Abstract

A high-efficiency multi-folded Nd:YVO4 slab amplifier is reported. Optimizing the mode overlapping carefully, high extraction efficiency was achieved using the wedged multi-folded configuration with two planar folding mirrors. The amplifier’s performance under different seed powers and different pulse repetition frequencies (PRF) is presented. The 52 W seed at the PRF of 152 kHz was amplified to 101.75 W with the pump power of 92 W, and the corresponding extraction efficiency is 54.3%. And no remarkable deterioration of the beam quality was observed.

©2008 Optical Society of America

1. Introduction

High power, high beam quality diode pumped solid-state lasers (DPSSL) are widely used in many applications, such as material processing, precise measurement, navigating, nonlinear frequency conversion, et al. And the master oscillator power amplifier (MOPA) structure is one of the most effective techniques used in the DPSSLs to get output with high power and high beam quality as well.

In recent years, big progress has been made on increasing the extraction efficiency in the 1064nm laser amplifier. Louis McDonagh, et al., used 888nm laser diode (LD) to pump a single-pass dual-end-pumping amplifier [1], and the high extraction efficiency of 52% was obtained thanks to the lower quantum loss and low heat load brought by the 888nm pumping. While more other groups prefer to use multi-pass or multi-folded structures to enhance the mode overlapping, hence to improve the extraction efficiency. In 1993, Henry Plaessmann, et al., reported a multi-pass amplifier based on the 1:1 confocally reimaging design [2], and optical-to-optical (O-O) efficiency reached 33.1%. In 2001 and 2004, K. M Du, et al., reported the multi-folded amplifiers using the off-axis hybrid resonator, and the O-O efficiencies of 33.6% and 28.1% were obtained respectively [3,4]. In 2005, Jeffrey G. Manni, et al., got a 13.25% O-O efficiency with a double-pass multi-folded amplifier using directly the faces of the wedged crystal as the folding mirrors [5]. From 2005 to 2007, M. J. Damzen, et al., and K. Nawata, et al., presented multi-pass bounce amplifiers, and reported the extraction efficiencies of 35.5% [6], 48% [7], 32.8% [8], 35% [9,10] respectively.

In this paper, we report a high-efficiency wedged multi-folded slab amplifier with two planar folding mirrors. In order to improve the filling factor to optimize the mode overlapping, the conditions to keep the full coverage of the active area in this configuration is discussed in detail, and the design equations are presented. With the help of these equations, 101.75 W output was obtained with a 52 W, 152 kHz seed and a 92 W pump, and the corresponding extraction efficiency reached 54.3%. The effects on pulse waveform and beam quality are also reported.

2. Mode overlapping optimization in a wedged multi-folded amplifier

In multi-folded amplifier, the extraction efficiency is greatly affected by the filling factor. This section we discuss about the design of the wedged multi-folded slab amplifier to maximize the filling factor.

The schematic of the wedged multi-folded amplifier in the X-Z plane is shown in Fig. 1. The blue dashed pane (with the width of hc) presents the air block equivalent to the amplifier crystal (with the refractive index of n and the width of h). The two planar folding mirrors M1 and M2 are on the opposite sides of the crystal. There is a little wedge angle β between the folding mirrors to avoid the parasitic oscillation and to suppress the amplified spontaneous emission (ASE). The minimum distance between the two folding mirrors is H.

 

Fig. 1. Schemetic of the wedged multi-folded amplifier in the X-Z plane

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The parallel seed with the beam width of ω is reflected by M1 and M2 alternately and hence passes the crystal several times. The initial incidence angle is α 0, and the number of reflections on M2 is N. In the k th reflection (1≤kN), αk, ϖ k,1 and ϖ k,2 are the incidence angle, the width of reflection area, and the blank between this and the next reflection areas respectively.

Define a parameter dk as

dk=ωk,1+ωk,2ωk,1

And hck is defined to be the maximum width of the air block equivalent to the crystal keeping the active area fully covered after the k th reflection on M2. The calculation using geometric optics shows that

dk2Hω(α0+2kβ)
hck2dkdkHk2dkdkH

To avoid the hard-edge diffraction at the edges of M2, considering the self aperture condition,

dk>1.5

is needed. To keep the active area fully covered which means hcN>0, we get from (2)~(4) that

1.5<dk<2

The length of active area covered by signal beam satisfies

LN2NH[α0+(N+1)β]=NωdN+½

It’s shown in (5), (6) that, when fully covering the same active area, more bounces are needed using narrower seed beam than using wider seed beam, which cause bigger intracavity loss and more restrict limit on the maximum value of β, and the latter is unfavorable to the avoidance of the parasitic oscillation and the suppression of the ASE. So expanding the beam width of the seed properly will benefit the extraction efficiency.

The amplifier parameters designed according to (2)~(6) are listed in Table. 1, where d 0 is defined like dk as

d0=ω0,1+ω0,2ω0,1
Tables Icon

Table 1. Optimized parameters

 

Fig. 2. Schemetic of the wedged multi-folded amplifier in the Y-Z plane

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In the Y-Z plane, a plano-plano cavity with two thermally induced lenses Mh1 and Mh2 inside is formed, as shown in Fig. 2. The seed beam goes through both of the two thermally induced lenses in each pass except for when the seed beam enters or the amplified beam exits from the amplifier. The simulation shows that, at the entrance of the amplifier, the beam waist of the seed beam is required to be a few millimeters away from M2 and with a proper size to ensure that the beam waists always fall on the folding mirrors with the same size in all passes, that is, to match the cavity mode.

3. Experimental setup

 

Fig. 3. Schematic of the experimental setup

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The experimental arrangement is shown in Fig. 3. A 0.3 at.% doped Nd:YVO4 slab crystal of 2mm×9mm×12mm was mounted between two water-cooled copper heat sinks. The two large unpolished faces were used for conduction cooling and the Indium foils were used between the crystal and the heat sinks for better thermal conduction. The two 2mm×12mm end faces were polished and coated with anti-reflective films at both 808nm and 1064nm for the laser going through. The c-axis of the Nd:YVO4 slab was perpendicular with the 9mm×12mm faces.

On the opposite sides of the 2mm×12mm faces, the two folding mirrors M1 and M2 were nearly parallel with a small wedge angle of about 2′ in the X-Z plane, and each of them was about 6mm from the crystal. The outer faces of M1 and M2 were coated the films antireflective at 808nm to let the pump beams going through, and the films coated on the faces of both M1 and M2 facing the crystal were highly reflective at 1064nm to reflect the signal beam and were anti-reflective at 808nm.

The pump sources were two fast-axis collimated 808nm TM-polarized LD bar (DILAS MY series) with the maximum output power of 46 W each, and were placed on the opposite sides of the crystal. The polarized directions of the pump beams were parallel with the c-axis of the Nd:YVO4 crystal. Two sheet shaped active regions were made in the crystal after the pump beams reshaped by cylinder lenses in the slow axis direction and the fast axis direction respectively and going through the folding mirrors. They were Mp11 and Mp12 for the LD1, and Mp21, Mp22 for the LD2. The dimensions of the left active region were 0.5mm in the Y direction and 7mm in the X direction, and those of the right active region were 0.5mm in the Y direction and 10mm in the X direction. The four cylinder lens Mp11, Mp12, Mp21 and Mp22 were coated with anti-reflective film at 808nm.

The seed source was another MOPA system with an acousto-optic Q-switched Nd:YVO4 laser as its seed, and it’s output was linearly polarized with the polarization direction parallel with the c-axis of the Nd:YVO4 crystal. The seed beam was first focused by a lens M4 with the focal length of 200mm and then collimated by a cylinder lens M3 with the focal length of - 100mm in the X-Z plane. Hence the beam width of the seed was expanded to about 1.6mm in the X direction, and the beam waist in the Y direction is about 0.45mm wide and located near M2. Finally it incidented into the crystal with a small incidence angle of about 4.3°and bounced back and forth between the folding mirrors and made a multi-folded geometry.

3. Experimental results and discuss

The amplifier’s performance under different PRF (50 kHz to 152 kHz) was measured with the seed’s average power of 52 W. Figure 4 shows the plots of the output power versus PRF and the pulse width versus PRF with the seed power of 52 W and amplifier’s pump power of 92 W. It’s shown that the output power and pulse width increase with the seed’s PRF. The former we thought was the combined result of the large simulated-emission cross section of the Nd:YVO4 which ensured the stored energy extracted efficiently and the relatively short upper level lifetime of the Nd:YVO4 (110 µs) which caused the saturation of the stored energy. And the variation in pulse width came from the seed, and the pulse widths of the output and the seed were consistent.

 

Fig. 4. The output power and pulse width with a 52 W seed and a 92 W pump

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The amplified beam was measured to be linearly polarized and its polarized direction was parallel with the c-axis of the Nd:YVO4 crystal.

The plots of both the amplifier’s output power and the corresponding extraction efficiency versus pump power with a 52 W, 152 kHz seed are shown in Fig. 5. The maximum output power of 101.75 W was achieved with the pump power of 92 W, and the corresponding extraction efficiency reached 54.3%. To our best knowledge, it’s the highest extraction efficiency in all the 1064nm solid state pulse amplifiers pumped by the 808nm LDs.

 

Fig. 5. The output power and extraction efficiency with a 52 W, 152 kHz seed

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Figure 6 shows the waveform of the pulses before and after the amplification, and no remarkable change can be observed on the pulse shape.

 

Fig. 6. The pulse waveform of (a) the seed; (b) the amplifier output

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The beam quality M2 factors before and after the amplification were measured with different seed powers (SPIRICON M2-200). Tab. 2 presents the variation of the beam quality with the 4 W, 50 kHz seed which was amplified to 32 W with an 80 W pump. And no remarkable deterioration was observed although the beam quality of the seed in the X direction deteriorated at the high power operation. Next we’ll focus on improving the performance of the preamplifier on the beam quality.

Tables Icon

Table 2. The beam quality of seed and amplifier output

4. Conclusion

A efficient multi-folded Nd:YVO4 slab amplifier is demonstrated. To optimizing the mode overlapping, the conditions to maximize the fill factor in the wedged multi-folded slab amplifier is discussed in detail and design equations are presented. With these equations, high extraction efficiency was achieved in our experiments. A 101.75 W output was obtained with a 52 W, 152 kHz seed and the 92 W pump, and the corresponding extraction efficiency reached 54.3%. And no remarkable change of the pulse waveform or deterioration of the beam quality was observed. Further more, there is no sign of saturation in the output power curve, so it can be believed that the scaling of the output while keeping the extraction efficiency as high as we now get requires only the increase of the pump power.

References and links

1. L. McDonagh and R. Wallenstein, “111 W, 110 MHz repetition-rate, passively mode-locked TEM00 Nd: YVO4 master oscillator power amplifier pumped at 888 nm,” Opt. Lett. 32, 1259–1261 (2007). [CrossRef]   [PubMed]  

2. H. Plaessmann, S. A. Rd, J. J. Alonis, D. L. Vecht, and W. M. Grossman, “Multipass diode-pumped solid-state optical amplifier,” Opt. Lett. 18, 1420–1422 (1993). [CrossRef]   [PubMed]  

3. J. Giesekus, T. Mans, K.-M. Du, B. Braun, P. Loosen, and R. Poprawe, “High power diode end pumped slab MOPA system,” in Lasers and Electro-Optics, 2001. CLEO ’01. Technical Digest.419, (2001).

4. B. Luther-Davies, V. Z. Kolev, M. J. Lederer, N. R. Madsen, A. V. Rode, J. Giesekus, K.-M. Du, and M. Duering, “Table-top 50-W laser system for ultra-fast laser ablation,” Appl. Phys. A 79, 1051–1055 (2004). [CrossRef]  

5. J. G. Manni, “Amplification of microchip oscillator emission using a diode-pumped wedged-slab amplifier,” Opt. Commun. 252, 117–126 (2005). [CrossRef]  

6. A. Minassian, B. A. Thompson, G. Smith, and M. J. Damzen, “High-Power Scaling (>100 W) of a Diode-Pumped TEM00 Nd:GdVO4 Laser System,” IEEE J. Sel. Top. Quantum Electron 11, 621–625 (2005). [CrossRef]  

7. D. J. Farrell and M. J. Damzen, “High power scaling of a passively modelocked laser oscillator in a bounce geometry,” Opt. Express 15, 4781–4786 (2007). [CrossRef]   [PubMed]  

8. Y. Ojima, K. Nawata, and T. Omatsu, “Over 10-watt pico-second diffraction-limited output from a Nd:YVO4 slab amplifier with a phase conjugate mirror,” Opt. Express 13, 8993–8998 (2005). [CrossRef]   [PubMed]  

9. K. Nawata, Y. Ojima, M. Okida, T. Ogawa, and T. Omatsu, “Power scaling of a pico-second Nd:YVO4 master-oscillator power amplifier with a phase-conjugate mirror,” Opt. Express 14, 10657–10662 (2006). [CrossRef]   [PubMed]  

10. T. Omatsu, K. Nawata, M. Okida, and K. Furuki, “MW ps pulse generation at sub-MHz repetition rates from a phase conjugate Nd:YVO4 bounce amplifier,” Opt. Express 15, 9123–9128 (2007). [CrossRef]   [PubMed]  

References

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  1. L. McDonagh and R. Wallenstein, “111 W, 110 MHz repetition-rate, passively mode-locked TEM00 Nd: YVO4 master oscillator power amplifier pumped at 888 nm,” Opt. Lett. 32, 1259–1261 (2007).
    [Crossref] [PubMed]
  2. H. Plaessmann, S. A. Rd, J. J. Alonis, D. L. Vecht, and W. M. Grossman, “Multipass diode-pumped solid-state optical amplifier,” Opt. Lett. 18, 1420–1422 (1993).
    [Crossref] [PubMed]
  3. J. Giesekus, T. Mans, K.-M. Du, B. Braun, P. Loosen, and R. Poprawe, “High power diode end pumped slab MOPA system,” in Lasers and Electro-Optics, 2001. CLEO ’01. Technical Digest.419, (2001).
  4. B. Luther-Davies, V. Z. Kolev, M. J. Lederer, N. R. Madsen, A. V. Rode, J. Giesekus, K.-M. Du, and M. Duering, “Table-top 50-W laser system for ultra-fast laser ablation,” Appl. Phys. A 79, 1051–1055 (2004).
    [Crossref]
  5. J. G. Manni, “Amplification of microchip oscillator emission using a diode-pumped wedged-slab amplifier,” Opt. Commun. 252, 117–126 (2005).
    [Crossref]
  6. A. Minassian, B. A. Thompson, G. Smith, and M. J. Damzen, “High-Power Scaling (>100 W) of a Diode-Pumped TEM00 Nd:GdVO4 Laser System,” IEEE J. Sel. Top. Quantum Electron 11, 621–625 (2005).
    [Crossref]
  7. D. J. Farrell and M. J. Damzen, “High power scaling of a passively modelocked laser oscillator in a bounce geometry,” Opt. Express 15, 4781–4786 (2007).
    [Crossref] [PubMed]
  8. Y. Ojima, K. Nawata, and T. Omatsu, “Over 10-watt pico-second diffraction-limited output from a Nd:YVO4 slab amplifier with a phase conjugate mirror,” Opt. Express 13, 8993–8998 (2005).
    [Crossref] [PubMed]
  9. K. Nawata, Y. Ojima, M. Okida, T. Ogawa, and T. Omatsu, “Power scaling of a pico-second Nd:YVO4 master-oscillator power amplifier with a phase-conjugate mirror,” Opt. Express 14, 10657–10662 (2006).
    [Crossref] [PubMed]
  10. T. Omatsu, K. Nawata, M. Okida, and K. Furuki, “MW ps pulse generation at sub-MHz repetition rates from a phase conjugate Nd:YVO4 bounce amplifier,” Opt. Express 15, 9123–9128 (2007).
    [Crossref] [PubMed]

2007 (3)

2006 (1)

2005 (3)

Y. Ojima, K. Nawata, and T. Omatsu, “Over 10-watt pico-second diffraction-limited output from a Nd:YVO4 slab amplifier with a phase conjugate mirror,” Opt. Express 13, 8993–8998 (2005).
[Crossref] [PubMed]

J. G. Manni, “Amplification of microchip oscillator emission using a diode-pumped wedged-slab amplifier,” Opt. Commun. 252, 117–126 (2005).
[Crossref]

A. Minassian, B. A. Thompson, G. Smith, and M. J. Damzen, “High-Power Scaling (>100 W) of a Diode-Pumped TEM00 Nd:GdVO4 Laser System,” IEEE J. Sel. Top. Quantum Electron 11, 621–625 (2005).
[Crossref]

2004 (1)

B. Luther-Davies, V. Z. Kolev, M. J. Lederer, N. R. Madsen, A. V. Rode, J. Giesekus, K.-M. Du, and M. Duering, “Table-top 50-W laser system for ultra-fast laser ablation,” Appl. Phys. A 79, 1051–1055 (2004).
[Crossref]

1993 (1)

Alonis, J. J.

Braun, B.

J. Giesekus, T. Mans, K.-M. Du, B. Braun, P. Loosen, and R. Poprawe, “High power diode end pumped slab MOPA system,” in Lasers and Electro-Optics, 2001. CLEO ’01. Technical Digest.419, (2001).

Damzen, M. J.

D. J. Farrell and M. J. Damzen, “High power scaling of a passively modelocked laser oscillator in a bounce geometry,” Opt. Express 15, 4781–4786 (2007).
[Crossref] [PubMed]

A. Minassian, B. A. Thompson, G. Smith, and M. J. Damzen, “High-Power Scaling (>100 W) of a Diode-Pumped TEM00 Nd:GdVO4 Laser System,” IEEE J. Sel. Top. Quantum Electron 11, 621–625 (2005).
[Crossref]

Du, K.-M.

B. Luther-Davies, V. Z. Kolev, M. J. Lederer, N. R. Madsen, A. V. Rode, J. Giesekus, K.-M. Du, and M. Duering, “Table-top 50-W laser system for ultra-fast laser ablation,” Appl. Phys. A 79, 1051–1055 (2004).
[Crossref]

J. Giesekus, T. Mans, K.-M. Du, B. Braun, P. Loosen, and R. Poprawe, “High power diode end pumped slab MOPA system,” in Lasers and Electro-Optics, 2001. CLEO ’01. Technical Digest.419, (2001).

Duering, M.

B. Luther-Davies, V. Z. Kolev, M. J. Lederer, N. R. Madsen, A. V. Rode, J. Giesekus, K.-M. Du, and M. Duering, “Table-top 50-W laser system for ultra-fast laser ablation,” Appl. Phys. A 79, 1051–1055 (2004).
[Crossref]

Farrell, D. J.

Furuki, K.

Giesekus, J.

B. Luther-Davies, V. Z. Kolev, M. J. Lederer, N. R. Madsen, A. V. Rode, J. Giesekus, K.-M. Du, and M. Duering, “Table-top 50-W laser system for ultra-fast laser ablation,” Appl. Phys. A 79, 1051–1055 (2004).
[Crossref]

J. Giesekus, T. Mans, K.-M. Du, B. Braun, P. Loosen, and R. Poprawe, “High power diode end pumped slab MOPA system,” in Lasers and Electro-Optics, 2001. CLEO ’01. Technical Digest.419, (2001).

Grossman, W. M.

Kolev, V. Z.

B. Luther-Davies, V. Z. Kolev, M. J. Lederer, N. R. Madsen, A. V. Rode, J. Giesekus, K.-M. Du, and M. Duering, “Table-top 50-W laser system for ultra-fast laser ablation,” Appl. Phys. A 79, 1051–1055 (2004).
[Crossref]

Lederer, M. J.

B. Luther-Davies, V. Z. Kolev, M. J. Lederer, N. R. Madsen, A. V. Rode, J. Giesekus, K.-M. Du, and M. Duering, “Table-top 50-W laser system for ultra-fast laser ablation,” Appl. Phys. A 79, 1051–1055 (2004).
[Crossref]

Loosen, P.

J. Giesekus, T. Mans, K.-M. Du, B. Braun, P. Loosen, and R. Poprawe, “High power diode end pumped slab MOPA system,” in Lasers and Electro-Optics, 2001. CLEO ’01. Technical Digest.419, (2001).

Luther-Davies, B.

B. Luther-Davies, V. Z. Kolev, M. J. Lederer, N. R. Madsen, A. V. Rode, J. Giesekus, K.-M. Du, and M. Duering, “Table-top 50-W laser system for ultra-fast laser ablation,” Appl. Phys. A 79, 1051–1055 (2004).
[Crossref]

Madsen, N. R.

B. Luther-Davies, V. Z. Kolev, M. J. Lederer, N. R. Madsen, A. V. Rode, J. Giesekus, K.-M. Du, and M. Duering, “Table-top 50-W laser system for ultra-fast laser ablation,” Appl. Phys. A 79, 1051–1055 (2004).
[Crossref]

Manni, J. G.

J. G. Manni, “Amplification of microchip oscillator emission using a diode-pumped wedged-slab amplifier,” Opt. Commun. 252, 117–126 (2005).
[Crossref]

Mans, T.

J. Giesekus, T. Mans, K.-M. Du, B. Braun, P. Loosen, and R. Poprawe, “High power diode end pumped slab MOPA system,” in Lasers and Electro-Optics, 2001. CLEO ’01. Technical Digest.419, (2001).

McDonagh, L.

Minassian, A.

A. Minassian, B. A. Thompson, G. Smith, and M. J. Damzen, “High-Power Scaling (>100 W) of a Diode-Pumped TEM00 Nd:GdVO4 Laser System,” IEEE J. Sel. Top. Quantum Electron 11, 621–625 (2005).
[Crossref]

Nawata, K.

Ogawa, T.

Ojima, Y.

Okida, M.

Omatsu, T.

Plaessmann, H.

Poprawe, R.

J. Giesekus, T. Mans, K.-M. Du, B. Braun, P. Loosen, and R. Poprawe, “High power diode end pumped slab MOPA system,” in Lasers and Electro-Optics, 2001. CLEO ’01. Technical Digest.419, (2001).

Rd, S. A.

Rode, A. V.

B. Luther-Davies, V. Z. Kolev, M. J. Lederer, N. R. Madsen, A. V. Rode, J. Giesekus, K.-M. Du, and M. Duering, “Table-top 50-W laser system for ultra-fast laser ablation,” Appl. Phys. A 79, 1051–1055 (2004).
[Crossref]

Smith, G.

A. Minassian, B. A. Thompson, G. Smith, and M. J. Damzen, “High-Power Scaling (>100 W) of a Diode-Pumped TEM00 Nd:GdVO4 Laser System,” IEEE J. Sel. Top. Quantum Electron 11, 621–625 (2005).
[Crossref]

Thompson, B. A.

A. Minassian, B. A. Thompson, G. Smith, and M. J. Damzen, “High-Power Scaling (>100 W) of a Diode-Pumped TEM00 Nd:GdVO4 Laser System,” IEEE J. Sel. Top. Quantum Electron 11, 621–625 (2005).
[Crossref]

Vecht, D. L.

Wallenstein, R.

Appl. Phys. A (1)

B. Luther-Davies, V. Z. Kolev, M. J. Lederer, N. R. Madsen, A. V. Rode, J. Giesekus, K.-M. Du, and M. Duering, “Table-top 50-W laser system for ultra-fast laser ablation,” Appl. Phys. A 79, 1051–1055 (2004).
[Crossref]

IEEE J. Sel. Top. Quantum Electron (1)

A. Minassian, B. A. Thompson, G. Smith, and M. J. Damzen, “High-Power Scaling (>100 W) of a Diode-Pumped TEM00 Nd:GdVO4 Laser System,” IEEE J. Sel. Top. Quantum Electron 11, 621–625 (2005).
[Crossref]

Opt. Commun. (1)

J. G. Manni, “Amplification of microchip oscillator emission using a diode-pumped wedged-slab amplifier,” Opt. Commun. 252, 117–126 (2005).
[Crossref]

Opt. Express (4)

Opt. Lett. (2)

Other (1)

J. Giesekus, T. Mans, K.-M. Du, B. Braun, P. Loosen, and R. Poprawe, “High power diode end pumped slab MOPA system,” in Lasers and Electro-Optics, 2001. CLEO ’01. Technical Digest.419, (2001).

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Figures (6)

Fig. 1.
Fig. 1. Schemetic of the wedged multi-folded amplifier in the X-Z plane
Fig. 2.
Fig. 2. Schemetic of the wedged multi-folded amplifier in the Y-Z plane
Fig. 3.
Fig. 3. Schematic of the experimental setup
Fig. 4.
Fig. 4. The output power and pulse width with a 52 W seed and a 92 W pump
Fig. 5.
Fig. 5. The output power and extraction efficiency with a 52 W, 152 kHz seed
Fig. 6.
Fig. 6. The pulse waveform of (a) the seed; (b) the amplifier output

Tables (2)

Tables Icon

Table 1. Optimized parameters

Tables Icon

Table 2. The beam quality of seed and amplifier output

Equations (7)

Equations on this page are rendered with MathJax. Learn more.

d k = ω k , 1 + ω k , 2 ω k , 1
d k 2 H ω ( α 0 + 2 k β )
h c k 2 d k d k H k 2 d k d k H
d k > 1.5
1.5 < d k < 2
L N 2 N H [ α 0 + ( N + 1 ) β ] = N ω d N + ½
d 0 = ω 0 , 1 + ω 0 , 2 ω 0 , 1

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