Spatiotemporal coherent noise in frequency-domain optical parametric amplification

Jing Wang; Jingui Ma; Peng Yuan; Daolong Tang; Bingjie Zhou; Guoqiang Xie; Liejia Qian

doi:10.1364/OE.26.010953

1. Introduction

The development of ultraintense lasers has progressed hand-in-hand with that of the laser technology for pulse contrast [1–3]. In order to launch clean laser-plasma interactions for applications such as plasma accelerators [4,5], multi-petawatt (PW) laser with intensity ~10²² W/cm² should be matched with a high contrast ratio of 10¹² between the laser pulse and temporal noise. Reducing the amplifier noise in chirped-pulse amplification (CPA) therefore becomes an urgent and crucial task. Besides the incoherent noise emitted spontaneously in conventional laser amplifiers, ultra-intense lasers based on CPA technique uniquely suffer from coherent noise that accompanies with the chirped pulse and is typically in the picosecond time scale [6]. State-of-the-art ultraintense lasers adopt the architecture of double CPA with intermediated pulse-cleaning devices that can isolate and suppress the incoherent noise in amplifiers [6–12]. As demonstrated by the Apollon 10-PW laser [7], the incoherent contrast in nanosecond range has been improved to the desired level of 10¹² in such a way. Nevertheless, the picosecond coherent-noise pedestal remains a poor contrast of only ~10⁸. In optical parametric chirped-pulse amplification (OPCPA), rich coherent noise is produced by the nonlinear mixing of the chirped signal pulse with the imperfections of pump laser [13–15]. Obviously, coherent noise has two notable features. Firstly, coherent noise has its evolution characteristic related strongly with the signal chirp [16]. Secondly, coherent noise has its magnitude inherently proportional to the signal magnitude, and thus could not be effectively suppressed by the double CPA architecture. How to improve coherent contrast has become an open question in the field of ultraintense lasers [17,18].

Since there is a strong relation between the signal chirp and coherent noise, it is pertinent to study the coherent noise in new amplification schemes that adopt different chirp strategies. Here, we present a first theoretical study on the coherent noise in a recently proposed scheme named frequency-domain optical parametric amplification (FOPA) [19,20]. In FOPA scheme, the femtosecond seed pulse is stretched by the strategy of spatial-spectral coupling [Fig. 1(a)], which tiles the laser frequencies (ω) over space (x) rather than time (as what does in conventional OPCPA). It is thus feasible to amplify different laser frequencies using different crystals, which enables extremely broadband amplification without the phase-matching limitation set by a single crystal. On the other hand, good performance on the picosecond contrast in FOPA has been suggested by experimental measurements [21]. Basically, the spatial-spectral coupling property of FOPA also results in an entirely different characteristic of coherent noise. Because the signal pulse is temporally unchirped (different from the OPCPA case), its spectrum no longer interacts with the pump pulse directly, and thus the pump pulse imperfections will not induce temporal coherent noise. Besides, since the signal does not present a spatial chirp at the gratings, the FOPA output is free of the kind of coherent noise originated from the finite aperture [22] and surface roughness [23,24] of the gratings in conventional stretchers and compressors. On the contrary, we find in this paper that the temporal coherent noise of an FOPA laser is originated from the spatial imperfections in amplification. As schematically illustrated in Fig. 1(b), a spatial modulation superimposed on the signal beam is equivalent to a spectral modulation owing to spatial-spectral coupling, which can get transformed into a series of temporal spikes encircling the compressed signal pulse. At the same time, the spatial modulation origin of these temporal spikes determines that they will also distinguish from the main signal in the spatial domain and present as beam side-lobes. Given the ease control over beam quality [25], FOPA scheme is likely to be more favorable toward high coherent-contrast.

Fig. 1 (a) FOPA setup with nonlinear crystals placed in the spectral Fourier-plane. G₁ and G₂, diffraction gratings; M₁ and M₂, concave mirrors with a focal length f; x and x' represents the transverse spatial coordinate in the spectral Fourier-plane and that in output near-field of FOPA, respectively, which links each other via a spatial-domain Fourier transformation; t and ω denotes time and frequency, respectively. (b) Diagram of the noise mechanism of FOPA. Spatiotemporal coherent noise is produced via three steps: (1) in the amplification stage, the pump beam modulation is nonlinearly imparted onto the signal beam and also spectrum; (2) in the pulse compression stage, the induced spectral modulation on the amplified signal gets transformed into pre- and post-pulses; (3) in the spatial Fourier-transform stage from the nonlinear crystals to grating G₂, the induced spatial modulation on the amplified signal beam gets transformed into beam side-lobes. Owing to the interdependence between x and ω introduced in the first step, the second and third steps jointly make the output noise exhibit spatiotemporal coupling.

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The paper is organized as follows. Section 2 illustrates the spatiotemporal evolution of signal laser in an FOPA system, where the spatial-spectral coupling property of FOPA amplifiers is quantified. This section also acts as a basis to understand the spatiotemporal coherent noise discussed in this paper. In Section 3 and Section 4, the spatiotemporal coherent noise originating from the optical imperfections of pump laser and crystal surfaces are theoretically characterized, respectively. Conclusions come in Section 5.

2. Spatiotemporal coupling property of FOPA

The centerpiece of an FOPA system is a 4-f image-relay line that links a pair of antiparallel diffraction gratings (G₁, G₂), as shown in Fig. 1(a). With grating G₁ placed at the object plane and grating G₂ placed at the image plane, the net dispersion of FOPA is zero. When an ultrashort pulse impinges on grating G₁, its local bandwidth at a position x will reduce with propagation owing to the angular dispersion. The local bandwidth is minimized in the spectral Fourier-plane where nonlinear crystals are placed, and accordingly the pulse duration reaches to its maximum (~10 ps typically). Finally, the amplified signal is restored to its original ultrashort duration and beam width after grating G₂ (i.e., the output near-field). Lens L₁ with a focal length f is placed behind grating G₂, and the output signal is focused at the rear focal plane of this lens (i.e., the output far-field).

In contrast to OPCPA, a distinct characteristic of FOPA scheme is that the signal pulse undergoes spatiotemporal couplings (STC) throughout the whole system including the amplification stage. Defined as the interdependence of temporal (or spectral) and spatial (or angular) coordinates, there are in principle eight forms of STC [26]. Although it is generally regarded as a distortion to ultrashort pulses, STC can also be quite useful if it is properly designed and controlled [26]. In what follows, we will address that the implementation of FOPA involves the four common forms of STC, including angular dispersion, spatial-spectral coupling, wave-front rotation, and pulse-front tilt. Let us consider an incident ultrashort pulse with its complex amplitude given by

a_{i n} (x, ω) = a_{0} \exp (- \frac{x^{2}}{2 D^{2}}) \exp [- \frac{ω^{2}}{2 {(Δ ω)}^{2}}] .

where D is the beam-width and Δω is frequency bandwidth (corresponding to a Fourier-transform-limited duration of τ₀ = 1/Δω) of the incident pulse. Based on the Fresnel-Kirchhoff diffraction theory [27], the signal amplitude in the spectral Fourier-plane (i.e., within the nonlinear crystals) can be derived as (see Appendix)

a_{F P} (x, ω) = \exp (- \frac{i k}{2 f} x^{2}) \exp (- \frac{i k β^{2} ω^{2} f}{2}) \exp [- \frac{ω^{2}}{2 {(Δ ω)}^{2}}] \exp (- i k β ω x) \exp [- \frac{{(x + β f ω)}^{2}}{2 σ^{2}}],

where β is the angular dispersion provided by grating G₁, k is the wave number; σ = f/k/D is the focal spot size for each monochromatic component of signal. The term exp(−ikβωx) in Eq. (2) represents STC in the form of angular dispersion, while the last term exp[−(x + βfω)²/2/σ²] indicates the appearance of spatial-spectral coupling, as another STC form. Different frequency components (ω) in the incident pulse have been focused onto distinct transverse positions (βfω) at the spectral Fourier-plane, with the characteristic coupling slope given by

ζ = \frac{d x}{d ω} = - β f .

The spatiotemporal amplitude of this signal pulse can be deduced as:

A_{F P} (x, t) = \exp [- \frac{x^{2}}{2 {(ζ Δ ω)}^{2}}] \exp [- \frac{t^{2}}{2 {(Δ T_{F P})}^{2}}] \exp (\frac{- i x t}{ζ}),

where the term exp(−ixt/ζ) manifests the STC in the form of wave-front rotation also present in the signal pulse at the spectral Fourier-plane. Equation (4) also indicates that the signal pulse is temporally unchirped. Thus its duration is determined by the spectral bandwidth at each spatial position. As indicated by Eq. (2), the spatial-spectral coupling property has reduced the local bandwidth to Δω_FP = σ/ζ. Consequently, the pulse duration is temporally stretched to

Δ T_{F P} = \frac{1}{Δ ω_{F P}} = k β D .

Figure 2 systematically illustrates the spatiotemporal evolution of a clean femtosecond pulse through FOPA. In the simulations, the incident signal pulse is assumed to have a Gaussian profile with a Fourier-transform-limited duration of τ₀ = 28 fs [Fig. 2(a-1)] at the central wavelength of 810 nm. The beam profile is also in Gaussian shape with a beam width of D = 1 mm [Fig. 2(a-2)]. All the widths refer to the half width at 1/e of the peak intensity. As for the FOPA setup, the gratings G₁ and G₂ with a groove density of d = 1480 lines/mm provide an angular dispersion of β = 0.64 mrad/THz under the Littrow configuration at 810 nm. Concave mirrors M₁ and M₂ have a same focal length of f = 500 mm. As guided by the red dashed line in the first row of Fig. 2, throughout the 4f optical path between G₁ and G₂, the signal pulse exhibits the STC effect of pulse-front-tilt, with the tilt angle continuously varying with propagation. Specifically, the tilt angle approaches to π/2 in the spectral Fourier plane [Fig. 2(d-1)], where the pulse duration gets to its maximum and the signal field is dominated by a spatial-spectral coupling [Fig. 2(d-2)]. In our case, the spatial-spectral coupling has a coupling slope of dx/dω = −0.3 mm/THz, and the pulse duration gets stretched to 5 ps in this plane. At the output of grating G₂, the signal is restored to an uncoupled pulsed-beam with τ₀ = 28 fs and D = 1 mm [Fig. 2(g-1) and 2(g-2)], which is exactly the same as the incident pulse.

Fig. 2 Spatiotemporal (the first row) and spatial-sp0ectral (the second row) intensity profiles of the signal at the planes of G₁ input (a), G₁ output (b), M₁ input (c), spectral Fourier-plane (d), M₂ input (e), G₂ input (f) and G₂ output (g), respectively. Dashed lines in the first row highlight the slope of spatiotemporal coupling.

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3. Spatiotemporal coherent noise originated from pump noise

In optical parametric amplification, signal is inherently disturbed by the imperfections of pump laser via the instantaneous parametric gain. Such a process of pump-noise transfer results in an important kind of coherent noise in OPCPA [13–15]. This section studies the spatiotemporal coherent noise caused by the pump-noise transfer in FOPA scheme, which evolves very differently from that in conventional OPCPA.

3.1 Nonlinear transfer of temporal pump-pulse modulation

Firstly, the transfer of the pump pulse imperfection is analytically deduced. For simplicity, we assume a sinusoidal intensity modulation on the pump pulse:

I_{p} (x, t) = I_{p 0} [1 + r_{t} \cos (Ω_{t} t)] = I_{p 0} [1 + \frac{r_{t}}{2} \exp (\pm i Ω_{t} t)],

where I_p₀ represents the pump intensity without the modulation; Ω_t and r_t denote the frequency and amplitude of a temporal modulation, respectively. In the regime of small-signal amplification where the pump-depletion is negligible and the gain is reasonably large, the amplitude of the amplified signal pulse can be directly written as

\begin{array}{l} A_{F P}^{a m p} (x, t) = A_{F P} (x, t) \times \cos h [Γ z \sqrt{I_{p 0} [1 + r_{t} \cos (Ω_{t} t)]}] \\ \approx A_{F P} (x, t) \times \sqrt{G_{0}} \times (1 + \sum_{m} {[\frac{\ln (4 G_{0})}{8}]}^{m} \frac{r_{t}^{m}}{m!} e^{\pm i m Ω_{t} t}), \end{array}

where Γ is the parametric gain coefficient defined as Γ² = 2ω_sω_id_eff²/(n_sn_in_pε₀c³) [28]; z is the crystal length, and G₀ = cosh²[

Γ z \sqrt{I_{p 0}}

] refers to the gain in signal intensity. Ideal FOPA condition with perfect phase-matching as well as where negligible dispersion and diffraction effects are assumed for Eq. (7). This equation clearly shows that the amplified signal experiences a series of temporal modulations, with the characteristic frequencies mΩ_t (m refers to arbitrary positive integer) linking with the pump pulse modulation. Note that, in FOPA scheme the signal pulse is temporally unchirped and does not show a correspondence between temporal and spectral coordinates. Thus the induced temporal modulations on amplified signal pulse, as usual, transform into multiple sidebands in the spectral domain. The spatial-spectral amplitude of amplified signal can be obtained by performing a temporal Fourier transform on Eq. (7) as

a_{F P}^{a m p} (x, ω) = a_{s} (x - ζ ω, ω) + \sum_{m} {[\frac{\ln (4 G_{0})}{8}]}^{m} \frac{r_{t}^{m}}{m!} a_{s} (x - ζ ω \mp ζ m Ω_{t}, ω \pm m Ω_{t}),

where a_s(x-ζω, ω) = a_FP(x, ω) × g₀ represents a clean signal spectrum that presents a spatial-spectral coupling described in Eq. (3); the second term indicates the production of multiple spectral sidebands. Specifically, the m-th sideband, with a frequency shift of mΩ_t with respect to the signal central wavelength, separates from the signal beam center by a spatial distance of Δx = ζ × mΩ_t. That is to say, these noise sidebands exhibit a same spatial-spectral coupling (dx/dω = ζ) with the signal. What we concern most is the distribution of noise at the FOPA output. For this purpose, we derive the spatial-spectral amplitude of amplified signal in the output near-field as

a_{o u t}^{N F} (x, ω) = a_{N F} (x, ω) + \sum_{m} {[\frac{\ln (4 G_{0})}{8}]}^{m} \frac{r_{t}^{m}}{m!} a_{N F} (x, ω \pm m Ω_{t}) \exp (\mp i k β m Ω_{t} x) .

Note that, as described by the term exp(ikßmΩ_tx) in this expression, each noise replica corresponds to a specific spatial-frequency. It implies that these noise replicas can get spatially separated from the signal after further propagation. In particular, the signal and noise amplitudes in the output far-field can be deduced as

a_{o u t}^{F F} (x, ω) = a_{F F} (x, ω) + \sum_{m} {[\frac{\ln (4 G_{0})}{8}]}^{m} \frac{r_{t}^{m}}{m!} a_{F F} (x \mp m β f Ω_{t}, ω \pm m Ω_{t}) .

The m-th noise sideband spatially separates from the signal beam by a distance of

Δ x (m Ω_{t}) = m β f Ω_{t} = m \times \frac{k β D}{\frac{1}{Ω_{t}}} \times σ .

This relation suggests that, if the temporal period (1/Ω_t) of pump pulse modulation is smaller than the duration of signal pulse in the spectral Fourier-plane (kβD), the induced noise sidebands can get completely separated from the signal in the output far-field, as Δx (mΩ_t)> σ in Eq. (11). In this case, the FOPA noise can be simply removed by spatial filtering. To address the impact on output pulse contrast, the spatiotemporal amplitude of the amplified signal in the output far-field is also deduced as

A_{o u t}^{F F} (x, t) = A_{F F} (x, t) + \sum_{m} {[\frac{\ln (4 G_{0})}{8}]}^{m} \frac{m_{t}^{m}}{m!} A_{F F} (x \mp m β f Ω_{t}, t) \exp (i m Ω_{t} t) .

Clearly, the m-th noise sideband, corresponding to the term exp(imΩ_tt) in Eq. (12), overlaps the signal pulse in time, but presenting as multiple spatial side-lobes that separate from the signal beam in space. Therefore, this kind of noise does not degrade the temporal contrast of amplified signal, even though without filtrating away. Equation (12) also indicates that the induced noise exhibits as temporal modulations with the frequencies mΩ_t on the final compressed signal pulse. Since the modulation period (1/Ω_t) is typically much longer than the compressed duration (τ₀), it has few impacts on the pulse profile.

Derivations from Eq. (7) to Eq. (12) constitute a full picture of the spatiotemporal evolution of noise induced by pump temporal modulations. Basically, the temporal pump modulation first produces a temporal gain variation across the amplified signal pulse in the spectral Fourier-plane [Eq. (7)], which then gets transformed into a spatial modulation of signal beam in the near-field of FOPA output [Eq. (9)] owing to the effect of STC, and appears as spatial side-lobes in the output far-field [Eq. (12)]. This is opposite to the conventional OPCPA case without the STC effect where a temporal gain variation induces significant temporal spikes and doses not lead to spatial modulations or side-lobes.

To verify the above analytical results, we run the numerical simulations based on the nonlinear coupled-wave equations. As depicted in Fig. 3(a), the Gaussian pump laser at 515 nm with a pulse duration of 8.3 ps and beam width of 20 mm is assumed to carry a temporal sinusoidal modulation with Ω_t = 6.28 THz and r_t = 0.1. The FOPA amplifier adopts a 2.5-mm-thick type-I β-BBO crystal used in the walk-off compensated non-collinear phase-matching configuration. At a pump intensity I_p₀ = 20 GW/cm², this crystal offers a parametric gain G₀ = ~10⁴ and a pump depletion ratio of η_p = 0.5% (within the small-signal amplification regime) to an incident signal at I_s₀ = 35 kW/cm². Figures 3(b)-3(d) present the intensity profiles of amplified signal and noise in the spectral Fourier-plane, output near-field and output far-field, respectively. To highlight the properties of STC, the profiles are characterized in two equivalent domains of (x, t) and (x, ω). In the spectral Fourier-plane, the induced noise exhibits as a temporal modulation on the amplified signal pulse in the (x, t) domain [Fig. 3(b-1)], and multiple sidebands with spatial-spectral coupling in the (x, ω) domain [Fig. 3(b-2)]. These noise sidebands are within the whole range of signal spectrum, indicating the ineffectiveness of spectral filtering. In the near-field of FOPA output, the induced temporal modulation does evolve to a spatial intensity modulation, as indicated by Figs. 3(c-1) and 3(c-2). By further focusing this amplified beam with a focal length of f = 500mm, the signal is focused to a spot size of ~50 μm, while the noise is converted to multiple spatial lobes with an equal interval of Δx = 2 mm [Figs. 3(d-1) and 3(d-2)]. This set of numerical results illustrates the very good qualitative and quantitative agreement with the analytical predictions given in Eqs. (7) to (10).There are no temporal spikes occurred around the signal pulse, indicating the temporal intensity modulations on pump pulse do not affect the contrast of amplified pulse in FOPA scheme. This represents a sharp feature of the FOPA noise performance.

Fig. 3 Spatiotemporal characterization of the FOPA noise originated from temporal pump-intensity modulation. (a) Intensity profile of the pump laser with a temporal sinusoidal modulation of Ω_t = 6.28 THz and r_t = 0.1. (b), (c), (d) Spatiotemporal and spatial-spectral intensity profiles of the amplified signal in the spectral Fourier-plane, the output near-field and output far-field, respectively. Red dotted line highlights the spatial-spectral coupling property of the induced noise. The FOPA is assumed in the small-signal amplification regime with G₀ = ~10⁴ and pump depletion η_p = 0.5%. (e) Spatial profile (temporally integrated) of the amplified signal beam in the far-field of FOPA output, calculated at a trivial pump-depletion η_p = 0.5% (black line) in comparison with that in a significant pump-depletion of η_p = 38% (blue line). (f) Temporal profiles of the amplified signal for the case of η_p = 0.5% (black line) and η_p = 38% (red line). Inset plots the signal spectrum before (black) and after amplification (red).

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We further extend the numerical simulations to the regime of saturated amplification. By increasing the incident signal intensity to 30 MW/cm² while keeping other parameters the same, a saturated amplification situation with a high pump depletion of η_p = 38% is simulated. As depicted by Fig. 3(e), the saturation of amplification helps to reduce the noise lobes induced by the temporal pump-intensity modulation. Regrading to the pulse contrast performance, Fig. 3(f) plots the intensity profile of the amplified pulse, which clearly shows that there is still no temporal spikes induced. The picosecond “wings” present around the main pulse are induced by the flat-top profile of amplified spectrum [inset of Fig. 3(f)] at a high pump depletion. These numerical results obtained in the saturated amplification regime further validate the advantages of FOPA in terms of high-contrast performance.

3.2 Nonlinear transfer of spatial pump-beam modulation

In this part, we characterize the transfer of the spatial imperfections of pump beam in FOPA. The incident pump laser is assumed to have an intensity profile given by

I_{p} (x, t) = I_{p 0} (x, t) [1 + \frac{r_{x}}{2} \exp (\pm i Ω_{x} x)],

where Ω_x and r_x represent the frequency and amplitude of a spatial sinusoidal modulation, respectively. With derivations similar to Eqs. (6)–(8), the spatial-spectral amplitude of the amplified signal pulse can be derived as

a_{F P}^{a m p} (x, ω) = a_{F P} (x - ζ ω, ω) \times {1 + \sum_{m} {[\frac{\ln (4 G_{0})}{8}]}^{m} \frac{r_{x}^{m}}{m!} \exp (\mp i m Ω_{x} x) \exp (\pm i \frac{m Ω_{x}}{ζ} ω)},

where the terms exp( ± imΩ_xx) and exp( ± imΩ_xω/ζ) indicate a simultaneous spatial and spectral modulation nonlinearly imposed onto the amplified signal. The signal amplitude in the output near-field of FOPA can be accordingly deduced as

A_{o u t}^{N F} (x, t) = A_{N F} (x, t) + \sum_{m} {[\frac{\ln (4 G_{0})}{8}]}^{m} \frac{r_{x}^{m}}{m!} A_{N F} (x \pm m \frac{f Ω_{x}}{k}, t \pm ζ m Ω_{x}),

where A_NF(x,t) refers to the clean signal pulse; while the second term represents multiple temporal spikes induced by the pump spatial modulation. Each temporal spike separates from the signal pulse with a specific temporal interval

Δ t (m Ω_{x}) = m Ω_{x} \times ζ .

At the same time, this kind of noise spikes separates from the signal beam also in the spatial domain, and the corresponding transverse separation is linearly proportional to the temporal interval as

Δ x (m Ω_{x}) = m Ω_{x} \times \frac{f}{k} .

As a combined result of Eq. (16) and Eq. (17), the induced noise exhibits the effect of STC in the output near-field, as predicted in Fig. 1(b), which is thus termed spatiotemporal coherent noise. Note that, the signal amplitude will further evolve into

A_{o u t}^{F F} (x, t) = A_{F F} (x, t) + \sum_{m} {[\frac{\ln (4 G_{0})}{8}]}^{m} \frac{r_{x}^{m}}{m!} A_{F F} (x, t \pm ζ m Ω_{x}) \exp [\mp i \frac{f}{k} m Ω_{x} x],

in the output far-field, where the STC effect of noise vanishes. The noise spikes become overlapped with the signal beam in space, which thus have a considerable impact on the pulse contrast. According to Eq. (18), the 1st order and 2nd order temporal spike (located at t = ± ζΩ_x = ± 2.5 ps and t = ± 2ζΩ_x = ± 5 ps, respectively) has its relative intensity given by

\frac{I_{s p i k e}^{(1)}}{I_{s i g n a l}} = {[\frac{\ln (4 G_{0})}{8} \times r_{x}]}^{2}, \frac{I_{s p i k e}^{(2)}}{I_{s i g n a l}} = {{[\frac{\ln (4 G_{0})}{8} \times r_{x}]}^{2} \times \frac{1}{2}}^{2} .

This result suggests that the coherent contrast degrades with the increase of parametric gain G₀.

In the numerical simulations, we assume the pump beam carries a sinusoidal intensity modulation with Ω_x = 7.85 × 10³ m⁻¹ and r_x = 0.01 [Fig. 4(a)], while other parameters keep the same with Fig. 3. Figures 4(c)-4(d) present the spatiotemporal and spatial-spectral intensity profiles of amplified signal calculated for a FOPA at a trivial pump-depletion (η_p = 0.5%). The pump spatial intensity modulation has induced multiple spatiotemporal spikes in the near-field of FOPA output, as illustrated in Fig. 4(c-1). These noise spikes, located at [t = ± 2.5 ps, x = ∓ 0.5 mm] and [t = ± 5 ps, x = ∓ 1 mm], exhibit STC along a slope of dt/dx = −5 ps/mm. While in the far-field of FOPA output, these spikes no longer show STC property explicitly in the (x, t) domain [Fig. 4(d-1)]. These results are consistent with the analytical predictions given by Eq. (15) and Eq. (18). These spatiotemporal spikes degrade the pulse contrast, as plotted in Fig. 4(e). Note that, the beam center in the far-field ‘sees’ more temporal spikes than the near-field axis, because there is an off-axis shift of the noise spikes in the near-field, dominated by the spatiotemporal coupling slope illustrated in Fig. 4(c-1). Under the condition of G₀ = ~10⁴ and η_p = 0.5%, the temporal spike at t = − 2.5 ps (i.e., 1st spike) and t = − 5 ps (i.e., 2nd spike) has a relative intensity of −40 dB and −90 dB, respectively, on the far-field axis.

Fig. 4 Characterization of the spatiotemporal coherent noise originated from spatial pump-intensity modulation. (a) Intensity profile of the pump laser with a sinusoidal spatial modulation of Ω_x = 7.85 × 10³ m⁻¹ and r_x = 0.01. (b), (c), (d) Spatiotemporal and spatial-spectral intensity profiles of the amplified signal and noise in the spectral Fourier-plane, the output near-field and output far-field, respectively. (e) Temporal profiles of the amplified signal pulse on the axis of FOPA output near-field (solid line) and far-field (dotted line), respectively. (f) The intensity of the temporal spike at t = −2.5ps (blue) and t = −5ps (red) versus the amplification parameters of parametric gain G₀ and pump-depletion ratio η_p. The seed-to-pump intensity ratio is fixed at 1.8 × 10⁻⁶. (g) The temporal spike intensities at t = −2.5ps (blue triangles) and t = −5ps (red triangles) versus the seed-to-pump intensity ratio.

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To extend the discussion further into the regime of saturation amplification, numerical simulations are performed with various gain values (achieved by changing crystal length). The calculated noise intensity (solid line), as plotted in Fig. 4(f), shows a good agreement with Eq. (19) (dotted line) not only in the regime of small-signal amplification but also in that of moderate saturation with η_p < 10%.Fig. 4(f) also indicates that the spatiotemporal coherent noise establishes quickly in the initial stage of amplification. As in our case, the coherent contrast at t = −5 ps degrades to −100 dB even at a relative small gain of G₀ = 50. For the above calculations, the intensity of seed pulse is fixed at I_s₀ = 35 kW/cm², corresponding to a seed-to-pump ratio of I_s₀/I_p₀ = 1.8 × 10⁻⁶. Figure 4(g) characterizes the temporal spike intensity as a function of I_s₀/I_p₀, under the condition of the fixed pump intensity (20 GW/cm²) as well as crystal length (2.5 mm), which clearly shows that, prior to significant pump-depletion, the temporal spike intensity keeps unchanged with the seeding intensity. This result agrees well with Eq. (19), which analytically describes that the noise intensity is completely determined by the parametric gain G₀ (i.e., a function of pump intensity and crystal length) and is unaffected by the seeding intensity under the condition of negligible pump-depletion. When pump-depletion is higher than 10%, the noise intensities will be decreased by a certain degree. For example, with a high seed intensity (I_s₀/I_p₀ = 10⁻²), the parametric amplification gets into the deep saturation with a pump-depletion ratio η_p = 38%. In this case, the temporal spikes at t = − 2.5 ps and t = − 5 ps have their relative intensities −52 dB and −100 dB, respectively, which are approximately one order of magnitude higher than those in small-signal amplification regime. The results shown in Figs. 4(f) and 4(g) give the confidence in using Eq. (19) to assess the output pulse contrast in the case of spatial pump-intensity modulation. The study in this section suggests that the spatial intensity modulation of pump laser should be viewed as a major factor of contrast degradation in FOPA systems. This is another essential difference of FOPA with an OPCPA, as the pump spatial modulation has no impacts on the temporal contrast of amplified pulse [13–15]. High quality of pump beam is then critically important in achieving high contrast.

It is worthwhile to address the influence of pump wave-front distortions. In principle, pump wave-front distortions will partly impart to the amplified signal due to the effect of walk-off [29], and will get transferred in the spectral domain due to the spatio-spectral coupling in FOPA. However, pump wave-front distortions, in reality, have little effect on the coherent noise by the following reason. The signal and pump have a feature of large beam sizes in the dimension of grating diffraction since the amplifier is set in the spectral Fourier-plane of FOPA, which makes the walk-off effect negligible. In our simulation case, the walk-off length between signal and pump is approximately 0.8 m, over two orders of magnitude larger than the crystal length (2.5 mm) used.

4. Spatiotemporal coherent noise originated from crystal surface roughness

Besides the spatial modulation transferred from the pump beam, the optical scattering due to surface roughness is another source of spatial phase imperfections. In this regard, the crystal surface roughness could also induce spatiotemporal coherent noise in FOPA lasers, which will be studied by numerical simulations in this section.

In the simulations, the surface roughness of crystal entrance is taken into consideration. A surface condition [Fig. 5(a)] with a root-mean-square (RMS) height of 0.3 nm is first considered. The pump laser is assumed to be spatiotemporally clean in both intensity and phase, and other simulation parameters keep the same with those for Fig. 3. Figure 5(b) shows the intensity profile of the amplified signal pulse subsequent to FOPA (i.e., on the axis of FOPA output plane in near-field), where a pedestal in 10 ps time range is induced due to the assumed surface roughness. The formation and spatiotemporal evolution of this kind of coherent noise throughout FOPA is illustrated in Figs. 5(c)-5(e). Basically, the surface roughness of crystal entrance first disturbs the signal phase in the spatial domain. Owing to the spatial-spectral coupling nature in the spectral Fourier-plane [Fig. 5(c)], this spatial phase imperfection is equivalent to a random spectral phase modulation. In the output plane of FOPA, the scattering background acquires the STC effect and thereby distinguishes from the main signal, as illustrated in Fig. 5(d-1). The coherent noise induced by crystal surface roughness exhibits a STC slope (dt/dx = −5 ps/mm) is exactly equal to that of the spatiotemporal coherent noise induced by spatial pump-beam modulation [Fig. 4(c-1)]. In essence, this STC slope is governed by the angular dispersion provided by the gratings, as described in Eq. (16) and Eq. (17). In the output far-field, the scattering background induced by crystal surface roughness does not exhibit the property of STC in the (x, t) domain [Fig. 6(e-1)].

Fig. 5 Characterization of the spatiotemporal coherent noise originated from the crystal surface roughness. (a) Crystal surface profile characterized with a RMS height of 0.3 nm. (b) Temporal profile of the amplified signal pulse profiles with a clean pump laser and the amplification parameters the same with Fig. 3. (c), (d), (e) Spatiotemporal and spatial-spectral intensity profiles of the amplified signal in the spectral Fourier-plane, the output near-field and output far-field, respectively. (f) Output coherent contrast at t = −5 ps (red line) and t = −10 ps (blue line) versus parametric gain G₀ and pump-depletion ratio η_p. (g) Coherent contrast values at t = −5 ps (red line) and t = −10 ps (blue line) versus the RMS height of crystal surface roughness, under the condition of a fixed amplification condition at G0 = 104 and pump-depletion ratio ηp = 0.5%.

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Quantitatively, a surface-roughness of 0.3 nm RMS has induced a degradation of the amplified pulse contrast at t = −10 ps to −110 dB [Fig. 6(b)]. In comparison, such a high picosecond contrast of −110 dB has been inaccessible in OPCPA systems so far. For example, the OPCPA frontend of APOLLON laser has its picosecond coherent contrast measured at t = −10 ps as −70 dB [8]. We move to characterize the coherent contrast degradation versus the amplification parameters of both parametric gain G₀ and pump-depletion ratio η_p. The results, as plotted in Fig. 5(f), indicate that a moderate saturation (i.e., η_p <20% in our case) helps to reduce the scattering noise from surface roughness. But if the amplifier is over saturated (with η_p > 20% or with back-conversion occurring), the noise pedestal reverses to increase. Besides, as shown by Fig. 5(g), such a degradation in coherent contrast due to the crystal surface roughness is primarily proportional to the square of the roughness value (in RMS height). This quantitative relation is similar to the conventional OPCPA situation where the coherent noise is induced by the surface roughness of gratings in the pulse stretchers and compressors [23].

5. Conclusion

The spatial-spectral coupling property of FOPA plays a significant role in the production and evolution of coherent noise. Unlike the conventional OPCPA situation, the coherent noise produced in FOPA amplifiers always exhibits spatiotemporal coupling in the output near-field. We have demonstrated that the spatiotemporal coupling slope of coherent noise is governed by the grating angular dispersion in FOPA configuration, and has nothing to do with the origins of noise. Two kinds of spatiotemporal coherent noise, originated from the spatial imperfections in pump laser and crystal surfaces, have been studied analytically and numerically. In the output near-field, the coherent noise distinguishes the signal in both time and space owing to the spatiotemporal coupling effect, which makes the noise reduction possible by using a suitable slit. While the noise contrast caused by pump beam modulation rapidly increases with the parametric gain, the noise contrast caused by crystal surface roughness remains approximately unchanged with the parametric gain. In the roughness case with a RMS height of typically 0.3 nm, the spatiotemporal noise limits a contrast ratio to approximately −80 dB at t = −5 ps and −110 dB at t = −10 ps. The results presented in this paper suggest a new perspective on the reduction of coherent noise: the various spatiotemporal coupling effects [26] of ultrafast lasers may enable the spatial separation and filtration of coherent noise from ultraintense pulses. Besides the FOPA scheme whose coherent noise has been studied in this paper, it is worthwhile to further exploit new amplification schemes with different spatiotemporal coupling effects for possibly improving the coherent contrast of ultraintense lasers.

Appendix

Here, we introduce the theoretical method to study the evolution of signal field in an FOPA system. For a clean incident pulse as given by Eq. (1), we give the amplitude a(x, ω) of each frequency (ω) component at each optics plane normal to the direction of propagation z. Firstly, after grating G₁, the pulse acquires an angular dispersion, and the signal amplitude evolves to

a_{G_{1}} (x, ω) = a_{i n} (x, ω) \exp (i k β ω x),

In Eq. (20), β is the angular dispersion defined by the grating density d as well as the angle of diffraction θ as

β = \frac{d θ}{d ω} = - \frac{λ^{3}}{2 π c d \cos θ},

where λ is the center wavelength of incident pulse, and c is the light speed in vacuum. The spatiotemporal amplitude of this angularly-dispersed signal pulse can be obtained by performing Fourier transform of Eq. (20) with respect to ω:

A_{G_{1}} (x, t) = \int_{- \infty}^{\infty} a_{G_{1}} (x, ω) \exp (i ω t) d ω = a_{1} \exp (- \frac{x^{2}}{2 D^{2}}) \exp [- \frac{{(t + k β x)}^{2}}{2 τ_{0}^{2}}] .

This expression clearly shows spatiotemporal coupling (STC) in the form of pulse-front-tilt. From Eq. (22), the STC slope is given by

ξ = \frac{d t}{d x} = - k β .

After outgoing from grating G₁, the signal field quickly spreads out in space and time since different frequency components propagate at different angles. The normalized amplitude of signal after a propagation distance z from grating G₁ can be obtained by performing an approximate Fresnel-Kirchhoff integral of the amplified given by Eq. (20) as

\begin{array}{l} a_{z} (x, ω, z) = \frac{i}{λ z} \int_{- \infty}^{\infty} a_{G_{1}} (x_{0}, ω) \exp [- \frac{i k}{2 z} {(x - x_{0})}^{2}] d x_{0} \\ \approx \exp [- \frac{ω^{2}}{2 {(Δ ω)}^{2}}] \exp (i k β ω x) \exp {- \frac{{(x + β ω z)}^{2}}{2 D^{2}}} \exp (\frac{i k}{2} β^{2} ω^{2} z) . \end{array}

The third and fourth exponential terms on the right hand side of Eq. (24) indicate a spatial chirp and a temporal chirp, respectively. For definiteness, we can define a spatial chirp coefficient u and a temporal chirp coefficient C as

u (z) = \frac{β Δ ω z}{D}, C (z) = k β^{2} {(Δ ω)}^{2} z .

The intensity profile of this spatiotemporally dispersed signal can thus be written as

I_{z} (x, t, z) = {| \int a_{z} (x, ω, z) e^{i ω t} d ω |}^{2} = \exp {- \frac{{[t - ξ (z) x]}^{2}}{\frac{{(1 + u^{2})}^{2} + C^{2}}{1 + u^{2}} τ_{0}^{2}}} \exp [- \frac{x^{2}}{(1 + u^{2}) D^{2}}],

where ξ(z) denotes a spatiotemporal coupling slope that varies with propagation distance z as

ξ (z) = - k β \times \frac{1}{1 + u^{2}} .

The local duration T_local(z) (i.e., pulse duration at a given transverse position x) increases with z accordingly as

T_{l o c a l} (z) = \sqrt{\frac{{(1 + u^{2})}^{2} + C^{2}}{1 + u^{2}}} τ_{0} .

Equation (28) indicates that the local duration of signal pulse will reach its maximum in the spectral Fourier-plane (equivalent to z → ∞). The spatial-spectral amplitude of signal in the spectral Fourier-plane can be deduced as

\begin{array}{l} a_{F P} (x, ω) = \frac{i}{λ f} \int_{- \infty}^{\infty} a_{z} (x_{0}, ω, z = f) \exp (\frac{i k}{2 f} x_{0}^{2}) \exp [- \frac{i k}{2 f} {(x - x_{0})}^{2}] d x_{0} \\ = \exp [- \frac{ω^{2}}{2 {(Δ ω)}^{2}}] \exp [- \frac{{(x + β f ω)}^{2}}{2 σ^{2}}] \exp (- \frac{i k}{2 f} x^{2}) \exp (- \frac{i k β^{2} ω^{2} f}{2}) \exp (- i k β ω x) . \end{array}

This signal field is featured by the property of spatial-spectral coupling, where the laser frequency (ω) varies with the transverse coordinate (x), with the coupling slope given by

ζ = \frac{d x}{d ω} = - β f .

With an inverse Fourier-transform of Eq. (29), the spatiotemporal amplitude of signal can be also derived as has been present in Eq. (4).

The evolution of signal field from the spectral Fourier-plane to grating G₂ is the reverse of that from grating G₁ to the spectral Fourier-plane. Thus, we give the analytical expressions for the signal field at the output plane of concave mirror M₂ and that in the output plane of grating G₂ directly as

a_{M_{2}} (x, ω) = \exp [- \frac{ω^{2}}{2 {(Δ ω)}^{2}}] \exp {- \frac{{(x + β ω f)}^{2}}{2 D^{2}}} \exp (- i k β ω x) \exp (- \frac{i k}{2} β^{2} ω^{2} f),

and

a_{G_{2}} (x, ω) = \exp [- \frac{ω^{2}}{2 {(Δ ω)}^{2}}] \exp (- \frac{x^{2}}{2 D^{2}}) .

Equation (32) clearly shows that the signal laser is restored to a Fourier-transform-limited pulse in the output near-field of FOPA.

Funding

National Basic Research Program of China (2013CBA01505); National Natural Science Foundation of China (61727820); China Postdoctoral Science Foundation (2016M601577, 2017T100294).

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Spatiotemporal coherent noise in frequency-domain optical parametric amplification

Abstract

1. Introduction

2. Spatiotemporal coupling property of FOPA

3. Spatiotemporal coherent noise originated from pump noise

3.1 Nonlinear transfer of temporal pump-pulse modulation

3.2 Nonlinear transfer of spatial pump-beam modulation

4. Spatiotemporal coherent noise originated from crystal surface roughness

5. Conclusion

Appendix

Funding

References and links

Cited By

Figures (5)

Equations (32)

Optics Express