.-C. Diels, Opt. Lett. 22 Ž1997. 642. w3x C. Wang, L. Chai, W. Zhang, K.M. Yoo, Appl. Opt. 37 Ž1998. 522. w4x M.R.X. de Barros, P.C. Becker, Opt. Lett. 18 Ž1993. 631. w5x J.M. Evans, D.E. Spence, D. Burns, W. Sibbett, Opt. Lett. 18 Ž1993. 1074. w6x D.R. Dykaar, S.B. Darack, Opt. Lett. 18 Ž1993. 634. w7x D.R. Dykaar, S.B. Darack, W.H. Knox, Opt. Lett. 19 Ž1994. 1058. w8x C. Furst, A. Leitenstorfer, A. Laubereau, IEEE J. Sel. Top. Quantum Electron. 2 Ž1996. 473. w9x W.H. Knox, Femtosecond intracavity dispersion measurements, Ultrafast Phenomena VIII, 1992, pp. 192–193. w10x R. Jason Jones, J. Jasapara, W. Rudolph, J.C. Diels, Group velocity modes versus phase velocity modes, to be published, 1999. w11x Th. Udem, J. Reichert, R. Holzwarth, T.W. Haensch, Phys. Rev. Lett. 82 Ž1999. 3568. w12x G.P. Agrawal, Nonlinear Fiber Optics, Academic Press, Boston, 2nd edition, 1989. w13x J.-C. Diels, W. Rudolph, Ultrashort Laser Pulse Phenomena, Academic Press, New York, 1996. w14x M. Sheik-Bahae, M. Ebrahimzadeh, Opt. Commun. 142 Ž1997. 294. w15x E.W. Van Stryland, M. Sheik-Bahae, Z-scan measurements of optical nonlinearites, in: CRC Handbook of Nonlinear Optics, CRC Press, Boca Raton, 1998. w16x T.D. Krauss, F.W. Wise, Appl. Phys. Lett. 65 Ž1994. 1739. w17x D.C. Hutchings, M. Sheik-Bahae, D.J. Hagan, E.W. Van Stryland, Opt. Quantum Electron. 24 Ž1992. 1. w18x C. Radzewicz, G.W. Pearson, J.S. Krasinski, Opt. Commun. 102 Ž1993. 464. w19x M.J. Bohn, J.-C. Diels, Opt. Commun. 141 Ž1997. 53. w20x T. Tsang, Opt. Lett. 18 Ž1993. 293. w21x M. Lai, J. Nicholson, W. Rudolph, Opt. Commun. 142 Ž1997. 45. w22x V.L. Kalashnikov, N.I. Zhavoronkov, I.G. Poloyko, V.P. Mikhailov, J. Opt. Soc. Am. B 14 Ž1997. 2705. w23x G.E. Stedman, Z. Li, C.H. Rowe, A.D. McGregor, H.R. Bilger, Phys. Rev. A 51 Ž1995. 4944. w24x M.J. Bohn, The Ti:sapphire Ring Laser Gyroscope, Ph.D. thesis, University of New Mexico, May 1998. w25x N. Bloembergen, P.S. Pershan, Phys. Rev. 128 Ž1962. 606. w26x M. Sheik-Bahae, A.A. Said, D.J. Hagan, M.J. Soileau, E.W. Van Stryland, Opt. Eng. 30 Ž1991. 1228.
Optics Communications 170 Ž1999. 85–92 www.elsevier.comrlocateroptcom
Mutual Kerr-lens mode-locking Matthew J. Bohn ) , R. Jason Jones, Jean-Claude Diels Department of Physics and Astronomy and Center for High Technology Materials, The UniÕersity of New Mexico, Albuquerque, NM 87131, USA Received 13 April 1999; received in revised form 5 August 1999; accepted 6 August 1999
Abstract Mutual Kerr-lens mode-locking is implemented to create synchronized, bidirectional pulse-trains in a mode-locked Ti:sapphire laser, in a linear as well as in a ring configuration. A simple ABCD matrix method is used to optimize the location of apertures in the cavity for preferential bidirectional mode-locked operation. q 1999 Elsevier Science B.V. All rights reserved.
1. Introduction When used for spectroscopy, ring cavities are preferred to linear operation, because of the absence of standing waves in the gain medium that cause gain competition between various modes. Bidirectional operation of a mode-locked laser is preferred for some specific applications such as rotation sensing, measurements of small displacements, fields, etc... w1x. Such an operation does not come easily in a Kerr-lens mode-locked Ti:sapphire ring laser, since the natural tendency of such a laser is to operate unidirectionally. We reported earlier bidirectional operation of a femtosecond Ti:sapphire laser through a combination of Kerr-lensing and saturable absorber w2x. For applications as a sensor, it is essential that there be no phase coupling between the oppositely directed waves, which precludes the use of the gain medium for mutual Kerr-lensing, as recently reported )
Corresponding author. DFMS Suite 6D2A, 2354 Fairchild Dr, USAFA, CO 80840. Tel.: q1-719-333-4470; fax: q1-719-3332114; e-mail: [email protected]
by Wang et al. w3x, as this would result in strong coupling of the oppositely travelling waves. In the work presented here, we analyze the operation of a bidirectional, mutually Kerr-lens mode-locked Ti:sapphire laser where we have separated the Kerr medium from the gain element. Because these elements are physically separated, the problems of the formation of a gain grating and subsequent gain competition are avoided. Three configurations are investigated: 1. a linear cavity with double pulses, which is a limiting case of an elongated ring laser; 2. a picosecond ring laser cavity without prisms for GVD compensation; and 3. a femtosecond ring laser with a sequence of 4 prisms for GVD compensation. 2. Theory The laser operating with counterpropagating pulses in the cavity is a particular case of a two-color laser w4–7x, since the central carrier frequency of each pulse is in general different Žthe purpose of the ring laser sensor being to observe a beat note between the two output pulse trains.. The important
0030-4018r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 Ž 9 9 . 0 0 4 4 1 - 1
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M.J. Bohn et al.r Optics Communications 170 (1999) 85–92
difference between our situation and that of Furst ¨ et al. w8x, is that, in the latter case, the modes of the laser outputs can overlap. In the ring laser sensor, where we are interested in measuring beat notes between the two pulse trains that are smaller than the mode spacing, there has to be two distinct combs of modes. To appreciate the difference, and to understand the role of mutual self-phase modulation versus mutual self-lensing, let us compare the output spectrum of a mode-locked laser with its longitudinal mode structure. The unequal spacing of the longitudinal mode structure can be measured by recording with a frequency counter the beating between adjacent longitudinal modes as a function of laser wavelength, while the laser is operating continuous-wave w9x. In the case of a continuous mode-locked train, the Fourier transform of a regularly spaced train of pulses of equal amplitude is a comb of equally spaced delta-functions, as we have verified experimentally with a 10 fs laser w10x. It is the phase modulation in the Kerr medium of the laser that is responsible for ‘shifting’ the modes into a regular comb. In the case of the two-color laser w8x, the cross phase modulation of co-propagating pulses, combined with dispersion, results in accelerating the pulse that is trailing, leading to perfect synchronization. In the frequency domain, this interpretation is equivalent to having the two combs of frequency components be part of a unique comb w11x. The situation is different in the case of the ring laser, where we want to minimize any frequency coupling between oppositely propagating pulses. On the other hand, the amplitude coupling has to be maximized. An ABCD matrix calculation of the cavity was made in order to optimize the mode-locked operation in bidirectional mode. The condition to be sought is that the loss be smaller for mode-locked bidirectional operation than for either continuous wave operation or mode-locked operation with two pulses in one direction. Because the lifetime of Ti:sapphire is much longer than the round-trip time, it is assumed that the total energy extracted from the cavity in one roundtrip is constant regardless of the mode of operation. Another approximation Žvalid within a factor of two. w2x is that the pulsewidths are equal. These combined assumptions require that the peak power in the bidirectional case be less than or equal to half the power in the single-pulse unidirectional case. The modifica-
tion to the nonlinear index of refraction due to two counter-propagating fields of direction i and j is: n i2 s n 2 Ž Ii q 2 Ij . .
Ž 1.
The factor two on the right hand side of Eq. Ž1. reflects the well known fact that cross-phase modulation is twice as effective as self-phase modulation for the same intensity w12x. Assuming equal intensity in the counter-propagating fields and similar waists for single pulse operation versus bidirectional operation yields the following approximate relationship for the nonlinear index of refraction in the ZnS crystal Žof length d . for single pulse Žunidirectional. versus double pulse Žbidirectional. operation: D n bidirectional ;
3 2
D n single
Leff d
Ž 2.
where Leff is the interaction length. Because the colliding pulses will only have an interaction length of approximately their pulsewidth, the mutual Kerrlens will be much smaller than the self Kerr-lens, if the length of the Kerr medium is much longer than the pulsewidth. Depending on the pulsewidth and crystal thickness, there may be enough nonlinearity to distinguish between single-pulse operation and bidirectional operation. For a laser gyro, we require that the pulses cross outside of the gain medium to prevent lock-in from scattering off a gain grating in the Ti:sapphire. To enhance the mutual Kerr-effect, we chose a high n 2 crystal, ZnS, which has a nonlinear index about 50 times larger than Ti:sapphire. We use the ABCD matrix method to determine where the slits should be located for bidirectional operation. We expect the simulation to yield only qualitative results because of the numerous approximations.
3. Numerical method for cavity analysis We use a simple ABCD matrix method to model the laser. The nonlinearity of the Ti:sapphire and the ZnS crystal is included by inserting a lens matrix of the appropriate focal length w13x at the location of the beam waist in the nonlinear medium: f nl s
1 n 0 p w 04 a 8 n 2 Pd
Ž 3.
M.J. Bohn et al.r Optics Communications 170 (1999) 85–92
where w 0 is the waist in the Ti:sapphire or ZnS, n 2 is the nonlinear index of refraction, P is the peak power, and d is the length of the crystal. The correction factor, a, ranging from 1 to 4, accounts for the use of a parabolic approximation for the beam profile. If a one-to-one correspondence is made in the Taylor series expansion, then the a-factor is 1. A recent empirical fit found the value of a to be 1.723 w14,15x for Gaussian beams. Eq. Ž3. is a good approximation for small nonlinearities Ž P - Pcr Žcritical power.. and thin Ž d - z 0 ŽRayleigh range.. samples. We are also assuming a TEM 00 mode, which was verified experimentally. For an infinitesimal slab, dz, a thin lens approximation can be used to determine the focal length of the nonlinear lens: d
1
ž /
s
f nl
a8 n 2 P n 0p w 4 Ž z .
dz
s
a8 n 2 P n 0p
f n l t o t al s
8 n2 P w 04
n 0p
Hz
z2
dz 4
1
Hz
w Ž z.
s
a8 n 2 P n 0p
f n l t o t al
w 04
z
z 2 2
ž ž // 1q
1
z
z0
2
ž
z0 z2 z 02 q z 22
yarctan
z1
q arctan
ž // z0
4
dz
4
dz
1 z 04
d eff
where z 0 is the confocal parameter and d eff is the effective length of the crystal defined by: z0
'
H wŽ z. H wŽ z.
n 0 pv 02rl,
d eff s
which means that the waist is at the left edge of the crystal. For lengths longer than the Rayleigh range, the nonlinearity approaches a constant. Typically the beam waist is near one edge of the crystal, such that small changes in the beam parameters can translate into large changes in the nonlinear lens. So far only the strength of the nonlinear lens has been discussed. The location, z, of the equivalent thin lens is an important parameter in determining the position and size of the beam waist in the cavity. It can be determined by calculating the centroid of the nonlinear lens:
dz
z2 1
Fig. 1. Plot of effective crystal length versus actual crystal length. Note for lengths longer than the Rayleigh range, the effective length is approximately a constant. a is an empirically determined constant w26x.
Ž 4.
The ABCD matrix method calculates the q beam parameter of the beam defined by 1rq Ž z . s 1rRŽ z . y i lrŽp w 2 Ž z ... From the beam parameter, we extract the radius of curvature of the beam, R 1 , and the beam waist, w1 Žthe 1re value of the field.. Since the crystals are not thin Žthe ZnS crystals used in these experiments were 3 mm and 1 mm., an integration is performed over the length of the crystal, d, from z 1 to z 2 : 1
87
z2
ž / z0
y
z 0 z1 z 02 q z 12
Ž 5.
It is instructive to plot d eff as a function of crystal length. Fig. 1 assumes w 0 s .001 cm and z 1 s 0,
z
'
2 Ž z 02 q z 12 .
y
z 04 2 Ž z 02 q z 22 .
d eff
Fig. 2 is a plot of Ž z y z . , the location of the nonlinear lens with respect to the center of the crystal, as a function of the location of the crystal with respect to the beam waist. Notice that the location of the nonlinear lens is always within the crystal. An iterative approach can be used where the waist is first calculated ignoring the nonlinearity and then the waist in the cavity is recalculated using the
88
M.J. Bohn et al.r Optics Communications 170 (1999) 85–92
Fig. 2. Location of the nonlinear lens plotted as a function of the location of the crystal with respect to the beam waist.
new waist. The waist can then be recalculated until the desired precision is reached. We have verified that, for our cavity, the process converges after a single iteration. The first iteration produces a correction of less than 0.5% from the waist calculated without the nonlinearity. After the first iteration, the change is less than the accuracy of the first order approximation used in Eq. Ž3.. Using this numerical method, we can calculate the waist in the cavity for continuous-wave operation, a single pulse in the cavity or double pulses in the cavity, by simply adjusting the strength of the nonlinearity in the ZnS crystal in accordance with Eq. Ž2. in the numerical calculation for the different regimes of operation. Fig. 3 is a plot of the beam waist as a function of position in the cavity. Notice that a slit at position B will increase the losses for continuous-wave operation Žpositive feedback. and slits at position A will limit single-pulse operation Žnegative feedback.. Judicious use of these two slits should yield bidirectional pulses. The distances from the Ti:sapphire crystal to the curved mirrors are included in Fig. 3 as d1 and d6. Likewise, d3 and d4 are the distances from the ZnS crystal to the curved mirrors. These distances are crucial to reproduce the results presented in this paper. 4. Experimental results ZnS is particularly attractive as a nonlinear crystal for the Ti:sapphire laser, because of its high nonlin-
earity Ž n 2 s 2.1 = 10y1 2 esu. combined with a low 2-photon absorption Ž b - .02 cmrGW. at l s 780 nm w16x. The nonlinear loss is low because the bandgap is at 3.6 eV which is beyond the 2-photon energy of the Ti:sapphire pulses Ž3 eV.. The close proximity of the two-photon energy to the bandgap enhances the effective x 3, although the interaction remains parametric Žno energy is transferred to the crystal. w17x. ZnS was first demonstrated as a Kerrlens mode-locker in the Ti:sapphire laser by Radwiecz et al. w18x. Because the ZnS is used to modelock the laser, the Ti:sapphire crystal can be tuned independently for maximum power w18x. Because the x 3 nonlinearity in ZnS is roughly 50 times larger than that of Ti:sapphire, the mode-locking threshold will be much lower. 4.1. Linear caÕity The gyroscopic response of a ring laser can be an undesirable noise factor in measuring a very small phase shift. Since the gyroscopic response is proportional to the ratio of area to perimeter, it can be eliminated by elongating the cavity. The limit of this elongation is a linear cavity, with two intracavity pulses per round-trip. Ultrasensitive phase measurements can be made by recording a beat note between the interwoven pulse trains of the laser output w19x.
Fig. 3. Beam radius as a function of position in the cavity. The calculation begins and ends in the Ti:sapphire crystal, TiS. Three different cases are plotted: continuous-wave, single-pulse and bidirectional pulses. d1 and d6 are the distances from the face of the Ti:sapphire to the curved mirrors. Likewise d3 and d4 are the distances from the ZnS crystal to the curved mirrors surrounding the ZnS crystal.
M.J. Bohn et al.r Optics Communications 170 (1999) 85–92
In the case of Kerr-lens mode-locking, a mechanism for creating multiple pulses w20x is enhanced Kerrlensing at the overlap of pulses in the Ti:sapphire crystal w21,22x. In the present case, mutual Kerr-lensing takes place in a ZnS crystal located at the center of the cavity, as shown in Fig. 4. For accurate centering of the ZnS crystal, one end mirror is positioned on a translation stage. Double pulses were observed over a broad range of end cavity placements Žup to 0.5 cm.. Although it is easier to achieve double pulses in this experimental setup Žin comparison to the dye jet w19x., they were less stable and the pulse train typically had approximately a 10–20% ripple. With careful alignment and the use of slits to control the spectrum, a very quiet mode of operation is achieved. The noise appears to be a result of the fluctuations of the Arq laser. The reflections off of an intracavity prism are used to extract the interwoven pulse trains, and, after appropriate delay line, make them interfere on a detector. A beat note of several kHz is observed, due to a small difference in intensity of the two intracavity pulses, resulting in a different nonlinear phase shift in the Ti:sapphire. There is significant second harmonic emission from the ZnS. Bidirectional operation is evident by the appearance of blue emission along the normal to the ZnS crystal. The various blue emissions will be discussed after the description of the ring laser configurations.
Fig. 4. Experimental setup for mutual Kerr-lens mode-locking in ZnS. The ZnS crystal is 3 mm long and at Brewster’s angle. The cavity fold mirrors all have a radius of curvature of 10 cm. The ZnS is equidistant from the end mirrors.
89
Fig. 5. Experimental arrangement for mutual Kerr-lens mode-locking. The bidirectional pulses meet in the ZnS crystal. The ZnS crystal is 3 mm in length.
4.2. Picosecond ring laser The bidirectional ring laser Žsee Fig. 5. is more difficult to align and typically requires a ‘kick’ to start the laser in the bidirectional mode, although self-starting operation was also observed. The laser is aligned by maximizing the blue light normal to the ZnS crystal. Once the bidirectional operation is achieved, slit A shown in Fig. 3 and in Fig. 5 is closed slightly to stabilize the bidirectional operation. According to the theory presented earlier Žsee Fig. 3., closing slit A should limit the peak power in the pulse and should therefore prevent single pulses. The two beams from the output coupler corresponding to the different directions of propagation in the cavity are made to interfere on a detector, after an appropriate delay. In Fig. 6 the top plot is the CCW intensity and the bottom is the beat signal. The beam intensity is modulated at the frequency of the beat signal, indicating that the beat frequency is close to the lock-in frequency. Auto- and cross-correlations performed on the laser indicate pulsewidths of 18.6 and 22.6 ps " 0.13 ps for the CCW and CW pulses, and 20 ps for the cross-correlation, which is close to the minimum expected for a Gaussian shape. Therefore the jitter of the crossing point is less than the precision of the autocorrelation measurement Ž0.13 ps.. Spectral measurements indicate that, even though the two pulses circulate in the same cavity, their average frequencies differ by as much as 2 nm. The widths of
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M.J. Bohn et al.r Optics Communications 170 (1999) 85–92
ity could be controlled by adjusting slits A and B as predicted by the theory presented in Fig. 3. The locations of slits A and B were qualitatively consistent with the theory presented earlier. From the two laser outputs a beat frequency of about 60 kHz is observed, which can be explained by the different powers in the CW and CCW directions at the ZnS. Assuming that the average power is different in the CW and CCW directions as a result of 50 mW of output coupled power, the beat frequency can be calculated to be: Fig. 6. CCW signal Žtop. and beat note Žbottom.. Notice that the beat frequency appears as an amplitude modulation on the CCW pulse train.
the spectra are: 2.65 and 2.5 nm " .04 nm for the CCW and CW pulse trains; however, the resolution of the monochromator was only 0.4 nm. The deconvolution of the spectra with the monochromator resolution yields a frequency bandwidth of: 1.24 and 1.17 THz for the CCW and CW pulses. The timebandwidth products are: 23.0 and 26.4 " .2 for the CCW and CW pulses. A possible explanation for the large time-bandwidth product is the large amount of SPM in the ZnS which would push the frequency away from line center.
Dn s
n 2 D Pave l
lp
w 02tp
F
2 n 2 D Pave
l2tp
f 285 kHz
Ž 6.
where l is the interaction length, n 2 is the nonlinear index of refraction, p w 02 is the spot size, D Pave is the average output coupled power and tp is the pulse width. Assuming that the ZnS crystal is centered in the beam waist leads to an upper bound of the beat frequency of 285 kHz. As the ZnS is translated beyond the beam waist, the beat frequency will decrease because the average spot size in the crystal will be greater. The Fourier transform of the beat note is plotted in Fig. 7, showing clearly the 2nd and 3rd harmonic in addition to the fundamental at 60 kHz. The lock-in frequency, l , can be calculated from the ratio, r, of the second harmonic amplitude to the fundamental
4.3. Femtosecond ring laser The introduction of a sequence of four dispersion compensating prisms into the ring laser cavity of Fig. 5 reduced the pulsewidth of the laser by roughly a factor of 100, down to 227 fs and 242 fs for the clockwise and counterclockwise pulses, respectively. Stable bidirectional pulse trains as short as 80 fs were also observed. Because the average power of the laser remained roughly constant, the peak power and thus the nonlinear drive are increased by a factor of 100. Several different operating regimes were observed including unidirectional operation, bidirectional unlocked operation, continuous wave operation and bidirectional operation that had asymmetrical powers and pulsewidths. Typically, the unidirectional operation did not favor any particular direction, but would alternate between CW and CCW directions after blocking the laser. The bidirectional-
Fig. 7. Fourier transform of the beat frequency. The geometric progression is predicted by the lock-in equation and yields information about the lock-in frequency.
M.J. Bohn et al.r Optics Communications 170 (1999) 85–92
Fig. 8. Wave vectors involved in SFG ŽSum Frequency Generation.. The beams are mixed in the intracavity ZnS crystal. The larger beams are the strong, intracavity fundamental beams. The weaker solid beams are the surface reflection of the fundamental. The dotted lines are the SFG beams.
amplitude of the beat signal, through the relation w23,24x:
ls
2 pr
Ž1yr 2 .
s 500 Hz
Ž 7.
Where p is the pulled Žor measured. frequency. From Eq. Ž7., the lock-in frequency is estimated to be of the order of 1.8 kHz. 4.4. Blue emission from the ZnS Blue radiation was observed emanating from the ZnS crystal in all configurations. Because ZnS has cubic symmetry, the index of refraction in the crystalline planes is nearly identical and therefore birefringent phase matching is not possible. We used two crystals for these experiments, the first is a 3 mm polycrystal and the second a 1 mm X-cut single crystal. For the picosecond laser without prisms, the 3 mm crystal is preferable because the nonlinear phase shifts accumulate over the full length of the crystal. The ZnS crystal is at Brewster’s angle inside the Ti:sapphire cavity. Collinear, non-phased matched second-harmonic is beyond the critical angle for total internal reflection and is trapped inside the crystal. A total of 8 beams of blue light are seen emerging from the ZnS, corresponding to reflected and transmitted surface second-harmonics w25x, and two beams normal to the ZnS surface, which appear only when the lasers are operating in bidirectional
91
mode. The latter are a result of the mixing of the fundamental in one direction with the Brewster’s reflection from the other direction, as illustrated in Fig. 8. The beams labeled SFG Žsum-frequency generation. are much brighter Žabout a factor of 100. in bidirectional mode-locked operation, as compared to continuous-wave lasing. Since the SFG beams completely disappear when the laser is running unidirectional, they provide a convenient means to monitor the bidirectionality of the laser. The SFG beams from the femtosecond laser were significantly weaker than the picosecond laser because of the reduced interaction length. To confirm that the mechanism is sum frequency generation, we compared the spectra of the surfacesecond harmonic to the SFG. Fig. 9 is a plot of the spectra of the surface second harmonic generation from the CW and CCW beams, and the spectrum of the SFG beam. Indeed, the SFG spectrum lies between the spectra of the two surface harmonic generation signals. This plot verifies that the beams are indeed a result of SFG. It is curious that the mean wavelength of the CW and CCW pulses can differ by several nanometers, yet the beat frequency differs by only a few kHz. The reason for the separation in wavelength is that the mechanism of sum frequency generation creates a loss that is frequency dependent. The frequency components that correspond to the shortest coherence length will be preferentially amplified in the cavity. Let v s s v 1 q v 2 be the sum frequency that is generated from the two counter propagating beams at v 1 and v 2 . Since the geome-
Fig. 9. Spectra of CW and CCW surface second-harmonic and the spectrum of sum frequency generated normal to the ZnS crystal.
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M.J. Bohn et al.r Optics Communications 170 (1999) 85–92
try is fixed, given an angle u between the two k vectors, the coherence length is given by l c s prw k v s y Ž k v 1 q k v 2 .cos u x or l c s p crw v s nŽ v s . y Ž v 1 nŽ v 1 . q v 2 n 2 Ž v 2 ..cos u x. If the phase mismatch is the smallest for v 1 s v 2 , one expects these two frequencies to drift away from each other, since this is a minimum loss configuration. As a precautionary measure, all possible fourwave mixing geometries were also considered. A total of 12 different possible Žnot probable. wavevectors were compared for energy and momentum conservation. k 4 s k1 q k 2 q k 3 Ž 8. The closest match had a wavevector mismatch of < D krk < ; 0.2, which is too large to be probable. 5. Conclusions We have demonstrated a new method of mutual Kerr-lens mode-locking which provides for a means to define the crossing point of counter-propagating waves in a ring or linear laser, without the use of a saturable absorber. Blue radiation emanating from the crystal has been identified as surface second harmonic and sum frequency generation. The latter, appearing only in condition of bidirectionality, provides a convenient means to verify that pulses are indeed crossing in the nonlinear crystal. This laser has promising applications for rotation sensing, motion sensing, electro-optic sampling and phase spectroscopy. This work was supported by the National Science Foundation under grant number ECS 9970082. References w1x S. Diddams, B. Atherton, J.-C. Diels, Appl. Phys. B 63 Ž1996. 473.
w2x M.J. Bohn, J.-C. Diels, Opt. Lett. 22 Ž1997. 642. w3x C. Wang, L. Chai, W. Zhang, K.M. Yoo, Appl. Opt. 37 Ž1998. 522. w4x M.R.X. de Barros, P.C. Becker, Opt. Lett. 18 Ž1993. 631. w5x J.M. Evans, D.E. Spence, D. Burns, W. Sibbett, Opt. Lett. 18 Ž1993. 1074. w6x D.R. Dykaar, S.B. Darack, Opt. Lett. 18 Ž1993. 634. w7x D.R. Dykaar, S.B. Darack, W.H. Knox, Opt. Lett. 19 Ž1994. 1058. w8x C. Furst, A. Leitenstorfer, A. Laubereau, IEEE J. Sel. Top. Quantum Electron. 2 Ž1996. 473. w9x W.H. Knox, Femtosecond intracavity dispersion measurements, Ultrafast Phenomena VIII, 1992, pp. 192–193. w10x R. Jason Jones, J. Jasapara, W. Rudolph, J.C. Diels, Group velocity modes versus phase velocity modes, to be published, 1999. w11x Th. Udem, J. Reichert, R. Holzwarth, T.W. Haensch, Phys. Rev. Lett. 82 Ž1999. 3568. w12x G.P. Agrawal, Nonlinear Fiber Optics, Academic Press, Boston, 2nd edition, 1989. w13x J.-C. Diels, W. Rudolph, Ultrashort Laser Pulse Phenomena, Academic Press, New York, 1996. w14x M. Sheik-Bahae, M. Ebrahimzadeh, Opt. Commun. 142 Ž1997. 294. w15x E.W. Van Stryland, M. Sheik-Bahae, Z-scan measurements of optical nonlinearites, in: CRC Handbook of Nonlinear Optics, CRC Press, Boca Raton, 1998. w16x T.D. Krauss, F.W. Wise, Appl. Phys. Lett. 65 Ž1994. 1739. w17x D.C. Hutchings, M. Sheik-Bahae, D.J. Hagan, E.W. Van Stryland, Opt. Quantum Electron. 24 Ž1992. 1. w18x C. Radzewicz, G.W. Pearson, J.S. Krasinski, Opt. Commun. 102 Ž1993. 464. w19x M.J. Bohn, J.-C. Diels, Opt. Commun. 141 Ž1997. 53. w20x T. Tsang, Opt. Lett. 18 Ž1993. 293. w21x M. Lai, J. Nicholson, W. Rudolph, Opt. Commun. 142 Ž1997. 45. w22x V.L. Kalashnikov, N.I. Zhavoronkov, I.G. Poloyko, V.P. Mikhailov, J. Opt. Soc. Am. B 14 Ž1997. 2705. w23x G.E. Stedman, Z. Li, C.H. Rowe, A.D. McGregor, H.R. Bilger, Phys. Rev. A 51 Ž1995. 4944. w24x M.J. Bohn, The Ti:sapphire Ring Laser Gyroscope, Ph.D. thesis, University of New Mexico, May 1998. w25x N. Bloembergen, P.S. Pershan, Phys. Rev. 128 Ž1962. 606. w26x M. Sheik-Bahae, A.A. Said, D.J. Hagan, M.J. Soileau, E.W. Van Stryland, Opt. Eng. 30 Ž1991. 1228.