## Abstract

655 nm laser radiation with power of >60 mW is generated by frequency doubling of a broadband randomly-polarized 1.31-μm Raman fiber laser (RFL). The red power appears to grow linearly with increasing RFL power up to 7 W at efficiency comparable with that for single-frequency lasers. It has been shown that multiple sum-frequency mixing processes involving different RFL modes provide the main contribution to the output, which is enhanced by 2 times due to the modes stochasticity.

©2009 Optical Society of America

## 1. Introduction

CW Raman fiber lasers (RFLs) are known as stable robust high-power fiber light sources providing almost any wavelength in the near-IR range (1.1-1.7 μm), see e.g. [1]. Recently, lasing at 2.1 μm has been demonstrated in GeO_{2} fiber [2]. Moreover, RFLs are able to emit multiple wavelengths simultaneously [3], and its output radiation can be tuned over a wide frequency range in an all-fiber configuration [4,5]. All these features turn the RFLs into useful tools which are already being applied in telecommunications as well as other fields such as supercontinuum generation [6] and optical coherence tomography [7].

A possibility to generate visible radiation by frequency doubling and thus to extend the range of RFLs applications is very attractive. A fiber-based yellow laser was proposed in [8]. The first experiment was performed in [9]. The bulk LBO crystal was placed inside the cavity of the RFL in order to increase efficiency. However, only ~10 mW of yellow power was obtained because of low conversion efficiency for the broadband unpolarized fundamental wave in this scheme. An impressive result has been obtained in [10]: 3 W at 589 nm was generated from a 23 W narrow-line linearly-polarized RFL with 8-mm long MgO-doped PPLN crystal. Recently, the yellow power of 14.5 W has been obtained with 20.7 W Raman amplifier [11]. A conversion of RFLs to red with tuning and power scaling capabilities is also possible [9] but not studied in spite of importance for bio-medical applications.

In the present paper, we report on the development of a robust CW red laser source based on the 1.31-μm phosphosilicate RFL. The obtained output power of >60 mW at 655 nm is the highest to our knowledge for frequency-doubled fiber lasers in the red range. The second harmonic generation (SHG) efficiency of ~1% obtained for the broadband fundamental wave appears comparable with that for single-frequency radiation at studied powers. A model taking into consideration sum-frequency mixing (SFM) processes inside the broad RFL spectrum has been developed. Computations show that SFM of multiple RFL modes with different frequencies provide significant changes in SH output power and spectral characteristics. A good qualitative and quantitative agreement of the model with the experimental data has been demonstrated.

## 2. Experiment

The experimental setup is shown in Fig. 1. We have utilized linear all-fiber scheme for the phosphosilicate RFL pumped by the Yb-doped fiber laser (YDFL), see e.g. [1, 12]. The YDFL is pumped by 3 laser diodes (LDs) and delivers up to 13.8 W at ~1.115 μm. Its cavity consists of fiber Bragg gratings (FBGs) with high reflection (HR_{1.1}) and high transmission (HT_{1.1}) at 1.11 μm. The RFL cavity is formed by FBGs HR_{1.3} and HT_{1.3} with reflection coefficients 99% and 23% at 1.31 μm, correspondingly, placed at the ends of 350-m long fiber produced by FORC (Moscow) having the Raman gain of 6.5 dB/km·W, the loss of ~1.1 dB/km and MFD of 5.9 μm at 1.31 μm. The second Stokes shift (~1330 cm^{-1}) associated with phosphorus secures conversion from 1.11 to 1.31 μm in one stage. Additional FBG HR_{1.1} (highly-reflective at 1.11 μm) is installed at the RFL output to implement a double pass pumping, see e.g. [12]. The RFL generates up to 7 W with *P _{RFL}*/

*P*~50% efficiency. Its output spectrum broadens significantly with increasing power, see Fig. 2(a), in spite of a narrowband (~0.2 nm) output FBG (HT

_{YDFL}_{1.3}). At high powers the spectrum acquires wide exponential tails with a central dip corresponding to the output FBG reflection. Mechanisms of the spectral broadening in RFLs have been clarified recently: multiple four-wave mixing processes involving numerous longitudinal modes (~10

^{6}in ~1-km cavity) induce stochastic evolution of their amplitudes and phases [13, 14]. The weak wave turbulence model [14] is applicable to RFLs having high-Q cavity with broadband HR Gaussian FBGs and describes their spectra well, i.e. exponential shape and square-root growth of the width with power. However, this analytical model is not applicable directly to our RFL with narrowband low-reflection FBG which behavior deviates from [14]: its linewidth grows almost linearly, see Fig. 2(c) (similar result has been obtained numerically [15]), while the power density tends to saturation at 3.5 W/nm level, see Fig. 2(a), that is important for frequency doubling.

Frequency doubling is performed in a simple and robust single-pass scheme with the 5% MgO-doped periodically poled LiNbO_{3} (PPLN) crystal with poling length L≈8 mm and specified poling period *∧*≈12.73 μm that provides quasi-phase matching (QPM) at ~49°C for 1.31-μm operation. Corresponding QPM bandwidth amounts to ~0.6 nm. The crystal has anti-reflection coatings for the fundamental and SH waves at both facets. RFL output radiation is collimated (see Fig. 1), then cleaned from 1.1 μm radiation by dichroic mirror DM and focused by lens L to beam waist radius w_{0}=22 μm into the PPLN crystal placed in the oven. The generated red spectrum has typical sidelobes, Fig. 2(b). The main SH peak is narrow, its width (FWHM) approaches ~0.3 nm at high powers, Fig. 2(c), in accordance with the QPM bandwidth. Although the RFL spectral width reaches ~1.6 nm (3 times broader than QPM bandwidth), the SH power (selected by filter F) grows linearly up to > 60mW at ~7 W RFL power, see Fig. 3.

The obtained ~1% efficiency is comparable with that for a single-frequency fundamental wave (results of calculations are shown by dashed line in Fig. 3). Moreover, in the low-power domain (*P _{RFL}*<2.5 W) the obtained efficiency is higher than that in the single-frequency model, while at

*P*>2.5 W it becomes lower. Thus we can conclude that not only direct frequency doubling contributes to the SH of the broadband RFL.

_{RFL}## 3. Theoretical model

To explain the observed behavior, let us take into account sum frequency mixing (SFM) processes. In our description we follow methods of [16]. If the fundamental wave spectrum consists of *N* equidistant modes with frequencies *ω _{1j}* (N~10

^{6}in the RFL), the second harmonic (SH) spectrum should consist of

*2N*-

*1*modes with frequencies

*ω*. Even SH modes are generated only due to SFM, but odd modes have also direct SH contribution. Inset in Fig. 4 shows a SHG example for

_{2j}*N*=

*4*. The power conversion is described by

*2N*-

*1*equations:

where *A _{in}* is the complex amplitude of

*n*-th spectral mode of

*i*-th harmonics,

*u*is the group velocity,

*σ*

_{2}is the nonlinear coefficient. The first term in the brackets describes the direct frequency doubling, the second one corresponds to the SFM. From Eq. (1) one can find an intensity of

*n*-th spectral component of the SH:

$$+2\sum \sum _{j\ne k}{\left({a}_{1j}{a}_{1k}\right)}^{2}+2\sum \sum \sum \sum _{j\ne k\ne p\ne q}{a}_{1j}{a}_{1k}{a}_{1p}{a}_{1q}\times \mathrm{cos}\left({\phi}_{1j}+{\phi}_{1k}-{\phi}_{1p}-{\phi}_{1q}\right)]$$

where *a _{ij}* and

*ϕ*are the real amplitudes and phases of corresponding modes, respectively. The equation can be simplified for free-running modes, when all the phases

_{ij}*ϕ*are random:

_{ij}Eqs. (1–3) are valid in the case of exact phase matching for all interacting waves. One can modify Eqs. (1–3) taking into account finite width of the phase matching (PM):

where *Δk _{jk}* =

*k*(

*ω*+

_{j}*ω*)—

_{k}*k*(

*ω*)-

_{j}*k*(

*ω*) is the wave vector mismatch.

_{k}It is also necessary to take into account specifics of the periodically-poled crystals (with period ∧). It is known, see e.g. [17], that the transition from PM to QPM can be done by:

One also needs to take into account Gaussian beam profile [18], which leads to modification of conversion coefficient (2σ_{2}
*L*/*π*)^{2} at single frequency and polarization to

where *d _{eff}* =2

*d*

_{33}/

*π*is the effective nonlinear coefficient,

*n*is the corresponding refractive index, ε

_{j}_{0}is the permittivity of vacuum,

*c*is the speed of light, and

*h*~1 is the Boyd-Kleinman focusing factor. For

*d*

_{33}= 27 pm/V and

*L*=8 mm

*η*= 1.24%/W at optimal focusing.

## 4. Comparison of the model with the experiment

Using Eqs. (4–6) we have calculated an expected SH spectrum using the measured RFL spectra (Fig. 2(a)). To obtain results at a reasonable computational time, we have reduced number of modes taking the effective mode spacing of 0.025 nm and checked that a further increase in number of modes does not lead to significant changes in SH power values (≤1%). Two cases have been compared: direct frequency doubling (first term in Eq. (4)) and frequency doubling assisted by SFM (both terms in Eq. (4)). The SH spectrum calculated for direct frequency doubling has a very low amplitude. It is normalized to 1 at maximum for a comparison (Fig. 4, dashed line). This spectrum “copies” the dip in the fundamental wave spectrum (Fig. 2(a)), that is absent in the measured SH spectrum. At the same time, calculations considering SFM (Fig. 4, solid line) provide a good coincidence with the experimental shape. Calculated second harmonic spectral width (FWHM) is also in a good agreement with the measured one, see Fig. 2(c). Thus, in presence of multiple frequencies in the fundamental wave the SFM makes the main contribution to the SH power.

For a quantitative comparison of the calculated and measured power values, we take into account that for the randomly polarized RFL laser a half of the power corresponds to one linear polarization component. We have also found a more exact value of the poling period (∧=12.54 μm) by fitting positions of the two first minima in the SH spectra. Slight difference with the specified period may be associated with temperature dependence and MgO doping while we use the refraction index data for pure lithium niobate [19]. Under these conditions, the SH power values calculated by summing up the power of individual modes (Eq. (4)) in frames of the SFM model (solid line in Fig. 3) are in very good agreement with the experimental points in low-power domain (≤2.5 W), confirming the fact that the multimode SH power is higher than single-frequency one. At higher powers one can see a significant deviation of the experimental points from the theory, which can be attributed to the poorer beam quality (M^{2}≈1.3) and non-uniform crystal heating induced by the SH light absorption (see [20] and references therein). Nevertheless, the SFM model describes the experimental data quite well, especially a character of the dependence both for the SH power and efficiency (see inset in Fig. 3), that is quite different from that for the single-frequency model.

The obtained results can be easily understood taking into account that the RFL modes are stochastic, i.e. they have random phases and amplitudes. Summing up the power of *N* random modes within QPM bandwidth results in factors *N* and *N*
^{2} for the first and the second terms in Eq. (4), correspondingly, both proportional to squared RFL mode power (*P _{ω}*/

*N*)

^{2}. So the relative contribution of the direct doubling 1/

*N*is vanishingly small at

*N*≫1. Moreover, the second term in Eqs. (3,4) describing SFM takes factor 2 after transition from Eq. (2) to the stochastic case (with gaussian statistics). The enhancement coefficient is (2–1/

*N*) and tends to 2 at

*N*≫1. Let’s focus our attention on this fact, which means that the conversion efficiency for the multi-frequency radiation is higher than that for single-frequency one. This interesting effect is discussed in theory starting from classical paper [21] (see also [16] and citation therein), here it is convincingly confirmed in the experiment with the RFL at low powers. At high powers the input spectrum becomes broader than QPM bandwidth, so an additional factor proportional to the ratio of the mode number within QPM and total mode number leads to relative reduction of the resulting SH power below the single-frequency curve. At linear growth of the RFL modes number with power, the SH power should grow nearly linearly.

Note that frequency doubling of fiber lasers generating multiple frequencies was studied in experimental works [10, 20] and was compared with calculations, but not accurately. In [10] a single-frequency model has been used with a correction of the input power according to the QPM bandwidth. In [20] the multi-frequency nature of Yb fiber lasers was taken into account, but without mode phases “randomization,” as a result factor 2 could not be obtained.

## 5. Conclusion

We proved that SHG with radiation comprising multiple modes, the sum-frequency mixing gives the main contribution to the SH power, while the direct frequency doubling gives a vanishing value at large number of modes. Thus, efficient SH generation is possible for broadband fiber lasers, which spectral width is sufficiently larger than QPM bandwidth of the crystal. The developed SFM model describes quite well the SH spectrum and power measured at frequency doubling of 1.31-μm Raman fiber laser. Laser radiation of >60 mW at 655 nm has been generated that corresponds to the ~1% SHG efficiency. To increase the efficiency further, one can use a linearly polarized RFL like in [10]. Another parameter is the crystal length, but calculations in frames of the SFM model have shown that there is no significant enhancement owing to a QPM bandwidth reduction at the lengthening.

The authors acknowledge financial support by the grants of the Presidium and the Department of Physical Sciences of the Russian Academy of Sciences, Russian Ministry of Education and Science and CRDF (RUP1-1509-NO-05). We also thank A. A. Vlasov for FBGs fabrication and E. V. Podivilov for fruitful discussions.

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