Noise reduction in third order cross-correlation by angle optimization of the interacting beams

V. A. Schanz; F. Wagner; M. Roth; V. Bagnoud

doi:10.1364/OE.25.009252

1. Introduction

Thanks to the chirped pulse amplification technique (CPA) [1] today’s high power laser facilities achieve intensities in excess of 10²¹ W/cm². This spans a range of twelve orders of magnitudes in intensity between the ionization threshold of matter and the maximal intensities achieved by lasers. The temporal structure of laser pulses, amplified using the chirped pulse amplification method, exhibits however a characteristic shape. This shape includes a nanosecond-long pedestal due to the temporally incoherent amplifier noise, pre and postpulses on the nanosecond and picosecond time scales, and a distortion of the pulse itself. This distortion is mostly due to the non-perfect pulse recompression, when spectral amplitude noise or a residual spectral phase are present. In addition, the energy and intensity temporal contrast [2] must be differentiated in order to precisely describe the pulse at stake and to predict its interaction dynamics. Because the intensity of the nanosecond pedestal usually lies above the ionization threshold of most materials and it yields a significant macroscopic pre-plasma expansion [3], different laser facilities around the world are working on increasing their temporal contrast – the ratio between the intensity at an arbitrary time before or after the maximum and the pulse maximum itself – to reduce pre-ionization effects. Established approaches are double CPA techniques, like cross-polarized wave generation [4], or the direct pre-amplification via optical parametric amplification pumped by short pulses, which for example is implemented in the PHELIX laser [5]. In any case, a precise knowledge of the temporal structure of the pulse is necessary to correctly interpret the experimental data created during the interaction of such pulses with targets.

Usually high dynamic measurements of such pulses are done by third order cross-correlation [6,7]. Yet the contrast levels estimated in modern high intensity lasers reportedly exceed eleven orders of magnitudes, which is however beyond the maximum dynamic range of scanning cross-correlators available for purchase. The signal to noise ratio (SNR) of these existing pulse measurement devices is given by the maximum signal to be generated, and the noise generated by the pulse to be measured. Different approaches are used to reduce this noise in third order cross-correlators, for example spectral filtering, like in Sequoia (Amplitude Technologies) and geometrical solutions to prevent scattered light to enter the detector [8,9]. All these are necessary to provide high dynamic range measurements but yet the limiting noise – noise photons whose frequency and propagation direction are equal to those of the real signal – remained untouched.

In this article, we propose and demonstrate that the optimization of the interaction angle between the beams of a scanning third order autocorrelator can reduce the noise level by up to four orders of magnitude, increasing the available dynamic range of such a device accordingly. In the following section the principle of pulse measurement by cross-correlation is summarized with emphasis on the origin of the measurement noise, the existing methods of noise reduction and their limits in third order cross-correlation. Later, we propose a technique to reduce the formation of such limiting noise signal in a third order cross-correlation measurement and show simulations for its impact when using β–barium borate (BBO) and monopotassium phosphate (KDP) crystals. In the fourth section, the design of an improved scanning cross-correlator is described, which implements this noise reduction scheme while maintaining a high signal. The validation of the method is made using pulses of the highest temporal contrast available at the PHELIX facility. This allows for a full characterization of the device, and in particular, by exploiting its full dynamic range, this proves the validity of our method to increase the signal to noise ratio.

2. Basic principle and noise origin

Temporal contrast measurements are standardly done by auto- or cross-correlators. Second order scanning auto-correlators split the pulse to be measured in two, delay one part and re-overlap the two of them in a nonlinear crystal where the interacting beams generate a third beam whose frequency is the sum of the two partial beams [10]. A measurement like this does not depend on the dynamic range of the detector, because in case of a saturation a defined attenuation of the overlapping beams is possible. By adjusting the delay the second beam can be scanned through the first one. The resulting signal, depending on the delay, shows the temporal profile of the incident pulse. Second order auto-correlators like this are still used for pulse profile measurements and achieve dynamic ranges in the order of 10⁵ to 10⁷ [11–13]. In such devices the generated signal has the frequency of the second harmonic of the incident beam. In addition both of the two partial beams perform a second harmonic generation (SHG) by themselves in the nonlinear crystal, which leads to a large noise level if these frequency doubled beams hit the detector. O. Konopolev [9] describes a reduction of this noise by using a large angle between the beams and blocking the direct line of sight of the detector onto the beams by a slit. What remains as noise is frequency doubled light of each beam which is scattered into the detector. With this O. Konopolev achieved a measurement over eleven orders of magnitudes.

By performing a SHG of one of the two partial beams in such a cross-correlator, the resulting frequency of the sum frequency generated signal beam becomes the third harmonic of the incident beam. This frequency does not match the second harmonic of any of the two partial beams and can be separated efficiently by frequency filters. One state of the art implementation of this measurement technique is the Sequoia from Amplitude Technologies, where frequency filtering is achieved by a diffraction grating. Therefore O. Konopolev’s noise reduction by a slit lacks of impact on the noise in third order cross-correlators. In several publications third-order scanning cross-correlators are described which perform measurements with dynamic ranges of 10⁶ to 10⁸ [8,14–18] and the commercial products Sequoia and Tundra (Ultrafast Innovations) achieve 10⁹ to 10¹¹, according to their data sheets.

The only origin of the noise in a device like this is light of the partial beam with the fundamental frequency, which is frequency tripled and scattered into the detector. Until now it was not examined, that a large angle between the two partial beams in a nonlinear crystal, which is adapted accordingly, also can be used to decrease the generation of this noise.

3. Increasing the signal to noise ratio

The idea proposed in this paper, namely to increase the SNR in a third-order cross-correlator, bases on a non-collinear overlap of the two beams which generate the beam with frequency of the third harmonic order. Therefore, in this section, we will be summarize the calculation of the necessary cutting angle of the crystal to maintain phase-matching conditions for a sum frequency generation (SFG). Afterwards the resulting phase-mismatch for noise-generation and therefore the resulting increase of the signal to noise ratio for an example of the crystals BBO and KDP is shown.

As it is well known, to support efficient frequency conversion the phasematching condition

0 = Δ \vec{k} = {\vec{k}}_{3 ω} - {\vec{k}}_{2 ω} - {\vec{k}}_{1 ω}

must be fulfilled, where

{\vec{k}}_{3 ω}, {\vec{k}}_{2 ω}, {\vec{k}}_{1 ω}

describe the wave vector of the beams with frequency 3ω, 2ω and 1ω, respectively. We define the two incoming beams, first with the frequency 1ω and the second with its second harmonic 2ω. These will propagate in the crystal with an angle to the crystal normal axis of α₁ and α₂, respectively. To simplify matters, the generated beam with the frequency 3ω shall be orthogonal to the surface of the crystal. A beam, generated by the process of SHG, is polarized perpendicular to the incoming beam. This set of perpendicular polarized 1ω and 2ω beams already matches the requirements for sum frequency generation with type II phase matching and no further change of polarization is needed. Therefore for the SFG a wave mixing of type II is standing to reason. Considering a type II phase matching with extraordinary polarization of the 1ω and 3ω beams and ordinary polarization of the 2ω beam (eoe) a separation of Eq. (1) into the parts parallel and perpendicular to the generated beam leads to the following two equations, where Θ is the angle between the optical axis (OA) of the crystal and the crystal-surface normal axis.

3 n_{3 ω} (Θ) = 2 n_{2 ω} \cos (α_{2}) + n_{1 ω} (Θ + α_{1}) \cos (α_{1})

0 = 2 n_{2 ω} \sin (α_{2}) + n_{1 ω} (Θ + α_{1}) \sin (α_{1})

Hereby the index of refraction, experienced by the 2ω-beam, is constant and the other indices of refraction can be calculated by the well known equation of an index ellipsoid:

n_{ω} (Θ + α_{1}) = {(\frac{\sin^{2} (Θ + α_{1})}{n_{ω, e}^{2}} + \frac{\cos^{2} (Θ + α_{1})}{n_{ω, o}^{2}})}^{- \frac{1}{2}} .

For the following calculations the values for the extraordinary refractive index n_ω,e and the ordinary refractive index n_ω,o were taken from the free software SNLO [19].

This set of equations can be solved for different geometries. The boundaries of possible solutions are given by the fact that a crystal just can be manufactured with |Θ| ⩽ 90°. Figure 1 shows a part of the possible solutions for BBO and KDP-crystals. Due to crystal symmetry conditions, there is another set of solutions with

- Θ (- α_{1}, α_{2}) = Θ (α_{1}, α_{2});

these shall not be further investigated.

Fig. 1 In this plots the blue lines show the calculated angles of incidence of the 2ω beam and the red lines the cutting angle of the crystal to provide phasematching conditions for SFG at wavelength λ = 1053 nm. All angles are defined inside of the crystal between the crystal-surface normal axis and the beams, or the optical axis in case of Θ. The Figs. a and b show the calculated values for BBO-crystal and KDP-crystal, respectively.

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For every set of values, the occurring noise can be calculated. In this analysis noise is considered to be light, which is originated by an interaction of one beam with itself, but whose frequency and propagation direction is equal to the signal, produced by SFG of the two incoming beams. This is illustrated in Fig. 2 and can occur in two ways. I: A part of the beam does a SHG, one of these photons is scattered in the direction of the 2ω-beam and perform a SFG with the remaining beam, or II: First two photons of the beam are scattered in the direction of the 2ω-beam, afterwards they perform a SHG and the new created photon does a SFG with the remaining beam.

Fig. 2 The left part of the picture shows the basic layout of the two beams entering a nonlinear crystal, their interaction with a large angle to each other and the resulting 3ω signal beam. The angles α₁ and α₂ are the angles between the crystal-surface normal axis (parallel to 3ω beam) and the 1ω and 2ω beam, respectively. The cutting angle Θ describes the angle between the crystal surface normal axis and the optical axis (OA). A small part of the picture is enlarged on the right side to show the effects which lead to 3ω noise: 1ω photons are scattered in every direction and frequency doubled. Frequency doubled photons which propagate parallel to the 2ω beam can interact with the 1ω beam and generate a 3ω photon, which propagates parallel to the 3ω signal beam.

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In general the intensity generated by a SFG process is given by the following equation [20]:

I_{ω 3} = \frac{^{128 π^{3} d^{2}} e f f^{I_{ω 1} I_{ω 2} ω_{3}^{2}}}{n_{ω 1} n_{ω 2} n_{ω 3} c^{3}} L^{2} s i n c^{2} (Δ k L / 2) .

I_ω₃ describes the intensity of the generated light, I_ω₁ and I_ω₂ are the intensities of the incoming beams,

n_{ω_{i}}

are the effective refractive indices, experienced by each corresponding beam. L means the crystal length, c the light speed in vacuum and d_eff is the nonlinear optical coefficient. For a phase-matching of type II d_eff can be calculated by

d_{e f f} = d_{22} \cos^{2} (Θ + β) \cos (3 Φ),

where d₂₂ is one of the nonlinear optical coefficients of the crystal and Φ is the azimuthal angle between the propagation vector and the x z crystalline plane (perpendicular to the OA) [20]. In our case, β describes the angle between the generated beam and the crystal-surface normal axis.

We apply Eq. (6) for both, the noise generation and the SFG of the signal. We assume the third order signal beam and the noise only differ by the amount of the intensity of light with the frequency 2ω. So the SNR can be calculated as follows:

S N R = \frac{I_{3 ω}}{I_{3 ω, n o i s e}} = \frac{I_{2 ω}}{I_{2 ω, n o i s e}} .

To calculate I₂_{ω, noise} again we use Eq. (6), but this time for Δk and d_eff the values for the SHG process have to be applied. Above we named the two cases of noise generation and because the SHG in these two occurs with different propagation directions the values of Δk and d_eff differ for both cases. Hence they will be named ∆k_I and d_eff,I for the first case and Δk_II and d_eff,II for the second case. Now the SNR resulting in both cases can be calculated to:

S N R_{I} = \frac{I_{2 ω} n_{1 ω}^{2} n_{2 ω} c^{3} {(Δ k_{I} L / 2)}^{2}}{^{128 π^{3} d^{2}} e f f, I^{I_{ω}^{2} S {(2 ω)}^{2} L^{2} \sin^{2} (Δ k_{I} L / 2)}}

\propto \frac{n_{1 ω} (Θ - α_{1}) \times {[Δ k_{I} (Θ - α_{1}) \times L / 2]}^{2}}{\cos^{2} (α_{1} - α_{2}) \cos^{4} (Θ - α_{1})},

S N R_{I I} = \frac{I_{2 ω} n_{1 ω}^{2} n_{2 ω} c^{3} {(Δ k_{I I} L / 2)}^{2}}{^{128 π^{3} d^{2}} e f f, I I^{{(I_{ω} S)}^{2} {(2 ω)}^{2} L^{2} \sin^{2} (Δ k_{I I} L / 2)}}

\propto \frac{n_{1 ω} (Θ - α_{1}) \times {[Δ k_{I I} (Θ + α_{2}) \times L / 2]}^{2}}{\cos^{4} (α_{1} - α_{2}) \cos^{4} (Θ + α_{2})} .

In this calculations I_ω is assumed not to be attenuated significantly by any effects and S names the distribution function that characterizes the scattered light direction. We assume a cos²(α₁ − α₂) scattering law which, as the scattering at a matte object, is the worst imaginable case for our investigation. More realistic distribution functions would prefer the forward direction and therefore increase the calculated SNR. Furthermore in the last step, leading to Eqs. (10) and (12) we did another worst case approximation and replaced sin(x) = 1 in the denominators. The Eqs. (4) and (7) and the fact that 2ω light is ordinary polarized were applied. The phase mismatches can be calculated to

Δ k_{I} = \frac{n_{2 w_{o}} 2 ω}{c} - \frac{2 n_{1 ω} (Θ - α_{1}) ω}{c} .

Δ k_{I I} = \frac{n_{2 w_{o}} 2 ω}{c} - \frac{2 n_{1 ω} (Θ + α_{2}) ω}{c}

With the Eqs. (10) and (12–14) the relative change of the SNR depending on the angle of incidence of the beam of fundamental frequency was calculated for an exemplary crystal thickness of 1 mm. The results are shown in Fig. 3 for BBO at wavelentgh λ = 1053 nm as well as in Fig. 4 for KDP at the same wavelength and BBO at λ = 800 nm. These show the normalized SNR over the angle between the beam and the crystal-surface normal axis, inside the crystal. It is notable that both kinds of noise-generation show a significant increase of the achievable SNR for larger angles. Also one can see the range of angles, where phase-matching can be achieved, is much smaller for KDP than for BBO. Therefore the maximum achievable SNR is higher for BBO, hence it supports larger angles. Comparing the two cases, one can see for case II the maximum noise-reduction is more effective in both crystal types. However normally the amount of light scattered in the bulk of a sub-milimeter thick crystal is five orders of magnitude lower than the amount scattered at a surface [9]. Because in case I light must be frequency doubled first, we assume this effect will not take place in the domain of the surface that frequent. Hence case II, which fully profits from the surface scattering, is the dominating and limiting process.

Fig. 3 The two curves show the normalized signal to noise ratio for different angles of incidence of the 1ω beam in case of a BBO crystal. For the blue curve case I was considered, which means photons of the 1ω beam are frequency doubled at first and scattered afterwards. The red curve shows the effect of the noise generation by case II.

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Fig. 4 The curves show the behaviour in case of a KDP crystal at a wavelength of 1053 nm and of a BBO crystal at 800 nm, accordingly to Fig. 3.

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For a first proof of principle experiment, a working point at α₁ = 15°, α₂ = −6.8° and Θ = 61° was chosen. Accordingly a BBO crystal was made and a scanning cross-correlator designed, suiting the needed geometry.

4. Experimental validation

The setup of the Enhanced Intensity Cross-correlator for High Energy Lasers (EICHEL), which was developed to work with the crystal parameters calculated in the section before, is shown in Fig. 5. This setup bases on the well known third order cross-correlation technique, as it was already described by S. Luan et al. [6] and later improved by F. Tavella et al. [8]. State of the art commercially available devices like Sequoia (Amplitude Technologies) or Tundra (UltraFast Innovations) use the techniques from the named publications and achieve dynamic ranges in the order of 10⁹ to 10¹¹. The EICHEL follows the known principle of scanning third order cross-correlators, but with three major differences: First, to attenuate the signal, when a saturation of the detector is reached, we use a combination of a wheel with variable filters, located after the beamsplitter, and two variable filters in the 3ω beam. This way we keep the intensity of the 2ω beam constant, which allows to drive the SHG, generating the 2ω beam, in the saturated regime without loosing on the linearity of the device. The according crystal length was simulated with SNLO. For an incoming beam energy of 2 mJ and a wave mixing of type I in a BBO-crystal, a length in the range of 0.8 mm to 1.5 mm will provide a saturated SHG. In the depicted setup, a BBO-crystal with a thickness of 0.8 mm is used, therefore the intensity of the generated 2ω beam is optimized, which directly maximizes the signal. The second difference is the large angle of 21.8° between the two beams entering the SFG-crystal. Current cross-correlators use a very small angle (e.g. Sequoia about 3°), just enough so the frequency doubled beam does not hit the detector. The third difference is a result of this large angle we choose to decrease the noise: Because of the spatial intensity distribution of the beams which interact in the SFG-crystal the generated beam shows a spatial distribution, containing the information of the temporal pulse profile. This effect is used in single-shot cross-correlators [21], but means an inefficient interaction of the beams resulting in a lower energy of the generated signal beam. Therefore this effect has to be minimized in scanning cross-correlators. To do so and to increase the intensities of the interacting beams we use two cylindrical lenses, which focus the beams into a line inside the crystal, which is perpendicular to the plane formed by the OA and the crystal-surface normal axis. On every point of this line the phase matching conditions are fulfilled, therefore efficient SFG is provided in the whole area of this line. Compared to spherical lenses or focusing mirrors, which are used in other scanning cross-correlators and also would solve the problem of the spatial distribution, a larger area of interaction is provided. In the geometry of this setup with a focal distance of 30 cm and a crystal height of 6 mm the cylindrical lens leads to 60 times the interaction area of a spherical lens with equal focal length and accordingly 60 times the energy of the signal beam. In this comparison of a spherical and a cylindrical lens, we assumed in both cases the focal length and incoming energy is limited by the damage threshold of the crystal, and the intensity in the focal plane shall be equal for both.

Fig. 5 Schematic representation of the EICHEL. The used components are iris apertures I1 and I2 a beamsplitter BS, mirrors M1-M9, mounted prism-mirrors for retroreflections in the delay-line, a dichroic mirror DcM high reflective for 526 nm, cylindrical lenses – the line focus is perpendicular to the drawing plane – L1 (f=30 cm) and L2 (f=1 m), a BBO-crystal for SHG and a second for sum-frequency-generation of the third harmonic and variable filters. The filters are mounted on a wheel and provide the transmissions T=1, T = 10⁻², T = 10⁻⁴, T = 10⁻⁶ and T = 10⁻⁸. Additional attenuation can provided by two ND filters with transmissions of the order of 10% and 1%, respectively. These can be moved in the blue beampath independently. A photomultiplier module H10722-210 from Hamamatsu is used as detector. Its response function for short laser pulses was cross calibrated with ND-filters. Highpass frequency filters (FF) are placed in front of the detector, to prevent scattered 1ω and 2ω light from entering the detector. The implemented linear stage is a OWIS Limes60–100 with travel range of 105 mm, which leads to a temporal range of 2.8 ns.

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With this setup the laser pulse of the femtosecond frontend of the PHELIX laser at the GSI Helmholtz Center for Heavy Ion Research in Darmstadt, Germany, was measured. With 10 Hz repetition rate, this system delivers a pulse with an ASE contrast better than 10¹¹ and an energy of up to 8 mJ in 350 fs at a central wavelength of 1053 nm. The noise level is measured by blocking the 2ω beam inside of the EICHEL and the dark noise of the detector is measured by blocking the beam before it enters the device. Results are shown in Fig. 6 and compared to a measurement performed earlier by a commercial cross-correlator. The EICHEL pulse profile was recorded with time-steps of 1000 fs over a 2.4 ns time window and 50 fs over a 20 ps time window around the maximum. For such a measurement, done with averaging over 10 pulses at each step, 117 min were necessary. The mean ASE level at 1.3·10⁻¹² below the peak of the pulse is close to the noise level at 5·10⁻¹³. When the measurement noise is substracted, the ASE level is estimated at 8·10⁻¹³, which is consistent and confirms prediction made earlier [22].

Fig. 6 The blue curve shows a first pulse profile measurement made with the EICHEL. Green crosses on the left side show the measured absolute values of the noise signal generated by the 1ω beam. Their mean value is 5·10¹³ below the peak intensity. Green dots indicate the area of the mean noise level plusminus the rms value of its measurement. Red dots indicate the dark noise of the detector. For better comparison, the black curve shows a measurement done with a Sequoia.

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The plot in Fig. 6 shows that the dark noise level (in red) and the detection noise level (including the optical noise) are very close, although we believe that the measurement dynamic range here is noise limited.

A pulse with identical contrast settings was measured earlier by the cross-correlator Sequoia. The Sequoia shows a noise floor between 10⁻¹⁰ and 10⁻¹¹, which is typical for a not brand-new system. For the chosen values of the angles α₁, α₂ and Θ an improvement of one order of magnitude is expected (see Fig. 3). The measured noise reduction exceeds the calculations by only half an order of magnitude. With the described differences between the two cross-correlation systems and the worst case assumption of an isotropic scattering distribution S in Eqs.(10) and (12) in mind, this measurement is in agreement to the calculations. Indeed in Fig. 6, one sees an experimentally measured noise level below 10⁻¹⁰ for the Sequoia, while our device resolves the ASE background of the laser at 1.3·10⁻¹².

In addition, when one compares the two plots, two differences between both curves can be seen regarding prepulses: The number of the prepulses and their relative values. The presence of a prepulse in one measurement but not the other can be attributed to measurement artifacts, which is a known issue of scanning cross-correlators. The relative lower value of the prepulses with the EICHEL can be attributed to the 1 ps temporal resolution used in this scan while the laser pulse is around 350 fs long. So it is unlikely that the peak of a prepulse is observed in scan with this parameters. In the case when the precise relative value of the prepulse is of interest, a scan with a fine step size should be performed around the prepulse position (not shown in Fig. 6).

5. Conclusion

In this paper, we have proposed a novel technique to reduce the noise level found in third order cross correlation. Calculations have shown that while performing the sum frequency generation, making the beams overlap with a large angle with respect to each other, reduces the noise by up to five orders of magnitude. Based on this we have shown a setup of a prototype and validated its functionality by a measurement. In the measurement a dynamic range of 5·10¹² was observed. With respect to the best measurements performed with state of the art techniques the signal to noise ratio was improved by a factor of 40, which substantiates previously named calculations. For further improvement it has to be kept in mind that with this technique the 1ω noise already is in the range of the dark noise. Therefore the dynamic range will be limited by the dark noise of the detector. With this improvement the dynamic range of scanning cross-correlators is sufficient to measure laser pulse profiles in the whole intensity regime from ionization threshold of matter until the maximum of the currently achievable laser intensities.

Funding

Eurotom 2014-2018 (633053).

Acknowledgments

This work has been carried out within the framework of the EUROfusion Consortium and has received partial funding from the Euratom research and training programme 2014 – 2018 under grant agreement No 633053. The views and opinions expressed herein do not necessarily reflect those of the European Commission.

References and links

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Noise reduction in third order cross-correlation by angle optimization of the interacting beams

Abstract

1. Introduction

2. Basic principle and noise origin

3. Increasing the signal to noise ratio

4. Experimental validation

5. Conclusion

Funding

Acknowledgments

References and links

Cited By

Figures (6)

Equations (14)

Optics Express