Basic considerations on coherent combining of ultrashort laser pulses

Arno Klenke; Enrico Seise; Jens Limpert; Andreas Tünnermann

doi:10.1364/OE.19.025379

1. Introduction

In recent decades multiple amplification schemes for ultrashort laser pulses have been investigated and a significant progress has been made in scaling their pulse energy and average power [1–4]. Furthermore, in research laboratories all these schemes have been pushed to their specific limits, given by e.g. thermo-optical or nonlinear effects, damage thresholds or available pump power. Hence, a new approach should be considered. Coherent combining of amplifiers is a well-known scaling concept in the continuous wave regime [5-6]. In the ultrashort pulse regime, coherent combination makes it possible to scale the pulse energy, as well as the average power independently from the amplification scheme. This concept has already been demonstrated for two single mode fiber amplifiers, delivering ultrashort pulses with a combining efficiency of over 95% [7]. Additionally, by using LMA fibers, combined pulse energies of 120µJ have been already achieved [8]. It should be noted, however, that the concept itself is not limited to fiber lasers but it can be used with any amplification scheme and it might even provide a way to go beyond the PW level. A related approach to the coherent combining of amplifiers is the use of an external enhancement cavity. In such a cavity, a circulating pulse interferes constructively with the next pulses emitted from the laser source. This results in an intracavity pulse with a drastically increased pulse energy. Using this approach a scaling of the peak power by a factor of > 1000 has been demonstrated [9]. However, in this setup, the high pulse energies are only available inside of the cavity. In spite of this, the effects described in this paper might also occur in enhancement cavities and they will have a similar detrimental impact.

The general setup for the coherent combination of N amplifiers is shown in Fig. 1 . The ultrashort pulses from one mode-locked laser are split into N channels to start with mutually coherent pulses. These pulses are then amplified in their specific channel and finally recombined. An additional element has to be added in N-1 channels to match and stabilize the optical path lengths in the channels and ensure a constructive interference. Additionally, a stretcher and a compressor can be added to the system to create a chirped-pulse amplification (CPA) system. Assuming an identical behavior of all channels and an ideal combining element, the performance of the system could be in principle scaled by a factor of N compared to a single amplifier. However, when using ultrashort pulses, effects like dispersion and self-phase modulation (SPM), as well as optical path length differences (OPD) between the pulses, will affect the temporal and spectral phases of the combined pulses and, thus, lead to a non-perfect combination of the pulses. In order to characterize the performance of the combination process a figure of merit has been defined. In this paper, the dependency of this figure of merit on the effects mentioned above will be explored, under the assumption that the spectral intensity profiles of the pulses, as well as the spatial intensity and phase profiles of the beams are identical. An in-depth analysis of the detrimental effects caused by an imperfect overlap of the beams has already been published [10].

Fig. 1 Schematic setup of coherent addition of ultrashort laser pulses; Δφ: element for path length matching

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2. General considerations on coherent beam combining of ultrashort pulses

The combining element plays an important role in the combining process, since it has to merge the beams coming from spatially separated amplification stages into one beam while preserving the beam quality and keeping losses as low as possible. Different elements have been successfully proposed such as partially reflective surfaces [11] or polarization dependent beam splitters [7]. Additionally, in the case of fiber lasers, fiber integrated couplers are another alternative. While all of these elements can in principle just combine two beams, by cascading them, the number of channels to be combined can be increased. Depending on the type of combining element, phase differences between the incoming pulses, e.g. OPDs, will have a different effect on the combined beam.

In the case of a partially reflective surface (Fig. 2a ), the power of the combined beam will depend on the constructive interference of the incident beams, while the rest of the power, i.e. that containing the non-interfering parts, will be emitted as a secondary beam in a different direction (see Fig. 2a). For every spectral component, the well-known interference formula can be used:

P (ω) = P_{0} (ω) (1 + \cos (Δ Φ (ω)))

with P₀(ω) being the spectral power of the beams to be combined (they are assumed to have an identical beam profile) and ΔΦ(ω) being the spectral phase difference between them. In order to define a figure of merit (FOM) suitable for characterizing the combining process, the power in the combined beam (P_comb) and the secondary beam (P_secondary) can be measured and used in the following calculation for the visibility:

Fig. 2 Combining process using (a) a partially reflective surface and (b) a polarization dependent cube.

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F O M = \frac{P_{c o m b} - P_{\sec o n d a r y}}{P_{c o m b} + P_{\sec o n d a r y}}

On the other hand, as shown in Fig. 2b, the combining process of two pulses can also be carried out using a polarization beam splitter. These elements are commercially available with excellent specifications, such as low losses and high damage thresholds. In this case, the two incident beams to be combined have to be one s- and the other p-polarized. This results in the combined beam being emitted exclusively from one port of the cube. Deviations from perfectly linearly polarized incident pulses and imperfections of the cube itself result in a power output at the so-called “dark port” and, therefore, in a reduction of the maximum achievable power of the combined beam. However, any spectral and/or temporal phase difference between the pulses will not be reflected in the power of the combined beam, but in its polarization state instead. Hence, a measurement of the polarization state is required to estimate the FOM of the combined beam.

Placing an analyzer at an angle of 45° behind the transmission port of the polarization beam splitter leads to an interference that can be expressed with Eq. (1). In this way the result is basically the same as for a partially reflective element. The definition of the figure of merit can also be used here by measuring the maximum and minimum average power obtained when rotating the analyzer. The figure of merit represents in this case the degree of linear polarization (DOLP) of the combined beam, and can be calculated as follows:

F O M = D O L P = \frac{P_{m a x} - P_{m i n}}{P_{m a x} + P_{m i n}}

For two pulses with identical spectral intensities and assuming a perfect spatial overlap of the beams, the figure of merit will deliver a value between 0 and 1. Coherent interference results in strong modulations of the FOM for OPDs in the subwavelength region and there are local maxima with the periodicity equal to the wavelength. This resembles the behavior in the continuous wave regime. However, in the pulsed regime, the frequency dependence of the spectral phase differences has to be taken into account.

For a spectral phase difference ΔΦ(ω), the figure of merit can be analytically calculated at each frequency. Thus, as seen in Eq. (1), if cos(ΔΦ(ω)) ≥ 0, then more power of this spectral component is in the combined beam than in the secondary beam. In this case, the maximum power of the combined beam P_max(ω) equals P(ω) and P_min(ω) = 2P₀(ω) - P_max(ω). For every spectral component, the figure of merit can now be calculated by using Eq. (2). For cos(ΔΦ(ω)) < 0, on the other hand, more power of this spectral component would instead be in the secondary beam. This results in a swap of the formula for P_max and P_min and corresponds to the negative solution in the following calculation:

F O M (ω) ​ = \frac{P_{m a x} (ω) - P_{m i n} (ω)}{P_{m a x} (ω) + P_{m i n} (ω)} = \pm \frac{2 P_{0} (ω) \cos (Δ Φ (ω))}{2 P_{0} (ω)} = \pm \cos (Δ Φ (ω))

To calculate the FOM for the whole pulse, this result has to be weighted with the spectral intensity profile before being finally integrated over frequency:

F O M ​ = C \int s (ω) F O M (ω) d ω = \pm C \int s (ω) \cos (Δ Φ (ω)) d ω

with the normalized spectral intensity s(ω) of the power P₀(ω) and the normalization factor

C \int s (ω) d ω = 1

. Please note that FOM(ω) can be negative for some spectral components in this calculation. It is important to note that the sign in Eq. (4) has to be equal for all spectral components, and it should be chosen in such a way that the result of Eq. (4) is positive. Thus, if the function ΔΦ(ω) can be calculated for a certain effect (e.g. SPM), it is now possible to estimate the FOM degradation that this effect causes. It is worth noting that the FOM is additionally a useful parameter to stabilize the system. In other words, the control loop can use the FOM as its feedback/error parameter and it can thus be locked to the best figure of merit. This can be done by using a phase modulation based locking system or with the Hänsch-Couillaud mechanism [12] in the case of polarization combining.

3. Influence of optical path length differences on the combining process

In this section the impact of OPDs between the pulses on the combining process is discussed.

An OPD between the two pulses is translated in a linear phase difference. The spectral phase shift of the OPD is thus given by:

Δ Φ (ω) = \frac{Δ l}{c_{0}} ω

with the OPD Δl and the speed of light c₀. As described in section 2, the interesting case is when Δl becomes a multiple of the wavelength λ₀, because in this case a local maximum for the combined power and, therefore, a local maximum for the FOM is reached. A decay of the FOM at these maxima is expected if the temporal overlap of the pulses is reduced. It should be noted that an OPD has the same impact for a chirped pulse as it would have for the corresponding transform limited pulse. This can be seen by interfering two pulses with the electric fields E₁ and E₂ and the common spectral intensity profile E₀(ω):

\begin{array}{l} E_{1} (t) = \frac{1}{\sqrt{2 π}} \int E_{0} (ω) e^{i Φ_{c h i r p} (ω)} e^{i ω t} d ω \\ E {}_{2}{(t)} = \frac{1}{\sqrt{2 π}} \int E_{0} (ω) e^{i Φ_{c h i r p} (ω)} e^{i Δ Φ_{d e l a y} (ω)} e^{i ω t} d ω \\ \Rightarrow E (t) = \frac{1}{\sqrt{2}} (E_{1} (t) + E {}_{2}{(t)}) \end{array}

The pulses have the same chirp phase Φ_chirp(ω) and there is an additional spectral phase difference ΔΦ_delay(ω) for the OPD. The fluence of the combined pulse can now be calculated using Parseval’s theorem to investigate the impact of the chirp on the FOM:

\begin{matrix} F ~ {\int | E (t) |}^{2} d t = {\int | E (ω) |}^{2} d ω = \frac{1}{2} \int {| E_{0} (ω) |}^{2} {| e^{i Φ_{c h i r p} (ω)} + e^{i (Φ_{c h i r p} (ω) + Δ Φ_{d e l a y} (ω))} |}^{2} d ω \\ = \int {| E_{0} (ω) |}^{2} (1 + \cos (Δ Φ_{d e l a y} (ω))) d ω \end{matrix}

The result shows that there is no dependency of the fluence of the combined pulse energy and therefore no dependency of the FOM on the chirp phase Φ_chirp(ω). For Gaussian pulses, the normalized spectral intensity is defined as:

s (ω) = e^{- 4 \ln 2 {(\frac{ω - ω_{0}}{ω_{F W H M}})}^{2}}

with the Full Width at Half Maximum (FWHM) bandwidth ω_FWHM. By setting

Δ l = k λ_{0}

and using Eqs. (5), (6), (9), the FOM at these points can be analytically calculated:

F O M = C \int s (ω) \cos (Δ Φ (ω)) d ω = e^{- \frac{{(\frac{2 π k ω_{F W H M}}{ω_{0}})}^{2}}{16 l n 2}}

In Fig. 3 , the result is depicted as a function of the OPD and of the spectral bandwidth. A center wavelength of 1030nm has been assumed to simulate a system working in the near infrared spectral region. The calculation has been done for bandwidths up to 10nm, which corresponds to pulses with transform limited pulse durations as short as ~150fs. A strong dependency of the acceptable delay on the signal bandwidth is immediately recognizable. For a system with a bandwidth of about 5nm, a OPD as large as 25 wavelengths (i.e. about 25µm in this case), is sufficient to keep the FOM above 90%. In comparison for shorter pulses (~10nm bandwidth) this value will drop to about 12µm. As can be seen from Eq. (10), the accuracy of the delay adjustment has to be increased by the same factor as the bandwidth to keep the FOM constant. It should be noted again that the OPD has to be stabilized to be as close as possible to one of the local maxima of the FOM. Only a variation of a fraction of wavelength is acceptable here.

Fig. 3 Dependency of the local maxima of the FOM on a OPD for Gaussian pulses with a bandwidth up to 10nm at a typical center wavelength of 1030nm. The white line defines the boundary where the FOM falls below 95%.

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4. Influences of SPM and dispersion on the combining process

When using ultrashort pulses special attention has to be paid to dispersion and nonlinear effects. If these effects impact the pulses in both channels in the same way, the spectral phase difference will be zero and according to Eq. (5) a perfect combination is still possible. However, in a realistic setup there will be small mismatches between the lengths of the dispersive elements of each channel (LDE), as well as input and output power differences caused by misalignments and time dependent fluctuations. These will result in spectral phase differences which will, in turn, reduce the achievable FOM. In general, dispersion acts in the spectral domain and self-phase modulation in the time domain, making it fairly difficult to calculate the overall phase differences. This can be done with simulations based on the Fourier split-step algorithm [13]. However, for strongly stretched pulses like in a CPA system, it is possible to give a formula for the spectral phase caused by second order dispersion and SPM when the center frequency is ω₀ [14]:

Φ (ω) = \frac{1}{2} β_{2} L {(ω - ω_{0})}^{2} + B s (ω - ω_{0})

with the B-Integral

B = \frac{2 π}{λ} \int n_{2} I (z) d z

[15], the LDE L and the second order dispersion coefficient β₂. Hence, the spectral phase difference for the case of equal spectral intensities s(ω) is:

Δ Φ (ω) = \frac{1}{2} β_{2} Δ L {(ω - ω_{0})}^{2} + Δ B s (ω - ω_{0})

which only depends on the B-Integral difference ΔB and LDE difference ΔL of the two pulses. The parameter ΔL is especially important for fiber lasers because of the large LDE typical of these devices. In reality, to achieve the best FOM, these phase differences will be partially compensated for by introducing an additional OPD between the pulses. This OPD is chosen to compensate the spectral phase difference for the center frequency:

\begin{array}{l} Δ Φ_{c} (ω_{0}) = \frac{Δ l}{c_{0}} ω_{0} = - Δ B \\ \Rightarrow Δ Φ_{c} (ω) = \frac{Δ l}{c_{0}} ω = - Δ B \frac{ω}{ω_{0}} = - Δ B (1 + \frac{ω - ω_{0}}{ω_{0}}) \approx - Δ B for | ω - ω_{0} | < < ω_{0} \end{array}

While it is possible to calculate the FOM using a simulation with Eq. (2), it is also interesting to find analytical solutions. However, solving the integral in Eq. (5) proves to be difficult due to the cosine term. For small phase differences ΔΦ(ω) < π/4, a Taylor expansion of the cosine function up to the second order turns out to result in an error of less than 2%. By using Eq. (5),(9),(12),(13), the FOM can now be calculated:

\begin{matrix} F O M = C \int s (ω) \cos (Δ Φ (ω) + Δ Φ_{c} (ω)) d ω \\ \approx 1 + (\frac{1}{\sqrt{2}} - \frac{1}{2} - \frac{1}{2 \sqrt{3}}) Δ B^{2} - \frac{3}{512 l n {(2)}^{2}} ω_{F W H M}^{4} β_{2}^{2} Δ L^{2} \\ + \frac{1}{16 l n {(2)}^{}} (1 - \frac{1}{2 \sqrt{2}}) ω_{F W H M}^{2} β_{2} Δ L Δ B \end{matrix}

So the quality of the combining process can be analytically calculated with just 4 parameters: ΔB, ΔL, the dispersion coefficient β₂ and the spectral bandwidth ω_FWHM. If we apply the condition ΔΦ(ω) < π/4 for all the spectral components of s(ω) with an intensity above 1/e, we can estimate the boundaries for the approximation:

Δ B < \frac{π}{4} {(1 - e^{- 1})}^{- 1} \approx 1.2 rad

The analytically calculated FOM depending on ΔB and ΔL is shown in Fig. 4 . Different bandwidths were chosen to show the dependence of the FOM on this parameter. The maximum deviation of the FOM between this analytically calculated solution and a simulation is 1%, which confirms the validity of the approximation taken to obtain Eq. (14). The graphs above show that for small deviations of the LDE, which should easily be realizable in a setup, a B-Integral difference ΔB of 0.5 rad still results in an excellent value for the FOM of over 95%. This means that even at a high absolute B-Integral of ~10 rad, i.e. in a highly nonlinear regime, a good FOM can still be realistically reached. However, the dispersion term in the equation has a fourth order dependency on the bandwidth. Hence, with broad bandwidths the match of the LDE in the channels becomes critical.

Fig. 4 Analytically calculated FOM for Gaussian pulses with a bandwidth of (a) 5nm, (b) 10nm, (c) 15nm propagating through fused silica, when a B-Integral or LDE difference is introduced.

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Influences of fluctuations of the input power and amplification coefficient

To calculate the impact that fluctuations of the input power and amplification factor have on the FOM (only taking into account the resulting phase fluctuations because the influence of the power fluctuations itself can be neglected), one has to derive an expression for the corresponding change of the B-Integral. Assuming exponential amplification (unsaturated case), the formula for the B-Integral is [13]:

B = ​ C P_{0} (g - 1) \ln {(g)}^{- 1} with C = \frac{8 n_{2} L}{λ_{0} M F D^{2} f_{r e p} τ}

with the nonlinear coefficient n₂, the propagation length L, the mode field diameter MFD, the repetition rate f_rep, the pulse duration τ, the input average power P₀ and the amplification coefficient g. The change of the B-Integral depending on fluctuations of the input power and amplification coefficient (ΔP₀, Δg) can then be approximated and a linear dependence on the absolute value of the B-Integral is found:

\frac{\partial B}{\partial P_{0}} Δ P_{0} = B \frac{Δ P_{0}}{P_{0}} and \frac{\partial B}{\partial g} Δ g \approx B \frac{Δ g}{g} for g > > 1

So while the FOM in Eq. (14) does not depend on the absolute value of B, for higher B-Integrals a relative fluctuation of the input power or the amplification coefficient will result in a larger change of the B-Integral and thus have a larger detrimental effect on the FOM.

This means that the sensitivity of a system will grow with higher B-Integrals, as shown in Fig. 5 . Hence, in a highly nonlinear regime, the stability of the amplifier input and output powers play a major role in determining the achievable FOM. In this case, additional stabilization of these factors might be required.

Fig. 5 Dependency of the absolute change of the B-Integral and of the FOM on the value of the B-Integral, if a fluctuation of the input power and amplification coefficient of 5% is introduced.

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5. Figure of merit for more than two channels

So far, the presented formulas are limited to calculate the FOM for two channels. To calculate the FOM for a larger number of channels, the combined intensity has to be calculated first. This can be done in the same way for cascaded combining elements as if the combination process is realized in one step [10]. It is assumed that the combining fractions of the beam combining system are the same for every channel. Hence, the combined electric field for N channels with the phases Φ_n(ω) can be written as follows if equal intensities are assumed:

E (ω) = \frac{1}{\sqrt{N}} \sum_{n = 1}^{N} E_{0} (ω) e^{i Φ_{n} (ω)} = E_{0} (ω_{0}) \sqrt{\frac{s (ω)}{N}} \sum_{n = 1}^{N} e^{i Φ_{n} (ω)}

The spectral intensity profile of the combined beam can now be calculated:

I (ω) ~ E^{*} (ω) E (ω) = {| E_{0} (ω_{0}) |}^{2} \frac{s (ω)}{N} (N + \sum_{i = 1, i \neq j}^{N} \sum_{j = 1}^{N} \cos (Δ Φ_{i j} (ω)))

with the spectral phase differences between two channels ΔΦ_i,j(ω). Using Eq. (3), the FOM can be calculated for the system:

\begin{matrix} F O M ​ = \frac{\int I (ω) - (N s (ω) - I (ω)) d ω}{\int N s (ω) d ω} = \frac{C}{N} \int s (ω) ((2 -N) + \frac{2}{N} \sum_{i = 1, i \neq j}^{N} \sum_{j = 1}^{N} \cos (Δ Φ_{i j} (ω))) d ω \\ = (\frac{2}{N} - 1) + \frac{2}{N^{2}} \sum_{i = 1, i \neq j}^{N} \sum_{j = 1}^{N} F O M_{i j} \end{matrix}

with the FOM for two channels FOM_ij. In reality, the FOM between two random channels will result in approximately the same value FOM₁₂, so the equation can be simplified further:

F O M ​ \approx (\frac{2}{N} - 1) + 2 (1 - \frac{1}{N}) F O M_{12}

6. Conclusion

In conclusion, we have investigated how different temporal and spectral effects impact a figure of merit introduced to characterize the coherent combination of two ultrashort pulses. These effects include SPM, dispersion and OPDs between the pulses. It has been shown that the detrimental effect of misalignments and fluctuations grow with increasing bandwidth of the pulses and with increasing B-Integral. However, even in those cases, an excellent FOM of > 90% should still be achievable with a carefully designed setup. This corresponds to a power loss of < 5% when the definition of the FOM is considered. In these cases, spatial effects, which were not considered in this paper, such as an imperfect overlap of the beams will have a larger impact on the combining process and might be the main factor in determining the viability of using the coherent combining approach. While the paper primarily deals with the combination of only two pulses, it is shown how the figure of merit can be calculated for a combining system with more than two channels.

Acknowledgements

This work has been partly supported by the German Federal Ministry of Education and Research (BMBF) and the European Research Council (ERC), SIRG 240460-PECS. A.K. acknowledges financial support by the Helmholtz-Institute Jena. E.S. acknowledges financial support by the Carl Zeiss Stiftung Germany.

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Basic considerations on coherent combining of ultrashort laser pulses

Abstract

1. Introduction

2. General considerations on coherent beam combining of ultrashort pulses

3. Influence of optical path length differences on the combining process

4. Influences of SPM and dispersion on the combining process

Influences of fluctuations of the input power and amplification coefficient

5. Figure of merit for more than two channels

6. Conclusion

Acknowledgements

References and links

Cited By

Figures (5)

Equations (22)

Optics Express