1. Introduction

Optical resonators have long been used to enhance the power and stabilize the frequency of laser radiation [1]. Coupling the output of a Mode-Locked Laser (MLL) to an optical resonator can offer similar benefits [2]. Frequency stabilization of the modes of an optical pulse train to the longtitudinal modes of a cavity corresponds to the coherent superposition of those pulses in the temporal or spatial domain inside the cavity. The resulting enhancement of pulse energy as well as of the peak power may be of use in applications such as nonlinear ultrafast spectroscopy [3], high harmonic generation [3] and resonant polarization interferometry [4]. In this paper we emphasize the energy- and power-enhancement benefits, rather than frequency stabilization or other applications such as repetition rate conversion. In contrast to reference [5] we consider a cavity constructed from two commercially available low dispersion mirrors, rather than making use of prism-based dispersion compensation techniques; this allows the construction of cavities which are mechanically simpler and more rigid, and with less intracavity loss.

We will present a general mathematical description of arbitrarily dispersive optics without resorting to the usual approximation of a power series expansion. Using this description, we will show how to determine the effect of mirror dispersion on a resonant laser pulse circulating in a cavity. These techniques, which allow calculation of both the frequency and time behavior of the pulse, can easily be extended to apply to more complex circumstances. It will be shown that there is an optimum input pulse duration which maximizes peak stored power and that current mirror technology allows for the generation of very high-power pulses under these conditions.

2. Optical frequency combs and resonators

An Optical Frequency Comb (OFC) is a signal source consisting of a large number of frequency components with a constant, common frequency separation Δν. The frequency of each comb component can thus be expressed as nΔν+ν ₀, where ν ₀ is termed the offset frequency. Current MLL technology enables the generation of optical frequency combs with extraordinarily uniform frequency distribution (to within 3 parts in 10¹⁷ [6]) and allows independent control of Δν and ν ₀.

In the time domain, an OFC corresponds to a train of pulses with a repetition rate f_rep=Δν. The complex amplitude envelope ℰ(t) of each individual pulse is the Fourier transform of the complex amplitude spectrum $\widetilde{ℰ}$(ω) of the comb (where the tilde denotes representation in the frequency domain). It is convenient to characterise these pulses in terms of their instantaneous power P(t), so for simplicity of notation, we define ℰ(t) such that P(t)=E_pulse|ℰ(t)|², where E_pulse is the total energy per pulse. This requires that ∫|ℰ(t)|² dt=1. Unchirped, Gaussian pulses of length τ at half their maximum power then have the following amplitude envelope in time:

By considering only the temporal envelope of the pulse and disregarding the rapid oscillations at the carrier frequency, we have effectively translated the central frequency ω ₀ of the comb spectrum to zero frequency, so the complex amplitude spectrum corresponding to (1) must be given not in terms of ω but in terms of the offset frequency Ω=ω-ω ₀. In addition, (1) describes only a single pulse and ignores the periodic nature of the pulse train, which amounts to discarding the comb structure in favor of a continuous amplitude spectrum given by the following expression:

We now turn our attention to the characteristics of the resonator into which an OFC is to be coupled. Consider a linear optical resonator of length L and optical length L_opt. If intracavity dispersion is neglected, the resonance frequencies must satisfy nλ=2L_opt; the n^th resonance frequency is then

If the intracavity dispersion is non-negligible, we define Δϕ(ω) to be the single-pass phase shift due to interaction with optical elements in the cavity, so that the optical length of the cavity is L_opt=L+Δϕ(ω)λ/(2π). Then the resonance frequencies of the cavity must satisfy

With no loss of generality we can write Δϕ(ω)=ϕ ₀+ωϕ′₀+δϕ(ω) such that δϕ contains only second- and higher-order variation with ω, and such that δϕ(ω ₀)=0. Substituting this into (4) and rearranging, one obtains the following expression for the n ^th resonance frequency:

Comparing equations (3) and (5), we see that the constant phase delay, ϕ ₀, has the effect of shifting each resonance frequency by a constant offset. The first-order frequency dependent phase delay ϕ′₀ changes the optical length of the cavity (and consequently the mode spacing) by an equal amount for all frequencies. This corresponds to the group delay of the pulse on reflection [7]. It is only the second- and higher-order terms contained in the dispersive phase delay δϕ(ω) that prevent perfect uniformity of mode spacing.

For a pulse train to couple to the resonator and create a steady-state, circulating pulse, each comb component must be in resonance with a cavity mode. In the absence of dispersion this is easily arranged: Setting Δν=c/(2L+2cϕ′₀) will ensure that each comb component is detuned equally from a resonance frequency¹; appropriate control of the comb offset frequency ν ₀ can then set this detuning to zero. Active control of Δν and ν ₀ is required to maintain this resonant condition. An effective technique for this purpose has been demonstrated by Jones and Diels [2], where the Pound-Drever-Hall technique [1] is used to generate error signals e ₁ and e ₂ which give information about the detuning from two subsets of resonator modes at opposite ends of the comb spectrum. The average of e ₁ and e ₂, which is approximately proportional to the average detuning across the comb, can be used as an error signal for feedback control of ν ₀. The difference between e ₁ and e ₂ can be used as an error signal for feedback control of Δν.

In a real resonator, the coupling of each comb component into the cavity depends on many factors, including impedance matching, beam alignment and mode matching. We assume that these factors can be separately optimised, and that fluctuations in Δν and ν ₀ are supressed in an active control system as explained above. Nonetheless, the ability of an OFC to couple to a resonator is fundamentally limited by the intracavity dispersion, to an extent which will be determined in the following sections.

3. Circulating comb characteristics

In the ideal case of a resonator with no group velocity dispersion, that is, with δϕ(ω)=0 for all frequencies, each component of an OFC can be made exactly resonant as described in the previous section. The circulating pulse power P_circ(t) is then identical to the incident pulse power P_inc(t) multiplied by F/π, where F is the resonator finesse. Only when δϕ(ω)=0 can the net round trip phase shift be zero for each comb component, allowing the incoming and circulating light to add coherently at the resonator input. Any dispersion in the cavity will result in an additional, frequency-dependent round-trip phase shift, lowering the efficiency of the superposition of the input and circulating light. We define the power coupling efficiency β(Ω) to be the ratio between the actual circulating power at a particular frequency, and the ideal (δϕ(ω)=0) case for which β=1:

In order to calculate and examine the broadening or distortion of circulating pulses due to the intracavity dispersion, the circulating complex amplitude spectrum at steady state must be determined. Reflection from a mirror of reflectance R results in multiplication of the complex amplitude by the complex reflectivity [7], which we denote h(Ω):

If the incident complex amplitude spectrum is $\widetilde{ℰ}$ _inc(Ω), then in a lossless and symmetric mirror system, the initial complex amplitude spectrum inside the cavity is $\sqrt{1-R}{\widetilde{ℰ}}_{\mathit{inc}}\left(\Omega \right)$ . The circulating complex amplitude spectrum is then

Taking the argument of $\widetilde{ℰ}$ _circ(Ω) gives the steady-state relative phase ϕ_circ(Ω) of the circulating comb. The power coupling efficiency can be determined from the cirulating power spectrum |$\widetilde{ℰ}$ _circ(Ω)|². For an unchirped incident OFC,

Mirrors for ultrafast pulsed laser applications are manufactured with dielectric coatings which are specifically designed to have flat dispersion characteristics near a specified wave-length; for example, CVI LGVD mirrors [8] have a central wavelength of 750 nm. A plot of δϕ(Ω) for these mirrors, obtained by integrating the published GVD (ϕ″(ω)) data, is given in Fig. 1(a). Henceforth in this paper we will assume the use of mirrors with these dispersion characteristics. The resulting circulating phase and coupling efficiency are given in Figs. 1(b) and 1(c) as calculated for R=99.8% (currently available as a stock component) and R=99.99%.

As can be seen in Fig. 1(c), coupling into the resonator is only possible for comb components in a certain bandwidth around Ω=0. This bandwidth can be seen to decrease as the cavity finesse F is increased, because increasing the number of passes increases the circulating phase shift, or equivalently, because the resonator modes are reduced in linewidth and thus the incoupled power is more sensitive to dispersive detuning of the cavity resonances. The limited circulating power bandwidth corresponds to a lower bound on the length of pulses which can be efficiently coupled into the cavity. Shortening the incident pulses will thus eventually result in relative broadening of the circulating pulses and reduced coupling efficiency.

4. Circulating pulse characteristics

Once the complex amplitude spectrum of the circulating pulse has been determined using equation (10), the complex amplitude ℰ _circ(t) of the pulse in the time domain can be obtained via the discrete Fourier transform, giving in turn the power

and relative phase

 

Fig. 1. (a) Dispersive phase shift δϕ(Ω) on reflection from CVI LGVD ultrafast mirrors. (b) Calculated circulating phase ϕ_circ(Ω). (c) Calculated power coupling efficiency β(Ω).

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of the circulating pulse in time. For example, Fig. 2 shows the calculated power and relative phase of the circulating pulse produced in a resonator by an incident OFC consisting of 20 fs Gaussian pulses, assuming a mirror reflectance of 99.8%. Also plotted, for comparison, is a 37 fs Gaussian pulse, demonstrating the extent to which the input pulse is broadened inside the resonator.

 

Fig. 2. (a) Calculated power P_circ(t) of the circulating pulse due to 20 fs Gaussian incident pulses. R=99.8%. A 37 fs Gaussian pulse (dashed) is shown for comparison. (b) Calculated circulating phase ϕ_circ(t).

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The relative broadening τ_circ/τ_inc of the circulating pulse depends on the input pulse length τ_inc. The form of this dependence can be examined by extracting τ_circ from the calculated circulating pulse power envelopes (by finding the time-between half-power points). The total power coupling efficiency,

can be determined for a particular input pulse length by numerically integrating over either the circulating pulse power envelope or the circulating power spectrum. The relative broadening and total power coupling efficiency, calculated as functions of the input pulse length τ_inc, are plotted in Fig. 3(a). For sufficiently long incident pulses, the circulating pulses are negligibly broadened, and the total power coupling efficiency approaches unity. For shorter input pulses, more of the comb spectrum lies outside of the efficient-coupling bandwidth, so the total power coupling efficiency decreases and the circulating pulse is relatively broadened. These effects become more pronounced as the finesse is increased because of the associated reduction in coupling bandwidth.

 

Fig. 3. (a) Total power coupling efficiency β_net and relative pulse broadening τ_circ/τ_inc. (b) Peak circulating power P_max (per µJ of energy per incident pulse).

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So long as the incident pulses are longer than the energy coupling turning point visible in Fig. 3(a), the energy per circulating pulse will be maximised and equal to F/π times the energy per incident pulse. Using a thin-disk gain element to facilitate the dissipation of heat from the pump beam, Brunner et al [9] constructed an MLL producing 0.9-µJ, 240-fs pulses at a repetition rate of 25 MHz; locking such an oscillator to a resonator built from CVI LGVD mirrors (reflectance 99.8%) would result in a circulating pulse energy of 1.8 mJ. If 99.99%-reflectance cavity mirrors with the same dispersion characteristics were used, a 36-mJ circulating pulse could be produced.

The maximum power P_max varies inversely with the circulating pulse length, so at first, shortening the incident pulses increases P_max. However, this can only continue until the comb bandwidth approaches the coupling bandwidth of the cavity, at which point the circulating pulse will begin to relatively broaden and the total power coupling β_net will decrease. Thus there exists an optimum incident pulse length which maximises P_max for a given finesse. The maximum instantaneous power P_max, obtained from the calculated circulating pulse power envelope, is plotted in Fig. 3(b) as a function of the incident pulse length τ_inc. The highest peak power that can be reached with R=99.8% is 8.3 GW per µJ of incident pulse energy, with 30 fs incident pulses. When R=99.99%, the optimal incident pulse length is 101 fs, producing a peak power in the cavity of 54 GW per µJ of incident pulse energy. This is only about 6.5 times higher than for the lower reflectance, despite a factor of 20 increase in finesse, because the effect of intracavity dispersion becomes more pronounced as the finesse is increased.

Currently, MLLs which produce pulses as short as the optimum pulse lengths illustrated in Fig. 3(b) tend to have lower average powers - and hence lower pulse energies - than the 240-fs oscillator of reference [9] discussed above. However, rapid advancements are being made in this field. For example, Beddard et al [10] have reported the construction of a 1.5-W average-power MLL which produced 13-fs pulses, using a standard brewster-cut Ti:sapphire crystal as the gain element. This oscillator had a pulse repetition rate of 110 MHz, but could have been modified (at the expense of some compactness) to operate at 25 MHz and produce 60-nJ pulses. It should also be possible for thin-disk MLLs of the type described in [9] to produce considerably shorter pulses without much reduction in average power. In light of these facts it is not unreasonable to expect 25-MHz repetition-rate oscillators with an average power of at least 20 W, over the entire range of pulse durations from 30 to 200 fs, to become available in the near future. Coupling such an oscillator into a resonator built from CVI LGVD mirrors (reflectance 99.8%) could result in a circulating pulse with a peak power as high as 6.6 GW. If 99.99%-reflectance cavity mirrors with the same dispersion characteristics were used, 43 GW of peak circulating power could be produced.

5. Conclusion

Current mirror technology enables the efficient coupling of ultrashort pulses into high-finesse cavities. Extremely high pulse energies and peak powers can be produced at repetition rates on the order of tens of MHz. The minimum pulse length for unity coupling, and the optimal pulse length to maximize peak power, can be determined in a straightforward way from the cavity mirror dispersion and reflectance characteristics. We have performed specific analyses for a particular model of mirror as an example, but have presented a technique that will handle any dispersion characteristics and could be extended to consider various dispersion-compensation schemes.