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Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 14, Iss. 6 — Mar. 20, 2006
  • pp: 2497–2511
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Carrier-envelope phase dynamics and noise analysis in octave-spanning Ti:sapphire lasers

Lia Matos, Oliver D. Mücke, Jian Chen, and Franz X. Kärtner  »View Author Affiliations


Optics Express, Vol. 14, Issue 6, pp. 2497-2511 (2006)
http://dx.doi.org/10.1364/OE.14.002497


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Abstract

We investigate the carrier-envelope phase dynamics of octave-spanning Ti:sapphire lasers and perform a complete noise analysis of the carrier-envelope phase stabilization. We model the effect of the laser dynamics on the residual carrier-envelope phase noise by deriving a transfer function representation of the octave-spanning frequency comb. The modelled phase noise and the experimental results show excellent agreement. This greatly enhances our capability of predicting the dependence of the residual carrier-envelope phase noise on the feedback loop filter, the carrier-envelope frequency control mechanism and the pump laser used.

© 2006 Optical Society of America

1. Introduction

Carrier-envelope phase control of femtosecond frequency combs has enabled major progress in frequency metrology, high-resolution laser spectroscopy, optical clocks and in high-field physics [1

1. T. Udem, J. Reichert, R. Holzwarth, and T.W. Hänsch, “Accurate measurement of large optical frequency differences with a mode-locked laser,” Opt. Lett. 24, 881–883 (1999). [CrossRef]

, 2

2. J. Ye, J. L. Hall, and S. A. Diddams, “Precision phase control of an ultrawide-bandwidth femtosecond laser: a network of ultrastable frequency marks across the visible spectrum,” Opt. Lett. 25, 1675–1677 (2000). [CrossRef]

, 3

3. D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, “Carrier-Envelope Phase Control of Femtosecond Mode-Locked Lasers and Direct Optical Frequency Synthesis,” Science 288, 635–639 (2000). [CrossRef] [PubMed]

, 4

4. R. Holzwarth, T. Udem, T.W. Hänsch, J. C. Knight, W. J. Wadsworth, and P. St. J. Russell, “Optical Frequency Synthesizer for Precision Spectroscopy,” Phys. Rev. Lett. 85, 2264–2268 (2000). [CrossRef] [PubMed]

, 5

5. J. von Zanthier, T. Becker, M. Eichenseer, A.Y. Nevsky, C. Schwedes, E. Peik, H. Walther, R. Holzwarth, J. Reichert, T. Udem, T.W. Hänsch, P.V. Pokasov, M. N. Skvortsov, and S. N. Bagayev, “Absolute frequency measurement of the In+ clock transition with a mode-locked laser,” Opt. Lett. 25, 1729–1731 (2000). [CrossRef]

, 6

6. M. Niering, R. Holzwarth, J. Reichert, P. Pokasov, T. Udem, M. Weitz, T.W. Hänsch, P. Lemonde, G. Santarelli, M. Abgrall, P. Laurent, C. Salomon, and A. Clairon, “Measurement of the Hydrogen 1S-2S Transition Frequency by Phase Coherent Comparison with a Microwave Cesium Fountain Clock,” Phys. Rev. Lett. 84, 5496–5499 (2000). [CrossRef] [PubMed]

, 7

7. T. Udem, S. A. Diddams, K. R. Vogel, C.W. Oates, E. A. Curtis, W. D. Lee, W. M. Itano, R. E. Drullinger, J. C. Bergquist, and L. Hollberg, “Absolute Frequency Measurements of the Hg+ and Ca Optical Clock Transitions with a Femtosecond Laser,” Phys. Rev. Lett. 86, 4996–4999 (2001). [CrossRef] [PubMed]

, 8

8. A. Baltus̆ka, T. Udem, M. Uiberacker, M. Hentschel, E. Goulielmakis, C. Gohle, R. Holzwarth, V. S. Yakovlev, A. Scrinzi, T.W. Hänsch, and F. Krausz, “Attosecond control of electronic processes by intense light fields,” Nature 421, 611–615 (2003). [CrossRef] [PubMed]

]. Active control of the carrier-envelope frequency, f CE, of modelocked lasers is a prerequisite for many applications in both time and frequency domain. For octave-spanning lasers which do not use intracavity prisms for dispersion compensation [9

9. L. Matos, D. Kleppner, O. Kuzucu, T. R. Schibli, J. Kim, E. P. Ippen, and F. X. Kaertner, “Direct frequency comb generation from an octave-spanning, prismless Ti:sapphire laser,” Opt. Lett. 29, 1683–1685 (2004). [CrossRef] [PubMed]

, 10

10. O. D. Mücke, R. Ell, A. Winter, J. Kim, J. R. Birge, L. Matos, and F. X. Kärtner, “Self-Referenced 200 MHz Octave-Spanning Ti:Sapphire Laser with 50 Attosecond Carrier-Envelope Phase Jitter,” Opt. Express 13, 5163–5169 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-13-5163. [CrossRef] [PubMed]

] this control can be achieved by utilizing the response of f CE to intracavity power, i.e., by controlling the intracavity pulse energy via modulation of the pump power [11

11. L. Xu, C. Spielmann, A. Poppe, T. Brabec, F. Krausz, and T.W. Hänsch, “Route to phase control of ultrashort light pulses,” Opt. Lett. 21, 2008–2010 (1996). [CrossRef] [PubMed]

, 12

12. S. T. Cundiff, J. Ye, and J. L. Hall, “Optical frequency synthesis based on mode-locked lasers,” Rev. Sci. Instrum. 72, 3749–3771 (2001). [CrossRef]

].

In order to include the gain dynamics in the noise analysis, a transfer function representation of the OSFC stabilization is given, which enables a quantitative noise analysis and eventually optimization of the overall system. It is shown that there is a major difference in the laser dynamics of the continuous-wave running laser and of the mode-locked laser, and that this dynamics has impact in the noise analysis. Measurement of the transfer function of the OSFC confirms the global behavior of the theoretical predictions and leads to excellent agreement between the computed and measured carrier-envelope phase dynamics and noise characteristics.

2. Carrier-envelope phase dynamics of mode-locked lasers

In this chapter we summarize the linear and nonlinear effects in the laser cavity that may lead to a carrier-envelope phase shift per round trip ΔϕCE and, therefore, contribute to the carrier-envelope frequency via

fCE=ΔϕCE2πTR.
(1)

Thus, in the frequency domain, the pulse train emitted by a mode-locked laser corresponds to a comb of frequencies equidistantly spaced by the repetition frequency frep and offset from zero by the carrier-envelope frequency f CE, i.e., fn = n f rep + f CE, with integer n [12

12. S. T. Cundiff, J. Ye, and J. L. Hall, “Optical frequency synthesis based on mode-locked lasers,” Rev. Sci. Instrum. 72, 3749–3771 (2001). [CrossRef]

]. If we assume that the laser operates at carrier frequency fc , then the complex carrier wave of the pulse is given by

ei2πfc(tzvp),
(2)

where v p is the phase velocity of the carrier wave in the cavity. In the absence of nonlinearities the phase velocity is simply the ratio between frequency and wavenumber due to the linear refractive index of the media in the cavity, i.e., v p = v p(fc ) = 2πfc =k(fc ). The envelope of a pulse that builds up in the cavity due to the mode-locking process will travel at the group velocity due to the presence of the linear media given by v g = v g(fc) = 2π[dk(fc)= dfc]-1. Therefore, after one round trip of the pulse over a distance 2L cav, which takes the time TR = 2L cav=v g, we obtain from Eq. (2) that the linear contribution to the carrier-envelope phase shift caused by the difference between phase and group velocities is

ΔϕCE=2πfc(1vg(fc)vp(fc))TR
(3)

and for the subsequent carrier-envelope frequency

fCE=fc(1vg(fc)vp(fc)).
(4)

In a dispersive medium, group and phase velocities depend on the carrier frequency. Therefore, if the carrier frequency shifts as a function of the intracavity pulse energy, the linear carrier-envelope frequency becomes energy and pump power dependent as found in Ref. [13

13. K.W. Holman, R. J. Jones, A. Marian, S. T. Cundiff, and J. Ye, “Detailed Studies and Control of Intensity- Related Dynamics of Femtosecond Frequency Combs From Mode-Locked Ti:Sapphire Lasers,” IEEE J. Sel. Top. Quantum Electron. 9, 1018–1024 (2003). [CrossRef]

].

In a mode-locked laser there are also nonlinear processes at work that may directly lead to an energy-depended carrier-envelope frequency. There are many effects that may contribute to such a shift. Here we re-derive briefly the effects due to the intensity-dependent refractive index as discussed by Haus and Ippen [14

14. H. A. Haus and E. P. Ippen, “Group velocity of solitons,” Opt. Lett. 26, 1654–1656 (2001). [CrossRef]

] for the case of a laser with strong soliton-like pulse shaping which can be evaluated analytically using soliton perturbation theory. We then argue that the same analysis holds for the general case where steady-state pulse formation is different from conventional soliton pulse shaping.

We start from the description of a mode-locked laser by a master equation of the form

TRAT=DirrevAiD22t2AA2A,
(5)

where we have already factored out the carrier wave [15

15. H. A. Haus, “Mode-Locking of Lasers,” IEEE J. Sel. Top. Quantum Electron. 6, 1173–1185 (2000). [CrossRef]

]. Here, D irrev is an operator that describes the irreversible dynamics occurring in a mode-locked laser such as gain, loss and saturable absorption. AA(T, t) is the slowly varying field envelope whose shape is investigated on two time scales: first, the global time T which is coarse grained on the time scale of the round-trip time TR , and second, the local time t which resolves the resulting pulse shape. A(T, t) is normalized such that |A(T, t)|2 is the instantaneous power and ∫dt |A(T, t)|2 the pulse energy at time T. D2=d2kdfc2Lcav8π2 is the group-velocity dispersion (GVD) parameter for the cavity. The Kerr coefficient is δ = (2π=λc)n 2 L/A eff, where λc is the carrier wavelength, n 2 is the nonlinear index in cm/W,L is the path length per round trip through the laser crystal, and A eff is the effective mode cross-sectional area. Strictly speaking Eq. (5) only applies to a laser with small changes in pulse shape within one round trip. Obviously this is not the case for few-cycle laser pulses where the pulse formation is governed by dispersion-managed mode locking [16

16. Y. Chen, F. X. Kärtner, U. Morgner, S. H. Cho, H. A. Haus, E. P. Ippen, and J. G. Fujimoto, “Dispersion-managed mode locking,” J. Opt. Soc. Am. B 16, 1999–2004 (1999). [CrossRef]

]. Nevertheless we want to understand this propagation equation as an effective equation of motion for the laser, where some of the parameters need to be determined self-consistently [15

15. H. A. Haus, “Mode-Locking of Lasers,” IEEE J. Sel. Top. Quantum Electron. 6, 1173–1185 (2000). [CrossRef]

].

Let us assume that the laser operates in the negative GVD regime, where a conventional soliton-like pulse forms, and that it is stabilized by the effective saturable absorber action against the filtering effects. Then the steady-state pulse solution is close to a fundamental soliton, i.e., a symmetric sech-shaped pulse that acquires an energy-dependent nonlinear phase shift per round trip due to the nonlinear index

A(T,t)=A0sech(tτ)eiϕsTTR,
(6)

see Ref. [14

14. H. A. Haus and E. P. Ippen, “Group velocity of solitons,” Opt. Lett. 26, 1654–1656 (2001). [CrossRef]

]. The nonlinear phase shift per round trip is

ϕs=12δA02.
(7)

A more careful treatment of the influence of the Kerr effect on the pulse propagation, especially for few-cycle pulses, needs to take the self-steepening of the pulse into account, i.e., the variation of the index during an optical cycle, by adding to the master equation the term [17

17. G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, 3rd edition, San Diego, 2001).

]

Lpert=δωct(A2A).
(8)

If the term in (8) is applied to a real and symmetric waveform, it generates an odd waveform. An odd waveform added as a perturbation to the symmetric waveform of the steady-state pulse leads, to first order, to a temporal shift of the steady-state pulse. For a soliton-like steady-state solution this timing shift can be evaluated with soliton perturbation theory, i.e., using the basis functions of the linearized operator, and results in a timing shift [14

14. H. A. Haus and E. P. Ippen, “Group velocity of solitons,” Opt. Lett. 26, 1654–1656 (2001). [CrossRef]

, 18

18. H. A. Haus and Y. Lai, “Quantum theory of soliton squeezing: a linearized approach,” J. Opt. Soc. Am. B 7, 386–392 (1990). [CrossRef]

]

TRΔt(T)T|selfsteep=Δ(1vg)=δωcA02=2ϕsωc.
(9)

In total, the compound effect of self-phase modulation, self-steepening, and linear dispersion on the pulse results in a carrier-envelope frequency of

fCE=fR2πΔϕCE=fRϕs2π+fcTΔt(T)selfsteep+fc(1vg(fc)vp(fc))
=fR4πδA02+2fR4πδA02+fc(1vg(fc)vp(fc)).
(10)

As the above expression shows, the term arising from the group delay change due to self-steepening is twice as large and of opposite sign compared with the one due to self-phase modulation. In total we obtain

fCE=fR4πδA02+fc(1vg(fc)vp(fc)).
(11)

We emphasize that soliton perturbation theory was only used in this derivation for analytical evaluation of timing shifts. If the pulse shaping in the laser is not governed by conventional soliton formation but rather by dispersion-managed soliton dynamics [16

16. Y. Chen, F. X. Kärtner, U. Morgner, S. H. Cho, H. A. Haus, E. P. Ippen, and J. G. Fujimoto, “Dispersion-managed mode locking,” J. Opt. Soc. Am. B 16, 1999–2004 (1999). [CrossRef]

] or a saturable absorber, the fundamental physics stays the same. If the steady-state solution has a real and symmetric component, the self-steepening term converts this component via the derivative into a real and odd term, which is to first order a timing shift in the autonomous dynamics of the free running mode-locked laser. Another mechanism that leads to a timing shift is, for example, the action of a slow saturable absorber, which absorbs only the front of the pulse. So care needs to be taken to include all relevant effects when a given laser system is analyzed.

The derivation above shows that the group velocity change due to self-steepening of the pulse leads to a change in sign of the energy-dependent contribution at fixed center wavelength of the pulse. We checked this prediction by observing the carrier-envelope frequency shift in a 200MHz repetition rate octave-spanning laser. We identified which of the peaks in the RF spectrum corresponded to the carrier-envelope frequency by inserting BaF2 material in the laser and observing which peak moved up in frequency (adding dispersion causes v g=v p to decrease, thus increasing the magnitude of the second term in Eq. (11)). We then varied the pump power and observed that the same peak also moved up in frequency, confirming the prediction by Eq. (11).

Fig. 1. (a) Carrier-envelope frequency shift (left axis) and relative change in intracavity power (right axis) as a function of pump power. Both curves make evident the presence of a cw breakthrough for pump powers above 6.3W, which is confirmed by the appearance of a cw component in the optical spectrum shown in (b). For clarity, the spectra are vertically offset by 10 dB.

Now, we can compare quantitatively the measured shift of f CE in Fig. 1 with the theoretically derived result from soliton perturbation theory. It turns out that the measurement and theory agree very well, despite the fact that the laser dynamics differ from the ideal conventional soliton operation regime.

The conversion coefficient

cfPintra=ΔfCEΔPintra,
(12)

where P intra is the intracavity power in mode-locked operation, can now be determined in terms of known cavity parameters under the assumption of a fixed pulse width

ΔfCEΔPintra=Ln24λcAeffτ,
(13)

where L = 4mm is the path length per round trip through the Ti:sapphire crystal, n 2 = 3 × 10-20m2/W is the nonlinear index of refraction for Ti:sapphire, λc = 800nm is the carrier wavelength, A eff = πw02 (w 0 = 16μm) is the mode cross sectional area, τ=τFWHM=1.76 with a pulse width of τFWHM = 5fs, and Pintra = 12W is the intracavity power. This expression gives, for a 5% change in intracavity power, a corresponding change in f CE of 9.6 MHz, which agrees well with the results shown in Fig. 1.

Despite this surprisingly good agreement, one has to be aware that the spot size and other parameters are rough estimates, which may easily change depending on cavity alignment. Also the pulse width is not constant in the crystal but rather stretching and compressing by more than a factor of 2. Nevertheless, the experimental observations in Fig. 1 agree well with the above theoretical estimate obtained from conventional soliton formation.

3. Noise analysis of carrier-envelope frequency stabilized lasers

The design and overall setup of carrier-envelope phase stabilized OSFC has been extensively discussed in Refs. [9

9. L. Matos, D. Kleppner, O. Kuzucu, T. R. Schibli, J. Kim, E. P. Ippen, and F. X. Kaertner, “Direct frequency comb generation from an octave-spanning, prismless Ti:sapphire laser,” Opt. Lett. 29, 1683–1685 (2004). [CrossRef] [PubMed]

, 10

10. O. D. Mücke, R. Ell, A. Winter, J. Kim, J. R. Birge, L. Matos, and F. X. Kärtner, “Self-Referenced 200 MHz Octave-Spanning Ti:Sapphire Laser with 50 Attosecond Carrier-Envelope Phase Jitter,” Opt. Express 13, 5163–5169 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-13-5163. [CrossRef] [PubMed]

]. We have built two of these OSFCs at 200MHz repetition rate with slightly different loop filter designs and pumped by different pump lasers. One is using a single-longitudinal-mode (slm) Nd:YVO4 pump laser (Verdi-V10, Coherent) and the other a multi-longitudinal-mode (mlm) Nd:YVO4 pump laser (Millennia Xs, Spectra-Physics). Fig. 2 depicts the measured relative intensity noise (RIN) for the mlm pump laser and slm pump laser. The mlm pump laser shows significantly higher RIN in the high-frequency range, whereas the slm pump laser has higher RIN at very low frequencies. Recently, S.Witte et al. [19

19. S. Witte, R. T. Zinkstok, W. Hogervorst, and K. S. E. Eikema, “Control and precise measurement of carrier-envelope phase dynamics,” Appl. Phys. B 78, 5–12 (2004). [CrossRef]

] also characterized the influence of RIN of these pump lasers on the residual carrier-envelope phase noise for a 10-fs Ti:sapphire laser employing chirped mirrors for intracavity dispersion compensation and external spectral broadening in a microstructure fiber, however, no rigorous noise analysis has been performed so far. The purpose of this section is to elucidate the impact of the RIN of different pump lasers on the finally achievable carrier-envelope phase noise and how the feedback mechanism and the design of the feedback loop employed impacts residual carrier-envelope phase noise.

Fig. 3 shows the corresponding spectrally-resolved and integrated carrier-envelope phase error measured for the two 200MHz OSFCs which are directly carrier-envelope phase locked. In agreement with the data measured by S. Witte et al., the high-frequency carrier-envelope phase noise is found to be larger for the mlm pump laser. The carrier-envelope phase fluctuations at lower frequencies, which are smaller for the mlm pump laser, are strongly suppressed by the large proportional-integral (PI) control loop gain and therefore do not contribute significantly to the residual carrier-envelope phase noise. The residual carrier-envelope phase fluctuations of the OSFC pumped by the mlm pump laser amount to 0.257 rad, compared to only 0.117 rad if the slm laser is used. First of all it is surprising that the mlm pumped system is only a factor of 2.2 worse than the slm pumped system despite the fact that the high-frequency noise of the mlm pump is worse. As we will see this is so because the feedback gain is large below 100 kHz. Obviously the system pumped by the mlm pump laser could easily do equally well if the feedback-loop bandwidth could be extended by one order of magnitude. The reasons why the high-frequency noise of the mlm pump can not be further suppressed will be elaborated further in the following feedback analysis.

Fig. 2. Relative intensity noise (RIN) of a Coherent Verdi-V6 (red curve) and a Spectra-Physics Millennia Xs (blue curve). The measurement noise floor is given by the black curve.
Fig. 3. Comparison of the carrier-envelope phase noise of a self-referenced 200MHz Ti:sapphire frequency comb pumped by a Coherent Verdi-V6 (red and green curves) and by a Spectra-Physics Millennia Xs (blue and orange curves).

From a control systems point of view, the f CE-stabilized laser is a phase-lock loop (PLL) [20

20. F. M. Gardner, Phaselock Techniques (John Wiley, 3rd edition, Hoboken, NJ, 2005). [CrossRef]

], where the voltage-controlled oscillator (VCO) is the carrier-envelope frequency controlled OSFC, which is the block indicated by the dashed frame in Fig. 4. When the laser is turned on, the carrier-envelope frequency f CE is determined by the cavity parameters and alignment, equivalent to the center frequency of oscillation of the VCO in a PLL. A voltage applied to the acousto-optic modulator (AOM) driver changes this frequency by an amount proportional to the equivalent VCO gain of the system. The model depicted in Fig. 4 includes all the electronic components used in the stabilization (phase detector, AOM and loop filter), whose transfer characteristics are easy to measure and to describe by an analytic model. Assuming an instantaneous response of the carrier-envelope frequency to the pump power via a constant C fPp, one is not able to reproduce the measured carrier-envelope phase noise spectrum. Therefore, the impact of the frequency response of the OSFC system must be taken into account in the analysis, which was done by considering the transfer function between the intracavity laser power (or pulse energy) and the pump power via the laser gain dynamics.

Fig. 4. Block diagram of the phase-lock loop (PLL) composed of the f CE-stabilized laser. The voltage-controlled oscillator (VCO) is depicted in the dashed box.

3.1. Transfer function representation for the pulse energy versus pump power dynamics

The starting point for derivation of the transfer function are laser rate equations for pulse energy and gain, which can be derived from the master equation (5) by proper elimination of the remaining degrees of freedom in the mode-locked laser as has been derived for example in the case of soliton lasers mode locked by slow saturable absorbers [21

21. F. X. Kärtner, L. R. Brovelli, D. Kopf, M. Kamp, I. Calasso, and U. Keller, “Control of solid state laser dynamics by semiconductor devices,” Opt. Eng. 34, 2024–2036 (1995). [CrossRef]

, 22

22. F. X. Kärtner, I. D. Jung, and U. Keller, ”Soliton Mode-Locking with Saturable Absorbers,” IEEE J. Sel. Top.Quantum Electron. 2, 540–556 (1996). [CrossRef]

]. One can write

TRdEdT=(glq(E))E
(14)
TRdgdT=gg0τLTRgEEsat,
(15)

where we have used:

  • T R = cavity round-trip time,
  • τL = upper state lifetime,
  • l = total non-saturable loss,
  • q(E) ≡ q ml(E) = effective energy-dependent saturable absorber and filter loss (for more details, e.g., see Ref. [22

    22. F. X. Kärtner, I. D. Jung, and U. Keller, ”Soliton Mode-Locking with Saturable Absorbers,” IEEE J. Sel. Top.Quantum Electron. 2, 540–556 (1996). [CrossRef]

    ]),
  • g 0 = small-signal gain, which is proportional to pump power,
  • E sat = saturation energy of the gain medium.

TREdT=qEsEsΔE+EsΔg
(16)
TRgdT=TRτstimΔggsEsatΔE+TRτLΔg0,
(17)
Fig. 5. Mode locking related energy-dependent loss q ml(E). At point 1, where dq ml=dE<0, mode locking stability is reduced. Stable mode locking occurs at operating point 2, where dq ml=dE > 0 or equal to 0, if the pulse energy is not yet too high leading to cw breakthrough or to multiple pulse operation.

where τstim is the stimulated lifetime given by τL (1+τL E s=T R E sat)-1. By taking the Laplace transform of the above equations, it is straightforward to derive a pump power to pulse energy transfer function, by writing g 0 as K g0 P p. Defining the pump parameter r = 1+τL E s=T R E sat, which indicates how many times the laser operates above threshold, we arrive at

ΔE˜=PsK0τLs2+(RτL+qEsPs)s+PsqEsRτL+R1τLτp(1+qsL)ΔP˜p,
(18)

where τp = T R=l is the photon decay time due to the linear cavity losses in cw operation. So far we have considered the mode-locked case, but a similar relation can be obtained for the intracavity power in cw operation simply by setting the saturable absorber terms in Eq. (18) to zero. However, the laser parameters, like mode cross section in the gain medium likely change values when the laser changes from cw operation to mode-locked operation, but since we have no way of measuring them in mode-locked operation, we have estimated them based on our knowledge of laser parameters in cw operation and used those to examine the effect of the inclusion of the saturable absorber into the transfer function. Fig. 6 shows the pump power to intracavity power transfer functions, in amplitude and phase, for cw and mode-locked operation for different values of the term qEsPs. It is obvious that the mode locking of the laser drastically changes the transfer characteristic between pump power and intracavity power due to the pulse operation, which introduces the term qEsPs into the rate equation (16). Depending on its sign this term enhances (for ∂q/∂E < 0 it may lead to Q-switching when large enough) or strongly
Fig. 6. Calculated amplitude and phase response of intracavity power with pump power for cw operation (red) and for different values of saturable absorption: qEsPs=10rτL(blue),qEsPs=50rτL(green), and qEsPs=150rτL(orange).
Fig. 7. Schematic of the transfer function measurement setup. In order to measure only the contribution from the OSFC laser itself, a first calibration measurement is performed with PD1 measuring the AOM and AOM driver response (PD1 and PD2 are the same photo detector). The OSFC laser transfer function is measured by detecting with PD2 the reflection from one intracavity BaF2 plate and subtracting the AOM and AOM driver response.
damps (for ∂q/∂E > 0) intracavity energy fluctuations. In cw operation its absence usually leads to pronounced relaxation oscillations, see Fig. 6. As we can see, the stronger the effective inverse saturable absorption, the more damped become the relaxation oscillations in the laser, and the weaker becomes the response at all frequencies. This result has to be expected, because the stronger the inverse saturable absorption the more clamped become the pulse energy and the average power.

Fig. 8. Measured (red and blue, measurement setup in Fig. 7) and modelled (green and orange) amplitude and phase response of OSFC laser, in cw and mode-locked operation.
Fig. 9. Block diagram describing the addition of intensity noise to the laser PLL.

3.2. Determination of the carrier-envelope phase error

To calculate the carrier-envelope phase noise spectrum of the OSFC, a linear noise analysis is performed. The block diagram in Fig. 9 shows the closed-loop system. The input noise source, characterized by the power spectral density of the pump noise SP(s), which is the RIN multiplied by the square of the pump power, is converted to the carrier-envelope phase noise spectral density S ϕ(s) in the laser and is partially suppressed in the feedback loop. The feedback path consists of the phase detector, the loop filter and the AOM, with transfer functions denoted by H PD(s), H LF(s) and H AOM(s), respectively. Table 1 shows the corresponding analytic expressions. The loop filter consists of a simple PI controller with time constants τ1, τ2, and τ3. The fused-silica AOM used in our experiments has a 5mm aperture, and the unfocused pump beam of 2.3mm diameter is placed as close as possible to the piezo-electric transducer. Then, to first order and up to a small drop in amplitude for higher frequencies, the AOM is equivalent to a delay line with a propagation delay given by the time it takes for the acoustic wave to travel from the piezo-electric transducer to the optical beam. We set this delay to 1.73μs to match the measured transfer function. The limitations of the bandwidth of the AOM are not taken into account here because the measurements show that it is approximately flat out to 1 MHz. The calculated and measured transfer functions for the loop filter and the AOM are shown in Fig. 10. We consider the phase detector (Analog Devices AD9901) transfer function to be flat based on its datasheet.

Table 1. Transfer functions of the Ti:sapphire laser, phase detector, loop filter, and AOM.

table-icon
View This Table
Fig. 10. Measured (blue) and calculated (red) transfer functions for loop filter (top) and AOM (bottom). Amplitude response is on the left and phase response on the right.

In the noise analysis we consider the case of the OSFC pumped by the multi-longitudinal-mode pump laser because of the increased high-frequency noise in such systems as discussed above. Given the RIN measurement in Fig. 2 as the input noise source, we calculate Sϕ(s) using the transfer functions in Table 1 to derive the closed loop transfer function that describes the conversion of pump noise S P(s) to carrier-envelope phase noise Sϕ(s),

HCL(s)=HTisa(s)1+HTisa(s)HPD(s)HLF(s)HAOM(s).
(19)

Then, Sϕ(s) is obtained by multiplying the intensity noise spectrum of the pump laser, SP(s), with the magnitude squared of H CL given by expression (19)

Sϕ(s)=HCL(s)2SP(s).
(20)

Fig. 11. Carrier-envelope phase noise spectrum of the Millennia-Xs-pumped OSFC. The red curve is the calculated spectrum, the blue curve is the measurement for phase locked carrier-envelope frequency.
HCLwn(s)=11+HTisa(s)HPD(s)HLF(s)HAOM(s),
(21)

where the magnitude of the noise source SPD was estimated based on the measurement. Improved noise performance could be obtained by increasing the closed-loop bandwidth, which is currently about 100 kHz. This bandwidth is dictated by the phase margin of the feedback loop, which according to the Nyquist theorem may run unstable when the gain is larger than 1 while the phase approaches 180° [23

23. G. F. Franklin, J. D. Powell, and A. Emami-Naeini, Feedback control of dynamic systems (Addison-Wesley, 3rd edition, Reading, MA, 1994).

]. From a control systems perspective, the VCO has integrating behavior, i.e., it integrates a frequency deviation into a phase deviation, thus causing the open-loop transfer function to start off with a -90° phase, as can be seen in Fig. 12. When the bandwidth of the gain medium is approached, additional phase accumulates from the gain dynamics and the time delay from the AOM (see Fig. 12), which renders the system unstable if the gain setting is not properly reduced. Of course, for a reasonably stable system and optimum operation of the feedback loop, large enough gain and phase margins are necessary. This imposes a limitation to the maximum loop gain since the jointly added phase from H AOM(s) and H Tisa(s) contribute to a decrease in the phase margin. However, now that the different phase contributions to the feedback loop are well understood, the noise suppression at high frequencies could be improved by custom design of the control electronics. For example, by adding a lead-lag compensator [23

23. G. F. Franklin, J. D. Powell, and A. Emami-Naeini, Feedback control of dynamic systems (Addison-Wesley, 3rd edition, Reading, MA, 1994).

], the limitation arising from the added phases could be reduced. Furthermore, the contribution to the open-loop phase from the time delay of the AOM could be further suppressed by replacing the AOM by an electro-optic modulator (EOM) or using an AOM/EOM combination. The intensity noise in the frequency range 1 kHz-10MHz amounts to 0.06%, which could easily be reduced by an EOM (e.g., made of rubidium titanyl phosphate (RTP)) with a ~1V modulation amplitude. It remains to be seen if an AOM/EOM combination indeed can further suppress the phase error.

Fig. 12. Amplitude (top) and phase (bottom) of open-loop transfer function of the laser PLL (red curves) corresponding to Fig. 9. The contribution from the loop filter (green), AOM (blue) and laser dynamics (orange) to the open-loop phase are also shown. The zoomed plots on the right-hand side confirm that the PLL satisfies the Nyquist stability criterion (gain margin 2 dB, phase margin 12°).

4. Conclusion

A complete noise analysis of a carrier-envelope phase stabilized OSFCs was performed. It was found that the inclusion of the pump power to intracavity power transfer function of the laser, that was derived from the linearized rate equations for the system, was essential to obtain correct predictions for the carrier-envelope phase noise behavior. Most importantly the finite response time of the gain introduces a phase delay that adds a phase to the closed-loop transfer function of the system which, together with the phase delay due to the AOM, decreases the phase margin and limits the loop bandwidth to ~100 kHz. This limited bandwidth is insufficient to fully suppress the intensity-dependent high-frequency carrier-envelope phase noise in the system. Because these different contributions are now well understood, the stabilization electronics can be further optimized to suppress the noise in OSFCs pumped by multi-longitudinal-mode lasers to the level of those pumped by single-longitudinal-mode lasers.

Acknowledgements

This research has been supported by ONR N00014-02-1-0717 and AFOSR FA9550-04-1-0011. O. D. Mücke acknowledges support from the Alexander von Humboldt Foundation. We thank E. P. Ippen for useful comments on the manuscript.

References and links

1.

T. Udem, J. Reichert, R. Holzwarth, and T.W. Hänsch, “Accurate measurement of large optical frequency differences with a mode-locked laser,” Opt. Lett. 24, 881–883 (1999). [CrossRef]

2.

J. Ye, J. L. Hall, and S. A. Diddams, “Precision phase control of an ultrawide-bandwidth femtosecond laser: a network of ultrastable frequency marks across the visible spectrum,” Opt. Lett. 25, 1675–1677 (2000). [CrossRef]

3.

D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, “Carrier-Envelope Phase Control of Femtosecond Mode-Locked Lasers and Direct Optical Frequency Synthesis,” Science 288, 635–639 (2000). [CrossRef] [PubMed]

4.

R. Holzwarth, T. Udem, T.W. Hänsch, J. C. Knight, W. J. Wadsworth, and P. St. J. Russell, “Optical Frequency Synthesizer for Precision Spectroscopy,” Phys. Rev. Lett. 85, 2264–2268 (2000). [CrossRef] [PubMed]

5.

J. von Zanthier, T. Becker, M. Eichenseer, A.Y. Nevsky, C. Schwedes, E. Peik, H. Walther, R. Holzwarth, J. Reichert, T. Udem, T.W. Hänsch, P.V. Pokasov, M. N. Skvortsov, and S. N. Bagayev, “Absolute frequency measurement of the In+ clock transition with a mode-locked laser,” Opt. Lett. 25, 1729–1731 (2000). [CrossRef]

6.

M. Niering, R. Holzwarth, J. Reichert, P. Pokasov, T. Udem, M. Weitz, T.W. Hänsch, P. Lemonde, G. Santarelli, M. Abgrall, P. Laurent, C. Salomon, and A. Clairon, “Measurement of the Hydrogen 1S-2S Transition Frequency by Phase Coherent Comparison with a Microwave Cesium Fountain Clock,” Phys. Rev. Lett. 84, 5496–5499 (2000). [CrossRef] [PubMed]

7.

T. Udem, S. A. Diddams, K. R. Vogel, C.W. Oates, E. A. Curtis, W. D. Lee, W. M. Itano, R. E. Drullinger, J. C. Bergquist, and L. Hollberg, “Absolute Frequency Measurements of the Hg+ and Ca Optical Clock Transitions with a Femtosecond Laser,” Phys. Rev. Lett. 86, 4996–4999 (2001). [CrossRef] [PubMed]

8.

A. Baltus̆ka, T. Udem, M. Uiberacker, M. Hentschel, E. Goulielmakis, C. Gohle, R. Holzwarth, V. S. Yakovlev, A. Scrinzi, T.W. Hänsch, and F. Krausz, “Attosecond control of electronic processes by intense light fields,” Nature 421, 611–615 (2003). [CrossRef] [PubMed]

9.

L. Matos, D. Kleppner, O. Kuzucu, T. R. Schibli, J. Kim, E. P. Ippen, and F. X. Kaertner, “Direct frequency comb generation from an octave-spanning, prismless Ti:sapphire laser,” Opt. Lett. 29, 1683–1685 (2004). [CrossRef] [PubMed]

10.

O. D. Mücke, R. Ell, A. Winter, J. Kim, J. R. Birge, L. Matos, and F. X. Kärtner, “Self-Referenced 200 MHz Octave-Spanning Ti:Sapphire Laser with 50 Attosecond Carrier-Envelope Phase Jitter,” Opt. Express 13, 5163–5169 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-13-5163. [CrossRef] [PubMed]

11.

L. Xu, C. Spielmann, A. Poppe, T. Brabec, F. Krausz, and T.W. Hänsch, “Route to phase control of ultrashort light pulses,” Opt. Lett. 21, 2008–2010 (1996). [CrossRef] [PubMed]

12.

S. T. Cundiff, J. Ye, and J. L. Hall, “Optical frequency synthesis based on mode-locked lasers,” Rev. Sci. Instrum. 72, 3749–3771 (2001). [CrossRef]

13.

K.W. Holman, R. J. Jones, A. Marian, S. T. Cundiff, and J. Ye, “Detailed Studies and Control of Intensity- Related Dynamics of Femtosecond Frequency Combs From Mode-Locked Ti:Sapphire Lasers,” IEEE J. Sel. Top. Quantum Electron. 9, 1018–1024 (2003). [CrossRef]

14.

H. A. Haus and E. P. Ippen, “Group velocity of solitons,” Opt. Lett. 26, 1654–1656 (2001). [CrossRef]

15.

H. A. Haus, “Mode-Locking of Lasers,” IEEE J. Sel. Top. Quantum Electron. 6, 1173–1185 (2000). [CrossRef]

16.

Y. Chen, F. X. Kärtner, U. Morgner, S. H. Cho, H. A. Haus, E. P. Ippen, and J. G. Fujimoto, “Dispersion-managed mode locking,” J. Opt. Soc. Am. B 16, 1999–2004 (1999). [CrossRef]

17.

G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, 3rd edition, San Diego, 2001).

18.

H. A. Haus and Y. Lai, “Quantum theory of soliton squeezing: a linearized approach,” J. Opt. Soc. Am. B 7, 386–392 (1990). [CrossRef]

19.

S. Witte, R. T. Zinkstok, W. Hogervorst, and K. S. E. Eikema, “Control and precise measurement of carrier-envelope phase dynamics,” Appl. Phys. B 78, 5–12 (2004). [CrossRef]

20.

F. M. Gardner, Phaselock Techniques (John Wiley, 3rd edition, Hoboken, NJ, 2005). [CrossRef]

21.

F. X. Kärtner, L. R. Brovelli, D. Kopf, M. Kamp, I. Calasso, and U. Keller, “Control of solid state laser dynamics by semiconductor devices,” Opt. Eng. 34, 2024–2036 (1995). [CrossRef]

22.

F. X. Kärtner, I. D. Jung, and U. Keller, ”Soliton Mode-Locking with Saturable Absorbers,” IEEE J. Sel. Top.Quantum Electron. 2, 540–556 (1996). [CrossRef]

23.

G. F. Franklin, J. D. Powell, and A. Emami-Naeini, Feedback control of dynamic systems (Addison-Wesley, 3rd edition, Reading, MA, 1994).

OCIS Codes
(120.3940) Instrumentation, measurement, and metrology : Metrology
(320.7090) Ultrafast optics : Ultrafast lasers
(320.7160) Ultrafast optics : Ultrafast technology

ToC Category:
Ultrafast Optics

History
Original Manuscript: November 22, 2005
Revised Manuscript: March 13, 2006
Manuscript Accepted: March 14, 2006
Published: March 20, 2006

Citation
Lia Matos, Oliver D. Mücke, Jian Chen, and Franz X. Kärtner, "Carrier-envelope phase dynamics and noise analysis in octave-spanning Ti:sapphire lasers," Opt. Express 14, 2497-2511 (2006)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-6-2497


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References

  1. T. Udem, J. Reichert, R. Holzwarth, and T.W. Hänsch, "Accurate measurement of large optical frequency differences with a mode-locked laser," Opt. Lett. 24, 881-883 (1999). [CrossRef]
  2. J. Ye, J. L. Hall, and S. A. Diddams, "Precision phase control of an ultrawide-bandwidth femtosecond laser: a network of ultrastable frequency marks across the visible spectrum," Opt. Lett. 25, 1675-1677 (2000). [CrossRef]
  3. D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, "Carrier-Envelope Phase Control of Femtosecond Mode-Locked Lasers and Direct Optical Frequency Synthesis," Science 288, 635-639 (2000). [CrossRef] [PubMed]
  4. R. Holzwarth, T. Udem, T.W. Hänsch, J. C. Knight, W. J. Wadsworth, P. St. J. Russell, "Optical Frequency Synthesizer for Precision Spectroscopy," Phys. Rev. Lett. 85, 2264-2268 (2000). [CrossRef] [PubMed]
  5. J. von Zanthier, T. Becker, M. Eichenseer, A.Y. Nevsky, C. Schwedes, E. Peik, H. Walther, R. Holzwarth, J. Reichert, T. Udem, T.W. H¨ansch, P.V. Pokasov, M. N. Skvortsov, and S. N. Bagayev, "Absolute frequency measurement of the In+ clock transition with a mode-locked laser," Opt. Lett. 25, 1729-1731 (2000). [CrossRef]
  6. M. Niering, R. Holzwarth, J. Reichert, P. Pokasov, T. Udem, M. Weitz, T.W. Hänsch, P. Lemonde, G. Santarelli, M. Abgrall, P. Laurent, C. Salomon, and A. Clairon, "Measurement of the Hydrogen 1S-2S Transition Frequency by Phase Coherent Comparison with a Microwave Cesium Fountain Clock," Phys. Rev. Lett. 84, 5496-5499 (2000). [CrossRef] [PubMed]
  7. T. Udem, S. A. Diddams, K. R. Vogel, C.W. Oates, E. A. Curtis, W. D. Lee, W. M. Itano, R. E. Drullinger, J. C. Bergquist, and L. Hollberg, "Absolute Frequency Measurements of the Hg+ and Ca Optical Clock Transitions with a Femtosecond Laser," Phys. Rev. Lett. 86, 4996-4999 (2001). [CrossRef] [PubMed]
  8. A. Baltuska, T. Udem, M. Uiberacker, M. Hentschel, E. Goulielmakis, C. Gohle, R. Holzwarth, V. S. Yakovlev, A. Scrinzi, T.W. H¨ansch, and F. Krausz, "Attosecond control of electronic processes by intense light fields," Nature 421, 611-615 (2003). [CrossRef] [PubMed]
  9. L. Matos, D. Kleppner, O. Kuzucu, T. R. Schibli, J. Kim, E. P. Ippen, and F. X. Kaertner, "Direct frequency comb generation from an octave-spanning, prismless Ti:sapphire laser," Opt. Lett. 29, 1683-1685 (2004). [CrossRef] [PubMed]
  10. O. D. Mücke, R. Ell, A. Winter, J. Kim, J. R. Birge, L. Matos, and F. X. Kärtner, "Self-Referenced 200 MHz Octave-Spanning Ti:Sapphire Laser with 50 Attosecond Carrier-Envelope Phase Jitter," Opt. Express 13, 5163-5169 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-13-5163. [CrossRef] [PubMed]
  11. L. Xu, C. Spielmann, A. Poppe, T. Brabec, F. Krausz, and T.W. Hänsch, "Route to phase control of ultrashort light pulses," Opt. Lett. 21, 2008-2010 (1996). [CrossRef] [PubMed]
  12. S. T. Cundiff, J. Ye, and J. L. Hall, "Optical frequency synthesis based on mode-locked lasers," Rev. Sci. Instrum. 72, 3749-3771 (2001). [CrossRef]
  13. K.W. Holman, R. J. Jones, A. Marian, S. T. Cundiff, and J. Ye, "Detailed Studies and Control of Intensity-Related Dynamics of Femtosecond Frequency Combs From Mode-Locked Ti:Sapphire Lasers," IEEE J. Sel. Top. Quantum Electron. 9, 1018-1024 (2003). [CrossRef]
  14. H. A. Haus and E. P. Ippen, "Group velocity of solitons," Opt. Lett. 26, 1654-1656 (2001). [CrossRef]
  15. H. A. Haus, "Mode-Locking of Lasers," IEEE J. Sel. Top. Quantum Electron. 6, 1173-1185 (2000). [CrossRef]
  16. Y. Chen, F. X. Kärtner, U. Morgner, S. H. Cho, H. A. Haus, E. P. Ippen, and J. G. Fujimoto, "Dispersion-managed mode locking," J. Opt. Soc. Am. B 16, 1999-2004 (1999). [CrossRef]
  17. G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, 3rd edition, San Diego, 2001).
  18. H. A. Haus and Y. Lai, "Quantum theory of soliton squeezing: a linearized approach," J. Opt. Soc. Am. B 7, 386-392 (1990). [CrossRef]
  19. S. Witte, R. T. Zinkstok, W. Hogervorst, K. S. E. Eikema, "Control and precise measurement of carrier-envelope phase dynamics," Appl. Phys. B 78, 5-12 (2004). [CrossRef]
  20. F. M. Gardner, Phaselock Techniques (John Wiley, 3rd edition, Hoboken, NJ, 2005). [CrossRef]
  21. F. X. Kärtner, L. R. Brovelli, D. Kopf, M. Kamp, I. Calasso, and U. Keller, "Control of solid state laser dynamics by semiconductor devices," Opt. Eng. 34, 2024-2036 (1995). [CrossRef]
  22. F. X. Kärtner, I. D. Jung, and U. Keller, "Soliton Mode-Locking with Saturable Absorbers," IEEE J. Sel. Top. Quantum Electron. 2, 540-556 (1996). [CrossRef]
  23. G. F. Franklin, J. D. Powell, and A. Emami-Naeini, Feedback control of dynamic systems (Addison-Wesley, 3rd edition, Reading, MA, 1994).

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