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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 16, Iss. 18 — Sep. 1, 2008
  • pp: 14186–14191
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Amplitude and envelope phase noise of a modelocked laser predicted from its noise transfer function and the pump noise power spectrum

Theresa D. Mulder, Ryan P. Scott, and Brian H. Kolner  »View Author Affiliations


Optics Express, Vol. 16, Issue 18, pp. 14186-14191 (2008)
http://dx.doi.org/10.1364/OE.16.014186


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Abstract

The amplitude and envelope phase noise of a modelocked Ti:sapphire laser are predicted based on the power spectral density of the pump laser and the noise transfer process. Pump laser noise is found to transfer directly to the modelocked laser’s amplitude and phase noise power spectra through the noise transfer function (NTF) which is independently measured. We find good agreement between the shapes and absolute values of the predicted and measured Ti:sapphire AM and PM noise spectra except in regions where additional environmental influences affect the Ti:sapphire laser. The experiments were conducted with both a single-mode and a multi-mode diode-pumped solid-state pump laser.

© 2008 Optical Society of America

1. Introduction

The stability of modelocked lasers has taken on tremendous importance in recent years, especially since the invention of the carrier-envelope offset (CEO)-stabilized, self-referencing technique which forms the basis of the new laser clockworks [1

1. 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, 2264–2267 (2000). [CrossRef] [PubMed]

, 2

2. 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]

]. In spite of the impressive results obtained with this new technique, the performance of feedback control systems depends, at least in part, on the performance of the object being controlled prior to closing the loop. Thus it is still important to study, quantify, and thereby predict the noise properties of the free-running modelocked laser. This may lend insight into better laser designs and what mechanisms determine the ultimate stability of laser clocks.

Today, the forefront of ultrafast laser performance is dominated by passive modelocking and, in particular, the fast saturable absorber-like processes such as Kerr-lens modelocking. The analytical approach to understanding and designing these types of lasers that is widely used today is based on the “master equation” of Haus [8

8. H. A. Haus, “Theory of mode locking with a fast saturable absorber,” J. Appl. Phys. 46, 3049–3058 (1975). [CrossRef]

] and was further developed to describe colliding pulse (CPM) [9

9. O. E. Martinez, R. L. Fork, and J. P. Gordon, “Theory of passively mode-locked lasers for the case of a nonlinear complex-propagation coefficient,” J. Opt. Soc. Am. B 2, 753–760 (1985). [CrossRef]

], additive pulse (APM) and Kerr-lens (KLM) modelocking [10

10. H. A. Haus, J. G. Fujimoto, and E. P. Ippen, “Analytic Theory of Additive Pulse and Kerr Lens Mode Locking,” IEEE J. Quantum Electron. 28, 2086–2096 (1992). [CrossRef]

]. The issue of applying the master equation to questions of noise and the stability of the solutions was addressed early on by Haus and Mecozzi [10

10. H. A. Haus, J. G. Fujimoto, and E. P. Ippen, “Analytic Theory of Additive Pulse and Kerr Lens Mode Locking,” IEEE J. Quantum Electron. 28, 2086–2096 (1992). [CrossRef]

] and more recently by Kapitula, et al. [11

11. T. Kapitula, J. N. Kutz, and B. Sandstede, “Stability of pulses in the master mode-locking equation,” J. Opt. Soc. Am. B 19, 740–746 (2002). [CrossRef]

] and Menyuk, et al. [12

12. C. R. Menyuk, J. K. Wahlstrand, J. Willits, R. P. Smith, T. Schibli, and S. T. Cundiff, “Pulse dynamics in mode-locked lasers: relaxation oscillations and frequency pulling,” Opt. Express 15, 6677–6689 (2007). [CrossRef] [PubMed]

]. What is common to most of these analyses is the goal of describing from first principles the mechanisms responsible for the intrinsic noise of the modelocked laser and the stability of the solutions when interrupted by a perturbation. A key finding from this work, and one that makes sense intuitively, is that fluctuations in the laser gain will have a direct impact on several of the measurable parameters of the modelocked laser.

With the advent of very low-noise diode-pumped solid-state (DPSS) pump lasers, which have become the norm for pumping KLM lasers, it seems reasonable that the modelocked laser might be simply treated as a linear system to which we can ascribe a transfer function that converts the noise of the pump to the important properties of the modelocked laser such as amplitude noise and timing jitter. These properties are, arguably, the most technically relevant and straightforward to measure.

The purpose of the study presented here is to demonstrate that there is a direct correlation between the noise properties of a modelocked laser’s pump source and the noise of the laser itself. Apart from environmentally-induced noise due to temperature fluctuations, air currents, acoustic vibrations, etc., and spontaneous emission noise, fluctuations in the pumping rate of any laser should be directly transferred to operating properties of the laser such as amplitude and timing stability, and therefore show up as additional noise. We can quantify the sensitivity of any laser to the fluctuations in its pump source by intentionally placing a small amount of modulation on the pump (stimulus) and measuring the effects on the laser (response) as we sweep the pump modulation frequency over the range of interest. By keeping track of both the amplitude and phase of the stimulus and response in this process we establish a complex noise transfer function (NTF) for the laser [13

13. R. P. Scott, T. D. Mulder, K. A. Baker, and B. H. Kolner, “Amplitude and phase noise sensitivity of modelocked Ti:sapphire lasers in terms of a complex noise transfer function,” Opt. Express 15, 9090–9095 (2007). [CrossRef] [PubMed]

]. This characterization can be applied to both amplitude noise and envelope phase noise (timing jitter) of a modelocked laser. Once the NTF is known, we can predict the amplitude and phase noise power spectral densities (PSDs) of the modelocked laser by considering the NTF as representing the transfer function of a linear system. For a random process passing through any linear system, the output power spectral density is Sout(ω)=|H(ω)|2 Sin(ω) where H(ω) is the transfer function of the linear system and Sin(ω) is the input power spectrum. Thus, we can predict the laser output noise power spectra (either AM or PM) once we have the corresponding noise transfer function characterized.

Fig. 1. Transfer of pump noise to AM and envelope PM noise of a modelocked Ti:sapphire laser.

The process of transferring pump noise to the AM and PM noise of a modelocked laser is shown schematically in Fig. 1. The baseband pump noise PSD, SP(ωm), is first measured over a wide range of frequencies (0.1 Hz–10 MHz). Next, the complex AM and PM noise transfer functions (NTFs) are measured over the same range. The NTF is defined as the ratio of the induced amplitude, L(ωm), or phase, β˜(ωm), modulation index to the pump modulation index, mp, at a given driving frequency, ωm [13

13. R. P. Scott, T. D. Mulder, K. A. Baker, and B. H. Kolner, “Amplitude and phase noise sensitivity of modelocked Ti:sapphire lasers in terms of a complex noise transfer function,” Opt. Express 15, 9090–9095 (2007). [CrossRef] [PubMed]

],

HAM(ωm)mL(ωm)mp,HPM(ωm)β(ωm)mp
(1)

The output amplitude noise PSD can now be predicted by taking the product of the pump noise PSD and the squared-modulus of the AM NTF

SAM(ωm)=HAM(ωm)2SP(ωm)
(2)

The single-sideband phase noise PSD is similarly given by

SPMSSB(ωm)=HPM(ωm)22SP(ωm)
(3)

where the factor of 1/2 is due to the noise being specified as single-sideband about the carrier.

2. Measurement of laser noise power spectral density

The apparatus for measuring both AM and PM PSD is shown in Fig. 2. Under test are a diode-pumped solid-state pump laser and a conventional Kerr-lens modelocked Ti:sapphire laser with intracavity prism-pair dispersion compensation running at a 100 MHz repetition rate. The acousto-optic modulator (AOM) in the pump beam path is used for characterizing the NTF of the Ti:sapphire laser and is disabled during noise tests. Photoreceivers AM1 and AM2 are conventional transimpedance amplifiers used for measuring amplitude noise at baseband and are detailed in [14

14. R. P. Scott, C. Langrock, and B. H. Kolner, “High dynamic range laser amplitude and phase noise measurement techniques,” IEEE J. Sel. Top. Quantum Electron. 7, 641–655 (2001). [CrossRef]

]. The envelope phase noise of the Ti:sapphire laser is measured at the fundamental of the 100 MHz pulse train using the quadrature mixing technique [14

14. R. P. Scott, C. Langrock, and B. H. Kolner, “High dynamic range laser amplitude and phase noise measurement techniques,” IEEE J. Sel. Top. Quantum Electron. 7, 641–655 (2001). [CrossRef]

] with a voltage-controlled crystal oscillator (VCXO) as the reference. Photoreceiver PM2 contains a silicon p-i-n diode followed by a broadband RF amplifier. The spectrum analyzer system is composed of an HP 3561A (FFT) for the frequency range 0.1 Hz–100 kHz and an HP 3585A (analog RF) for the frequency range 100 kHz–10 MHz.

Fig. 2. Simplified block diagram of the setup used to measure laser noise and the complex noise transfer function. VSA, Agilent 89410 Vector Signal Analyzer; AOM, acousto-optic modulator; PD, photodiode; PLL, phase-locked loop; VCXO, voltage-controlled crystal oscillator; AM1 and AM2, baseband AM noise photoreceivers (DC-40 MHz); PM1, phase noise RF photoreceiver (80 MHz-120 MHz); LNA, low noise amplifier; Spectrum Analyzer, HP 3561A+HP 3585A.

For this experiment we chose to compare the performance of the modelocked Ti:sapphire laser using two different pump lasers: a single-longitudinal mode laser (Coherent Verdi V5 hereafter designated as Pump 1) and a multi-longitudinal mode laser (Spectra-Physics Millenia Vs, Pump 2). Although both are commercial frequency-doubled Nd:YVO 4 lasers, their noise spectra differ substantially as shown in Fig. 3(a). The additional noise seen in Pump 2 is attributed to mode competition and beating effects in the multi-longitudinal mode laser. (Note: a comparison between similar pump lasers was recently reported by Matos, et al. [15

15. L. Matos, O. D. Mücke, C. Jian, and F. X. Kärtner, “Carrier-envelope phase dynamics and noise analysis in octave-spanning Ti:sapphire lasers,” Opt. Express 14, 2497–2511 (2006). [CrossRef] [PubMed]

]).

3. Measurement of the laser noise transfer function

Like any transfer function, measurement of the laser noise transfer function requires a stimulus and measurement of a coherent response. This is accomplished using a vector signal analyzer (VSA, Agilent 89410A) which covers the frequency range 0.1 Hz–10 MHz. The VSA provides a drive signal that is applied to the AOM placed between the pump and Ti:sapphire laser. The VSA detects the amplitude and phase modulations induced on the Ti:sapphire laser (Ch. 2) as well as those on a sample of the pump beam (Ch. 1). The ratio of these establishes the NTF according to (1). Further details of the NTF measurement procedure can be found in [13

13. R. P. Scott, T. D. Mulder, K. A. Baker, and B. H. Kolner, “Amplitude and phase noise sensitivity of modelocked Ti:sapphire lasers in terms of a complex noise transfer function,” Opt. Express 15, 9090–9095 (2007). [CrossRef] [PubMed]

].

Fig. 3. (a) Pump laser noise PSD for a single-longitudinal mode DPSS laser (Pump 1) and a multi-longitudinal mode DPSS laser (Pump 2). (b) Magnitudes of AM and PM NTFs of the Ti:sapphire laser using Pump 1 and Pump 2.

4. Results

Fig. 4. Measured and predicted (a) AM and (b) PM noise of a KLM Ti:sapphire laser pumped with a single-mode DPSS laser (Pump 1). (Note the different scales).
Fig. 5. Measured and predicted (a) AM and (b) PM noise of a KLM Ti:sapphire laser pumped with a multi-mode DPSS laser (Pump 2). (Note the different scales).

5. Conclusions

The AM and PM noise spectra of a free-running modelocked laser appear to be dominated by the influence of pump noise and can therefore be predicted by characterization and application of the noise transfer functions to the AM noise spectrum of the pump. We applied this procedure to a single- and multi-longitudinal mode DPSS pump laser. The predicted AM and PM noise spectra are in reasonably good agreement with the measured spectra for both pumps.

Acknowledgment

This research was supported in part by NSF Grant ECS-0622235 and the David and Lucile Packard Foundation. The authors thank Bob Temple of Agilent Technologies for many insightful discussions and technical support.

References and links

1.

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, 2264–2267 (2000). [CrossRef] [PubMed]

2.

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]

3.

A. L. Schawlow and C. H. Townes, “Infrared and Optical Masers,” Phys. Rev. 112, 1940–1949 (1958). [CrossRef]

4.

H. Haken, “Analogy Between Higher Instabilities in Fluids and Lasers,” Phys. Lett. 53A, 77–78 (1975).

5.

E. N. Lorenz, “Deterministic Nonperiodic Flow,” J. Atmos. Sci. 20, 130–141 (1963). [CrossRef]

6.

H. Haken, Light; Laser Light Dynamics (North-Holland, Amsterdam, 1985) Vol. 2. [PubMed]

7.

C. O. Weiss and R. Vilaseca, Dynamics of Lasers (Weinheim, New York, 1991).

8.

H. A. Haus, “Theory of mode locking with a fast saturable absorber,” J. Appl. Phys. 46, 3049–3058 (1975). [CrossRef]

9.

O. E. Martinez, R. L. Fork, and J. P. Gordon, “Theory of passively mode-locked lasers for the case of a nonlinear complex-propagation coefficient,” J. Opt. Soc. Am. B 2, 753–760 (1985). [CrossRef]

10.

H. A. Haus, J. G. Fujimoto, and E. P. Ippen, “Analytic Theory of Additive Pulse and Kerr Lens Mode Locking,” IEEE J. Quantum Electron. 28, 2086–2096 (1992). [CrossRef]

11.

T. Kapitula, J. N. Kutz, and B. Sandstede, “Stability of pulses in the master mode-locking equation,” J. Opt. Soc. Am. B 19, 740–746 (2002). [CrossRef]

12.

C. R. Menyuk, J. K. Wahlstrand, J. Willits, R. P. Smith, T. Schibli, and S. T. Cundiff, “Pulse dynamics in mode-locked lasers: relaxation oscillations and frequency pulling,” Opt. Express 15, 6677–6689 (2007). [CrossRef] [PubMed]

13.

R. P. Scott, T. D. Mulder, K. A. Baker, and B. H. Kolner, “Amplitude and phase noise sensitivity of modelocked Ti:sapphire lasers in terms of a complex noise transfer function,” Opt. Express 15, 9090–9095 (2007). [CrossRef] [PubMed]

14.

R. P. Scott, C. Langrock, and B. H. Kolner, “High dynamic range laser amplitude and phase noise measurement techniques,” IEEE J. Sel. Top. Quantum Electron. 7, 641–655 (2001). [CrossRef]

15.

L. Matos, O. D. Mücke, C. Jian, and F. X. Kärtner, “Carrier-envelope phase dynamics and noise analysis in octave-spanning Ti:sapphire lasers,” Opt. Express 14, 2497–2511 (2006). [CrossRef] [PubMed]

16.

R. P. Scott, B. H. Kolner, C. Langrock, R. L. Byer, and M. M. Fejer, “Ti:sapphire laser pump-noise transfer function,” in Proceedings of the Conference on Lasers and Electro-optics, Paper CFB2 (Baltimore, MD, 2003).

OCIS Codes
(140.4050) Lasers and laser optics : Mode-locked lasers
(140.3425) Lasers and laser optics : Laser stabilization

ToC Category:
Lasers and Laser Optics

History
Original Manuscript: July 11, 2008
Revised Manuscript: August 20, 2008
Manuscript Accepted: August 22, 2008
Published: August 26, 2008

Citation
Theresa D. Mulder, Ryan P. Scott, and Brian H. Kolner, "Amplitude and envelope phase noise of a modelocked laser predicted from its noise transfer function and the pump noise power spectrum," Opt. Express 16, 14186-14191 (2008)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-18-14186


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References

  1. R. Holzwarth, T. Udem, T. W. H¨ansch, J. C. Knight, W. J. Wadsworth, and P. St. J. Russell, "Optical frequency synthesizer for precision spectroscopy," Phys. Rev. Lett. 85, 2264-2267 (2000). [CrossRef] [PubMed]
  2. 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]
  3. A. L. Schawlow and C. H. Townes, "Infrared and Optical Masers," Phys. Rev. 112, 1940-1949 (1958). [CrossRef]
  4. H. Haken, "Analogy Between Higher Instabilities in Fluids and Lasers," Phys. Lett. 53A, 77-78 (1975).
  5. E. N. Lorenz, "Deterministic Nonperiodic Flow," J. Atmos. Sci. 20, 130-141 (1963). [CrossRef]
  6. H. Haken, Light; Laser Light Dynamics (North-Holland, Amsterdam, 1985) Vol. 2. [PubMed]
  7. C. O. Weiss and R. Vilaseca, Dynamics of Lasers (Weinheim, New York, 1991).
  8. H. A. Haus, "Theory of mode locking with a fast saturable absorber," J. Appl. Phys. 46, 3049-3058 (1975). [CrossRef]
  9. O. E. Martinez, R. L. Fork, and J. P. Gordon, "Theory of passively mode-locked lasers for the case of a nonlinear complex-propagation coefficient," J. Opt. Soc. Am. B 2, 753-760 (1985). [CrossRef]
  10. H. A. Haus, J. G. Fujimoto, and E. P. Ippen, "Analytic Theory of Additive Pulse and Kerr Lens Mode Locking," IEEE J. Quantum Electron. 28, 2086-2096 (1992). [CrossRef]
  11. T. Kapitula, J. N. Kutz, and B. Sandstede, "Stability of pulses in the master mode-locking equation," J. Opt. Soc. Am. B 19, 740-746 (2002). [CrossRef]
  12. C. R. Menyuk, J. K. Wahlstrand, J. Willits, R. P. Smith, T. Schibli, and S. T. Cundiff, "Pulse dynamics in modelocked lasers: relaxation oscillations and frequency pulling," Opt. Express 15, 6677-6689 (2007). [CrossRef] [PubMed]
  13. R. P. Scott, T. D. Mulder, K. A. Baker, and B. H. Kolner, "Amplitude and phase noise sensitivity of modelocked Ti:sapphire lasers in terms of a complex noise transfer function," Opt. Express 15, 9090-9095 (2007). [CrossRef] [PubMed]
  14. R. P. Scott, C. Langrock, and B. H. Kolner, "High dynamic range laser amplitude and phase noise measurement techniques," IEEE J. Sel. Top. Quantum Electron. 7, 641-655 (2001). [CrossRef]
  15. L. Matos, O. D. M¨ucke, C. Jian, and F. X. K¨artner, "Carrier-envelope phase dynamics and noise analysis in octave-spanning Ti:sapphire lasers," Opt. Express 14, 2497-2511 (2006). [CrossRef] [PubMed]
  16. R. P. Scott, B. H. Kolner, C. Langrock, R. L. Byer, and M. M. Fejer, "Ti:sapphire laser pump-noise transfer function," in Proceedings of the Conference on Lasers and Electro-optics, Paper CFB2 (Baltimore, MD, 2003).

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