We demonstrate a general procedure for varying the repetition rate of a modelocked Ti:sapphire laser using an asymmetric focusing geometry. Using this procedure, we have made an extended length cavity with a repetition rate of 45 MHz, and a reduced length cavity with a repetition rate of 275 MHz, each of which generates sub-20 fs pulses. This procedure allows the repetition rate of the laser to be more precisely tailored for a variety of applications without compromise in performance.
© 2004 Optical Society of America
Since the first demonstration of self-modelocking , ultrafast Ti:sapphire lasers have become the workhorse devices for generating ultrafast pulses in the near infrared. Typical oscillators operate at a repetition rate between 70 and 100 MHz, with an output power of about 500 mW when pumped with 5 W of CW green laser radiation. This corresponds to a pulse energy of about 6 nJ. Since the gain bandwidth of Ti:sapphire is so large, pulse durations of 10 fs and shorter are routinely achievable . Thus the typical peak output power of the standard oscillator is about 0.6 MW.
Several techniques have been demonstrated for increasing the peak power directly from an oscillator, without the use of extracavity amplification. Cavity dumping has been shown to generate peak powers of up to 5 MW [3,4]. However, cavity dumping is complex, expensive, and generally more difficult to operate than a simple laser oscillator. Recently, alternative approaches to increasing the peak power from oscillator have been developed by increasing the cavity length [5,6,7]. The lowered repetition rate allows for more energy in a single pulse for a given average power. Simply increasing the length of the arms of a Ti:sapphire laser cavity quickly results in an unstable geometry. However, by inserting a telescope, or multiple telescopes, into the cavity, the overall physical length of the cavity can be increased without changing the round-trip “ABCD” matrix  that determines the stability of the laser. This procedure has been demonstrated for repetition rates as low as 5.85 MHz with pulse durations as short as 43 fs . In some of these designs, the use of saturable Bragg mirror reflectors (SBR) to prevent multiple pulsing limits the pulse duration obtainable from the laser.
Alternatively, there has also been interest in the development of compact, high repetition rate lasers, which have applications in frequency metrology, high speed communications, and biological imaging . Several examples of high repetition rate oscillators have been demonstrated [10–14], some of which have repetition rates of 1 GHz or more. However, these designs are also more specialized and expensive, requiring custom coatings and optics, and are not tunable. Most are forced to sacrifice bandwidth, average output power, or experimental robustness. The Kerr lens modelocking mechanism that results in ultrashort pulse generation is simple in principle, but is complex and incompletely understood in practice. Extensive work has been done in modeling KLM laser resonators using ABCD formalisms [15–17], and also using more rigorous approaches including full space-time propagation models . However, complex dynamics such as space-time focusing and self-phase modulation in the Ti:sapphire crystal  are complex enough that modeling gives limited guidance on real world usability and stability issues. As a consequence, experimentation is necessary for each new cavity design, and few variations have a robustness comparable to the “standard” modelocked Ti:sapphire cavity design. 
In this paper, we present a very simple approach that can be applied to both extending and shortening the length of a Ti:sapphire cavity. We use a simple asymmetric focusing design that alters the cavity length while preserving the overall cavity propagation, that provides the exceptional stability and ease of use inherent in the standard cavity configuration.
The basic cavity configuration of the modelocked Ti:sapphire laser is shown in Fig. 1, along with definitions of various cavity parameters. Calculations were done to determine how the stability could be maintained as the cavity length was altered. The dynamics of modelocking are neglected in the calculation, which is relevant only to cw operation of the oscillator. As in previous research , the goal of these calculations is to construct a round-trip ABCD matrix that is effectively the same as the standard cavity, even as the overall cavity length is altered. We can do this by changing an arm length L and the radius of curvature R, of one or both “arms” of the cavity, as referenced to the Ti:sapphire crystal. If the q-parameter is the same for the asymmetric cavity as for a standard cavity in cw operation, the idea is that the modelocking initiation procedure and behavior will be the same as well.
Ignoring initially the astigmatism in the cavity introduced by the off-axis reflection, the round trip ABCD matrix for one arm of the cavity, starting from the edge of the crystal, reflecting from the curved mirror and to the plane mirror and back, is given by
where L is the distance from the curved mirror to the output coupler, d is the distance from the curved mirror to the crystal, and R is the radius of curvature (ROC) of the mirror. Generally in the cw Ti:sapphire cavity, R ~2d. Setting R=2d simplifies the matrix considerably to
By equating the value for C for two different cavities, we determine cavity parameters (Lb,Rb), that give an overall ray matrix similar to the standard cavity (La,Ra). Lb corresponds to
Using the parameters from the standard cavity, Table 1 lists the corresponding values for two possible asymmetric cavities, having altered the output coupling arm, which give the same ABCD matrix in this approximation.
Without changing the q parameter of the Gaussian beam, and therefore without changing the laser stability, the distance to the output coupler may be nearly quadrupled by using a 20 cm ROC as one of the focusing optics, and it may be nearly quartered by using a 5 cm ROC optic. The length of the prism arm of the cavity could be altered in a similar way.
The radii of curvature of the two focusing mirrors are different, and therefore the incidence angles must also be different in order to compensate for astigmatism. The incidence angles for this laser were calculated using ABCD matrices [20, 21].
Here lc is the length and nL is the refractive index of the Brewster angled crystal, R1 and R2 are the radii of curvature of the mirrors as indicated in Fig. 1, and q is the incidence angle. The angles of incidence for each of the curved mirrors were calculated using Eq. (4) and are listed in Table 2. In all cases, nL=1.75.
3. Low repetition rate design
Our design for extending the cavity length of the Ti:sapphire laser requires changing only two optics in a standard Ti:sapphire laser: the output coupler and one of the curved mirrors. Instead of using two identical mirrors with the same radius of curvature to focus the beam into the Ti:sapphire crystal, this laser uses one mirror with twice the ROC (20 cm) of the other (10 cm). This is the first time to our knowledge that such an asymmetric cavity has been employed for the purpose of changing the repetition rate of a modelocked Ti:sapphire laser, although past ABCD analyses for modelocked Ti:sapphire lasers have included the possibility of an asymmetric cavity . Asymmetric cavities have been demonstrated [22, 23] in modelocked Cr:LiSAF systems so that they may be directly pumped by a single diode laser and have a small footprint. In this case, the larger ROC optic allows the distance to the output coupler to be increased significantly, while maintaining cavity stability as discussed above. The length of the prism arm remains unchanged. This results in a laser with approximately twice the cavity length and therefore half the repetition rate of the standard Ti:sapphire laser.
The setup, shown in Fig. 2, is quite similar to that of a standard Ti:sapphire laser. The laser has two flat end mirrors, three flat fold mirrors, two curved mirrors focusing into a Ti:sapphire crystal, and two prisms for intracavity dispersion compensation. The Ti:sapphire crystal is 4.75 mm long, doped at 0.15%. The three fold mirrors and one of the end mirrors are dielectric and >99% reflective at center wavelength 800 nm. A larger output coupler of 20% is used to suppress multiple pulsing and to increase the output power. The curved mirrors, with radii of curvature 10 cm and 20 cm, are aligned in an astigmatism compensated X configuration about the crystal. The 10 cm mirror was set to an astigmatism compensation angle of 7.5±1.0°, and the 20 cm was set to an angle of 5.3±1.0°. The Ti:sapphire crystal is pumped by 4.75 W from a frequency doubled Nd:Vanadate laser (Coherent Verdi) at 532 nm. The pump beam is focused into the cavity with a lens of focal length 10.5 cm through the R=10 cm optic. The flat side of the mirror was AR coated for 532 nm to minimize loss. The prisms are Brewster cut fused silica, at a separation of ~63 cm. The total length of the single pass cavity is ~333 cm, resulting in a pulse repetition rate of 45±1 MHz, as measured by a fast photodiode.
4. Performance of low repetition rate laser
An extracavity prism pair compensated for the dispersion of the output coupler, crystal, and pulse measurement optics. Calculations indicated that the equilateral, fused silica prisms should be separated by about 100 cm in order to correct for GVD. The compressed pulse was measured using SHG FROG . The FROG trace was deconvolved using Femtosoft FROG3 software to yield a FWHM pulse duration of 15±2 fs, shown in Fig. 3(a). The pulse duration was limited by uncompensated third order dispersion, as can be seen as a “ringing” effect at the leading edge of the pulse. This and the 1 fs resolution of the FROG setup were the main sources of uncertainty in the measurement of the pulse duration. The transform limited pulse duration, determined by an inverse Fourier transform of the experimental spectrum shown in Figure 3(b), is ~13 fs.
When modelocked in the standard negative dispersion regime, stable, ~15 fs pulses were obtained at an average output power of 400 mW, with a typical power discrimination of 50 mW. At a repetition rate of 45 MHz, this corresponds to a pulse energy of 10 nJ, or twice that usually obtained from a standard Ti:sapphire laser operating at such short pulse widths.
In order to initiate modelocking, we decrease the length d until the intracavity power reaches a value that favors stably modelocked operation, then introduce a brief disturbance to initiate a pulse. This is the same procedure used for a standard Ti:sapphire laser. However, for a similar bandwidth, the extended cavity laser operates at a slightly lower average output power of ~400 mW, likely the result of a somewhat lower than optimum output coupling Modelocking is achieved as a result of gain-aperturing, as in the standard Kerr lens modelocked Ti:sapphire laser. Stability is limited mainly by environmental factors- in our case from air currents and significant temperature fluctuations. The laser would persist in modelocked operation for several hours uncovered in the lab. The absence of multiple pulse instabilities, self Q-switching and double pulsing, was confirmed by observation of the pulse train by a fast photodiode and a spectrometer.
5. High repetition rate design
To shorten the cavity length, we incorporated asymmetric focusing mirrors along with other techniques. Since the main limit on the length of the cavity is determined by the prism pair, two steps were taken to minimize the prism separation. First, a shorter Ti:sapphire crystal was used, introducing less dispersion. Second, chirped mirrors were added to the cavity, adding negative chirp and thereby reducing the prism separation. This allowed us to shrink both cavity arm lengths. We also further shortened the cavity by shrinking the arm lengths while maintaining the same arm-length ratio, until the laser operation became noticeably less stable. The output coupling was also reduced to 3.5%, to compensate for a lower intracavity peak intensity.
The optimal length for the output coupling arm (L in figure 1) is smaller than that predicted by Eq. (2). The value of this distance in the cavity demonstrated was 11.5 cm, compared with the predicted 16.5 cm. According to ray-propagation simulations, the astigmatism is quite large (a factor of 1.5 difference between the two dimensions of the beam waist) at stable resonator values of the ABCD matrix when L=16.5 cm. This astigmatism caused the unstable modelocking observed with this configuration. Another reason for this discrepancy is that the dispersive arm of the cavity was also altered. Had the dispersive arm of the cavity remained similar to that of the standard cavity, the prediction of Eq. (2) results in a stable, non-astigmatic beam in the cavity, according to the ray-propagation calculations.
The setup for the high repetition rate laser is shown in Fig. 4. The laser has two flat end mirrors, two flat chirped mirrors, two curved mirrors focusing into a Ti:sapphire crystal, and two prisms for intracavity dispersion compensation. The Ti:sapphire crystal is 3 mm long, doped at 0.25%. One of the end mirrors is dielectric and >99% reflective at center wavelength 800 nm. The chirped mirrors are a dispersion compensated pair with a GVD of ~-60 fs2 each, introducing a total of ~-480 fs2 GVD with eight bounces round-trip. Output coupling is achieved by a 3.5% transmissive, flat end mirror. The curved mirrors, with ROC 7.5 cm and 5 cm, are aligned in an astigmatism compensated X configuration about the crystal. The 7.5 cm mirror was set to an astigmatism compensation angle of 7.1±1.0°, and the 5 cm was set to an angle of 8.7±1.0°. The Ti:sapphire crystal is pumped by 4.0 W from a frequency doubled Nd:Vanadate laser (Spectra-Physics Millennia) at 532 nm. The pump beam is focused into the cavity with a lens of focal length 10.5 cm through the R=7.5 cm optic. The flat side of the mirror was AR coated for 532 nm to minimize loss. The prisms are Brewster cut fused silica, at a separation of about 25 cm. The total length of the single pass cavity is ~54.5 cm, resulting in a pulse repetition rate of 275±1 MHz.
6. Performance of high repetition rate laser
The pulse was compressed as described for the low repetition rate cavity, and the pulse duration was again measured by SHG FROG. The FROG trace was deconvolved to yield a FWHM pulse duration of 14±2 fs, shown in Fig. 5(a), limited by uncompensated third order dispersion. The transform limited pulse duration, determined by a Fourier transform of the experimental spectrum shown in Fig. 5(b), is ~10.3 fs. When modelocked in the standard negative dispersion regime, stable, ~14 fs pulses were obtained at an average output power of 400 mW, with a typical power discrimination of 50 mW. At a repetition rate of 275 MHz, this corresponds to a pulse energy of about 1.5 nJ.
Modelocking is again achieved as a result of gain-aperturing. The procedure for initiating and maintaining modelocking is also nearly identical to that of a standard laser, except that the spatial mode of the output is not TEM00 at bandwidths greater than ~40nm. These larger bandwidths are likely caused by single-pass self-phase modulation, which is not governed by the spatial modes of the cavity. However, stability is again limited mainly by environmental factors. The laser would persist in modelocked operation for several hours uncovered in the lab. The group of spectral peaks around 875 nm is due to unidentified cavity dynamics, but contributes only a small amount to the spectrum and does not affect the cavity perfomance. Although the spectra in Figs. 3(b) and 5(b) have different structure, Fourier analysis shows that they will each support a short pulse (13 fs and 10.3 fs respectively). The fact that the pulses shown in the time domain, Figs. 3(a) and 5(a), appear similar, is a result of the fact that the same compression scheme, with limiting third order dispersion, was used. The shape of the pulses presented is dominated by the spectral phase rather than the shape of the spectrum. The particular shapes of the spectra are likely determined by the dispersion of the respective optics, particularly the chirped mirrors in the high repetition rate cavity, and the coating of the output couplers. The absence of multiple pulse instabilities, self Q-switching and double pulsing, was confirmed by observation of the pulse train by a fast photodiode and a spectrometer.
We also investigated a symmetric cavity of comparable repetition rate which employs two R=5 cm curved mirrors in place of the asymmetric pair. This cavity was observed experimentally to be unstable in modelocked operation and overall quite sensitive to alignment. ABCD matrix models confirmed this alignment sensitivity. The asymmetric cavity did not have these stability issues, and operated with generally the same stability observed in a standard 100 MHz laser.
In conclusion, we have demonstrated the operation of Ti:sapphire oscillators with novel geometries. By constructing cavities with asymmetric pairs of curved mirrors, we have demonstrated a simple method of altering the repetition rate of a Ti:sapphire oscillator while maintaining the simplicity of the standard cavity.
This technique is useful especially for applications in which a repetition rate within a factor of 2–3 of the standard 80–100 MHz is needed. Only slight changes to a standard Ti:sapphire oscillator are needed to effect this result. For repetition rates outside this range, other techniques, such as cavity-dumping or chirped mirror based cavities, may be appropriate. However, the cavities described above maintain characteristics similar to a standard 100 MHz oscillator. Our results indicate that for low- and high-repetition rate asymmetric cavities, both laser geometries are stable, easy to construct and to use, and can support the generation of sub-20 fs pulses, limited only by dispersion and the onset of multiple pulse instabilities.
This work was supported by the the Multidisciplinary Research Program of the University Research Initiative (MURI) for the development of optical atomic clocks, administered by the Office of Naval Research, and by the National Science Foundation. D.M. was supported in part by a grant from the National Institute of Standards and Technology.
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