Supercontinuum generation in bulk and photonic crystal fibers

Transcription

1 Supercontinuum generation in bulk and photonic crystal fibers Denis Seletskiy May 5, 2005 Abstract In this paper we overview the concept of supercontinuum generation (SCG) and take a brief look at several mechanisms that are believed to give rise to this not yet fully understood phenomenon. We will emphasize why photonic crystal fiber (PCF) is a more desirable medium for observation of SCG and will conclude with a discussion of several emerging applications. 1 Introduction to SCG 1.1 What is supercontinuum generation? In 1970s [1] it has been observed that intense picosecond pulses can exhibit spectral broadening upon nonlinear propagation through transparent glasses and crystals. The observed continuum covered visible and near-infrared wavelengths. The following years, similar broadening has also been observed in other solids, like semiconductors [2] and also in liquids [4] and gases [3]. Spectral broadening covering ultraviolet to near-infrared has been observed by femtosecond excitation of ethylene glycol films [5]. This general behavior, i.e. spectral broadening of light, has been termed supercontinuum generation (SCG) (or white light generation), as it can typically reach 200 nm broadening or more in the optical frequency range (Figure 1) [6]. As was noted by Gaeta [7], shapes of observed supercontinuum generated spectra for various nonlinear media and under a wide variety of experimental conditions are similar, which suggests that SCG is a universal feature of the light-matter interaction. Most general characteristics of SCG spectra are highlighted by Brodeur et al. [6] as follows: spectral width is dependent on the host medium, output polarization is preserved with input pulse, anti-stokes frequency components of spectra are temporally lagging its Stokes components and are asymmetric with respect to each other, broad blue-shifted pedestal with sharp cutoff appears as distinct feature of SCG [7](see Figure 1). Experimental investigations also noted that threshold power to generate supercontinuum spectrum is always higher than critical power needed for self-focusing in given material. In spite of wealth of experimental data investigating SCG phenomena, at the time of writing of this report, no well-agreed upon theoretical explanation has been given to the observed spectral behavior. Nonetheless, let us try to qualitatively summarize approaches currently existing in the literature. 1

2 Figure 1: Typical white-light continuum generated in water, UV-grade fused silica and NaCl [6] 1.2 Proposed mechanisms of SCG Initial attempts to theoretically explain spectral features of SCG included processes of selfphase modulation (SPM) [1], ionization-enhanced SPM [4], and four-wave mixing [1],[9]. As the most agreed upon mechanism in the literature is SPM, we will center our discussion on that. To qualitatively understand how SPM might be responsible for spectral broadening, consider a pulse given by E(r, t) = A(r, t) exp[ik o z iω o t + i φ(t)] + c.c. - where ω o is a carrier frequency of the pulse, φ(t) is a phase change accumulated upon propagation through an optical medium. If that propagation is through a non-linear Kerr medium of length L, with quasi-instantaneous response time (typical electronic nonlinearity has response time of sec) then induced phase change will be of the form (Eqn 1): φ(t) = L 0 n 2 I(z, t)k o dz (1) where I(z, t) is intensity of the pulse and n 2 = 12π 2 χ (3) /(n 2 oc). Based on Equation 1 we can redefine carrier frequency of the pulse such that: ω o ω o + ω(t), where ω(t) = d φ(t)/dt. If we now assume A(z, t) = A o cosh 1 (t/τ p ), then normalized broadening ω(t)/ω o will be given by Eqn 2: ω(t) ω o = Q sinh(t/τ p) cosh 2 (t/τ p ) (2) where τ p is pulsewidth, Q = 2n 2 I o L/(cτ p ) and I o = n o c A o 2 /2π. This basic consideration shows qualitatively how SPM leads to spectral broadening of the pulse. If one now includes self-focusing nonlinearity, SCG behavior is evident: initial pulse will broaden due to SPM, but self-focusing will tend to increase magnitude of I o term, and also ω(t) (by Eqn 2), leading to enhancement of SPM and hence further broadening. As was argued, 2

3 self-focusing wouldn t lead to singularity (damage), but processes like free electron ionization, multi-photon excitation (MPE), stimulated Raman scattering (SRS) would limit process of self-focusing [6]. These processes are mentioned here for completeness only and won t be covered. One has to note however that, in the spirit of early explanations (e.g. [1]), treatment above ignores, among other things, envelope-shape distortion (which leads to self-steepening of the pulse). It is also very important to note that above treatment, although illustrative of the broadening, is not adequate for describing more complex features of supercontinuum spectra. Note that ω(t) in Eqn 2 is symmetric about ω o, and that feature is drastically different from the experimentally observed shape of spectra, where Stokes and anti-stokes components are asymmetric about carrier frequency. Asymmetry in the spectrum was for some time attributed to an interplay of group velocity dispersion (GVD) and SPM [10]. However, researchers soon realized the importance of before-ignored fact of self-steepening. It was shown that asymmetry of the spectra can be explained if Equation 1 was modified to the following [8]: [ z + n o c ( 1 + n 2 E 2) ] φ = n 2ω o E 2 (3) n o t c Note that to obtain Eqn 1 from more general Eqn 3 one merely sets second term in square brackets to zero. Treatment of Ref. [8] relaxes slow-varying envelope approximation and achieves asymmetric broadening, by solving Eqn 3 with above assumption of electric field distribution in the pulse. Solution yields asymmetric broadening (Eqn 4) which is more consistent with observed features of SCG spectra (Figure??): ω(t) [ ] = 1 + (Q 2 2Q sinh(t/τ p ))/ cosh 2 1 (t/τ p ) 2 1 (4) ω o where Q is defined above. Note, for Q 1 Eqn 4 reduces to Eqn 2. For Q 1 effects of spectral asymmetry are clearly observed, leading to effect of self-steepening (SS) of the pulse in the time domain (see Figure 2). Self steepening, as shown by Yang et al. ([8]) more accurately describes SCG spectra, by highlighting asymmetry of the spectral broadening. However, treatment given by Ref.[8] misses another important point in the spectrum of SCG - feature of broad blue-shifted pedestal [7]. To finish presentation of SCG proposed mechanisms let us present a model by A.Gaeta. Starting point of his analysis (Ref.[7]) is the thought that collapse of the pulse, and not necessarily self-steepening is a more correct way of describing SCG. His formalism starts with nonlinear envelope equation (NEE), which has the following form: u ζ = i 4 ˆD 1 2 u i z o L ds 2 u τ 2 + i ˆDp nl (5) where u = A(r, t)/a o, z o = kwo/2 2 - Rayleigh range, ζ = z/z o - normalized distance, L ds = τp 2 /β 2 - dispersion length with β 2 - group velocity dispersion, τ = (t z/v g )/τ p - retarded time, p nl - normalized nonlinear polarization and ˆD = 1 + i/(ωτ p )( / τ) - differential operator responsible for space-time focusing and self-steepening [7]. NEE is a normalized equation that is readily obtained from wave equation in frequency domain, where k(ω) dependence is Taylor expanded to second order and then converted back into 3

4 Figure 2: SPM is compared with high value of Q, leading to self-steepening of the pulse in time domain (asymmetric spectra), based on [8] Figure 3: Numerically simulated spectra of SCG with respect to carrier frequency, (a) - P/P cr = 1.7; (b) and (c) - P/P cr = 1.8 and ζ = z/z o = 1.63 and 1.7 respectively [7]. 4

5 the time domain, with retarded time coordinate transformation. Gaeta (Ref. [7]) also goes on to include Kerr index change, multiphoton absorption and electron plasma contributions to the nonlinear polarization term. The main result of Ref. [7] is reproduced in Figure 3. It shows simulations at the numerically determined power threshold of SCG; i.e. when P/P cr > 1.7 where P cr = π(0.61) 2 λ 2 o/(8n o n 2 ) - is a critical power required for self-focusing to occur [11]. Results successfully show asymmetric spectra along with blue-shifted spectral pedestal and agree with the experimentally observed onset of SCG at the power threshold of self-focusing. To conclude this overview it would be instructive to note several things that emerged from the presented succession of theoretical understanding of SCG spectra. Models (Ref. [7]) seem to explain qualitative shape of SCG spectra, and further agree with experiment about the onset of SCG - that it is intimately tied to the self-focusing of the pulses and transformation of collapse into SCG spectrum. Note that power requirement to reach P cr can be quite high. That power is even higher for ultrashort pulses than for CW pump. Obviously temporal dispersion will be a negating mechanism for self-focusing (balance of Kerr index change and diffraction), and hence higher P cr for ultrashort pulse. One can lower power threshold of SCG by engineering dispersion properties of material, and one very powerful way of achieving just that is through emerging technology of photonic crystal fibers. 2 Introduction to PCF 2.1 What is photonic crystal fiber? Photonic crystal fibers (PCF) is a whole family of fibers which can be classified as having inhomogeneously engineered core. As a comparison with conventional fibers, where typically homogeneous higher index of refraction cylinder (core) is surrounded by lower index shell (cladding), PCF is a solid cylinder of material with microstructured air holes. Transverse confinement of light in conventional fiber is achieved through total internal reflection at the core-cladding interface, while in PCF transverse confinement is achieved by two different methods. In both methods PCF cross-section looks similar to Figure 4, with one crucial difference: center of the fiber (core) has either a material (as in figure) or an air pocket. Fiber with material center is termed microstructured fiber (MF) and guidance mechanism is similar to conventional fiber, light being confined to material center and the presence of surrounding air holes reduces effective index, acting as cladding. Fiber with air hole center is termed photonic bandgap fiber (PBF) and it guides light by having two-dimensional transverse bandgap, i.e. 2D Bragg condition on the geometry of the air hole lattice. Since we are interested in nonlinear optical processes in PCF we are mainly going to concentrate on MF, as requirement of light propagating in nonlinear material is easily satisfied. Current methods of manufacturing of PCF involve 2 processing stages. During first, preform is stacked with capillary tubes and rods of different material and geometry. Second stage involves controlled heating of preform glass to the so called super laminar flow point, at which cross section of the preform, as being stretched, only scales in size, while preserving packed geometry [13]. Due to complex geometry, analytical solution for PCF parameters (e.g. dispersion 5

6 Figure 4: SEM images of microstructured fibers. (A) - multimode fiber in visible wavelengths, (B) single mode fiber at 800 nm (same scale) [12] curves) can not be obtained and numerical techniques are therefore routinely implemented. Numerical methods used to solve PCF geometries are approximate (multipole and plane wave expansions) and exact (finite-difference time and frequency domain approaches). These are outside of the scope of this review and are only mentioned here for completeness. 2.2 General properties of PCF favorable for SCG Modal properties In conventional waveguide, one can define a normalized frequency parameter: V = 2πa n λ 2 c n 2 cl (6) where a - core radius, n c and n cl - core and cladding indices of refraction. For cylindrical waveguide to support single mode one requires V < [14]. For conventional fiber it means that once a structure is specified (a,n c,n cl ), the single mode bandwidth is determined as well. In contrast, effective n cl of the MF is wavelength dependent, and hence adjusting photonic crystal geometry (period and pitch) one can create endlessly single mode fibers [13]. Naturally single mode operation means higher nonlinearity and hence reduced power threshold for SCG. Dispersion Due to the above mentioned n cl (λ) in MF, dispersion properties can also be engineered. For example, increasing the air-filling fraction and reducing the size of the effective core allows for drastic increase of the waveguide s dispersion. Zero-dispersion wavelength can therefore be shifted to comfortable wavelengths for Ti:Sa oscillators around 800nm [13] ( Figure 5). Note that having zero-dispersion wavelength right at the excitation wavelength for SCG can mean that part of the broadening spectrum will be in the anomalous dispersion region and hence can support solitonic propagation. This would imply that treatment given in previous section would be inadequate for case of SCG via MF. Moreover, treatment presented above relied mainly on the fact that excitation peak powers were higher than P cr to induce self-focusing (i.e. short pulses). That also would require focusing pump beam into the medium and hence limiting interaction length of the nonlinear processes. In MF nonlinearity can be induced with much less peak power requirement and interaction length is not sacrificed, enabling researchers to investigate other weaker nonlinearities than 6

7 Figure 5: Figure shows modification of dispersion in 2 different geometry MF. S1 and S3 have very similar profile so not shown together for brevity [13]. SPM, like stimulated Raman scattering (SRS) and four-wave mixing (FWM), which are also candidates for mechanisms of SCG. Nonlinearity It is obvious that strength of nonlinearity scales inversely with the area of the interaction, i.e. more confined modes in a fiber can effectively increase induced nonlinearity. MF allows both large area and small area single-mode operation, hence controlling nonlinearity better than conventional fiber ( Eqn 6). This again is important for SCG as peak power threshold can be lowered. 2.3 SCG via MF In previous sections we have introduced concepts of supercontinuum generation and discussed why it is beneficial to use photonic crystal fibers for these purposes. Now we are finally ready to talk about SCG using MF. As was hinted above, MF allows SCG to be observed in different peak power regime than SPM (i.e. if SPM was done with femtosecond pulses, SCG in MF can be achieved with nano- and picosecond excitations). This also implies that SPM is not a leading mechanism of SCG at those powers and further new array of mechanisms is proposed, including stimulated Raman scattering (SBS), four-wave mixing (FWM). However work with SCG in MF has been started fairly recently [15] and therefore no widely accepted opinion has been reached on the leading mechanisms of SCG. For that reason, the rest of this report will try to explain only qualitatively observations and proposed mechanisms leading to SCG in MF. Two distinct power regimes are quickly reviewed below, one is femtosecond excitation of MF leading to SCG [15], and another is picosecond excitation proposing different mechanisms mentioned above for SCG [16]. Very first realization that dispersion in MF can be controlled to the benefit of SCG was realized by Ranka et al. in Researchers realized that if zero-dispersion wavelength is matched in a MF with the excitation wavelength, SPM will have lower threshold. Excitation centered at zero-dispersion wavelength thus will be larger than P cr longer (as compared to bulk) and hence will lead to broader spectra being generated. Figure 6 shows SCG extending 7

8 Figure 6: SCG in 75cm NF fiber. Dashed curve is initial 100fs pulse. [15] from 400nm to 1.6 µm, which corresponds to 550 THz spectral width - a dramatic result hinting at the possibilities of MF to produce SCG. Researchers concluded that the main mechanism of SCG in their setup was SPM [15], although spectral shape looks different from the bulk and the question of mechanism is still an open topic in modern research. Following work of Ranka et al., another research group in 2002 realized that picosecond excitation can also be used to generate SCG in MF [16]. They argued that femtosecond pumping is a brute force method of SCG and conducted careful studies investigating SCG below P cr necessary for appreciable SPM. They concluded that the main mechanisms responsible for SCG in the lower power regime are SRS and FWM. The limitations of testing their hypothesis came from the phase-matching requirement of the FWM process. They solved it through an engineering of GVD in the MF such that phase-matching condition was satisfied around the excitation wavelength. This is similar idea of the conventional quasiphase matching, where structural, not material dispersion dictates optical properties. Since both mechanisms have gain (which researchers estimated to be g f 2g s 100/m [16] being FWM and SRS respectively), one can expect strong effects due to these NL mechanisms for fibers of few centimeters or more. For fiber lengths of 10 meters researchers observed continuum broadening, and argued that long interaction length eventually ensures merging of generated spectral components from both mechanisms (Figure 7). From spectral shape of SCG in Figure 7 several features are observed. First of all, there s a strong component of the initial excitation frequency. This suggests that SPM is not the leading mechanism of SCG for this experiment. The P cr threshold is not reached even at highest peak power as excitation feature is still strongly visible. One can qualitatively understand the broadening as follows: initially SRS broadens excitation pulse by small amount. Since DFWM process ( ωo + ω o ω s + ω as ) is phase-matched by the GVD engineering especially well around the zero-dispersion wavelength (pump wavelength), new frequencies are quickly generated. This cascading nonlinear effect is further enhanced by the contribution of nondegenerate FWM (which, according to calculations is also phase matched [16]). Researchers also changed fiber length and determined that reduction of length by factor of 3 didn t affect spectral 8

9 Figure 7: SCG spectra with 10m long MF for input powers P = 120, 225, 675W (bottom to top)[16]. composition of SCG leading them to believe that observed spectral broadening is limited by interaction length, i.e. efficiency of their phase-matched GVD profile. To summarize, SCG is phenomena that has been discovered in bulk material in 1970s. SCG was initially generated with high peak power pulsed excitations leading to asymmetrically broad spectrum when peak power was higher than critical self-focusing power in the material. These observations has led to theoretical models of self-phase modulation mechanisms combined with self-steepening followed by collapse of the pulse. The point of collapse was determined to give SCG and several mechanisms of that transition have been proposed (but not covered in this review). In 2000 a realization of importance of dispersion control in fibers leads researchers to experimentally produce a 1200nm bandwidth SCG via MF in optical frequency range. Dispersion control also allows researchers to create phase-matching conditions for FWM and by probing MF below the self-focusing threshold new mechanisms of SCG are discovered, such as SRS and FWM. Currently, new experiments are being done to further push understanding of SCG mechanisms. As this is done, numerous research groups are using the already discovered properties of SCG for real world applications. 3 Applications In spite of difficulty of theoretical explanations for SCG, experimental findings of empirical rules and engineering strategies have already exploited some aspects of SCG for applications such as time resolved spectroscopy, optical frequency metrology and optical coherence tomography to mention a few. Let us summarize this report by brief introduction to some of these interesting applications. 9

10 3.1 Time resolved spectroscopy Relative high brightness and spectral width allows spectroscopists to collect signals using SCG illumination with about 10 fs resolution and in one-shot. This is done in a emerging standard of Pump/Supercontinuum probe (PSCP) arrangement, where a monochromatic pump excites the sample, and delayed SC probe looks at the transmission spectra. Another attraction of using SCG in spectroscopy is ability to use relatively inexpensive nanosecond pulse laser for SCG via MF. 3.2 Optical frequency metrology Current time standard are laser-cooled cesium atoms that emit vibrational radiation at 9GHz. Up until few years ago there was a big problem to translate that frequency standard into optical domain, i.e. 5 orders of magnitude higher. With discovery of SCG in MF routine experiments are done to get better measurement of absolute time than from cesium clocks by two orders of magnitude. Idea is very simple: a Ti:Sa oscillator generates a pulse train in time domain, with a jitter error of at least 1 part in [17]. In frequency domain this train of pulses is represented as a comb of frequencies, separated by free spectral range of the oscillator s cavity. If MF can support SCG that is at least wide as a carrier frequency (i.e. if ω scg = ω o, then absolute frequency of the oscillator can be measured. If initial frequency comb is: ν m = mν fsr + ν o, where ν o is some offset frequency and m is integer, then after one octave broadening (via SCG in MF) one can compare initial ν m passed through a χ (2) medium to generate SHG with the SCG broadened 2ν m, and by interfering the two can determine beat frequency, i.e. ν o. Once ν o and ν fsr are known, any ν m is known to the absolute value. Precise determination of time is needed not only in basic research but also in modern optical communications, and global positioning systems. 3.3 Optical Coherence Tomography Optical coherence tomography (OCT) is used in biomedical imaging to get axial as well as transverse visual detailed information, similar to confocal microscopy. Reflected light from sample depth is being correlated by scanning Michelson interferometer to produce a lowcoherence interferogram. Signal is then reconstructed by taking inverse Fourier transform of the obtained interferogram. Broadband is essential for getting axial resolution, as can be seen from ωτ p which is a transform pair and getting good resolution in z-axis (hence short τ p ) requires high bandwidth. SCG is a natural candidate for this application. In fact, using MF is an economical way of getting SCG since pulses as long as nano- and/or picosecond can be used. References [1] R.R.Alfano and S.L. Shapiro Phys. Rev. Lett. 24, 592 (1970) [2] P.B. Corkum et al. Opt. Lett. 10, 624 (1985) [3] P.B. Corkum et al. Phys. Rev. Lett. 57, 2268 (1986) 10

11 [4] W.L. Smith and N. Bloembergen Phys. Rev. A 15, 2396 (1977) [5] R.L. Fork et al. Opt. Lett. 8, 1 (1983) [6] A. Brodeur and S.L. Chin J. Opt. Soc. Am. B 16, 637 (1999) [7] A.L. Gaeta Phys. Rev. Lett. 84, 3582 (2000) [8] G. Yang and Y.R. Chen Opt. Lett. 9, 510 (1984) [9] A. Penzkofer et al. Phys. Rev. Lett. 31, 863 (1973) [10] R.A. Fisher and W.K. Bischel J. Appl. Phys. 46, 4921 (1975) [11] R.W. Boyd Nonlinear Optics, 2nd ed., Academic Press (2003) [12] - application note on scg in pcf [13] G. Genty SCG in MF and novel optical measurement techniques, Ph.D. thesis, Helsinki Univ. of Tech. (2004) [14] J.D. Jackson Classical Electrodynamics 3rd ed., John Wiley & Sons (1999) [15] J.K. Ranka et al. Opt. Lett. 25, 25 (2000) [16] S. Coen et al. J. Opt. Soc. Am. B 19, 753 (2002) [17] 11