All-fiber Otical Parametric Oscillator Chengao Wang Otical Science and Engineering, Deartment of Physics & Astronomy, University of New Mexico Albuquerque, NM 87131-0001, USA Abstract All-fiber otical arametric oscillator (FOPO) has received considerable attention throughout these years. In this aer, we review the basic theory four wave mixing (FWM) in the otical fibers, which leads to the arametric amlification. Phase matching condition is then discussed, with emhasis on the hase matching near the zero disersion wavelength. In the end, recent develoments on FOPO are introduced, with two reorts on the demonstration of cw FOPOs. Index Terms fiber otical arametric oscillator, arametric amlification, hase matching 1
I. Introduction Parametric amlification is a well-known henomenon in materials roviding () χ nonlinearity and otical arametric oscillators that use χ () nonlinear crystals have become widely available tools for use in otical research and develoment. However, arametric amlification and oscillation can also be obtained in otical fibers exloring the (3) χ nonlinearity. In FOPOs, crystals are relaced with otical fibers and the configurations can be made totally fiber integrated, virtually eliminating the need for alignment and allowing for increased comactness [1]. Thus, it has become a romising alternative to solid-state lasers. Since the nonlinearity of commonly available fibers is fairly low, obtaining sufficient gain for making a arametric oscillator with such fibers was difficult for a long time. Recently, with the availability of highly-nonlinear fibers and new high ower light sources, interest has been increasing in such FOPOs. II. Theory[] Parametric amlification with (3) χ nonlinearity relies on highly efficient four wave mixing (FWM) henomenon. We will describe it from three view angles. In an intuitive aroach, the non-degenerated rocess starts with two waves at frequencies 1 and that co-roagate together through the fiber. As they roagate, they will continuously beat with each other. The intensity modulated beat note at frequency 1will modulate the intensity deendent refractive index of the fiber. When a third wave at frequency 3 is added, it will become hase modulated with frequency 1, due to the modulated n. From the hase modulation, the wave at 3 will develo sidebands at the frequencies 3 ± ( 1 ). The amlitude of the sidebands will be roortional to the amlitude of the signal at 3. In the same way, 3 will beat with 1 and hase modulate, which will
generate sidebands at ± ( 3 1 ), where + ( 3 1 ) will coincide with the reviously mentioned 3 + ( 1). It should be noted that from a FWM rocess including three incident waves, nine new frequencies will be generated. Fig. 1 shows all nine frequencies. Fig.1 Frequency comonents generated due to FWM for two um at frequency w and w3 and a weak signal at w1 It also shows that some FWM-roducts will overla with the signal frequency. These roducts will result in a gain for the signal, i.e., rovide arametric amlification. In general, the remaining weaker frequencies are usually neglected excet the frequency comonent at 4 = 3+ ( 1) = + ( 3 1). The two overlaing comonents are referred to as generated idler. In the degenerated case with only one um, and 3 will coincide. For the rest of the aer, we will focus on the degenerated case including one um at, one signal at s and one idler at i. And also we assume the um ower is high enough to be un-deleted. Fig. shows the arametric amlification in the degenerated case. In this figure, the wave at frequency s is the idler and it can be noticed that the idler and the signal are ositioned symmetrical to the um. Another wave at frequency s is weak enough to be neglected. 3
Fig. Parametric amlification in degenerated FWM Parametric amlification can be viewed from a quantum mechanical icture. Here, the degenerated arametric amlification is manifested as the conversion of two um hotons at frequency to a signal and an idler hoton at frequencies s and i. The conversion needs to satisfy the energy conservation and the quantum-mechanical hoton momentum conservation relation, which is here interreted as hase-matching condition. From an electromagnetic oint of view, we may consider the interaction of three stationary co-olarized waves at angular frequencies, s and i characterized by the slowly varying electric fields with comlex amlitudes A, As and A i, resectively. This theory about FWM in otical fibers has been exlored fully by Stolen and Bjorkholm[3] and is summarized along with exerimental results by Agrawal[4]. One obtains the following couled amlitude equations that describe FWM in the fiber[5]: A A α = A ( ) (1 ) + iγ A a A z α = A + i [ A A + A A ex( i βz)](1 b) si () * si () γ s si () is ( ) z In Eqs. (1-a) and (1-b), α is the attenuation coefficient of the fiber according to P( z) = P(0)ex( α z), where P is ower and z is the roagation distance. β is the linear hase mismatch, which is given by β = β ( ) + β ( ) β ( ).The s i 4
nonlinear coefficient, γ, is related to the nonlinear refractive index, n, by γ n A c A = / eff, where eff is the effective mode area of the fiber. We have assumed here the um is much stronger than the inut signal. Furthermore, the frequencies are assumed to be similar such that γ are equal for the three light waves. By rewriting Eqs. (1) in terms of owers and hases of the waves, insight can be gained. Let φ ( z), where A ( z) = P ex( iφ ) j j j j dps 1/ = γ( P PP s i) sinθ dz ( a) dpi 1/ = γ( P PP s i) sin θ dz ( b) dθ 1/ 1/ = β + γ( P Ps Pi) + γ[( PPi / Ps) 4( PP s i) ]cos θ dz ( c) Here, θ ( z) describes the relative hase difference between the four involved light waves. θ ( z) = βz+ φ ( z) φ ( z) φ ( z) (3) s i As can be observed from (-a)-(-c), by controlling the hase relation, θ, we have the oortunity to control the direction of the ower flow from the um to the signal and the idler ( θ = π /, arametric amlification) or from the signal and the idler to the um ( θ = π /, arametric attenuation). For the general alication when the um is very intense and the idler is initially zero, we have θ = π / at the fiber inut ort [6]. Thus, if we somehow satisfy the condition β + γ( P P P) = 0 (4) s i θ will remain π /along the fiber and both the signal and the idler will grow exonentially. III. Phase Matching As discussed above, only when (4) is satisfied, can arametric amlification haen effectively. Thus, this is just the hase matching condition for arametric amlification. When it is oerating in an un-deleted mode ( P Ps ), 5
β + γ( P P P) β + γp = κ (5) s i The hase mismatch arameter κ is first introduced by Stolen and Bjorkholm in [3]. We may see that it contains both the linear hase mismatch and the nonlinear hase mismatch. In order to achieve hase matching, a lot of methods are used, such as using fiber modes, Birefringence matching, or hase matching near the zero disersion wavelength[3]. The last technique is the most widely used. Exanding β ( ) in Taylor series around the zero-disersion frequency 0, ( β( 0) = 0), the wavelength deendent art, β of the hase mismatch arameter κ can be rewritten as β = β 3( 0)( s) (6) Here, β and β 3 is the second and third derivative of the roagating constant β ( ) at 0. (6) may be transformed to the more generally used wavelength domain π cdd β = β + β β = λ λ λ λ ( s) ( i) ( ) ( 0)( s) (7) λ0 dλ π c Here, D = β, and dd / dλ is the sloe of the disersion at the λ zero-disersion wavelength. 0 Now, the hase matching condition can be written as γp = β ( λ λ0 )( λ λ s ) (8) By ositioning the um wavelength in the anomalous disersion regime ( λ > λ0 ), it is ossible to comensate for the nonlinear hase mismatch γ by the linear P hase mismatch β. Or, we can say, the arametric rocess establishes a balance between grou velocity disersion and the nonlinear Kerr-effect. 6
IV. Recent Develoment Since the first FOPO was reorted more than a decade ago[7], it has been attracting much interest. However, to date FOPOs have resented two major limitations: As the nonlinearity of conventional fibers is considerably lower than that of crystals, most FOPOs reorted used ulsed ums such that high eak um owers could be achieved and the arametric threshold reached; In addition, the hase matching between the um and the signal waves is achieved with the um near the fiber s zero-disersion wavelength, which limits the oeration of FOPOs to wavelength linger than 1300nm, where the zero disersion of silica is located[1]. Recently, FOPO configurations were reorted that somewhat overcome one or both of the abovementioned limitations. In one exeriment[8], a cw FOPO was demonstrated at ~1550nm through the use of fiber with a small core diameter that resulted in increased um intensities and therefore in a decreased arametric threshold. Last year, A team from Imerial College UK and the Danish firm Crystal Fibre claims to have made the first continuous wave (cw) all-fiber otical arametric oscillator (OPO) using a holey fiber[1]. The oscillator they made oerates at 1550nm and can yield an oscillation arametric signal that consists of a single line with a 30 db extinction ratio and a 10-m linewidth or that consists of multile lines. The source reaches threshold for a um ower of 1.8W and saturates for um owers in excess of ~1.6W. Fig. 3 shows their exerimental configuration. 7
Fig. 3 Exerimental configuration of the all-fiber arametric oscillator using holey fiber 8
References [1]C. J. S. de Matos, J. R. Taylor and K. P. Hansen, Continuous-wave, totally fiber integrated otical arametric oscillator using holey fiber, Otics Letters, Vol. 9,. 983-985, 004 [] Jonas Hansrd, Peter A.Andrekson, Mathias Westlund, et al. Fiber-Based Otical Parametric Amlifiers and Their Alications, IEEE Journal of Selected Toics in Quantum Electronics, Vol.8,No.3, P506-50, May/June00 [3]R. H. Stolen and G. D. Bjorkholm, Parametric Amlification and Frequency Conversion in Otical Fibers, IEEE J. Quantum Electron. 18,.106, 1981 [4] G. P. Agrawal, Nonlinear Fiber Otics, (Otics and Photonics, 3rd ed. (Academic Press, SanDiego, 001). [5]J. E. Sharing, M. Fiorentino, A. Coker and P. Kumar, Four-wave Mixing in micrestructure fiber, Otics Letters, Vol. 6,. 1048-1050,001 [6]K. Inoue and T. Mukai, Signal Wavelength Deendence of gain Saturation in a Otical Fiber Parametric Amlifier, Otics Letters, Vol. 6,.10-1, 001 [7]K. O. Hill, B. S. Kawasaki, Y. Fujii and D. C. Johnson, Efficient sequence-frequency generation in a arametric fiber-otic oscillator, Al. Phys. Lett. 36(11),. 888, 1980 [8]M. E. Marhic, K. K. Y. Wong and L. G. Kazovsky, Continuous-wave fiber otical arametric oscillator, Otics Letter, Vol. 7,. 1439, 00 9