1. Introduction

The interaction of dispersion and nonlinearity typically affects the pulse-quality at the output of ultrafast laser-amplifiers. The effect of dispersion can be well compensated; however, the influence of nonlinearity generally causes degradation of the pulse-quality, and in the worst case, wavebreaking of the pulse. The most dominant nonlinear effect is the optical Kerr-effect in form of self-phase modulation (SPM). This nonlinear effect is proportional to the peak-power and the shape of the optical pulse. Thus, the use of stretched pulses, i.e. with reduced peak-powers, during amplification lowers the impact of the Kerr-nonlinearity. This technique is known as chirped-pulse amplification [1]. Furthermore, the modification of the pulse-shape allows for the control of the generally uncompressible nonlinear phase, e.g. [2].

In fiber-amplifiers the onset of nonlinear effects is particularly low since light propagates ’diffraction-less’ in form of modes, i.e. high intensities are confined over long lengths. As SPM is inversely proportional to the mode-area and proportional to the fiber-length, novel large mode area fibers of short-length, such as the rod-type waveguide, can significantly reduce nonlinear effects in pulsed amplifiers of the mJ energy-class [3, 4].

In this contribution, the use of circularly polarized light during nonlinear amplification in optically isotropic gain media, such as fused silica or YAG, is proposed. It will be experimentally shown that circularly polarized light reduces the Kerr-effect. The technique can be employed in large mode area fibers of short-length, which must be non polarization maintaining and non-polarizing [5]. In such fibers the impact of birefringence is negligible since the beat-length is typically at least an order of magnitude longer than the actual fiber length (~1 m).

Nonlinear effects, such as self-phase modulation, cross-phase modulation, self-focussing, four-wave mixing etc., can be lowered. Although the different nonlinear refraction coefficients n ₂ for circular and linear polarization are theoretically understood [6], to the best of our knowledge, the advantage of circularly polarized light over linear polarization has not been exploited in real-world ultrafast laser-amplifiers.

2. Nonlinear refraction coefficients for linearly and circularly polarized light

The polarization P(t) of a medium under the influence of an applied electric field can be expressed in terms of a power series [7]

where P ⁽¹⁾ is linear in the field and represents the non-instantaneous response of matter to stimulation by light, and it is related to the frequency dependence of the refractive index of an optical medium. The term P ⁽²⁾ is quadratic, and P ⁽³⁾ is cubic in the field, and so on. The term P ⁽⁰⁾ is independent of the field, and corresponds to a static polarization as found in some crystals. However it is insignificant for fused silica, which is the medium of interest in this work. Furthermore, fused silica is an isotropic material, and thus, the third-order term is the lowest nonlinear component [7].

In the following the electric field of the pulse is expressed as $\mathbf{E}\left(t\right)=\genfrac{}{}{0.1ex}{}{1}{2}\hat{\mathbf{E}}\left(t\right)e^{-{\mathrm{i\omega }}_{0}t}+c.c$., where Ê is the slowly varying envelope. The wave has a spectral bandwidth centered around ω0.

The nonlinear effect of interest is independent of phase-matching. Furthermore, the third-order polarization has its characteristic oscillation also at the frequency ω ₀, and it can be expressed as ${\mathbf{P}}^{\left(3\right)}\left(t\right)=\genfrac{}{}{0.1ex}{}{1}{2}{\hat{\mathbf{P}}}^{\left(3\right)}\left(t\right)e^{-{\mathrm{i\omega }}_{0}t}+c.c$ . It is assumed that the amplitude variations of the electric field are much slower than the relaxation of the polarization. In this adiabatic limit the response of the medium is instantaneous, and the cartesian components of the vector of the slowly varying envelope P̂⁽³⁾ are given by [6, 7]

where µ ∊ {x,y, z}. The vacuum permittivity is denoted as ε ₀, and χ ⁽³⁾ is the third-order susceptibility tensor (evaluated at the center frequency of the signal ω ₀). In general, this fourth-rank tensor has 81 components. However, because of the isotropy of the medium and the situation of the laser wavelength far from any resonance of the medium, there is only one remaining (independent) element [7].

Suppose the light is linearly polarized in the x direction, Ê=(Ê_x, Ê_y)=Ê(1,0)=Ê e _x, then the third-order polarization (of Eq. (2)) can be re-written as follows

For left-hand (+) and right-hand (-) circularly polarized light, the Jones vector of the electric envelope is given by $\hat{\mathbf{E}}=\hat{E}\genfrac{}{}{0.1ex}{}{1}{\sqrt{2}}(1,i)=\hat{E}{\mathbf{e}}_{}$. Thus, the third-order polarization is given by

Equations (3) and (4) show that nonlinear propagation does not change the state of polarization for both linearly and circularly polarized light. The polarization remains parallel to the electric field. It is worth noting that for elliptically polarized light this is not the case, and rotation of the polarization ellipse is a consequence [8]. Thus, for linearly and circularly polarized light, it is possible to define a scalar nonlinear susceptibility, from which the nonlinear refractive index can be derived [7]. The first order correction describes the intensity dependence of the refractive index (Kerr-effect) and is given by n=n ₀+n ₂ I (t), where n ₀ is the linear refractive index, n2 is the nonlinear refraction coefficient, and I(t) denotes the intensity-shape of the pulse. For linearly polarized light the nonlinear refraction coefficient is given by

For circularly polarized light the nonlinear refraction coefficient is given by

Thus, the ratio of n ₂,C to n ₂,L is 2/3 [6]. This difference constitutes the basic idea to reduce the Kerr-effect by using circularly instead of linearly polarized light.

3. Passive peak-power scaling

During ultrafast amplification the pulse can acquire a phase due to Kerr-nonlinearity. The magnitude of this phase is denoted as the B-integral. It is directly proportional to the nonlinear refraction coefficient [9]

where Î(z) is the z-dependent peak-intensity of the temporal pulse, and L is the length of the amplifier. Since the nonlinear refraction coefficient is reduced for circularly polarized light, the corresponding B-integral is also decreased.

 

Fig. 1. (a) Strehl-ratio of the output pulse after nonlinear chirped pulse amplification as a function of the B-integral. (b) relative peak-powe enhancement if circularly polarized light is used instead of linearly polarized light for different pulse shapes.

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The Kerr-nonlinearity reduces the performance (i.e. the Strehl-ratio) of CPA-systems. The cause for the decrease of the output peak-power is the uncompressible part of the phase due to nonlinearity. The spectral phase due to self-phase modulation can be approximated by [10]

where s (Ω) is the power-spectrum of the optical pulse (max[s (Ω)]=1) and B is the value of the B-integral of the nonlinear amplifier. It is important to stress that the Kerr-effect describes the intensity-dependent refractive index in time-domain. If dispersion is negligible during amplification, then the intensity shape of the pulse is unchanged. Thus, Eq. (8) must be seen as an approximate description of the impact of Kerr-nonlinearity on the strongly stretched pulse in frequency-domain. This spectral phase can be partially compensated with the positive parabolic phase from a stretcher-compressor-mismatch [10]. In the case of an initial sech² pulse, Fig. 1(a) shows the decrease in performance (in terms of the Strehl-ratio) as a function of the B-integral. Where the Strehl-ratio is the highest possible output peak-power (i.e. at best compression) referred to the peak-power of the transform-limited pulse [10]. In Fig. 1(a) the curves are identical for linearly and circularly polarized light, however, in the case of circularly polarized light, the data refers to the upper B-integral axis, which is scaled by a factor of 2/3 compared to the lower axis. Where the factor 2/3 corresponds to the ratio (n _2,C/n ₂,L). Thus, if circularly polarized light is used instead of linearly polarized light, one expects a peak-power enhancement as shown in Fig. 1(b). This increase in peak-power is dependent on the pulse-shape. The sech² is compared to a Gaussian shape. A Gauss pulse allocates less energy to the regions far from the center (i.e., lower kurtosis), and thus, SPM can be better compensated with a parabolic phase. Consequently, this diminishes the relative peak-power enhancement in the case of a Gaussian shape. In any case, circularly polarized light increases the self-focussing threshold.

4. Experiment and results

In the following, the ratio of the B-integrals, corresponding to the amplification with linearly and circularly polarized light, is experimentally measured. For this purpose, a pulse-shaper is inserted in a fiber-based CPA-system. In particular, the stretcher and compressor are perfectly matched in the linear operation of the CPA-system. This configuration is kept for the nonlinear regime of the CPA-system. For such a configuration, the residual spectral phase due to self-phase modulation is described by Eq. (8). The B-integral is determined by phase-only shaping. In particular, nearly transform-limited pulses can be produced when a pulse-shaper generates a phase φ*_ps=-B s(Ω). The spectrum s (Ω) is obtained with an optical spectrum analyzer, however, the exact value of the B-integral is unknown. So, in practice the phase φ_ps=Δϕ _s (Ω) is generated with the pulse-shaper, and the parameter Δϕ is varied. For the different values of Δϕ, and thus, different phase-shapes realized with the pulse-shaper, the pulse-energy at the input of the amplifier is kept constant, and as a consequence, the output pulse-energy remains the same. In particular, the growth of energy is independent of the pulse-shape [11]. Thus, at the output of the nonlinear CPA-system the peak-power will change for the different values of the parameter Δϕ. The change of peak-power is monitored at the output of the CPA-system. If (-Δϕ) agrees with the B-integral, then the (normalized) autocorrelation corresponds to the autocorrelation of the linear CPA-system. In this way, nearly transform-limited pulses are produced from a nonlinear CPA-system. Furthermore, the B-integrals for the two states of polarizations can be accurately determined.

 

Fig. 2. Schematic of the experimental setup of the fiber CPA-system with a pulse-shaper for phase-only shaping. AOM, acousto-optical modulator; OSA, optical spectrum analyzer; SHG, second harmonic stage; AC, autocorrelator.

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Fig. 3. Results from the measurement with the polarimeter: representation of the states of polarization on the Poincare-sphere for the case of circularly polarized light (a), and linearly polarized light (b).

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Fig. 4. (a) The spectrum measured at the output of the main-amplifier. (b) The SLM produces phases that show different maximum phase-shifts Δϕ. The autocorrelation traces measured at the output of the fiber CPA-system for the different values of the phase-compensation parameter Δϕ : (c) for the case of circularly polarized light and (d) linearly polarized light.

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The schematic of the fiber-based CPA-system is shown in Fig. 2. It consists of a passively mode-locked oscillator, a stretcher, an acousto-optical modulator, a first pre-amplifier, a pulse-shaper, a second pre-amplifier, a main amplifier, and a compressor. The mode-locked laser emits soliton-pulses with a FWHM-bandwidth of 3nm at a repetition rate of 10MHz and an average power of 3.5W. The stretcher-compressor unit employs two 1740 lpmm dielectric diffraction gratings and stretches to about 1.5ns (FWHM). The quartz-based acousto-optical modulator reduces the pulse repetition rate to 50kHz. The first pre-amplifier is a 1.2m-length of bendable photonic crystal fiber, its mode-field diameter (MFD) is 33µm. The pulse-shaper is a spatial light modulator (SLM) (640 pixels covering a spectral window of about 10nm), which is placed in the Fourier-plane of a 4-f zero-dispersion stretcher [12]. The SLM is a double mask spatial light modulator which permits independent control of the spectral amplitude and the spectral phase. However, for the results herein after presented the phase is solely shaped. The first pre-amplifier raises the average power after pulse-picking in order to compensate for the moderate efficiency (30%) of the 4-f zero dispersion stretcher. High-efficiency diffraction gratings in this 4-f setup will render the first pre-amplifier unnecessary. A seed of ~4mW is launched into the second pre-amplifier. The second pre-amplifier is a 1.3m-length of bendable photonic crystal fiber with a MFD of 33µm. The main amplifier is a 1.2m-length of rod-type photonic crystal fiber with a MFD of 71µm [3]. An optical isolator prevents feedback between the second amplifier and the main amplifier. To amplify circularly polarized light, a quarter-wave plate is placed before the main-amplifier. At this position, the states of polarizations are measured with a polarimeter. The results are shown in Fig. 3.

 

Fig. 5. Peak of the autocorrelation traces at the output of the fiber CPA-system for the different values of the phase-compensation parameter Δϕ. The blueish and redish line correspond to linearly and circularly polarized light, respectively. They represent the mean values of the experimental data (grey curves). The vertical lines mark the position of Δϕ=-B.

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The state of polarization can be represented by an ellipse. The parameter ellipticity is an angle, which is calculated from the atan of the ratio of the semi-minor to the semi-major axis. For the case of circularly polarized light, the ellipticity of the state of polarization is measured to be 43°, and for linearly polarized light it is experimentally determined to be 0.3°. After the main-amplifier the circular polarization is changed to linear polarization with a second quarter-wave plate. For both cases, a half-wave plate is used to achieve the 70%-efficiency of the compressor. Nonlinear polarization rotation is not an issue. Since the gratings of the compressor are polarization sensitive, nonlinear polarization rotation would cause a drop in compressing efficiency. However, it is experimentally observed that the compressor efficiency is the same for both the nonlinear and linear regime of the CPA-system. For both kinds of polarization states, 70mW of average power is launched into the main-amplifier at a repetition rate of 50 kHz, and the output power is 31W corresponding to 620µJ of pulse-energy (before compression). The nonlinearity is solely accumulated in the main-amplifier since only moderate pulse-energies are extracted from the pre-amplifiers, which feature large-mode areas.

The spectrum is measured behind the main amplifier and it is shown in Fig. 4(a). The spectrum is normalized (i.e., the peak equals 1) and multiplied with the phase-compensation parameter Δϕ. In the experiment, the parameter Δϕ is varied between 0 and -9 rad. The resulting spectral phases, which are produced by the SLM, are shown in Fig. 4(b). For sake of visibility, in the figure the step-size is 1 rad; in the experiment the actual step is 0.5 rad. For the case of circularly polarized light, the step is refined to 0.25 rad in the region of -4 rad. For linearly polarized light, the steps are reduced to 0.25 rad around Δϕ=-6rad. For each Δϕ the autocorrelation traces are measured at the output of the nonlinear CPA-system. To minimize the error of the measurement, the sweep of the parameter Δϕ is is repeated three times and two times for the circular polarization and the linear polarization, respectively. For the case of circularly polarized light, one set of autocorrelation traces is shown in Fig. 4(c). For sake of clarity, only steps of 1 rad are shown. The (average) ’ridge’ of the three sets of autocorrelations (full resolution) is mapped via the blue line in Fig. 4(c). The autocorrelation trace with maximum peak agrees with the (normalized) autocorrelation trace that was recorded for the linear system, i.e nearly transform-limited pulses are produced at this point. Thus, at the maximum peak the absolute value of the phase-compensation parameter equals the B-integral, i.e. Δϕ*=-B. For the case of circularly polarized light, the B-integral is 4.0 rad. The precision is limited by the step-size of 0.25 rad. However, a finer step-size does not improve the measurement because of little flucutation of the autocorrelation traces. This can be seen in Fig 5. The case of linearly polarized light is shown in Fig. 4(d). The B-integral is determined to be 5.8 rad. To better compare the results for the two states of polarizations, the both sets are presented in Fig. 5. Thus, from both measurements the ratio of the B-integrals is found to be in good agreement with the theoretical value of 2/3.

5. Conclusion

Using circularly polarized light instead of linearly polarized light, the reduction of Kerrnonlinearity has been experimentally demonstrated. The relative decrease corresponds to a factor of 2/3, which is in accord with theoretical findings. In an ultrafast fiber-based CPA-system, the B-integrals of circular and linear polarization have been measured using phase-shaping. The technique requires an optically isotropic gain medium. The method can be used to increase the peak-power from ultrafast oscillators, and to reduce the impact of modulation instability in twisted-mode resonators. The technique is particularly relevant for ultrafast fiber-systems, which are susceptible to nonlinear effects. The method can be combined with techniques that control self-phase modulation by using active pulse-shaping in order to reduce requirements on the shaping dynamics [2]. In this context, the application of polarization maintaining as well as polarizing fibers seems not ideal for peak-power scaling. In general, the beat-length (due to birefringence) must be longer compared to the fiber-length.

Moreover, the use of circularly polarized light increases the self-focussing threshold from 4MW to about 6MW (at a wavelength around 1.03µm) [13].