Parameter space for the collective laser coupling in the laser fusion driver based on the concept of fiber amplification network

Zhihua Huang; Honghuan Lin; Dangpeng Xu; Mingzhong Li; Jianjun Wang; Ying Deng; Rui Zhang; Yongliang Zhang; Xiaocheng Tian; Xiaofeng Wei

doi:10.1364/OE.21.016494

1. Introduction

The history of the research of the laser driven inertial confinement fusion (ICF) is almost as long as that of the laser technology. The output energy, peak power and intensity of the laser beam have been enhanced profoundly in the last few decades due to the construction of the large-scale laser drivers such as NIF [1], LMJ [2] and SG-III [3]. Large aperture Nd:Glass bulk medium is applied in these facilities which can produce nanosecond-pulse with energy of around twenty kilo-joules per beamlet at fundamental harmonic wavelength (1ω). Therefore, only around 200 beamlets are required to generate pulse energy as high as 2MJ at third harmonic wavelength (3ω) which was thought to be enough for the indirect drive central ignition method [4] but proven to be too optimistic [5]. However, direct drive or other ignition modes may also be feasible at this energy level [5].

A disadvantage of current Xenon lamp pumped bulk laser driver is that its energy conversion efficiency is too low which results in high thermal load and very low repetition rate. Replacing the pump source to laser diode while keeping the laser amplifiers unchanged is one option which guides the design of LIFE [6]. The fiber amplification network (FAN) proposed by G. Mourou et al. [7, 8] provides us a promising alternative which takes advantage of the rapid development of the fiber laser technology. In the last few years, the output energy of from the single-mode photonic crystal fiber has been enhanced to larger than 4.3mJ for 1ns pulse [9] while the output energy from the multimode fiber has reached 33mJ for 10ns pulse [10]. However, due to the megajoule-class energy requirement in the ICF laser driver, millions of fiber amplifiers are still required in the FAN configuration which makes laser coupling a difficult problem. The requirements of output capabilities from a single fiber amplifier are prior issues to be confirmed in order to assess the feasibility of this concept and guide the research of fiber amplifiers based on this concept.

In this work, the feasible parameter space for the collective laser coupling of the laser fusion driver based on the FAN concept is researched. Beam quality factor conservation law is applied by neglecting the influence of the frequency conversion on the beam quality. The output beam from each fiber amplifier is assumed to be diffraction-limited with a lower limit of center-to-center pitch.

2. Collective laser coupling

Collective laser coupling means that a number of fiber sources (fiber array) use the same laser coupling optics, as is shown in Fig. 1. The integration level is higher in collective laser coupling while the output beam quality of the fiber array is degraded compared to that of a single fiber source. Frequency conversion unit is omitted in this configuration.

Fig. 1 Configuration of collective laser coupling optics.

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2.1. Integrated parameters for the ICF laser driver

Part of the integrated parameters for the ICF laser driver are given in table 1 which follow the design of NIF [1] and LMJ [2] while replacing indirect drive method with direct drive method. For comparison, laser energy requirements at the fundamental harmonic (1ω), second harmonic (2ω) and (3ω) are assumed. The chamber opening proportion ζ determines the area that can be used for laser coupling, and therefore the average energy fluence through the coupling area for given laser energy which is shown in Fig. 2(left). ζ is approximately 0.1 for NIF [1]. ζ = 0.3 is chosen here to accommodate numerous fiber sources which is almost the upper limit of the opening proportion considering the mechanical stability of the chamber.

Table 1. Part of the integrated parameters for the ICF laser driver

View Table

Fig. 2 (Left) variation of average energy fluence with the chamber opening proportion, (right) variation of the output pulse energy with the mode field diameter for the in-core fluence of 200J/cm².

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Another important parameter is the output energy fluence E_F from a single circular core fiber. In the large aperture Nd:glass laser system, the saturation fluence J_s is around 4.5J/cm². The output energy fluence at fundamental harmonic wave is approximately 20J/cm² which is 4 ∼ 5 times of J_s[4]. In Ytterbium-doped fibers, J_s at 1053nm is around 50J/cm²[11]. The extracted energy fluence can be as large as 10 times of J_s [12] in small mode area fiber amplifiers. However, based on the experimental demonstrations of large mode area fiber (LMAF) amplifiers, the output energy fluence of around 200J/cm² (4J_s) for 10ns pulse is expectable. Therefore, J_F = 200J/cm² is used in this paper to estimate the output pulse energy for a given mode filed diameter (MFD). The variation of output pulse energy with the MFD is shown in Fig. 2 (right).

2.2. Arrangement of the fiber array

Two kinds of typical two-dimensional coplanar arrangement of the fiber array is shown Fig. 3, i.e. the rectangular and hexagonal arrangements. The number of rings k is chosen to describe the scale of the fiber array. The relation between k and the total number of fiber sources is as follows:

N_{F} = {\begin{array}{l} 4 k^{2} & rectangular \\ 3 k (k - 1) + 1 & hexagonal \end{array}

The relation between the diagonal of the fiber array D_B and the center-to-center pitch Λ is given as

D_{B} = {\begin{array}{l} (m - 1) \sqrt{2} Λ + D_{F} & rectangular \\ (m - 1) Λ + D_{F} & hexagonal \end{array}

where m is the number of fiber sources across the diagonal. The relation between m and k is

m = {\begin{array}{l} 2 k & rectangular \\ 2 k - 1 & hexagonal \end{array}

A key quantity related to the fiber array arrangement is the spatial duty cycle ς which is defined as the ratio of the total fiber mode field area to the area of the fiber array

ς = {\begin{array}{l} \frac{N_{F} \times {(D_{F} / 2)}^{2}}{D_{B}^{2} / 2} & rectangular \\ \frac{N_{F} \times π {(D_{F} / 2)}^{2}}{3 \sqrt{3} D_{B}^{2} / 8} & hexagonal \end{array}

For the densest arrangement, i.e. D_F = Λ, ς approaches 78.5% and 90.7% for large scale rectangular and hexagonal array (i.e. k ≫ 1), respectively. ς determines the reduction of the energy fluence of the synthetic beam J_B compared to that of a single fiber source J_F, i.e. J_B = ςJ_F. Therefore, hexagonal arrangement is assumed for the analysis in the following section because of its higher spatial duty cycle. For rectangular arrangement, the analysis is also straightforward.

Fig. 3 (Left) rectangular and (right) hexagonal arrangement of the fiber array.

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3. Beam quality factor conservation

As is shown in Fig. 4, the beam parameter product (BPP) is defined as the product of the beam waist radius w and half divergence angle θ defined by the second order intensity moment within which over 86.5% energy is enclosed [13]

BPP = w \times θ

For diffraction-limited beam, the minimum of beam parameter product BPP₀ is

{BPP}_{0} = λ / π

where λ is the central wavelength. For λ = 1053nm, BPP₀ = 0.335mm · mrad. Beam quality factor M² of an arbitrary beam is defined as the ratio of its BPP to BPP₀, therefore, M² equals to 1.0 for the diffraction-limited Gaussian beam. Generally, beam quality becomes worse for larger M². In the context of laser coupling, worse beam quality means that the focused spot size is larger for given beam aperture and focusing lens. In the LMFA, M² is larger for higher order modes [14].

Fig. 4 Conservation of the beam parameter product in the focusing process.

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The conservation law of the beam quality factor says that M² is a constant in an ABCD-optical system where only free space propagation and parabolic phase element (ideal lens) are involved [13]. In the laser-driven ICF, incoherent beam combining is an advantage to realize uniform laser intensity distribution on the target. For an incoherent fiber array, the summation of all the beams can be viewed as a single beam which also must follow the conservation law if incoherent condition is assumed. For simplicity, the impact of the frequency conversion to M² factor is neglected (BPP varies because the wavelength is different). This is close to be true if the combined near field intensity distribution is flattened which results in an equal nonlinear phase shift across the lateral section. Only a frequency conversion efficiency is assumed for the calculation of the effective energy at the fundamental harmonic wavelength. The maximum 1-D M² (along the longest diagnoal) is applied to guarantee high coupling efficiency althought 2-D arrangement is assumed for analysis.

4. Analyzing procedure of the collective laser coupling

The analyzing procedure is shown in Fig. 5, where the quantities within the round angle boxes are the fixed input parameters, the quantities within the rectangular boxes are intermediate quantities to be calculated and the quantities within the slashed round angle boxes are variable input parameters. The purpose of this analyzing procedure is to search for the parameter space within which the collective laser coupling is feasible. The analyzing procedure goes from left to right following the direction of the arrows. 11 steps are involved which are described separately as follows:

Step 1: Calculate the size of the focusing lens. Assume that the beam quality factor of the synthetic beam for laser coupling $M_{B}^{2}$ is a variable parameter. For wavelength of λ^xω where x = 1, 2, 3 corresponds to the fundamental, second and third harmonic wave, respectively, the BPP of the laser beam after frequency conversion is given as ${BPP}_{B}^{x ω} = M_{B}^{2} \frac{λ^{x ω}}{π}$ The target locates at the waist of focused beam and beam waist diameter equals to the spherical target diameter D_T. The half divergence angle of beam is ${BPP}_{B}^{x ω} / (D_{T} / 2)$ , corresponding to the Rayleigh range of ${(D_{T} / 2)}^{2} / {BPP}_{B}^{x ω}$ . Following the results of Gaussian optics [13], the beam size after the free space propagation across the chamber with a given diameter of D_C is $D_{W} = D_{T} \sqrt{1 + {(D_{C} / 2 Z_{R})}^{2}}$ where D_W is the diameter of the focusing lens which has been assumed to be equal to the beam size on the chamber surface if the vignetting effect is neglected.
Step 2: Calculate the number of fiber arrays. The opening area on the chamber surface for laser coupling is determined by the opening proportion ζ and the chamber diameter D_C. The total number of fiber arrays N_B can be obtained as follows $N_{B} = \frac{ζ \times π D_{C}^{2} / 4}{A_{W}} = ζ \times {(\frac{D_{C}}{D_{W}})}^{2}$ where $A_{W} = π D_{W}^{2} / 4$ has been applied for circular lens.
Step 3: Calculate the energy requirement for a single fiber array. For a given laser energy requirement $E_{D}^{x ω}$ at xω, the energy requirement for a single fiber array $E_{B}^{x ω}$ is given as $E_{B}^{x ω} = E_{D}^{x ω} / N_{B}$
Step 4: Calculate the equivalent fundamental harmonic energy. Because the output laser wavelength of the fiber amplifier is near-infrared, therefore, the equivalent fundamental harmonic energy for each fiber array $E_{B}^{1 ω}$ must be calculated according to the second harmonic conversion (η₁₂) or third harmonic conversion (η₁₃) efficiency $E_{B}^{1 ω} = E_{B}^{x ω} / η_{1 x}$ where x = 1, 2, 3 and η₁₁ = 1.0 if the fundamental harmonic wave is used for laser coupling directly.
Step 5: Calculate the half divergence angle. Assume that the output beam quality is diffraction-limited for each fiber source, i.e. the beam quality factor $M_{F}^{2} = 1.0$ , corresponding to the beam parameter product of ${BPP}_{F}^{1 ω} = λ^{1 ω} / π$ . For given mode field diameter of D_F, the half divergence angle $θ_{F}^{1 ω}$ is given as $θ_{F}^{1 ω} = \frac{{BPP}_{F}^{1 ω}}{D_{F} / 2}$
Step 6: Calculate the BPP of the fiber array. If the effect of the frequency conversion on the beam quality can be neglected, i.e. the beam quality factor conserves, the BPP of the fiber array ${BPP}_{B}^{1 ω}$ is ${BPP}_{B}^{1 ω} = M_{B}^{2} \times \frac{λ^{1 ω}}{π}$ Because the divergence angle of the fiber array is the same with that of a single fiber source, the diameter of the fiber array can be evaluated as follows $D_{B} = 2 {BPP}_{B}^{1 ω} / θ_{F}^{1 ω}$
Step 7: Calculate the output energy from a single fiber source. For a given in-core energy fluence J_F, the output energy from a single fiber source $E_{F}^{1 ω}$ can be obtained according to variable of the mode field diameter D_F $E_{F}^{1 ω} = J_{F} \times π D_{F}^{2} / 4$
Step 8: Calculate the number of fiber sources within a fiber array. For known $E_{B}^{1 ω}$ and $E_{F}^{1 ω}$ , the total number of the fiber sources within a fiber array is $N_{F} = E_{B}^{1 ω} / E_{F}^{1 ω}$
Step 9: Calculate the pitch between neighboring fiber sources. As is shown in Fig. 3, the pitch between neighboring fiber sources for hexagonal arrangement is as follows $Λ = \frac{D_{B} - D_{F}}{m - 1}$ where m is the number of fiber sources across the diagonal which is solved in subsection 2.2.
Step 10: Calculate the lower limit of pitch. For any fiber array arrangement, there is a lower limit for the pitch Λ_lim which is at least the core diameter D_co of fiber source. Conventionally, D_F is a little larger than D_co in small core fiber and D_co is a little larger than D_F in large core fiber. Assume that Λ is χ times of D_F, we use χ = 2.0 for preliminary analysis considering that there should be proper space between the fiber core. Λ_lim is given as $Λ_{\lim} = χ D_{F}$
Step 11: Compare Λ and Λ_lim. Compare Λ calculated from subsection 4 with Λ_lim, if Λ > Λ_lim, the preset variable $M_{B}^{2}$ and D_F fall into the feasible parameter space for collective laser coupling.

Fig. 5 Analyzing procedure of the collective laser coupling under the beam quality factor conservation condition (notations are explained in the text).

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5. Results

Following the steps of the previous section, the feasible parameter space can be obtained by searching the proper parameter range. The results for 2MJ@3ω is shown in Fig. 6. The lower limit for $M_{B}^{2}$ is chosen to be 10 rather than 1 (i.e. there is only one fiber source within a fiber array, and Λ has no meaning in this case) due to the collective characteristic although it’s not very strict. As is shown in Fig. 6, the ratio Λ/Λ_lim has a strong dependence on $M_{B}^{2}$ only in a small range near 10. For larger ratio, the range of dependence is also larger which is shown clearly in Fig. 6(b). The contour marked by 1 is the boundary of feasible parameter space. The critical value of D_F is 494 μm for $M_{B}^{2} = 10$ , and it increases gradually and approaches saturation to 588 μm for $M_{B}^{2} = 200$ . The corresponding output pulse energy is 0.38J and 0.54J, respectively. Because the diameter of the coupling window increases with $M_{B}^{2}$ , i.e. the energy of a single fiber array increases, N_B decreases with $M_{B}^{2}$ for given total energy, as is shown in Fig. 6(c). However, even for $M_{B}^{2}$ as large as 200, N_B is still over 2000 which is one order of magnitude larger than the number of beamlets of the traditional ICF laser driver. As is shown in Fig. 6(d), because $E_{B}^{1 ω}$ increases with $M_{B}^{2}$ , more fiber sources are required which can be a few hundreds or thousands for smaller D_F.

Fig. 6 For laser coupling of 2MJ@3ω, (a) 3-dimensional and (b) contour plot of Λ/Λ_lim, variation of the number of (c) fiber arrays and (d) fiber sources.

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For comparison, the results for the laser coupling of 4MJ@1ω and 3MJ@2ω are shown in Fig. 7. Similar trends can be found in Fig. 7. However, for same value of $M_{B}^{2}$ , the critical values of D_F for the coupling of 1ω and 2ω wave are 3.7 and 1.7 times of that for the coupling of 3ω wave, respectively. Therefore, D_F is close to or even larger than 1mm and the corresponding energy requirements can be as large as 1.13J and 5.38J for 2ω and 1ω wave, respectively, which approaches the level of bulk lasers. The higher energy requirement results from the stronger diffraction effect for longer wavelength and larger coupling window is required for the same focal spot size. Therefore, based on the consideration of reducing output energy requirement (i.e. mode filed diameter) from the single fiber, frequency tripling is preferred under the assumption that the beam quality degradation in the frequency conversion process is neglected.

Fig. 7 For laser coupling of (top) 4MJ@1ω and (bottom) 3MJ@2ω, (a)(c) are the contour plot of Λ/Λ_lim, (b)(d) are the variation of number of fiber arrays N_B with beam quality factor $M_{B}^{2}$ .

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6. Conclusions and discussions

According to the results given above, for large scale collective laser coupling, the output energy requirement from a single fiber for 3ω laser coupling is around 0.4J which corresponds to a mode field diameter of around 500μm. This is more than one order of magnitude higher than what can be achieved using the state-of-the-art technology. Novel waveguide structure needs to be developed which can operate in the fundamental mode while providing ultra-large-mode-area size. Waveguide for mitigation of the peak power limitation imposed by the self-focusing effect, such as multi-core fiber, needs to be developed. Related technologies should be developed to solve the mode instability and realize the low loss laser coupling in the large-mode-area fiber. Coherence is not discussed in detail in this paper which may bring a few new phenomena and possibilities. For example, if cophased coherent combining of the fiber array can be realized, beam shaping employing phase element is possible and the difficulty of the laser coupling can be relieved.

Acknowledgments

This study was funded by China Academy of Engineering Physics (grant no. 2012B0401060 and 2011B0401063).

References and links

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Parameter	Value
Drive method	Direct drive
Ignition method	Central ignition
Fundamental harmonic wavelength (λ¹^ω)	1053nm
Laser energy@1ω ( $E_{D}^{1 ω}$ )	4MJ
Laser energy@2ω ( $E_{D}^{2 ω}$ )	3MJ
Laser energy@3ω ( $E_{D}^{3 ω}$ )	2MJ
Frequency doubling efficiency (η₁₂)	90%
Frequency tripling efficiency (η₁₃)	80%
Chamber diameter (D_C)	10m
Chamber opening proportion (ζ)	30%
Target diameter (D_T)	2mm

Parameter space for the collective laser coupling in the laser fusion driver based on the concept of fiber amplification network

Abstract

1. Introduction

2. Collective laser coupling

2.1. Integrated parameters for the ICF laser driver

2.2. Arrangement of the fiber array

3. Beam quality factor conservation

4. Analyzing procedure of the collective laser coupling

5. Results

6. Conclusions and discussions

Acknowledgments

References and links

Cited By

Figures (7)

Tables (1)

Equations (18)

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