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We present a simple and quick, yet accurate method to measure the dispersion of high finesse optical cavities. By exciting the cavity with a femtosecond frequency comb and measuring the resonance condition as a function of optical frequency, the cavity’s dispersion can be determined with minimal uncertainty. Measurement results are presented from an evacuated reference cavity with low group delay dispersion as well as several differential, intra-cavity measurements of well known optical materials demonstrating the dynamic range and accuracy of this technique.
©2009 Optical Society of America
1. Introduction
Precision control and stabilization of femtosecond (fs) lasers has enabled advancement in fields ranging from optical frequency metrology to attosecond science to arbitrary optical waveform generation [1]. Such control has also enabled ultrashort pulse lasers to be efficiently coupled into high finesse optical cavities [2]. As substantial intracavity average (peak) powers on the order of 500 W (50 MW) can be achieved, this approach offers an alternative route to amplify pulses with widths ranging from several picoseconds to roughly 50 fs. By locating a gas jet inside the enhancement cavity, production of extreme ultra-violet (EUV) radiation produced via high harmonic generation (HHG) has been observed [3, 4]. As the full 100 MHz repetition rate of the oscillator is preserved with this technique, optical frequency combs in the EUV can also be produced. Moreover, the high repetition rate could offer alternative spectroscopic detection schemes in the EUV and higher photon flux than HHG via traditional chirped pulse amplification schemes, which typically run at repetition rates of 10 Hz to 10 kHz. In addition to EUV generation, other applications of fs enhancement cavities are being developed such as the use of the multiple resonant comb lines in a cavity for a massively parallel, cavity-enhanced spectroscopy with significantly enhanced sensitivity [5, 6].
For success in these and other applications, the fs pulse train must be efficiently coupled into the enhancement cavity. More specifically, to obtain the highest enhancement across the broadest bandwidth, the cavity must have a dispersion that is as close to zero as possible. Otherwise, the resulting frequency-dependent variation of the cavity’s free spectral range (FSR) will limit the number of equidistant spaced comb elements of the fs laser that can be simultaneously resonant in the cavity. Accordingly, it is critical that the dispersion of fs enhancement cavities be well characterized and subsequently minimized.
Several techniques have been developed over the years to measure the dispersive properties of optical elements and laser cavities as discussed in ref. [7]. Most of the methods that focus on cavity measurements examine changes in the resonance condition to infer the dispersive behavior. With recent work on fs enhancement cavities new methods have emerged that use femtosecond frequency combs (FFC) to make rapid, precision measurements of cavity dispersion [8, 9].
In this manuscript we approach measurement of dispersion of optical cavities via FFCs in a different manner. Our approach is intuitive and technically straightforward, and since it only relies on properties of the enhancement cavity, can be applied with any sufficiently stable FFC. It does not require the FFC to be locked to the enhancement cavity under test, and it only requires the FFC’s offset frequency to be stable; its actual value is immaterial. Moreover, this technique has a high dynamic range and is capable of measuring the group delay dispersion (GDD) as a function of optical frequency (and hence higher orders of dispersion as well). And it is capable of measuring GDD ranging from few fs2 to more than 103 fs2 with an uncertainty of < 5%. We present the background for our technique and describe our experiment, as well as some of its advantages and limitations relative to existing methods. We employ our method to minimize the GDD of low-loss mirrors in a fs enhancement cavity and confirm its capability by measuring the dispersion of several well-characterized optical materials placed inside the cavity.
2. GDD cavity measurement method
2.1. Background
Our technique for measuring the frequency dependence of the spectral phase, and hence the GDD, of the cavity hinges on the recognition that GDD and higher order dispersive effects cause different optical frequencies to be resonant in the cavity at different physical cavity lengths. The resonance condition states that the optical phase in a round trip through the cavity precesses an integer multiple of 2π or specifically,
where m is an integer, L is the physical length of the cavity, and Φ(ω) includes the remaining spectral phase terms (such as GDD, etc.) due to cavity mirrors or other elements. Specifically, the GDD can be calculated via [10],
Our task is to measure Φ(ω) and then calculate the GDD via the above expression. First, consider a pair of frequencies including a constant reference frequency, ωref and a variable frequency, ω. Their resonance conditions are,
Taking the difference in physical length between the two cases, Δd ≡ dref - d gives,
Solving for Φ(ω) and taking the second derivative with respect to frequency yields the GDD of the cavity in terms of this extra physical length Δd(ω). In principle, one could measure Δd(ω) using a fixed CW laser at a frequency ωref and a second tunable laser, at a frequency ω, and determine the change in resonance condition, Δd, as ω is tuned away from ωref. Given that an FFC has, in effect, many CW laser lines it is also possible to use such a source to quickly perform the measurement. This method is discussed below.
2.2. GDD measurement using a femtosecond frequency comb
When using an FFC to measure the resonance condition the two frequencies can be expressed as,
where ωrep is the repetition rate of the laser, ω 0 is the offset frequency, and m′ref and q′ are the laser mode numbers and at this point are general integers.
Substituting Eqs. (6) and (7) into the first term of Eq. (5) gives,
In Eq. (8) the term (mref - q) is only dependent on the enhancement cavity modes, while (mref q′ - m′ref q) is a cross term of both the cavity and laser modes. In order to simplify this expression it is helpful to first consider a cavity with a null phase response (Φ(ω) = Φ(ωref) = 0) and an FFC with no offset frequency (ω 0 = 0). In this case, there exists some enhancement cavity length that will tune the free spectral range ωFSR to match the ωrep of the laser. Since ωFSR is now independent of ω, all of the modes of the cavity will simultaneously align with the FFC’s comb elements meaning mref = m′ref and q = q′. Moreover, Δd = 0 for all ω and thus
Fig. 1. (color online) Simulation of (top) the resonance condition as a function of frequency ω and enhancement cavity length d, and (bottom) the resulting cavity reflection signal power P as a function of d. The cavity resonances are assumed to have a Lorentzian lineshape and the laser spectrum has a gaussian envelope. The overall Lorentzian lineshape of the reflection (in dotted red) as a function of d is derived in ref. [11], where a Gaussian lineshape was assumed for the cavity resonance and did not include dispersion, both of which lead to a disagreement with the simulation results. (a) The simple case where the incident FFC has zero offset frequency, ω 0 = 0, and the cavity has a null phase response, Φ(ω) = 0. At one unique enhancement cavity length d, all of the FFC’s comb elements are aligned to the enhancement cavity’s resonance peaks. In this case Δd = 0 and is labelled as the central fringe. (b) The effect of having dispersion in the enhancement cavity. Now even at the central fringe not all of the comb elements can simultaneously align to the cavity resonances at one particular cavity length d leading to a decrease in the cavity reflection amplitude. The curvature of the parabolic shape of a particular fringe number shown in the upper right pane is directly proportional to the GDD.
This condition is represented graphically in Fig. 1(a) and corresponds to the central fringe as denoted in ref. [11]. Since our goal is to achieve the largest intracavity power which occurs at this fringe, we focus on the effect of dispersion on this particular fringe. Any additional phase introduced by the cavity will change the cavity length required to bring the cavity to resonance, but not the specific cavity mode number with which each FFC mode is resonant. This results in a curvature of the central fringe, as shown in Fig. 1(b), but Eq. (9) still holds. For measurements of Δd (ω) on the central fringe, we can simplify the general form of Eq. (8) (now including cavity phase variations and a non-zero offset frequency) to obtain
Thus when GDD is present in the cavity, the resonance condition varies as a function of ω even on the central fringe as shown in Fig. 1(b). In particular, the GDD causes the fringes to have a parabolic curvature as a function of ω. From Δd, the GDD can now be calculated via
3. Experimental procedure and cavity measurements
3.1. Experimental setup
To measure the enhancement cavity’s GDD through the change in resonance condition Δd(ω), the cavity is excited with an FFC generated by a mode-locked Ti:Sapphire laser. The laser’s repetition rate is 50 MHz and its spectrum has a full width at half maximum (FWHM) bandwidth of Δλ ≈ 20 nm. The experimental setup is shown in Fig. 2.
Fig. 2. (color online) Experimental setup. A mode-locked Ti:Sapphire laser (with ω 0, the offset frequency, stabilized through the f - 2f interferometer) is coupled into a six mirror enhancement cavity which is under vacuum. The cavity reflection from the input coupler (IC) is separated into two spectrally resolved branches with gratings. PD1 is the reference branch photodetector; PD2 the measurement branch; IC cavity input coupler (0.25%); the small mirror attached to the PZT is used to sweep the cavity length. The scope is used to measure the (time) delay between the resonance conditions of ωref and ω as the cavity length is swept.
The signal used in the measurement is derived from the cavity reflection off the input coupler (IC) mirror and is separated into two spectrally resolved branches with one being the reference branch and the other the measurement branch. Each branch is spectrally dispersed with a grating at 1200 lines/mm. Following the gratings, a portion of the beam coupled into an optical spectrum analyzer (OSA) for measurement of the optical frequency. The remaining signals in each branch are individually focused onto two photodetectors. The cavity length is scanned with a PZT that is driven open loop with a linear ramp over a total of three FSRs in one cycle, and the two cavity reflection signals are displayed on an oscilloscope. It is important to ensure the PZT moves linearly through its sweep for proper calibration. The time delay between the reference and measurement resonance peaks appearing on the oscilloscope is converted to a distance Δd by calibrating the time delay against one FSR (i.e., one optical wavelength) at ωref. In our case, the scan rate results in a calibration factor of 1.728 ± 0.008 nm μs -1.
3.2. Cavity measurements
Initially, an “empty” enhancement cavity was measured. It consisted of five highly reflecting (R = 99.99%) and low GDD mirrors in an evacuated (< 2 mTorr) chamber. With an input coupler of T = 0.25% this cavity had a finesse of ℱ ≈ 2000. For this case, a representative data set for Δd is shown in Fig. 3 on the left axis. Also shown in this figure is the resulting GDD, which is calculated by stitching together polynomial fits (over a 5 nm bandwidth) of overlapping ≈ 0.4 nm intervals via Chebyshev differentiation matrices using spectral collocation methods [12]. We call this a Chebyshev fit. As expected by design this cavity gave an extremely low and flat GDD of (≤ 5fs2) for a bandwidth of Δλ ~ 40 nm, centered at 790 nm. We measured the GDD over a total bandwidth of 60 nm, or 20 dB down from ωref (the central wavelength of the pulse spectrum). While not necessary in our application, if a larger spectral range needs to be characterized, a broadened comb can be used to excite the cavity.
Fig. 3. (color online) Group delay dispersion (GDD) measurement of an evacuated six mirror cavity. The measured delay (black points, left axes) represent the raw data collected via an oscilloscope and is converted to a path length distance through a calibration of the free spectral range. The resulting GDD (red curve, right axis) is calculated via Eq. (11) and spectral collocation methods in numerical analysis [12]. The spectral limits of the measurement are due to the finite width of the FFC. The cavity mirrors were designed for low GDD centered at 790 nm.
As indicated in Eq. (10), the change in resonant condition is dependent on both the offset frequency and repetition rates of the FFC. At first we simply measured ω 0 and ωrep but fluctuations in the free-running ω 0 (over the duration of the measurement) were found to corrupt the GDD measurement, particularly when net cavity GDD was close to zero. To minimize this error ω 0 is detected and subsequently stabilized using a prism-based f to 2f referencing interferometer. A key point is the actual value of ω 0 is not necessary to determine the GDD. If the FFC used to probe the cavity is constructed from a fiber laser oscillator, an f to 2f interferometer is likely not necessary due to the inherent stability of the offset frequency in these systems [13]. The repetition rate of our FFC is sufficiently stable (with a free running drift of about 10 Hz, or 0.2 ppm) that it does not significantly affect the error in Δd. After adjusting the repetition rate to locate the cavity on the central fringe (via a PZT-mounted mirror) it remains free running during the measurement.
3.3. Additional GDD measurements
To confirm the accuracy of our GDD measurement technique we performed an additional set of measurements of optical elements with known values of GDD. First a 0.45 mm piece of sapphire was inserted into the cavity at the Brewster angle. To determine the GDD of the sapphire plate, the difference in Δd(ω) between the cavity with and without the sapphire plate was measured which was then fit to a polynomial where the GDD was calculated via Eq. (11). As the GDD of the sapphire is a smooth function of λ and has a value significantly greater than the empty cavity, both a third order polynomial fit to the entire data set as well as the previously used Chebyshev differentiation matrices were employed. These results are shown in Fig. 4(a). Both data analysis methods show excellent agreement with the Sellmeier equation prediction. As a second check, we evaluated the GDD of a 2.2 mm piece of fused silica in a similar fashion and again obtained excellent agreement with predicted values of GDD with results shown in Fig. 4(b).
As with nearly all GDD measurement techniques, the GDD itself is found indirectly from the raw data by curve fitting and/or taking numerical derivatives. With these processing procedures it is important to evaluate the resulting error on the obtained value for GDD. In Fig. 4(b), we show the upper and lower confidence intervals of the GDD arising from the uncertainty in the calculation via the Chebyshev differentiation matrices, equivalent to less than ±2 fs2 over ≈ 50 nm of bandwidth. The Sellemeir prediction falls within the uncertainty interval except at the extreme high wavelength limit of our measurement.
Fig. 4. (color online) (a) An intra-cavity measurement of a 0.45 mm thick piece of sapphire. The GDD is calculated both by a polynomial fit to the entire data set (right axis, red) and a Chebyshev fit (right axis, green). Both show excellent agreement with the Sellmeier prediction (blue). (b) An intra-cavity measurement of a 2.2 mm thick piece of fused silica. The uncertainty in the Chebyshev fit is displayed as the gray area.
As a further demonstration of the capabilities of this dispersion measurement approach, we also measured a cavity mirror with oscillating dispersion and an intra-cavity optic with large GDD. In the former, we selected one mirror from a pair of oscillation compensated GDD mirrors fabricated by Layertec (P/N102267) and it was used to replace a mirror from the cavity measured in Fig. 3. Shown in Fig. 5 is our measurement result compared with a theoretical prediction and a conventional GDD single pass measurement. These latter two curves are provided by the mirror manufacturer for a normal incidence angle. Accounting for the slight off-normal incident angle(> 5°) for our measurement this result agrees favorably with the expected value. Moreover, it indicates our technique can resolve changing in rapidly oscillating values of GDD. Lastly, a 6.6 mm thick piece of BK7 glass and a 14 mm piece of SF11 glass with predicted GDD of ≈ 360 fs2 and ≈ 3200 fs2 respectively were measured. Over a bandwidth of 50 nm, our measurement differed from the Sellmeier prediction by at most 5%.
3.4. Sources of error
Once the FFC’s offset frequency is stabilized, the largest source of error in determining the GDD is uncertainty in the measurement of the relative time delay between the cavity resonances at ωref and ω. This includes factors such as our ability to locate the peak of the resonance lineshapes amid residual dispersion due to the finite spectral resolution at the detectors, resolution of the oscilloscope, and proper calibration measurement. The latter could potentially be improved by using an encoded PZT for shorter distance calibration. As indicated in Fig. 3, for small GDD the relative delay changes on the order of 500 ns (a corresponding distance of 0.8 nm). Thus even at this minimal extreme, fluctuations in the differential physical path between the two branches in the experimental setup are insignificant. As the GDD increases, larger relative delays can be easily determined. However, once the relative delay reaches 25 μs (a distance of 42 nm), the delay measurements begin to suffer some error due to the finite bandwidth (0.5 nm) that is incident on the photodetectors which in manifested by a broadening of the cavity reflection signal. Aside from the GDD value itself, the cavity’s finesse also influences the delay measurement as a lower finesse causes the cavity reflection signal to broaden thereby making it harder to resolve the reflection peak at ω. For the finesse values considered here (ℱ > 500), this effect was not an issue.
Fig. 5. (color online) GDD of an cavity mirror with known oscillations in the GDD. The theoretical and experimentally specifications given by the manufacturer are shown as blue dashed and red dotted lines respecively. Also shown is our differential measurement with a Chebyshev fit as a solid black line
4. Conclusion
The experimental method presented in this paper can accurately measure the frequency dependence of the physical path length change of the resonance condition in an optical cavity. By exciting the optical cavity under test with a FFC, the change in resonance condition can be rapidly measured to provide the GDD over a large spectral region. The measurement is simple, robust, and has a high dynamic range with minimal error. Moreover, it only requires the FFC to have a stable offset frequency (its actual value is not needed) nor does the FCC need to be locked to the cavity under test. We have found it to be a valuable tool in optimizing fs enhancement cavities by minimizing the intracavity GDD. It could also be used in the design and fabrication of dispersion compensating mirrors.
Acknowledgments
We thank Rob Stead and Steve Plotkin for useful discussions. This research is supported by Natural Science and Engineering Research Council (NSERC), Canadian Foundation for Innovation, and British Columbia Knowledge Development Fund.
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