ASE and parasitic lasing in thin disk laser with anti-ASE cap

Hiroaki Furuse; Haik Chosrowjan; Junji Kawanaka; Noriaki Miyanaga; Masayuki Fujita; Yasukazu Izawa

doi:10.1364/OE.21.013118

ASE and parasitic lasing in thin disk laser with anti-ASE cap

Hiroaki Furuse, Haik Chosrowjan, Junji Kawanaka, Noriaki Miyanaga, Masayuki Fujita, and Yasukazu Izawa

Optics Express
Vol. 21,
Issue 11,
pp. 13118-13124
(2013)
•https://doi.org/10.1364/OE.21.013118

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Hiroaki Furuse, Haik Chosrowjan, Junji Kawanaka, Noriaki Miyanaga, Masayuki Fujita, and Yasukazu Izawa, "ASE and parasitic lasing in thin disk laser with anti-ASE cap," Opt. Express 21, 13118-13124 (2013)

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Abstract

The amplified spontaneous emission (ASE) and parasitic lasing (PL) effects in thin disk laser with an anti-ASE cap have been investigated in detail by measuring both time-resolved radiated intensity at longer axis of elliptical pump profile (dominant ASE direction) and small signal gain (SSG) in laser amplifier. A cryogenically-cooled total-reflection active-mirror laser consisting of 9.8 at.% doped, 0.6-mm thick Yb:YAG and un-doped YAG trapezoidal ceramics cap was used as a sample. The phased transitions from spontaneous emission (SE) to ASE and from ASE to PL have been unambiguously observed. For several pump beam diameters, the ASE gain parameter g₀l_ASE at ASE threshold was about 3, and the SSG coefficient was down to about 65% until PL started. To the best of our knowledge, this is the first quantitative characterization of the ASE/PL effects in the thin disk laser with an anti-ASE cap.

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A. Giesen, H. Hügel, A. Voss, K. Wittig, U. Brauch, and H. Opower, “Scalable concept for diode-pumped high-power solid-state-lasers,” Appl. Phys. B 58(5), 365–372 (1994).
[Crossref]
A. Giesen and J. Speiser, “Fifteen years of work on thin-disk lasers: results and scaling laws,” IEEE J. Sel. Top. Quantum Electron. 13(3), 598–609 (2007).
[Crossref]
D. Harvilla and R. Brockmann, “Latest advances in high power disk lasers,” Proc. SPIE 7578, 75780C (2010).
[Crossref]
J. Speiser, “Scaling of thin-disk lasers – influence of amplified spontaneous emission,” J. Opt. Soc. Am. B 26(1), 26–35 (2009).
[Crossref]
G. I. Peters and L. Allen, “Amplified Spontaneous Emission 0.1. Threshold Condition,” J. Phys. A 4(2), 238–243 (1971).
[Crossref]
L. Allen and G. I. Peters, “Amplified spontaneous emission and external signal amplification in an inverted medium,” Phys. Rev. A 8(4), 2031–2047 (1973).
[Crossref]
R. J. Beach, E. C. Honea, S. A. Payne, H. Powell, W. F. Krupke, and S. B. Sutton, “High average power scalable thin-disk laser,” U.S. patent 6,347,109 (12 February 2002).
T. J. Carrig and A. K. Hankla, “Lasers and amplifiers based on hybrid slab active mirrors,” U.S. patent 6,658,036 (2 December 2003).
D. Kouznetsov and J. F. Bisson, “Role of undoped cap in scaling of thin-disk lasers,” J. Opt. Soc. Am. B 25(3), 338–345 (2008).
[Crossref]
D. Albach, J. C. Chanteloup, and G. Touzé, “Influence of ASE on the gain distribution in large size, high gain Yb3+:YAG slabs,” Opt. Express 17(5), 3792–3801 (2009).
[Crossref] [PubMed]
S. Banerjee, K. Ertel, P. D. Mason, P. J. Phillips, M. Siebold, M. Loeser, C. Hernandez-Gomez, and J. L. Collier, “High-efficiency 10 J diode pumped cryogenic gas cooled Yb:YAG multislab amplifier,” Opt. Lett. 37(12), 2175–2177 (2012).
[Crossref] [PubMed]
D. C. Brown, High-Peak-Power Nd:Glass Laser Systems (Springer-Verlag, New York, 1981).
D. Kouznetsov, J. F. Bisson, and K. Ueda, “Scaling laws of disk lasers,” Opt. Mater. 31(5), 754–759 (2009).
[Crossref]
P. Peterson, A. Gavrielides, T. C. Newell, N. Vretenar, and W. P. Latham, “ASE in thin disk lasers: theory and experiment,” Opt. Express 19(25), 25672–25684 (2011).
[Crossref] [PubMed]
H. Furuse, J. Kawanaka, K. Takeshita, N. Miyanaga, T. Saiki, K. Imasaki, M. Fujita, and S. Ishii, “Total-reflection active-mirror laser with cryogenic Yb:YAG ceramics,” Opt. Lett. 34(21), 3439–3441 (2009).
[Crossref] [PubMed]
J. Kawanaka, Y. Takeuchi, A. Yoshida, S. J. Pearce, R. Yasuhara, T. Kawashima, and H. Kan, “Highly efficient cryogenically-cooled Yb:YAG laser,” Laser Phys. 20(5), 1079–1084 (2010).
[Crossref]
H. Furuse, J. Kawanaka, N. Miyanaga, H. Chosrowjan, M. Fujita, K. Takeshita, and Y. Izawa, “Output characteristics of high power cryogenic Yb:YAG TRAM laser oscillator,” Opt. Express 20(19), 21739–21748 (2012).
[Crossref] [PubMed]
W. Koechner, Solid State Laser Engineering (Springer, 2006).
A. Antognini, K. Schuhmann, F. D. Amaro, F. Biraben, A. Dax, A. Giesen, T. Graf, T. W. Hänsch, P. Indelicato, L. Julien, C. Y. Kao, P. E. Knowles, F. Kottmann, E. Bigot, Y. W. Liu, L. Ludhova, N. Moschüring, F. Mulhauser, T. Nebel, F. Nez, P. Rabinowitz, C. Schwob, D. Taqqu, and R. Pohl, “Thin-disk Yb:YAG oscillator-amplifier laser, ASE, and effective Yb:YAG lifetime,” J. Quantum Electron. 45(8), 993–1005 (2009).
[Crossref]

2012 (2)

S. Banerjee, K. Ertel, P. D. Mason, P. J. Phillips, M. Siebold, M. Loeser, C. Hernandez-Gomez, and J. L. Collier, “High-efficiency 10 J diode pumped cryogenic gas cooled Yb:YAG multislab amplifier,” Opt. Lett. 37(12), 2175–2177 (2012).
[Crossref] [PubMed]

H. Furuse, J. Kawanaka, N. Miyanaga, H. Chosrowjan, M. Fujita, K. Takeshita, and Y. Izawa, “Output characteristics of high power cryogenic Yb:YAG TRAM laser oscillator,” Opt. Express 20(19), 21739–21748 (2012).
[Crossref] [PubMed]

2011 (1)

P. Peterson, A. Gavrielides, T. C. Newell, N. Vretenar, and W. P. Latham, “ASE in thin disk lasers: theory and experiment,” Opt. Express 19(25), 25672–25684 (2011).
[Crossref] [PubMed]

2010 (2)

J. Kawanaka, Y. Takeuchi, A. Yoshida, S. J. Pearce, R. Yasuhara, T. Kawashima, and H. Kan, “Highly efficient cryogenically-cooled Yb:YAG laser,” Laser Phys. 20(5), 1079–1084 (2010).
[Crossref]

D. Harvilla and R. Brockmann, “Latest advances in high power disk lasers,” Proc. SPIE 7578, 75780C (2010).
[Crossref]

2009 (5)

J. Speiser, “Scaling of thin-disk lasers – influence of amplified spontaneous emission,” J. Opt. Soc. Am. B 26(1), 26–35 (2009).
[Crossref]

D. Kouznetsov, J. F. Bisson, and K. Ueda, “Scaling laws of disk lasers,” Opt. Mater. 31(5), 754–759 (2009).
[Crossref]

D. Albach, J. C. Chanteloup, and G. Touzé, “Influence of ASE on the gain distribution in large size, high gain Yb3+:YAG slabs,” Opt. Express 17(5), 3792–3801 (2009).
[Crossref] [PubMed]

H. Furuse, J. Kawanaka, K. Takeshita, N. Miyanaga, T. Saiki, K. Imasaki, M. Fujita, and S. Ishii, “Total-reflection active-mirror laser with cryogenic Yb:YAG ceramics,” Opt. Lett. 34(21), 3439–3441 (2009).
[Crossref] [PubMed]

A. Antognini, K. Schuhmann, F. D. Amaro, F. Biraben, A. Dax, A. Giesen, T. Graf, T. W. Hänsch, P. Indelicato, L. Julien, C. Y. Kao, P. E. Knowles, F. Kottmann, E. Bigot, Y. W. Liu, L. Ludhova, N. Moschüring, F. Mulhauser, T. Nebel, F. Nez, P. Rabinowitz, C. Schwob, D. Taqqu, and R. Pohl, “Thin-disk Yb:YAG oscillator-amplifier laser, ASE, and effective Yb:YAG lifetime,” J. Quantum Electron. 45(8), 993–1005 (2009).
[Crossref]

2008 (1)

D. Kouznetsov and J. F. Bisson, “Role of undoped cap in scaling of thin-disk lasers,” J. Opt. Soc. Am. B 25(3), 338–345 (2008).
[Crossref]

2007 (1)

A. Giesen and J. Speiser, “Fifteen years of work on thin-disk lasers: results and scaling laws,” IEEE J. Sel. Top. Quantum Electron. 13(3), 598–609 (2007).
[Crossref]

1994 (1)

A. Giesen, H. Hügel, A. Voss, K. Wittig, U. Brauch, and H. Opower, “Scalable concept for diode-pumped high-power solid-state-lasers,” Appl. Phys. B 58(5), 365–372 (1994).
[Crossref]

1973 (1)

L. Allen and G. I. Peters, “Amplified spontaneous emission and external signal amplification in an inverted medium,” Phys. Rev. A 8(4), 2031–2047 (1973).
[Crossref]

1971 (1)

G. I. Peters and L. Allen, “Amplified Spontaneous Emission 0.1. Threshold Condition,” J. Phys. A 4(2), 238–243 (1971).
[Crossref]

Albach, D.

D. Albach, J. C. Chanteloup, and G. Touzé, “Influence of ASE on the gain distribution in large size, high gain Yb3+:YAG slabs,” Opt. Express 17(5), 3792–3801 (2009).
[Crossref] [PubMed]

Allen, L.

L. Allen and G. I. Peters, “Amplified spontaneous emission and external signal amplification in an inverted medium,” Phys. Rev. A 8(4), 2031–2047 (1973).
[Crossref]

G. I. Peters and L. Allen, “Amplified Spontaneous Emission 0.1. Threshold Condition,” J. Phys. A 4(2), 238–243 (1971).
[Crossref]

Amaro, F. D.

Antognini, A.

Banerjee, S.

Bigot, E.

Biraben, F.

Bisson, J. F.

D. Kouznetsov, J. F. Bisson, and K. Ueda, “Scaling laws of disk lasers,” Opt. Mater. 31(5), 754–759 (2009).
[Crossref]

D. Kouznetsov and J. F. Bisson, “Role of undoped cap in scaling of thin-disk lasers,” J. Opt. Soc. Am. B 25(3), 338–345 (2008).
[Crossref]

Brauch, U.

A. Giesen, H. Hügel, A. Voss, K. Wittig, U. Brauch, and H. Opower, “Scalable concept for diode-pumped high-power solid-state-lasers,” Appl. Phys. B 58(5), 365–372 (1994).
[Crossref]

Brockmann, R.

D. Harvilla and R. Brockmann, “Latest advances in high power disk lasers,” Proc. SPIE 7578, 75780C (2010).
[Crossref]

Chanteloup, J. C.

D. Albach, J. C. Chanteloup, and G. Touzé, “Influence of ASE on the gain distribution in large size, high gain Yb3+:YAG slabs,” Opt. Express 17(5), 3792–3801 (2009).
[Crossref] [PubMed]

Chosrowjan, H.

Collier, J. L.

Dax, A.

Ertel, K.

Fujita, M.

Furuse, H.

Gavrielides, A.

P. Peterson, A. Gavrielides, T. C. Newell, N. Vretenar, and W. P. Latham, “ASE in thin disk lasers: theory and experiment,” Opt. Express 19(25), 25672–25684 (2011).
[Crossref] [PubMed]

Giesen, A.

A. Giesen and J. Speiser, “Fifteen years of work on thin-disk lasers: results and scaling laws,” IEEE J. Sel. Top. Quantum Electron. 13(3), 598–609 (2007).
[Crossref]

A. Giesen, H. Hügel, A. Voss, K. Wittig, U. Brauch, and H. Opower, “Scalable concept for diode-pumped high-power solid-state-lasers,” Appl. Phys. B 58(5), 365–372 (1994).
[Crossref]

Graf, T.

Hänsch, T. W.

Harvilla, D.

D. Harvilla and R. Brockmann, “Latest advances in high power disk lasers,” Proc. SPIE 7578, 75780C (2010).
[Crossref]

Hernandez-Gomez, C.

Hügel, H.

A. Giesen, H. Hügel, A. Voss, K. Wittig, U. Brauch, and H. Opower, “Scalable concept for diode-pumped high-power solid-state-lasers,” Appl. Phys. B 58(5), 365–372 (1994).
[Crossref]

Imasaki, K.

Indelicato, P.

Ishii, S.

Izawa, Y.

Julien, L.

Kan, H.

J. Kawanaka, Y. Takeuchi, A. Yoshida, S. J. Pearce, R. Yasuhara, T. Kawashima, and H. Kan, “Highly efficient cryogenically-cooled Yb:YAG laser,” Laser Phys. 20(5), 1079–1084 (2010).
[Crossref]

Kao, C. Y.

Kawanaka, J.

J. Kawanaka, Y. Takeuchi, A. Yoshida, S. J. Pearce, R. Yasuhara, T. Kawashima, and H. Kan, “Highly efficient cryogenically-cooled Yb:YAG laser,” Laser Phys. 20(5), 1079–1084 (2010).
[Crossref]

Kawashima, T.

J. Kawanaka, Y. Takeuchi, A. Yoshida, S. J. Pearce, R. Yasuhara, T. Kawashima, and H. Kan, “Highly efficient cryogenically-cooled Yb:YAG laser,” Laser Phys. 20(5), 1079–1084 (2010).
[Crossref]

Knowles, P. E.

Kottmann, F.

Kouznetsov, D.

D. Kouznetsov, J. F. Bisson, and K. Ueda, “Scaling laws of disk lasers,” Opt. Mater. 31(5), 754–759 (2009).
[Crossref]

D. Kouznetsov and J. F. Bisson, “Role of undoped cap in scaling of thin-disk lasers,” J. Opt. Soc. Am. B 25(3), 338–345 (2008).
[Crossref]

Latham, W. P.

P. Peterson, A. Gavrielides, T. C. Newell, N. Vretenar, and W. P. Latham, “ASE in thin disk lasers: theory and experiment,” Opt. Express 19(25), 25672–25684 (2011).
[Crossref] [PubMed]

Liu, Y. W.

Loeser, M.

Ludhova, L.

Mason, P. D.

Miyanaga, N.

Moschüring, N.

Mulhauser, F.

Nebel, T.

Newell, T. C.

P. Peterson, A. Gavrielides, T. C. Newell, N. Vretenar, and W. P. Latham, “ASE in thin disk lasers: theory and experiment,” Opt. Express 19(25), 25672–25684 (2011).
[Crossref] [PubMed]

Nez, F.

Opower, H.

A. Giesen, H. Hügel, A. Voss, K. Wittig, U. Brauch, and H. Opower, “Scalable concept for diode-pumped high-power solid-state-lasers,” Appl. Phys. B 58(5), 365–372 (1994).
[Crossref]

Pearce, S. J.

J. Kawanaka, Y. Takeuchi, A. Yoshida, S. J. Pearce, R. Yasuhara, T. Kawashima, and H. Kan, “Highly efficient cryogenically-cooled Yb:YAG laser,” Laser Phys. 20(5), 1079–1084 (2010).
[Crossref]

Peters, G. I.

L. Allen and G. I. Peters, “Amplified spontaneous emission and external signal amplification in an inverted medium,” Phys. Rev. A 8(4), 2031–2047 (1973).
[Crossref]

G. I. Peters and L. Allen, “Amplified Spontaneous Emission 0.1. Threshold Condition,” J. Phys. A 4(2), 238–243 (1971).
[Crossref]

Peterson, P.

P. Peterson, A. Gavrielides, T. C. Newell, N. Vretenar, and W. P. Latham, “ASE in thin disk lasers: theory and experiment,” Opt. Express 19(25), 25672–25684 (2011).
[Crossref] [PubMed]

Phillips, P. J.

Pohl, R.

Rabinowitz, P.

Saiki, T.

Schuhmann, K.

Schwob, C.

Siebold, M.

Speiser, J.

J. Speiser, “Scaling of thin-disk lasers – influence of amplified spontaneous emission,” J. Opt. Soc. Am. B 26(1), 26–35 (2009).
[Crossref]

A. Giesen and J. Speiser, “Fifteen years of work on thin-disk lasers: results and scaling laws,” IEEE J. Sel. Top. Quantum Electron. 13(3), 598–609 (2007).
[Crossref]

Takeshita, K.

Takeuchi, Y.

J. Kawanaka, Y. Takeuchi, A. Yoshida, S. J. Pearce, R. Yasuhara, T. Kawashima, and H. Kan, “Highly efficient cryogenically-cooled Yb:YAG laser,” Laser Phys. 20(5), 1079–1084 (2010).
[Crossref]

Taqqu, D.

Touzé, G.

D. Albach, J. C. Chanteloup, and G. Touzé, “Influence of ASE on the gain distribution in large size, high gain Yb3+:YAG slabs,” Opt. Express 17(5), 3792–3801 (2009).
[Crossref] [PubMed]

Ueda, K.

D. Kouznetsov, J. F. Bisson, and K. Ueda, “Scaling laws of disk lasers,” Opt. Mater. 31(5), 754–759 (2009).
[Crossref]

Voss, A.

A. Giesen, H. Hügel, A. Voss, K. Wittig, U. Brauch, and H. Opower, “Scalable concept for diode-pumped high-power solid-state-lasers,” Appl. Phys. B 58(5), 365–372 (1994).
[Crossref]

Vretenar, N.

P. Peterson, A. Gavrielides, T. C. Newell, N. Vretenar, and W. P. Latham, “ASE in thin disk lasers: theory and experiment,” Opt. Express 19(25), 25672–25684 (2011).
[Crossref] [PubMed]

Wittig, K.

A. Giesen, H. Hügel, A. Voss, K. Wittig, U. Brauch, and H. Opower, “Scalable concept for diode-pumped high-power solid-state-lasers,” Appl. Phys. B 58(5), 365–372 (1994).
[Crossref]

Yasuhara, R.

J. Kawanaka, Y. Takeuchi, A. Yoshida, S. J. Pearce, R. Yasuhara, T. Kawashima, and H. Kan, “Highly efficient cryogenically-cooled Yb:YAG laser,” Laser Phys. 20(5), 1079–1084 (2010).
[Crossref]

Yoshida, A.

J. Kawanaka, Y. Takeuchi, A. Yoshida, S. J. Pearce, R. Yasuhara, T. Kawashima, and H. Kan, “Highly efficient cryogenically-cooled Yb:YAG laser,” Laser Phys. 20(5), 1079–1084 (2010).
[Crossref]

Appl. Phys. B (1)

A. Giesen, H. Hügel, A. Voss, K. Wittig, U. Brauch, and H. Opower, “Scalable concept for diode-pumped high-power solid-state-lasers,” Appl. Phys. B 58(5), 365–372 (1994).
[Crossref]

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

A. Giesen and J. Speiser, “Fifteen years of work on thin-disk lasers: results and scaling laws,” IEEE J. Sel. Top. Quantum Electron. 13(3), 598–609 (2007).
[Crossref]

J. Opt. Soc. Am. B (2)

D. Kouznetsov and J. F. Bisson, “Role of undoped cap in scaling of thin-disk lasers,” J. Opt. Soc. Am. B 25(3), 338–345 (2008).
[Crossref]

J. Speiser, “Scaling of thin-disk lasers – influence of amplified spontaneous emission,” J. Opt. Soc. Am. B 26(1), 26–35 (2009).
[Crossref]

J. Phys. A (1)

G. I. Peters and L. Allen, “Amplified Spontaneous Emission 0.1. Threshold Condition,” J. Phys. A 4(2), 238–243 (1971).
[Crossref]

J. Quantum Electron. (1)

Laser Phys. (1)

J. Kawanaka, Y. Takeuchi, A. Yoshida, S. J. Pearce, R. Yasuhara, T. Kawashima, and H. Kan, “Highly efficient cryogenically-cooled Yb:YAG laser,” Laser Phys. 20(5), 1079–1084 (2010).
[Crossref]

Opt. Express (3)

P. Peterson, A. Gavrielides, T. C. Newell, N. Vretenar, and W. P. Latham, “ASE in thin disk lasers: theory and experiment,” Opt. Express 19(25), 25672–25684 (2011).
[Crossref] [PubMed]

D. Albach, J. C. Chanteloup, and G. Touzé, “Influence of ASE on the gain distribution in large size, high gain Yb3+:YAG slabs,” Opt. Express 17(5), 3792–3801 (2009).
[Crossref] [PubMed]

Opt. Lett. (2)

Opt. Mater. (1)

D. Kouznetsov, J. F. Bisson, and K. Ueda, “Scaling laws of disk lasers,” Opt. Mater. 31(5), 754–759 (2009).
[Crossref]

Phys. Rev. A (1)

L. Allen and G. I. Peters, “Amplified spontaneous emission and external signal amplification in an inverted medium,” Phys. Rev. A 8(4), 2031–2047 (1973).
[Crossref]

Proc. SPIE (1)

D. Harvilla and R. Brockmann, “Latest advances in high power disk lasers,” Proc. SPIE 7578, 75780C (2010).
[Crossref]

Other (4)

W. Koechner, Solid State Laser Engineering (Springer, 2006).

R. J. Beach, E. C. Honea, S. A. Payne, H. Powell, W. F. Krupke, and S. B. Sutton, “High average power scalable thin-disk laser,” U.S. patent 6,347,109 (12 February 2002).

T. J. Carrig and A. K. Hankla, “Lasers and amplifiers based on hybrid slab active mirrors,” U.S. patent 6,658,036 (2 December 2003).

D. C. Brown, High-Peak-Power Nd:Glass Laser Systems (Springer-Verlag, New York, 1981).

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Fig. 1

Schematic diagram of the measured TRAM sample.

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Fig. 2

Experimental setup for time-resolved ASE and small signal gain measurements.

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Fig. 3

Experimentally obtained time-dependent profiles of ASE signal intensity and small signal gain. The inset shows the results within 0 - 10 msec time interval.

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Fig. 4

(a) Peak ASE signal intensity in logarithmic scale as a function of the pump power. The dotted lines represent linear and exponential fitting curves. (b) Peak value of small signal gain in logarithmic scale as a function of the pump power and ASE gain parameter g₀l_ASE (upper scale). The dot-dashed line shows the numerical results, dashed and dotted lines show the exponential fitting curves.

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Equations (8)

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d I = \frac{E_{st}}{τ_{f}} \frac{Δ Ω}{4 π} d x .

g_{0} = \frac{E_{st}}{I_{sat}},

\begin{array}{l} I_{ASE} = \frac{E_{st}}{τ_{f}} \frac{Δ Ω}{4 π} \int_{0}^{l_{ASE}} \exp [g_{0} (l_{ASE} - x)] d x \\ = \frac{I_{sat}}{τ_{f}} \frac{Δ Ω}{4 π} [\exp (g_{0} l_{ASE}) - 1] . \end{array}

I_{ASE} \sim \frac{E_{st}}{τ_{f}} \frac{Δ Ω}{4 π} l_{ASE},

I_{ASE} \sim \frac{I_{sat}}{τ_{f}} \frac{Δ Ω}{4 π} \exp (g_{0} l_{ASE}) .

g_{0} (P) = \frac{σ_{emi} τ_{f} η_{t} η_{Q} η_{S} η_{a} η_{B}}{h ν} \frac{P}{V} (1 - \exp (- \frac{t}{τ_{f}})),

\begin{array}{l} G_{ASE} (P) = \exp [g_{0} (165 W) l + η_{ASE} g_{0} (P - 165 W) l] \\ = \exp [g_{ASE} (P) l], \end{array}

\begin{array}{l} G_{PL} (P) = \exp [g_{ASE} (300 W) l + η_{PL} g_{0} (P - 300 W) l] \\ = \exp [g_{PL} (P) l] . \end{array}