Fiber Lasers E-Book

• Herausgeber: Wiley-VCH
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 536

Veröffentlichungsjahr: 2012

Beschreibung

A comprehensive account of the latest developments and applications in this rapidly developing field, covering a wide range of topics, such as power scaling and short pulse generation, dispersion management and modeling, broadband supercontinuum generation and wavelength tailoring.
The book brings together contributions from the world's leading experts at major collaborative research centers throughout Europe, Australia, Russia and the USA. Each chapter presents a tutorial style introduction to the selected topic suitable for scientists, researchers and experts, as well as graduate and postgraduate students with a basic background in optics.

Leseprobe

Contents

Cover

Related Titles

Title Page

Copyright

Preface

List of Contributors

Chapter 1: Introduction

References

Chapter 2: High-Power Fiber Lasers and Amplifiers: Fundamentals and Enabling Technologies to Enter the Upper Limits

2.1 Introduction

2.2 High-Power Fiber Design

2.3 Theoretical Description and Nonlinear Effects in Laser Fibers

2.4 Fiber Components for High-Power Fiber Lasers

2.5 High-Power Experiments

2.6 Summary

References

Chapter 3: Supercontinuum Sources Based on Photonic Crystal Fiber

3.1 Introduction and Brief History

3.2 Photonic Crystal Fibers and Tapers

3.3 Modeling Nonlinear Pulse Propagation in Optical Fiber

3.4 Ultrafast Pumped Supercontinuum Sources

3.5 Conclusion

References

Chapter 4: Dissipative Soliton Fiber Lasers

4.1 Introduction

4.2 Theory: Analytic Approach

4.3 Theory: Simulations

4.4 Physical Limits

4.5 Practical Extensions

4.6 Giant-Chirp Oscillators

4.7 Summary

References

Chapter 5: Modeling and Technologies of Ultrafast Fiber Lasers

5.1 Overview of Short Pulse Fiber Lasers

5.2 Modeling of Ultrafast Fiber Lasers

5.3 Implementation and Control of Advanced Components

5.4 Conclusions and Future Outlook

References

Chapter 6: Tapered Fiber Lasers and Amplifiers

6.1 Introduction

6.2 Theoretical Model and Experimental Results

6.3 Lasers and Amplifiers with Active Tapered Fibers

6.4 Summary

References

Chapter 7: Fiber Lasers that Bridge the Shortwave to Midwave Regions of the Infrared Spectrum

7.1 Introduction

7.2 Survey of the Power and Efficiency of Long-Wavelength Fiber Lasers

7.3 Shortwave Infrared Fiber Lasers Employing Silicate Glass

7.4 Infrared Fiber Fabrication

7.5 Shortwave and Midwave Infrared Fiber Lasers Employing Fluoride Glass

7.6 Exotic Glasses for Fiber Lasers

7.7 Conclusions

References

Chapter 8: Outlook

Index

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Preface

This book is a contemporary overview of selected topics in fiber lasers. The recent and swift development of these systems triggered the writing and publication of this text. The book offers the reader a wide and critical overview of the state-of-the-art within this practical – as well as important and interesting – field of quantum electronics.

I am extremely satisfied with the result achieved through the collective efforts of the wonderful team of authors, who have gathered and shared their knowledge to make this project possible. The authors are leading-edge scientists in the field of fiber lasers. Accomplishing what we have was no easy task since writing such an extensive work is very time-consuming, making the outcome of this effort even more valuable.

I am firmly aware that the present edition does not address all the branches of fiber technology. However, the chapters presented to you in this book offer an insight into the most interesting and developed aspects in the field; such is my opinion.

We still expect new opinions and reviews, as well as other books dedicated to fiber laser technology. This, however, does not diminish the worth of this particular book as a document that reflects and fixes the current situation within this important field.

I wish to thank all the contributors for their enthusiasm and effort. A special thank you goes to the staff of Wiley-VCH Verlag GmbH for their patience and practical guidance.

Oleg G. Okhotnikov

December 2011Tampere

List of Contributors

Brandon G. Bale

Aston University

School of Electronic Engineering and Applied Science

Photonics Research Group

Aston Triangle

Birmingham B4 7ET

United Kingdom

Ramona Eberhardt

Fraunhofer Institute for Applied Optics and Precision Engineering

Albert-Einstein Str. 7

07745 Jena

Germany

Stuart D. Jackson

University of Sydney

Optical Fibre Technology Centre

Australian Photonics

206 National Innovation Centre

6 Q2 Australian Technology Park, Eveleigh

Sydney, NSW 1430

Australia

Nicolas Y. Joly

Max Planck Institute for the Science of Light

Günther-Scharowsky Str. 1/Bau 24

91058 Erlangen

Germany

Valery Filippov

Tampere University of Technology

Optoelectronics Research Center

Korkeakoulunkatu 3

333101 Tampere

Finland

Juho Kerttula

Tampere University of Technology

Optoelectronics Research Center

Korkeakoulunkatu 3

333101 Tampere

Finland

David G. Lancaster

University of Adelaide

Institute for Photonics and Advanced Sensing

School of Chemistry and Physics

Adelaide, SA 5005

Australia

Jens Limpert

Friedrich-Schiller-University

Institute of Applied Physics

Albert-Einstein-Str. 15

07745 Jena

Germany

Oleg G. Okhitnikov

Tampere University of Technology

Optoelectronics Research Center

Korkeakoulunkatu 3

333101 Tampere

Finland

William H. Renninger

Cornell University

Department of Applied Physics

271 Clark Hall

Ithaca, NY 14853

USA

Philip St. J. Russell

Max Planck Institute for the Science of Light

Günther-Scharowsky Str. 1/Bau 24

91058 Erlangen

Germany

Thomas Schreiber

Fraunhofer Institute for Applied Optics and Precision Engineering

Albert-Einstein Str. 7

07745 Jena

Germany

Sebastian Stark

Max Planck Institute for the Science of Light

Günther-Scharowsky Str. 1/Bau 24

91058 Erlangen

Germany

John C. Travers

Max Planck Institute for the Science of Light

Günther-Scharowsky Str. 1/Bau 24

91058 Erlangen

Germany

Andreas Tünnermann

Fraunhofer Institute for Applied Optics and Precision Engineering

Albert-Einstein Str. 7

07745 Jena

Germany

and

Friedrich-Schiller-University

Institute of Applied Physics

Albert-Einstein-Str. 15

07745 Jena

Germany

Sergei K. Turitsyn

Aston University

School of Electronic Engineering and Applied Science

Photonics Research Group

Aston Triangle

Birmingham B4 7ET

United Kingdom

Frank W. Wise

Cornell University

Department of Applied Physics

271 Clark Hall

Ithaca, NY 14853

USA

Chapter 1

Introduction

Oleg G. Okhotnikov

Nowadays fiber technology is a mature and vast sector of industry that has advanced remarkably during a few decades, firstly owing to the rapid development of optical communications. It is recognized that there are no alternatives to optical technologies in communication because non-optical methods cannot cope with the modern demand for information transfer.

A different situation exists in the field of fiber lasers. Based largely on the technology developed for optical communications, the practical value of fiber lasers should be critically examined before application to determine the actual impact of their implementation. This assessment is needed because alternative laser technologies, for example, solid-state and semiconductor, are well developed and could provide competitive solutions. Nevertheless, intensive R&D in fiber laser technology has radically changed the market situation in the scientific and industrial lasers sector. Nowadays, fiber lasers hold firmly the leading position in some applications by forcing out other laser systems. In particular, fiber lasers are starting to dominate in applications related to high-power lasers, sources with high brightness and several areas dealing with pulsed oscillators and spectral manipulation (http://www.ipgphotonics.com/) [1]. Combined with advances in glass technology and nonlinear optics, fiber systems are now available commercially over an extended spectral range.

Being a guided-wave system, fiber lasers allow us to use approaches that are not available with systems where the modes are determined firmly by the laser cavity, for example, solid state lasers. The guiding properties of the fiber establishing the mode structure allow us to prevent constraints induced by thermal lensing and implement specific methods of mode and dispersion control, for example, axially non-uniform waveguides and photonic crystal structures. The operation of an ordinary fiber system relies on the classical principle known as total internal reflection. Regardless of the significance of this physical mechanism, it imposes some limitations in the tailoring mode area, nonlinearity, and waveguide dispersion. Extensive efforts have been made in recent decades to improve the methods of light control. Among them, photonic crystal fibers (PCFs), which are rapidly developing the research field in optical science, represent a significant breakthrough both in research and applications [2–4]. The high potential of these waveguides in tailoring optical parameters permits diverse applications in various areas of photonics, particularly in nonlinear optics, ultrafast fiber oscillators, and high-power amplifiers. The correct structure of a PCF can guarantee that only the fundamental mode is guided, resulting in “endless single-mode” behavior. Photonic crystal fibers can be designed to have a large single-mode area essential for high-energy lasers. Very large mode-area PCFs have been demonstrated that reveal their superiority in high-power delivery, amplifiers, and lasers [5–7]. This development is discussed in Chapter 2.

PCFs with small glass cores and a high air-filling factor can generate peculiar chromatic dispersion and offer high optical density. The determined enhancement of different nonlinear processes in micro-structured optical fibers can be achieved through manipulation of the dispersion characteristics of the fiber [8, 9]. Chapter 3 is devoted to one of the most successful applications – supercontinuum generation, which takes advantage of the high nonlinearities and accurate control of chromatic dispersion provided by PCFs. Using PCFs and chirped fiber Bragg gratings contributes essentially to the dispersion management techniques that maintain the all-fiber format of ultrafast oscillators. In particular, exploiting PCFs that could generate anomalous dispersion over extended spectral band has resulted in all-fiber dispersion-managed soliton lasers operation around 1 µm [10–13]. With the net anomalous group-velocity dispersion (GVD), the nonlinearity balances GVD, resulting in a soliton-like pulse shaping, which implies that these fibers could support short pulse propagation with neither temporal nor spectral distortion as optical solitons. Net anomalous GVD compensates the accumulated pulse phase shift in a cavity consisting of segments with normal and anomalous GVD, representing a so-called dispersion map. With an increase in pulse energy, however, the excessive nonlinear phase shift cannot be eventually compensated by the dispersion, giving rise to the phenomenon known as pulse breaking. Though the strong dispersion map was shown to increase the threshold for multiple pulse operation, the wave-breaking instability still prevents energy scaling using this laser concept. Recent studies show that laser cavities with large normal dispersion tend to support highly-chirped pulses that can reach unprecedented energies and peak powers, while avoiding wave-breaking despite the accumulation of large nonlinear phase shifts [14–16]. The performance of such normal-dispersion lasers is presented in Chapter 4.

Chapter 5 is devoted to experimental and modeling aspects of ultrafast fiber systems. It overviews recent experimental results obtained for fiber lasers passively mode-locked with saturable absorbers and describes the methods for their modeling and computing. Various techniques used for dispersion compensation are discussed, including chirped fiber Bragg gratings and microstructure fibers.

As optically pumped devices, fiber lasers depend critically on the performance of the pumping sources. Progress in fiber lasers became possible owing to unprecedented development of optical pumping systems based on semiconductor lasers. The power available now commercially reaches the multi-kW level [17]. Fiber lasers and amplifiers as wave-guiding systems offer a unique opportunity to exploit pumping sources of low brightness. The so-called double clad pumping concept allows high-power, large numerical aperture, and large area sources to be efficiently used in cost-effective high-power fiber systems [18]. This approach exploits broad-area semiconductor lasers or bars usually coupled to multimode fibers that are optically matched with pumping cladding of double-clad fiber. Axially non-uniform, tapered (flared) amplifiers combined with a cladding pumping scheme can be utilized in high-power technology. Semiconductor [19] and fiber [20] gain media provide a practical solution for power scaling. This method, described in Chapter 6, is particularly valuable for all-fiber systems because it allows the achievement of high power while maintaining the diffraction-limited beam characteristics [21, 22].

Some fiber systems, however, require a core-pumping scheme. The most important example is Raman fiber devices, which represent one of the key technologies in modern optical communications. Although light generation covering a large wavelength range (895–1560 nm) has been reported using neodymium, ytterbium, bismuth, and erbium fiber systems, Raman fiber lasers and amplifiers offer an interesting opportunity for flexible wavelength tailoring [23]. Raman gain exists in every optical fiber and could provide amplification in every fiber optic link. Raman gain is available over the entire transparency region of the silica fiber, ranging from approximately 0.3 to 2 µm provided that an appropriate pump is used. The wavelength of the Raman peak gain is shifted from the pumping wavelength by the frequency of the optical phonons and therefore it can be tailored by tuning the pump wavelength. Another advantage of Raman amplification is that it has a relatively broad-band bandwidth of 5 THz, and the gain is reasonably flat over a wide wavelength range [23]. Mode-locked Raman fiber lasers with high-quality pulses are obtained both at normal and anomalous dispersion. Raman lasers and amplifiers are basically core-pumped devices since the cladding pumping scheme offers low gain efficiency. Consequently, a relatively large pump power launched into a single-mode fiber core is required to achieve noticeable gain. The relatively high pump power in a single-mode fiber required for Raman amplifiers is a serious challenge for communication technology. The development of high pump power sources has resulted in a broad deployment of Raman amplifiers in fiber-optic transmission systems, making them one of the first widely commercialized nonlinear optical devices in telecommunications [24]. Broadband non-resonant gain positioned by pump wavelength selection can be further extended by using multiple-wavelength pumping and improves the gain flatness. Distributed Raman amplifiers are demonstrated to improve the noise figures and reduce the nonlinear penalty of fiber systems, allowing for longer amplifier spans, higher bit rates, closer channel spacing, and operation near the zero-dispersion wavelength [25]. Available commercial laser diodes, however, produce single-mode fiber coupled power up to 1 W only and at very few wavelengths. Alternative pumping with powerful fiber lasers comes at a high cost and high power consumption. The fast response time of Raman gain could cause additional noise due to transfer of pump fluctuations to the signal. Pump–signal interaction in a long-length fiber exhibits an averaging effect of noise transfer dependent on the pumping direction. When a co-propagating pumping scheme is used, the averaging effect is low compared with counter-propagating geometry due to small walk-off between pump and signal and, consequently, tighter requirements on the noise level of pump lasers should be applied. Co-pumping pumping is, however, advantageous over the technique using only a counter-propagating scheme because the signal can be maintained at low level throughout each span of the transmission line. It is expected that co-propagating Raman pumping could improve system performance, significantly increasing the amplifier spacing under the condition that pumping sources have low-noise characteristics. The availability of low-noise pumping sources is critical for further improvement of the links using Raman amplification. Currently, due to a shortage of efficient low-noise pump sources, a counter-propagating pumping scheme for Raman amplifiers is preferred.

A promising pumping approach for Raman fiber amplifiers could utilize a semiconductor disk laser (SDL), which was demonstrated to offer low-noise and high power with diffraction-limited beam characteristics [26]. It has been shown that the relative intensity noise (RIN) of semiconductor lasers can reach an extremely low level, close to shot noise limit, provided that the laser operates in the so-called class-A regime. This regime is attained when the photon lifetime in the laser cavity becomes much longer than the carrier lifetime in the active medium. A laser operating under this condition exhibits a relaxation-oscillation-free flat spectral noise density. The emergence of low-noise, high-power disk lasers operating in the wavelength range 1.2–1.6 µm could radically change the conventional technology of Raman fiber amplifiers and lasers [27–30].

Extension of the operation wavelength towards the mid-infrared range has been triggered by numerous applications. Thulium- and thulium–holmium-doped fibers have been demonstrated to be a major candidate for high-power sources operating around 2 µm [31]. Silica-based fiber lasers producing outputs in the shortwave infrared (SWIR) region of the spectrum are fast becoming a mature technology [32]. Most of the important demonstrations of highly efficient and high-power SWIR fiber lasers involved the Tm3+ ion, because of the favorable ion interactions that produce high quantum efficiencies, and the compatibility of this laser with commercial diode laser excitation is presented in Chapter 7. Pushing the emission wavelength of silica-based fiber lasers further into the SWIR spectrum is of current interest for a range of applications, including atmospheric light transmission, Si photonics, and nonlinear optics. Thulium fiber has a broad amplification bandwidth, between 1.65 and 2.1 µm, and is, therefore, suitable for short pulse generation and wide spectral tuning [33]. A specific feature of optical fiber operating at 2 µm is a large anomalous dispersion that causes the operation in the soliton pulse regime. Using concept presented in Chapter 4, a 2-µm net normal-dispersion regime of the cavity consisting solely of anomalous-dispersion fiber has been demonstrated recently using dispersion offset set by the chirped fiber Bragg grating, which could be a practical solution for power scaling of long-wavelength lasers [34].

References

1. Fianium Ltd. (2011) Product Datasheet, FemtoPower1060 & FP532: High-Power Ultrafast Lasers, available at http://www.fianium.com/pdf/fp-1064-532(v1.1).pdf (accessed on 21.03.2012).

2. Russell, Ph.St.J. (2003) Photonic crystal fibers. Science, 299, 358–362.

3. Knight, J.C. (2003) Photonic crystal fibres. Nature, 424, 847–851.

4. Russell, Ph.St.J. (2006) Photonic-crystal fibers. J. Lightwave Technol., 24, 4728–4749.

5. Limpert, J., Roeser, F., Schreiber, T., and Tuennermann, A. (2006) High-power ultrafast fiber systems. IEEE J. Sel. Top. Quantum Electron., 12, 233–244.

6. Limpert, J., Roeser, F., Klingebiel, S., Schreiber, T., Wirth, Ch., Peschel, T., Eberhardt, R., and Tuennermann, A. (2007) The rising power of fiber lasers and amplifiers. IEEE J. Sel. Top. Quantum Electron., 12, 537–545.

7. Tuennermann, A., Schreiber, T., and Limpert, J. (2010) Fiber lasers and amplifiers: an ultrafast performance evolution. Appl. Opt., 49, F71–F78.

8. Birks, T.A., Wadsworth, W.J., and Russell, P.S.J. (2000) Supercontinuum generation in tapered fibers. Opt. Lett., 25, 1415–1417.

9. Ranka, J.K., Windeler, R.S., and Stentz, A.J. (2000) Visible continuum generation in air–silica microstructure optical fibers with anomalous dispersion at 800nm. Opt. Lett., 25, 25–27.

10. Isomäki, A. and Okhotnikov, O.G. (2006) All-fiber ytterbium soliton mode-locked laser with dispersion control by solid-core photonic bandgap fiber. Opt. Express, 14, 4368–4373.

11. Isomäki, A. and Okhotnikov, O.G. (2006) Femtosecond soliton mode-locked laser based on ytterbium-doped photonic bandgap fiber. Opt. Express, 14, 9238–9243.

12. Gumenyuk, R., Vartiainen, I., Tuovinen, H., Kivistö, S., Chamorovskiy, Yu., and Okhotnikov, O.G. (2011) Dispersion compensation technologies for femtosecond fiber system. Appl. Opt., 50, 797–801.

13. Chamorovskiy, A., Chamorovskiy, Yu., Vorob'ev, I., and Okhotnikov, O.G. (2010) 95 fs suspended core ytterbium fiber laser. IEEE Photon. Technol. Lett., 22, 1321–1323.

14. Renninger, W.H., Chong, A., and Wise, F.W. (2008) Dissipative solitons in normal-dispersion fiber lasers. Phys. Rev. A, 77, 023814.

15. Wise, F.W., Chong, A., and Renninger, W.H. (2008) High-energy femtosecond fiber lasers based on pulse propagation at normal dispersion. Laser & Photon. Rev., 2, 58–73.

16. Kieu, K. and Wise, F.W. (2008) All-normal-dispersion femtosecond laser. Opt. Express, 16, 11453–11458.

17. Laserline (2011) Fiber-coupled Diode Lasers - Mobile Power. http://www.laserline-inc.com/high-power-diode-lasers-fiber-coupled-diode-lasers.php (accessed on 21.03.2012).

18. Zenteno, L. (1993) High-power double-clad fiber lasers. J. Lightwave Technol., 11, 1435–1446.

19. Wenzel, H., Paschke, K., Brox, O., Bugge, F., Frocke, J., Ginolas, A., Knauer, A., Ressel, P., and Erbert, G. (2007) 10 W continuous-wave monolithically integrated master-oscillator power-amplifier. Electron. Lett., 43, 160–161.

20. Okhotnikov, O.G. and Sousa, J.M. (1999) Flared single-transverse-mode fibre amplifier. Electron. Lett., 35, 1011–1013.

21. Kerttula, J., Filippov, V., Chamorovskii, Yu., Golant and, K., and Okhotnikov, O.G. (2010) Actively Q-switched 1.6 mJ tapered double-clad ytterbium-doped fiber laser. Opt. Express, 18, 18543–18549.

22. Filippov, V., Kerttula, J., Chamorovskii, Yu., Golant, K., and Okhotnikov, O.G. (2010) Highly efficient 750W tapered double-clad ytterbium fiber laser. Opt. Express, 18, 12499–12512.

23. Headley, C. III and Agrawal, G.P. (2004) Raman Amplification in Fiber Optical communication Systems, Academic Press, EUA.

24. Agrawal, G.P. (2002) Fiber-Optic Communication Systems, 3rd edn, Wiley-Interscience, New York.

25. Faralli, S., Bolognini, G., Sacchi, G., Sugliani, S., and Di Pasquale, F. (2005) Bidirectional higher order cascaded Raman amplification benefits for 10-Gb/s WDM unrepeated transmission systems. J. Lightwave Technol., 23, 2427–2433.

26. Okhotnikov, O.G. (ed.) (2010) Semiconductor Disk Lasers, Physics and Technology, Wiley-VCH Verlag GmbH, Weinheim.

27. Chamorovskiy, A., Rantamäki, J., Sirbu, A., Mereuta, A., Kapon, E., and Okhotnikov, O.G. (2010) 1.38-µm mode-locked Raman fiber laser pumped by semiconductor disk laser. Opt. Express, 18, 23872–23877.

28. Chamorovskiy, A., Rautiainen, J., Lyytikäinen, J., Ranta, S., Tavast, M., Sirbu, A., Kapon, E., and Okhotnikov, O.G. (2010) Raman fiber laser pumped by semiconductor disk laser and mode-locked by a semiconductor saturable absorber mirror. Opt. Lett., 35, 3529–3531.

29. Chamorovskiy, A., Rautiainen, J., Rantamäki, J., and Okhotnikov, O.G. (2011) Low-noise Raman fiber amplifier pumped by semiconductor disk laser. Opt. Express, 18, 6414–6419.

30. Chamorovskiy, A., Rautiainen, J., Rantamäki, J., Golant, K., and Okhotnikov, O.G. (2011) 1.3 µm Raman-bismuth fiber amplifier pumped by semiconductor disk laser. Opt. Express, 18, 6433–6438.

31. Jackson, S.D. (2008) Efficient Tm3+, Ho3+-co-doped silica fibre laser diode pumped at 1150 nm. Opt. Commun., 281, 3837–3840.

32. Jackson, S.D. (2009) The spectroscopic and energy transfer characteristics of the rare earth ions used for silicate glass fibre lasers operating in the shortwave infrared. Laser & Photon. Rev., 3, 466–482.

33. Kivistö, S. and Okhotnikov, O.G. (2011) 600-fs mode-locked Tm-Ho-doped fiber laser synchronized to optical clock with optically driven semiconductor saturable absorber. IEEE Photon. Technol. Lett., 23, 477–479.

34. Gumenyuk, R. and Okhotnikov, O.G. (2011) Dissipative dispersion-managed soliton 2 µm thulium/holmium fiber laser. Opt. Lett., 36, 609–611.

Chapter 2

High-Power Fiber Lasers and Amplifiers: Fundamentals and Enabling Technologies to Enter the Upper Limits

Thomas Schreiber, Ramona Eberhardt, Jens Limpert, and Andreas Tünnermann

2.1 Introduction

It is understood that several enabling technologies have been responsible for the rapid performance scaling of fiber laser systems and their huge market success in different applications. Firstly, as for most solid state lasers, highly efficient and reliable diode lasers are one of the most essential parts for delivering pump photons. Secondly, the generation and control of light inside a fiber laser would not be possible without the use of low loss rare-earth doped optical fibers. Additionally, the application of integration technologies as well as different fiber-based optical components into alignment-free laser systems enabled the usability outside the laser laboratory.

Originally, the development of optical fibers utilized the potential of optical communication technology to allow transmission of vast amounts of data over huge distances. A major breakthrough was achieved when low loss glass fibers were manufactured for the first time by Corning in 1970 [1]. In the years that followed, these fibers replaced copper wires and almost all commercial telecommunication and network systems worldwide were built on fiber technology, thereby revolutionizing the method of information delivery and processing on a global scale. The fabrication technology allowing the production of fibers with the required properties had not been possible before the 1970s, when the chemical vapor deposition technique for producing fused silica was adapted. Previously, the large attenuation of fibers only enabled short light path transmission in instruments such as medical endoscopes or those used for illumination. Nevertheless, with the work on medical endoscopes driven by van Hell in 1954 and L. Curtiss in 1956, fibers had been made where guiding was achieved by a lower refractive index cladding surrounding a higher index core. The theoretical description of fibers as cylindrical waveguides, especially single mode fibers, was brought forward by E. Snitzer [2] in the early 1960s after he recognized the first waveguide modes found accidentally by W. Hicks. This theoretical basis together with the rapidly improving fabrication technology led to the realization of low loss single mode fibers that were applicable to information transmission.

Beside the fact that the first flash-lamp pumped fiber laser was realized in 1961 and used a neodymium (Nd)-doped crown glass as the core surrounded by an ordinary soda-lime-silicate glass, giving three inch long fiber samples [3], the telecommunication industry promoted its development by requiring an amplification scheme for intercontinental undersea transmission lines. Such amplification in a fiber has been shown by a rigid Nd-doped fiber wound around a flash lamp [4]. The replacement of flash lamps for pumping solid state laser was recognized and demonstrated in the early 1960s by the use of diode lasers, but they had no practical appearance until room temperature operation was possible in these semiconductor devices in the 1980s [5]. Diode lasers and erbium-doped fibers have been the basis of the optical amplifier now used for most long haul transmission lines, developed in 1987 [6].

Despite the success of optical fibers in telecommunication a second revolution started just after the twenty-first century began. With the availability of high power, high brightness diode lasers to pump double clad fibers doped with rare-earth elements to provide the active laser medium, the race for high power and energies from single mode fiber lasers began [7].

In the following sections the principles of high-power fiber lasers are reviewed. The design of double clad fibers that enable highest power levels are discussed in Section 2.2. Included is a review of the tailored optical properties of photonic crystal fibers, which have huge potential in future scaling efforts. Section 2.3 gives the basics required to theoretically describe the fiber lasers. This is followed by a brief discussion of additional high-power components and examples of high-power experiments in Sections 2.4 and 2.5, respectively.

2.2 High-Power Fiber Design

2.2.1 Double Clad Fiber Design

In its simplest form, a high-power laser fiber is designed as a double clad structure. The first cladding surrounds a doped core, where the core's refractive index is typically set to only guide the fundamental mode. The second cladding is the pump cladding that is formed by a multimode waveguide with a much larger area and numerical aperture. The pump light that is typically delivered by diode lasers can be coupled to the pump cladding with a high efficiency but is only weakly absorbed in this case. However, the small signal absorption along the fiber is not described by an exponential decay (Beers law) [8] due to the existence of modes having a quite low spatial overlap with the doped core and which are therefore negligibly absorbed. To avoid such modes the fibers may be coiled or the symmetry within the double clad structure can be broken. Symmetry breaking ensures an intrinsic absorption independent of bending and is typically favored. Figure 2.1 shows examples with an offset core or modified shapes.

Figure 2.1 (a) Double clad fiber design and refractive index distribution; (b) symmetry breaking by geometry for a D-shaped and hexagonal pump core as well as offset-core fiber.

The core is typically doped with rare earth elements to provide the laser ions. Figure 2.2 gives an overview of some rare earth elements and their possible lasing wavelength in the visible and near-infrared region. The most prominent example is erbium, which is used to realize erbium-doped fiber amplifiers (EDFAs) for telecommunication networks [9]. Currently, the highest power levels out of a fiber laser are obtained with ytterbium (Yb3+ by Yb2O3) doped fused silica fibers, which will be focused on in the following.

Figure 2.2 Accessible wavelengths by rare earth doping.

The refractive index increase Δn of the fiber's core is determined by the rare earth content and also by the concentration of other co-dopants. In most cases, a linear addition rule is valid with respect to the molar composition with constant increments [10]. Figure 2.3 shows the molar index changes of rare earth and co-dopants typically used. Apart from B2O3 and SiF4 all conventional rare earth and other co-dopants for silica fibers exhibit a positive value for the molar index change. Thus, high dopant concentrations generally increase the core index significantly with respect to the surrounding cladding and alter the guiding properties of the fiber. The molar refractivity change is very high for Yb2O3 doping, and it is therefore difficult to design a fiber core with a low index increase Δn without co-doping the core with negative index materials such as SiF4 or B2O3. Moreover, when higher concentrations are used to achieve high absorption values, the addition rule no longer holds and more complicated design rules have to be determined [11].

Figure 2.3 Molar refractive index changes for commonly used rare earth dopants and co-dopant materials in silica glass.

The waveguide properties of a fiber core can theoretically be described by eigenvalue analysis of the propagation equation (2.1). The equation is known as the scalar Helmholtz equation and is an approximation assuming no external sources and currents for the material properties with no transverse dependence of the dielectric function ε(x,y,ω) = n(ω)2, weak guidance, and no favor of a polarization state. It is not applicable to all fiber designs; however, it leads to good insight into the physics [12]:

(2.1)

As a result, different modes F(x,y) and their propagation constants β are calculated, as can be seen schematically in Figure 2.4 for a circular core with a radius aeff and the first two modes LP01 and LP11. The propagation constant β is related to the effective index of the guide's mode by β(ω) = neff(ω) · ω/c, with a value that is smaller than the core index nc and larger than the index of the surrounding cladding nclad. For step index fibers approximations have been made and a normalized frequency parameter V as well as the parameters U and W can be defined as:

(2.2)

Figure 2.4 Schematic representation of the refractive index profile in a step index fiber and the resulting effective index for the first two modes shown with their typical intensity profiles |F(x,y)|2.

These parameters can be used to provide a unified way to describe basic guiding properties. For weakly guiding step index fibers, these values can be expressed by semi-analytical equations [13] and it has been shown that only the fundamental mode is guided, if V < 2.405. To achieve this, the core index has to be controlled carefully. The resulting nearly Gaussian mode has a 1/e2 mode field diameter (MFD) that is related to the V parameter by the Marcuse equation [14] to a good approximation within a range of V ~ 0.8–2.5 by Equation (2.3). To describe nonlinear effects, it is more convenient to use the effective area Aeff of the mode, with its definition in Equation (2.4) based on the actual profile field F(x,y) of the mode:

(2.3)

(2.4)

2.2.2 Large Core Design in Special Fibers

2.2.2.1 Motivation

As will be discussed below, nonlinear effects limit the achievable power out of fiber laser systems. The use of larger cores reduces nonlinear effects due to the reduction of propagating intensity along the fiber. For single-mode fibers (V < 2.4), the enlargement of the core requires the reduction of the core NA according to Equation (2.2). This is practically limited for several reasons. Firstly, the characterization and knowledge and control of the refractive index before drawing the fiber is limited to 10−4 by measurement technology, which might be overcome by characterizing the mode quality of the drawn fiber. Secondly, lowering the refractive index might be difficult due to the required concentration level of the rare-earth ions to provide sufficient absorption. As a result, typical so-called large mode area (LMA) fibers with core diameters above 15 µm are able to guide several transversal modes.

Several techniques have been implemented to provide single-mode operation in these LMA fibers, such as modified matching [15], differential bend loss for higher-order modes (HOMs) [16, 17], resonant out-coupling of HOMs [18], mode filtering with tapers [19], confined doping [20], and gain–guiding index–anti-guiding [21].

However, the step index fiber discussed briefly in the last section is by no means the only method available to define a large single-mode core inside a fiber. By including additional refractive index structures in the fiber, one can greatly tailor additional optical parameters. One successful example of such microstructuring is the inclusion of regularly spaced air-filled holes that run along the length of the fiber. These fibers are also known as photonic crystal fibers (PCFs). Indeed, up to now the largest mode field diameter combined with the highest average power has been demonstrated using PCF [22–24]. The advantage of this type of fiber is based on the geometrical design freedom and control in addition to the doping profile used in step index fibers.

2.2.2.2 Core Design in Photonic Crystal Fibers

The geometrical parameters for standard step index fibers and PCFs, which are defined by a regular hexagonal array of air-holes, are compared in Figure 2.5 with the dimensional parameters aeff as the core radius, Λ as the hole to hole center distance (pitch), and d as the hole diameter. The core of the PCF is defined by one missing air-hole in the center of the structure.

Figure 2.5 Geometrical parameters for a step index fiber (a) and a photonic crystal fiber (b); (a – core radius, d – hole diameter, Λ – pitch).

To investigate the modal properties of such PCFs, the parameters V, U, and W can be employed, if the effective indices of the actual structure are used. For PCFs, both, neff and nclad and therefore V, U, and W have to be evaluated numerically. Numerous attempts have been made to improve the accuracy and speed of the computations based on different approaches like beam propagation, finite difference schemes, and different functional expansion methods [25]. For a wide range of parameters empirical relations can be used, which have been fitted to these numerical results and have been used here for solid core PCFs to simplify the design in terms of calculating the fundamental properties [26].

The parameter aeff is equal to the core radius for step index fibers (SIF) and corresponds to Λ/ for PCFs with one-hole missing. With the definition of Equation (2.2) one can show that a PCF becomes single-mode for V < 2.405, just as for a SIF. The main difference between a SIF and PCF is that for PCF the cladding index is strongly dependent on λ/Λ (Figure 2.6), where the refractive index of the solid matrix of fused silica nSiO2 (assumed constant over wavelength) is set to 1.42. In the limit of large spacing Λ or vanishing hole diameters compared to the wavelength λ. the effective cladding index reaches the core index.

Figure 2.6 Effective cladding index of a photonic crystal fiber versus normalized wavelength λ/Λ for different relative hole sizes (d/Λ).

Figure 2.7 shows examples of effective indices neff one obtains for the fundamental mode in typical SIFs and PCFs, where neff is plotted against the normalized wavelength, which is λ/aeff for standard step index fibers and λ/Λ for photonic crystal fibers. For the SIF, the cladding index nclad is that of fused silica and the core index nC is raised to provide the necessary index step for guidance. The cladding index nclad is almost constant with wavelength for step index fibers (neglected chromatic dispersion), but in the case of a PCF corresponds to the effective index of the fundamental space-filling mode of the holey cladding. The results become more interesting when plotting the single mode boundary V = 2.405 of the normalized wavelength over the relative hole diameter as carried out in Figure 2.8 not only for a PCF with only one hole removed to form the core (PCF-1) but also a three hole missing design (PCF-3) and a seven hole missing design (PCF-7) [27].

Figure 2.7 Effective index neff of the fundamental mode with respect to the normalized wavelength λ/Λ (λ/aeff) for (a) a step index fiber with 2a = 6.6 µm and ncore-SiO2 = 1.2 × 10−3, (b) single mode PCF with Λ = 10 µm, d/Λ = 0.4, and (c) highly nonlinear PCF with two zero dispersion wavelengths Λ = 1 µm, d/Λ = 0.8.

Figure 2.8 Modal characteristics of a one-, three-, and seven-hole-missing photonic crystal fiber; gray: multi-mode region.

In PCF-1 the fiber turns from multi-mode operation into single mode operation for d/Λ > 0.424 if the normalized wavelength is large enough. For d/Λ < 0.424 this fiber becomes single mode for all wavelengths λ/Λ because the wavelength dependence of nclad keeps the NA low enough to stay single mode. This behavior is called endlessly single mode. It was observed in the first photonic crystal fibers and was investigated theoretically soon after that [28, 29]. Such an endlessly single mode operation is not known in SIFs and leads to the concept of scaling the core diameter of a photonic crystal fiber, since the pitch Λ. and thereby the mode field diameter, could in principle be arbitrary scaled up in the regime. The boundaries for the other designs are different – in particular the endlessly single mode condition is d/Λ < 0.165 and d/Λ < 0.046 for PCF-3 and PCF-7, respectively.

Nonetheless, the advantage of an additional control of how the light is captured within the holes remains true even for more sophisticated designs. Even though such PCFs have played a fundamental role in enabling this scaling in mode area and power/pulse energy in fiber laser systems, a strict analytical consideration shows that these fibers are actually not single-mode [30]. This is due to the finite outer cladding, so that all PCF are indeed leaky waveguides and the fundamental mode has to be discriminated to the next higher order mode of the hole structure. By proper design, it is possible to achieve effective single-transverse-mode operation by offering higher confinement losses for HOMs compared to the fundamental mode (FM), that is, by exploiting mode discrimination. This concept was introduced by P. Russell as “modal sieve” in the context of endlessly single-mode fibers [31]. Very large mode area fibers using this effect are also known as leakage channel fibers [32]. However, this label is misleading for double clad structures and, therefore, the more general definition of large-pitch photonic crystal fiber (LPF) is preferred [33, 34]. This term makes reference to the fact that the hole-to-hole distance (pitch Λ) is at least ten-times larger than the wavelength to be guided. Novel developments aim for even higher mode discrimination by exploring lower symmetries for the arrangement of holes [35], as shown schematically in Figure 2.9. Finally, a complex analysis of the whole waveguide including the pump core and, for example, its bending has to be considered to maintain a stable output mode [36].

Figure 2.9 Example of non-hexagonal fiber design: building instructions for a pentagonal two-ring large pitch fiber core region.

2.2.3 Pump Core Design

The pump core of a double clad fiber is usually defined by a low index polymer coating or a low index doped glass as the outer cladding. Depending on the index difference, numerical apertures for the pump core of up to 0.45 are possible by such methods. It is also possible to form an air-cladding region to create double-clad fibers. Such an air-cladding can be achieved by surrounding the inner cladding with a web of silica bridges (Figure 2.10). A simple calculation evaluating the effective index nb of the fundamental mode in the silica bridge (slab waveguide with the diameter of the bridge width) to determine the numerical aperture is shown in Figure 2.10b. This simplified calculation is in excellent agreement with experiments and full calculations [37]. If the bridge width is substantially narrower than the wavelength of the guided radiation, a higher numerical aperture compared to conventional polymer or glass double clad fibers (NA ≈ 0.4) can be achieved. This allows the diameter of the inner pump cladding to be reduced, while maintaining sufficient numerical aperture for efficient pumping. The advantage of shrinking the inner cladding is that the overlap ratio of the core to the inner cladding increases, leading to shorter absorption lengths – and thus higher thresholds for nonlinear effects. Alternatively, the large index step between the inner and outer claddings can lead to very large numerical apertures (≈0.8) [38]. These large numerical apertures greatly reduce the need for sophisticated coupling optics of high-power diode laser stacks into the active fiber if the diameter of the inner cladding is not reduced. Furthermore, no radiation has direct contact with the coating material (and therefore cannot burn it), which avoids another common problem with conventional fibers. Figure 2.11 shows some realizations of air-clad photonic crystal fibers: a three-missing hole core design (Figure 2.11a), a 19-missing hole design (Figure 2.11b), and also advanced designs with different hole diameters to provide single-mode guidance (Figure 2.11c) as well as a seven-hole missing design but with a web-type air-cladding (left inset Figure 2.11d).

Figure 2.10 (a) Schematic comparison of a step-index double clad fiber (left) and a microstructured double clad fiber consisting of a microstructured core and an air-cladding region (right); (b) theoretical dependence of the NA of the air-cladding on the bridge width. The measured values for different bridge width are also shown.

Figure 2.11 Various realizations of double clad PCFs.

2.2.4 Polarization Control

Beside the innovative inclusion of the double clad design by means of air-holes and the realization of large single-mode cores by a microstructured cladding, further optical properties can be included while maintaining all other advantages and properties. For instance, polarization control is added to a waveguide by sufficient birefringence. For isotropic fibers, the modal birefringence (the difference in the effective index for both polarization states) almost vanishes and is of the order of 10−6 due to fabrication irregularities. Enough birefringence can be obtained by form birefringence or material anisotropy. Form birefringence can be realized, for instance, by elliptical cores in step index fibers. Despite the realization of elliptical cores in photonic crystal fibers, small variations in the symmetry of the cladding structure can also break the degeneracy of the two polarization states of the fundamental mode. Using this technique, a high birefringence can simply be obtained with different air-hole sizes in twofold symmetry surrounding the core [39]. Especially for small core PCF, a higher level of form birefringence (B ≈ 10−3) can be obtained compared to step index fibers, which can again be attributed to the higher index contrast of fused silica to the air-holes [40].

Nevertheless, the implementation of form birefringence suffers from the disadvantage that it decreases rapidly for larger cores [Noda86]. Thus, as an alternative, material anisotropy can be introduced to the fiber core. This can be achieved by external forces or by the well-known technique of stress-applying parts (SAP) inside the fiber, where the elasto-optical effect introduces anisotropy and therefore birefringence. The latter offers the advantage of an intrinsically permanent birefringence. In addition, there is the advantage of a relatively low wavelength dependence of the stress-induced birefringence [41], which results from the low dispersion of the stress-optical coefficient. In combination with the large single-mode wavelength range of photonic crystal fibers, stress-induced birefringent fibers can provide a large highly birefringent bandwidth [42].

To achieve this, the stress applying elements, which are usually placed apart from the core, (or even the inner photonic cladding) are moved close to the core. At the same time, the guiding properties of the single-mode core should not be affected. Figure 2.12 shows the basic idea.

Figure 2.12 (a) Design of a polarization maintaining large mode area (seven missing holes to form the core) photonic crystal fiber consisting of index-matched stress applying elements (largely-filled circles) as part of the photonic cladding; (b) microscopic image of a realized fiber.

The stress applying elements consist of a material with a different thermal expansion coefficient α to that of the surrounding cladding material [fused silica (FS)]. Using boron-doped silica (BS) with αBS = 5 × 10−7 K−1 compared to αFS = 10 × 10−7 K−1, a permanent stress field can be generated when cooling the fiber below the softening temperature during the drawing process. Because the refractive index of this material is lower than that of fused silica (Δn =− 0.008), a similar periodic inner cladding compared to that of the air-hole cladding can be constructed (Figure 2.12a) by matching the effective cladding indices so that they are equal. Figure 2.12b shows an image of a fabricated fiber.

It turns out that the birefringence is high enough to reduce the effective index of the fast axis to a value at which the polarization mode is no longer guided. The fiber only guides one single polarization mode within a certain wavelength region depending on the structural parameters. This polarization window is not only affected by the guiding strength (~d/Λ) but also by the bending of such a fiber. (Figure 2.13). It shows that the polarizing window can be tailored to the user's requirements, for instance to overlap with the laser bandwidth of an actively doped fiber.

Figure 2.13 Dependence of the polarizing window on the bending diameter.

Figure 2.14a shows an actively doped polarizing fiber with an air-cladding to obtain a double clad fiber. The corresponding mode of the core is shown by a near-field image in Figure 2.14b. It combines the low nonlinearity single-mode core with the double clad design, but also features single-polarization properties in the maximum gain wavelength region of short-length Yb-doped fibers. Such a fiber has been applied to laser and amplifier configurations to prove these properties in stable single-mode, high-power laser output [43].

Figure 2.14 (a) Cross-section of the polarizing large-mode-area PCF; (b) corresponding near-field image of the fundamental mode.

2.3 Theoretical Description and Nonlinear Effects in Laser Fibers

2.3.1 Propagation and Rate Equation Description

The laser process in fiber amplifiers is described theoretically by combining the local rate equation for the laser process and the power flow (propagation equation) for the fields along the fiber. They have been developed to predict and optimize erbium-doped fiber amplifiers used for telecommunication application [44].

The local rate equation describes the dynamic of the emission and absorption processes of the rare earth ion within its host material by using its spectroscopic properties. Figure 2.15 shows a model for the energy level system for the ytterbium-ion, along with some of the most important emission and absorption lines, which result from the Stark splitting of the upper and lower energy lines. It has been argued that for erbium and ytterbium fibers a reduced two level model for the emission and absorption process can be used so that the effective emission and absorption cross section include the population density and cross section values [44, 45].

Figure 2.15 Energy level model for ytterbium-ion in fused silica.

Figure 2.16 shows the effective cross sections for emission and absorption σem/abs of an Yb-doped fiber. Clearly, for a specific fiber, these parameters have to be measured because they are to some extent influenced by the actual core composition – different co-dopants lead to changes in the resulting curve. Especially, phosphor and aluminium influence the absorption peak around 960–980 nm and have been actually used to tailor this property.

Figure 2.16 Typical emission and absorption cross section of an Yb-doped fiber.

Approximations to the full equations given in Reference [44] lead to simplified models. Here we only show a very basic system of equations for the steady-state solution of a single-mode double clad fiber laser or amplifier. Four equations for the forward and backward propagating signal and pump powers remain:

(2.5)

where the total ion density n0 = n1 + n2 is the sum of upper and lower population density and αP/S is an additional loss for the fields. Amplified spontaneous emission has been neglected and it is assumed that the pump absorption can be described by a simple overlap factor Γp, which is the ratio of doped core area to pump core area (Γs = 1). The upper population density for steady-state conditions is:

(2.6)

with τ as the upper state lifetime and the total power at a given position P(z) = P+(z) + P− (z). The inversion level is defined as n2/n0. Equations (2.5) and (2.6) can be solved not only for fiber amplifiers but also for lasers, if the boundary conditions at the laser mirrors (typically z = 0, L) are included in the description for forward and backward propagating fields.

As an example, Figure 2.17 shows the result of such a simulation for a fiber laser with a length of L = 10 m. The boundary conditions of an out-coupling mirror with a reflectance of R1 = 4% at z = 0 and a highly reflectance mirror R2 = 99% at z = L has been realized by setting and . The signal wavelength is 1060 nm. The fiber has a pump core diameter of 500 µm and a doped core of 30 µm with an ytterbium doping concentration of 3600 ppm. The fiber is pumped at 976 nm from both sides with and . The background loss has been set to 0.02 dB m−1. The output power result is .

Figure 2.17 (a) Propagating signal and pump powers inside a fiber laser cavity; (b) inversion level and power dissipation along the fiber.

Beside the power distribution inside the fiber, Figure 2.17b shows the inversion level along the fiber. It is quite uniform for this example, with an average inversion of ~7%. By calculating the dissipated power, which is also shown in the figure, one can realize that on average 25 W m−1 of thermal load dΦTL is generated in this example. The processes involved in this power loss are the quantum defect, non-radiative decays, and the linear loss introduced into the simulation. It might be that the loss does not contribute to heat generation directly in the core but the level of dissipated power is worth considering when dealing with thermal issues in high-power fiber lasers (see next section).

2.3.2 Thermo-optical Effects

To analyze the thermo-optical properties of fibers heat transfer considerations can be performed [46]. Thermal conduction is the heat transfer process in solid materials, that is, in the fused silica and coating part of a fiber. It is described by dΦCOND in Equation (2.7), where k is the thermal conductivity of the material, L the length of the heat flow, dA the cross area of the heat flow, and ΔΤ the temperature difference. For the heat flow through the cross section of a cylinder with R2 as the outer and R1 as the inner radius, which a fiber can be approximated to, Equation (2.8) can be used. For a photonic crystal fiber with an air-clad the conductive heat flow dΦCOND through the silica bridges of the air-clad is simply given by Equation (2.9) with N as the number of silica bridges, WBridge as the bridge width, and LBridge as the bridge length:

(2.7)

(2.8)

(2.9)

In addition there might be convective and radiative heat flow in the chambers of the air-clad, described by Equation (2.10) and Equation. (2.11), where αk is an empirical value of the thermal conductivity for natural (not forced) convection, with its temperature dependent coefficient C1 and d as the diameter of the cylinder [47]. Equation (2.11) describes the radiative heat flow by the well-known Stefan–Boltzmann law with σ = 5.6705 × 10−8 W m−2 K−4 and a typical emission factor for fused silica of ε = 0.95. Actually, due to the small size of the air-chambers, convection is negligible here, but, indeed, the conduction in air of 0.023 W mK−1 leads to good heat transport through the air-clad [48]:

(2.10)

(2.11)

Heat dissipation from the coating to the ambient air also involves convective and radiative mechanisms. Therefore, the balance between the thermal load in the fiber's core and the convective and radiative heat flow given in Equation (2.12) determines the temperature difference between the fiber surface and the ambient air (see Figure 2.18b for temperature difference definition ΔT):

(2.12)

Figure 2.18 Temperature increase within the fiber's cross section for different thermal loads.

A full finite element analysis of a fiber, which basically solves the same heat transfer equations as discussed, is shown in Figure 2.18 for a fiber with an outer diameter of 850 µm, a pump core of 500 µm, and a core of 30 µm, as well as an acrylate coating with a thickness of 150 µm. The heat load is varied from 100 to 300 W m−1 but it is assumed that the heat is efficiently removed on the outer surface to keep a constant temperature. Nevertheless, the outer temperature will not change the gradient inside the fiber and so a temperature increase in the fiber's core above 100 K is obtained for such geometry and at such heat load levels. Is also reveals that the air-clad adds some thermal resistance but its influence also in terms of mechanical stability is negligible to first order [49].

From Figure 2.18 it can also be seen that because the heat source is the fiber core a gradient appears within the core itself. Owing to the temperature dependence of the refractive index, this might influence the guiding properties of the fiber. Figure 2.19 shows the situation schematically. An analysis can be made by simulating the guided mode under the influence of such an index deformation [50]. For this analysis it is necessary to introduce an alternative condition for single mode operation as the V-parameter definition does not hold for such a shape. Calculating the overlap of the intensity distribution of the first higher order mode with the core region (Icore/Ioverall) is more useful – indeed this parameter is constant for SIF (=0.33) and one-hole missing PCF (=0.52) at the cut-off condition independent of the actual core design (radius, NA). Therefore, this criterion is used as single-mode to multi-mode transition in the analysis of a step index fiber with NA = 0.03 (MFD = 30 µm) shown in Figure 2.20. Illustrated is the overlap of the first higher order mode with the core region (–––) and the change of the mode field diameter (– – –) subject to the thermal load. As shown, with increasing the thermal load the overlap of the LP11 mode increases (the confinement increases) and reaches the single-mode limit at a certain value. The MFD changes to smaller values with a nearly linear slope. Lowering the V-parameter makes the fiber more insensitive to thermally induced refractive index profile deformations. On the other hand, there is a trade-off on lowering the V-parameter at a constant NA because it also means a lower mode-field diameter and therefore higher nonlinear effects, which will be discussed in the next section. In conclusion of this analysis, for a fiber with given parameters, the thermal lensing might influence the fiber performance in terms of higher order modes above a thermal load of 160 W m−1. According to the rate-equations simulation carried out before, this should happen at power levels well above 10 kW.

Figure 2.19 (a) Undisturbed index profile of a step index fiber; (b) typical shape of the temperature gradient in a fiber core under laser conditions; (c) influence on fibers core refractive index profile.

Figure 2.20 Simulated overlap of the LP11 mode with the core region (solid line) and the MFD of the LP01 (– – –) for a SIF with a NA of 0.03. The horizontal dotted line is the single-mode condition described in the text.

2.3.3 Inelastic Scattering

With its confinement of light inside a small core and the long propagation length nonlinear optical effects might increase when operating at high power levels. One sort of these nonlinear effects often occurring in continuous wave but also pulsed high-power fiber lasers is stimulated inelastic scattering processes. Brillouin scattering is the interaction of photonic with acoustic phonons while Raman scattering involves optical phonons. These inelastic scattering processes lead to an energy loss of the photon and therefore wavelength shift (Stokes shift). For fiber laser systems, this means a reduction in spectral purity but also power loss and an additional heat load. Here we consider stimulated Raman scattering (SRS), which is described by the propagation equations (2.13), assuming identical effective areas for the different wavelength, where PS is the power in the Stokes wavelength λS, PP the original signal power at λP (pump power for the Stokes field), gR the Raman gain coefficient, and αS and αP the losses at the Stokes and signal wavelength, respectively:

(2.13)

The equations for stimulated Brillouin scattering (SBS) are similar, except that the generated Stokes field is propagating in the other direction. This can be explained by keeping in mind that the acoustic wave (sound) is a local deviation of pressure and therefore density and leads to an optical modulation thanks to the elasto-optical effect. This modulation can be seen as a moving grating Doppler-shifting and reflecting the optical field. Furthermore, the Brillouin gain is 500 times smaller than the SRS gain.

The physical origin of inelastic Raman scattering is the excitation of molecular vibrations. Practically, the wavelength of the propagating signal is shifted and power is lost. With some photons already generated at the Raman wavelength, the process can be stimulated and converts (depletes) a large amount of the signal (the term ~gRPS is large in Equation (2.13)). From the above equations and some approximations, one can derive a threshold power at which this conversion manifests [51]:

(2.14)

Solving the Equation (2.13) numerically visualizes this situation. The results are shown in Figure 2.21, where a passive lossless fiber is assumed. For a power below the threshold, the light can propagate undisturbed through the fiber (Figure 2.21a). Above the threshold, the signal is converted into the first Stokes wave at a position determined by the precise power level (Figure 2.21b). At higher power levels, this first Stokes might generate a second Stokes with respect to the original signal as shown in Figure 2.21c. In addition power is lost due to the quantum defect even in this lossless situation.

Figure 2.21 Simulation of SRS in a lossless fiber with signal input at z = 0: (a) signal power below threshold, (b) above threshold for generating 1st Stokes, and (c) strong signal to generate 2nd Stokes.

Figure 2.22 shows an experimental proof of this. A CW fiber laser at 1062 nm with an average power of 10 W is coupled into a 1.6-km long passive single mode fiber. The spectrum versus the input power is shown in Figure 2.22a. As one can see, at a relatively low power of 0.75 W, a new spectral component at 1120 nm is generated. Increasing the power to 5 W leads to the generation up to the fifth Stokes and a power generated up to a wavelength around 1500 nm. Above that power level, no individual lines can be observed due to the interplay of further linear and nonlinear effects. The spectral picture is again analyzed in Figure 2.22b, which show a good qualitative agreement with the simulation above with respect to total conversion to the next higher Stokes.

Figure 2.22 (a) Experimental spectrum at the output of the fiber, depending on input power, and (b) its integrated powers for each spectral line.

For the design of fiber laser systems, SRS has to be considered and the equations have to be combined with the propagation equations (2.13) to account for additional gain due to the laser process and spontaneous photons through ASE. One then can end up with better prediction of the Raman threshold [52].

2.3.4 Self-Phase Modulation

Another nonlinear effect encountered in fibers is the Kerr effect – the intensity dependence of the refractive index covered by Equation (2.15). There, n2 is the nonlinear refractive index that is of the order of 3 × 10−20 m2 W−1 for fused silica:

(2.15)

Owing to the instantaneous nature of this effect, an intensity variation over time will lead to a phase change and, consequently, to an instantaneous frequency shift. With the scalar optical field amplitude A(T) = , the frequency change dω/dz is covered by Equation (2.16), where γ is the nonlinear coefficient:

(2.16)

While this effect plays an important role for short pulse lasers and amplifiers due to the temporal profile of the pulses and their high peak powers, it will also lead to spectral changes for CW radiation. Practically, there are two kinds of spectral changes that are usually recognized experimentally. Firstly, a spectral broadening appears on the output signal with increasing power and, secondly, the generation of sidebands. Both situations are not directly obvious by taking a CW field with ϕ(T) = 0 in Equation (2.16)