Abstract

For future generations of gravitational wave detectors, it is proposed to use the helical Laguerre–Gaussian ${{\rm{LG}}_{3,3}}$ mode to reduce thermal noise, which limits the detector sensitivity. At the same time, this requires the efficient generation of squeezed vacuum states in the ${{\rm{LG}}_{3,3}}$ mode for quantum noise reduction. Since this technique includes the process of second harmonic generation (SHG), we experimentally compare the conversion efficiency and harmonic output field of the ${{\rm{LG}}_{0,0}}$ and ${{\rm{LG}}_{3,3}}$ modes in a cavity-enhanced SHG using the same 7% doped ${\rm{MgO}}\!:\! {{\rm{LiNbO}}_3}$ crystal. Conversion efficiencies of 96% and 45% are achieved, respectively. The influence of mode mismatches and astigmatism is analyzed to estimate the ratio of the pump mode-dependent effective nonlinearities to be ${d_{0,0}}/{d_{3,3}} \sim 5$. Furthermore, we show that absorption loss in the crystal is more relevant for the ${{\rm{LG}}_{3,3}}$ mode.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

Full Article  |  PDF Article

Corrections

22 September 2020: Typographical corrections were made to the title, a figure caption, and an equation.


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References

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2019 (2)

The Virgo Collaboration, Phys. Rev. Lett. 123, 231108 (2019).
[Crossref]

M. Mehmet and H. Vahlbruch, Classical Quantum Gravity 36, 015014 (2019).
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2018 (1)

T. D. Huang and T. H. Lu, Appl. Phys. B 124, 72 (2018).
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2017 (1)

2016 (1)

H. Vahlbruch, M. Mehmet, K. Danzmann, and R. Schnabel, Phys. Rev. Lett. 117, 110801 (2016).
[Crossref]

2015 (3)

The LIGO Scientific Collaboration, Classical Quantum Gravity 32,074001 (2015).
[Crossref]

The Virgo Collaboration, Classical Quantum Gravity 32, 024001 (2015).
[Crossref]

A. Allocca, A. Gatto, M. Tacca, R. A. Day, M. Barsuglia, G. Pillant, C. Buy, and G. Vajente, Phys. Rev. D 92, 102002 (2015).
[Crossref]

2014 (2)

Z.-Y. Zhou, Y. Li, D.-S. Ding, W. Zhang, S. Shi, B.-S. Shi, and G.-C. Guo, Opt. Express 22, 23673 (2014).
[Crossref]

A. Gatto, M. Tacca, F. Kéfélian, C. Buy, and M. Barsuglia, Phys. Rev. D 90, 122011 (2014).
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2013 (2)

B. Sorazu, P. J. Fulda, B. W. Barr, A. S. Bell, C. Bond, L. Carbone, A. Freise, S. Hild, S. H. Huttner, J. Macarthur, and K. A. Strain, Classical Quantum Gravity 30, 035004 (2013).
[Crossref]

L. Carbone, C. Bogan, P. Fulda, A. Freise, and B. Willke, Phys. Rev. Lett. 110, 251101 (2013).
[Crossref]

2011 (3)

C. Bond, P. Fulda, L. Carbone, K. Kokeyama, and A. Freise, Phys. Rev. D 84, 102002 (2011).
[Crossref]

T. Hong, J. Miller, H. Yamamoto, Y. Chen, and R. Adhikari, Phys. Rev. D 84, 102001 (2011).
[Crossref]

S. Ast, R. M. Nia, A. Schönbeck, N. Lastzka, J. Steinlechner, T. Eberle, M. Mehmet, S. Steinlechner, and R. Schnabel, Opt. Lett. 36, 3467 (2011).
[Crossref]

2010 (1)

P. Fulda, K. Kokeyama, S. Chelkowski, and A. Freise, Phys. Rev. D 82, 012002 (2010).
[Crossref]

2009 (1)

S. Chelkowski, S. Hild, and A. Freise, Phys. Rev. D 79, 122002 (2009).
[Crossref]

2008 (2)

H. Vahlbruch, M. Mehmet, S. Chelkowski, B. Hage, A. Franzen, N. Lastzka, S. Goßler, K. Danzmann, and R. Schnabel, Phys. Rev. Lett. 100, 033602 (2008).
[Crossref]

T. Südmeyer, Y. Imai, H. Masuda, N. Eguchi, M. Saito, and S. Kubota, Opt. Express 16, 1546 (2008).
[Crossref]

2007 (2)

P. Kwee, F. Seifert, B. Willke, and K. Danzmann, Rev. Sci. Instrum. 78, 073103 (2007).
[Crossref]

J.-Y. Vinet, Classical Quantum Gravity 24, 3897 (2007).
[Crossref]

1997 (1)

J. Courtial, K. Dholakia, L. Allen, and M. J. Padgett, Phys. Rev. A 56, 4193 (1997).
[Crossref]

Adhikari, R.

T. Hong, J. Miller, H. Yamamoto, Y. Chen, and R. Adhikari, Phys. Rev. D 84, 102001 (2011).
[Crossref]

Allen, L.

J. Courtial, K. Dholakia, L. Allen, and M. J. Padgett, Phys. Rev. A 56, 4193 (1997).
[Crossref]

Allocca, A.

A. Allocca, A. Gatto, M. Tacca, R. A. Day, M. Barsuglia, G. Pillant, C. Buy, and G. Vajente, Phys. Rev. D 92, 102002 (2015).
[Crossref]

Arai, K.

K. Arai, “On the accumulated round-trip gouy phase shift for a general optical cavity,” Technical Report LIGO-T1300189–v1 (LIGO Scientific Collaboration, 2013).

Ast, S.

Barr, B. W.

B. Sorazu, P. J. Fulda, B. W. Barr, A. S. Bell, C. Bond, L. Carbone, A. Freise, S. Hild, S. H. Huttner, J. Macarthur, and K. A. Strain, Classical Quantum Gravity 30, 035004 (2013).
[Crossref]

Barsuglia, M.

A. Allocca, A. Gatto, M. Tacca, R. A. Day, M. Barsuglia, G. Pillant, C. Buy, and G. Vajente, Phys. Rev. D 92, 102002 (2015).
[Crossref]

A. Gatto, M. Tacca, F. Kéfélian, C. Buy, and M. Barsuglia, Phys. Rev. D 90, 122011 (2014).
[Crossref]

Bell, A. S.

B. Sorazu, P. J. Fulda, B. W. Barr, A. S. Bell, C. Bond, L. Carbone, A. Freise, S. Hild, S. H. Huttner, J. Macarthur, and K. A. Strain, Classical Quantum Gravity 30, 035004 (2013).
[Crossref]

Bogan, C.

A. Noack, C. Bogan, and B. Willke, Opt. Lett. 42, 751 (2017).
[Crossref]

L. Carbone, C. Bogan, P. Fulda, A. Freise, and B. Willke, Phys. Rev. Lett. 110, 251101 (2013).
[Crossref]

Bond, C.

B. Sorazu, P. J. Fulda, B. W. Barr, A. S. Bell, C. Bond, L. Carbone, A. Freise, S. Hild, S. H. Huttner, J. Macarthur, and K. A. Strain, Classical Quantum Gravity 30, 035004 (2013).
[Crossref]

C. Bond, P. Fulda, L. Carbone, K. Kokeyama, and A. Freise, Phys. Rev. D 84, 102002 (2011).
[Crossref]

Buy, C.

A. Allocca, A. Gatto, M. Tacca, R. A. Day, M. Barsuglia, G. Pillant, C. Buy, and G. Vajente, Phys. Rev. D 92, 102002 (2015).
[Crossref]

A. Gatto, M. Tacca, F. Kéfélian, C. Buy, and M. Barsuglia, Phys. Rev. D 90, 122011 (2014).
[Crossref]

Carbone, L.

B. Sorazu, P. J. Fulda, B. W. Barr, A. S. Bell, C. Bond, L. Carbone, A. Freise, S. Hild, S. H. Huttner, J. Macarthur, and K. A. Strain, Classical Quantum Gravity 30, 035004 (2013).
[Crossref]

L. Carbone, C. Bogan, P. Fulda, A. Freise, and B. Willke, Phys. Rev. Lett. 110, 251101 (2013).
[Crossref]

C. Bond, P. Fulda, L. Carbone, K. Kokeyama, and A. Freise, Phys. Rev. D 84, 102002 (2011).
[Crossref]

Chelkowski, S.

P. Fulda, K. Kokeyama, S. Chelkowski, and A. Freise, Phys. Rev. D 82, 012002 (2010).
[Crossref]

S. Chelkowski, S. Hild, and A. Freise, Phys. Rev. D 79, 122002 (2009).
[Crossref]

H. Vahlbruch, M. Mehmet, S. Chelkowski, B. Hage, A. Franzen, N. Lastzka, S. Goßler, K. Danzmann, and R. Schnabel, Phys. Rev. Lett. 100, 033602 (2008).
[Crossref]

Chen, Y.

T. Hong, J. Miller, H. Yamamoto, Y. Chen, and R. Adhikari, Phys. Rev. D 84, 102001 (2011).
[Crossref]

Courtial, J.

J. Courtial, K. Dholakia, L. Allen, and M. J. Padgett, Phys. Rev. A 56, 4193 (1997).
[Crossref]

Danzmann, K.

H. Vahlbruch, M. Mehmet, K. Danzmann, and R. Schnabel, Phys. Rev. Lett. 117, 110801 (2016).
[Crossref]

H. Vahlbruch, M. Mehmet, S. Chelkowski, B. Hage, A. Franzen, N. Lastzka, S. Goßler, K. Danzmann, and R. Schnabel, Phys. Rev. Lett. 100, 033602 (2008).
[Crossref]

P. Kwee, F. Seifert, B. Willke, and K. Danzmann, Rev. Sci. Instrum. 78, 073103 (2007).
[Crossref]

Day, R. A.

A. Allocca, A. Gatto, M. Tacca, R. A. Day, M. Barsuglia, G. Pillant, C. Buy, and G. Vajente, Phys. Rev. D 92, 102002 (2015).
[Crossref]

Dholakia, K.

J. Courtial, K. Dholakia, L. Allen, and M. J. Padgett, Phys. Rev. A 56, 4193 (1997).
[Crossref]

Ding, D.-S.

Eberle, T.

Eguchi, N.

Franzen, A.

H. Vahlbruch, M. Mehmet, S. Chelkowski, B. Hage, A. Franzen, N. Lastzka, S. Goßler, K. Danzmann, and R. Schnabel, Phys. Rev. Lett. 100, 033602 (2008).
[Crossref]

Freise, A.

L. Carbone, C. Bogan, P. Fulda, A. Freise, and B. Willke, Phys. Rev. Lett. 110, 251101 (2013).
[Crossref]

B. Sorazu, P. J. Fulda, B. W. Barr, A. S. Bell, C. Bond, L. Carbone, A. Freise, S. Hild, S. H. Huttner, J. Macarthur, and K. A. Strain, Classical Quantum Gravity 30, 035004 (2013).
[Crossref]

C. Bond, P. Fulda, L. Carbone, K. Kokeyama, and A. Freise, Phys. Rev. D 84, 102002 (2011).
[Crossref]

P. Fulda, K. Kokeyama, S. Chelkowski, and A. Freise, Phys. Rev. D 82, 012002 (2010).
[Crossref]

S. Chelkowski, S. Hild, and A. Freise, Phys. Rev. D 79, 122002 (2009).
[Crossref]

Fulda, P.

L. Carbone, C. Bogan, P. Fulda, A. Freise, and B. Willke, Phys. Rev. Lett. 110, 251101 (2013).
[Crossref]

C. Bond, P. Fulda, L. Carbone, K. Kokeyama, and A. Freise, Phys. Rev. D 84, 102002 (2011).
[Crossref]

P. Fulda, K. Kokeyama, S. Chelkowski, and A. Freise, Phys. Rev. D 82, 012002 (2010).
[Crossref]

Fulda, P. J.

B. Sorazu, P. J. Fulda, B. W. Barr, A. S. Bell, C. Bond, L. Carbone, A. Freise, S. Hild, S. H. Huttner, J. Macarthur, and K. A. Strain, Classical Quantum Gravity 30, 035004 (2013).
[Crossref]

Gatto, A.

A. Allocca, A. Gatto, M. Tacca, R. A. Day, M. Barsuglia, G. Pillant, C. Buy, and G. Vajente, Phys. Rev. D 92, 102002 (2015).
[Crossref]

A. Gatto, M. Tacca, F. Kéfélian, C. Buy, and M. Barsuglia, Phys. Rev. D 90, 122011 (2014).
[Crossref]

Goßler, S.

H. Vahlbruch, M. Mehmet, S. Chelkowski, B. Hage, A. Franzen, N. Lastzka, S. Goßler, K. Danzmann, and R. Schnabel, Phys. Rev. Lett. 100, 033602 (2008).
[Crossref]

Guo, G.-C.

Hage, B.

H. Vahlbruch, M. Mehmet, S. Chelkowski, B. Hage, A. Franzen, N. Lastzka, S. Goßler, K. Danzmann, and R. Schnabel, Phys. Rev. Lett. 100, 033602 (2008).
[Crossref]

Heinze, J.

J. Heinze, H. Vahlbruch, and B. Willke, “Numerical analysis of LG33 second harmonic generation in comparison to the LG00 case,” arXiv:2008.10243 (2020).

Hild, S.

B. Sorazu, P. J. Fulda, B. W. Barr, A. S. Bell, C. Bond, L. Carbone, A. Freise, S. Hild, S. H. Huttner, J. Macarthur, and K. A. Strain, Classical Quantum Gravity 30, 035004 (2013).
[Crossref]

S. Chelkowski, S. Hild, and A. Freise, Phys. Rev. D 79, 122002 (2009).
[Crossref]

Hong, T.

T. Hong, J. Miller, H. Yamamoto, Y. Chen, and R. Adhikari, Phys. Rev. D 84, 102001 (2011).
[Crossref]

Huang, T. D.

T. D. Huang and T. H. Lu, Appl. Phys. B 124, 72 (2018).
[Crossref]

Huttner, S. H.

B. Sorazu, P. J. Fulda, B. W. Barr, A. S. Bell, C. Bond, L. Carbone, A. Freise, S. Hild, S. H. Huttner, J. Macarthur, and K. A. Strain, Classical Quantum Gravity 30, 035004 (2013).
[Crossref]

Imai, Y.

Kéfélian, F.

A. Gatto, M. Tacca, F. Kéfélian, C. Buy, and M. Barsuglia, Phys. Rev. D 90, 122011 (2014).
[Crossref]

Kokeyama, K.

C. Bond, P. Fulda, L. Carbone, K. Kokeyama, and A. Freise, Phys. Rev. D 84, 102002 (2011).
[Crossref]

P. Fulda, K. Kokeyama, S. Chelkowski, and A. Freise, Phys. Rev. D 82, 012002 (2010).
[Crossref]

Kubota, S.

Kwee, P.

P. Kwee, F. Seifert, B. Willke, and K. Danzmann, Rev. Sci. Instrum. 78, 073103 (2007).
[Crossref]

Lastzka, N.

S. Ast, R. M. Nia, A. Schönbeck, N. Lastzka, J. Steinlechner, T. Eberle, M. Mehmet, S. Steinlechner, and R. Schnabel, Opt. Lett. 36, 3467 (2011).
[Crossref]

H. Vahlbruch, M. Mehmet, S. Chelkowski, B. Hage, A. Franzen, N. Lastzka, S. Goßler, K. Danzmann, and R. Schnabel, Phys. Rev. Lett. 100, 033602 (2008).
[Crossref]

N. Lastzka, “Numerical modelling of classical and quantum effects in non-linear optical systems,” Ph.D. thesis (Leibniz Universität Hannover, 2010).

Li, Y.

Lu, T. H.

T. D. Huang and T. H. Lu, Appl. Phys. B 124, 72 (2018).
[Crossref]

Macarthur, J.

B. Sorazu, P. J. Fulda, B. W. Barr, A. S. Bell, C. Bond, L. Carbone, A. Freise, S. Hild, S. H. Huttner, J. Macarthur, and K. A. Strain, Classical Quantum Gravity 30, 035004 (2013).
[Crossref]

Masuda, H.

Mehmet, M.

M. Mehmet and H. Vahlbruch, Classical Quantum Gravity 36, 015014 (2019).
[Crossref]

H. Vahlbruch, M. Mehmet, K. Danzmann, and R. Schnabel, Phys. Rev. Lett. 117, 110801 (2016).
[Crossref]

S. Ast, R. M. Nia, A. Schönbeck, N. Lastzka, J. Steinlechner, T. Eberle, M. Mehmet, S. Steinlechner, and R. Schnabel, Opt. Lett. 36, 3467 (2011).
[Crossref]

H. Vahlbruch, M. Mehmet, S. Chelkowski, B. Hage, A. Franzen, N. Lastzka, S. Goßler, K. Danzmann, and R. Schnabel, Phys. Rev. Lett. 100, 033602 (2008).
[Crossref]

Miller, J.

T. Hong, J. Miller, H. Yamamoto, Y. Chen, and R. Adhikari, Phys. Rev. D 84, 102001 (2011).
[Crossref]

Nia, R. M.

Noack, A.

Padgett, M. J.

J. Courtial, K. Dholakia, L. Allen, and M. J. Padgett, Phys. Rev. A 56, 4193 (1997).
[Crossref]

Pillant, G.

A. Allocca, A. Gatto, M. Tacca, R. A. Day, M. Barsuglia, G. Pillant, C. Buy, and G. Vajente, Phys. Rev. D 92, 102002 (2015).
[Crossref]

Saito, M.

Schnabel, R.

H. Vahlbruch, M. Mehmet, K. Danzmann, and R. Schnabel, Phys. Rev. Lett. 117, 110801 (2016).
[Crossref]

S. Ast, R. M. Nia, A. Schönbeck, N. Lastzka, J. Steinlechner, T. Eberle, M. Mehmet, S. Steinlechner, and R. Schnabel, Opt. Lett. 36, 3467 (2011).
[Crossref]

H. Vahlbruch, M. Mehmet, S. Chelkowski, B. Hage, A. Franzen, N. Lastzka, S. Goßler, K. Danzmann, and R. Schnabel, Phys. Rev. Lett. 100, 033602 (2008).
[Crossref]

Schönbeck, A.

Seifert, F.

P. Kwee, F. Seifert, B. Willke, and K. Danzmann, Rev. Sci. Instrum. 78, 073103 (2007).
[Crossref]

Shi, B.-S.

Shi, S.

Sorazu, B.

B. Sorazu, P. J. Fulda, B. W. Barr, A. S. Bell, C. Bond, L. Carbone, A. Freise, S. Hild, S. H. Huttner, J. Macarthur, and K. A. Strain, Classical Quantum Gravity 30, 035004 (2013).
[Crossref]

Steinlechner, J.

Steinlechner, S.

Strain, K. A.

B. Sorazu, P. J. Fulda, B. W. Barr, A. S. Bell, C. Bond, L. Carbone, A. Freise, S. Hild, S. H. Huttner, J. Macarthur, and K. A. Strain, Classical Quantum Gravity 30, 035004 (2013).
[Crossref]

Südmeyer, T.

Tacca, M.

A. Allocca, A. Gatto, M. Tacca, R. A. Day, M. Barsuglia, G. Pillant, C. Buy, and G. Vajente, Phys. Rev. D 92, 102002 (2015).
[Crossref]

A. Gatto, M. Tacca, F. Kéfélian, C. Buy, and M. Barsuglia, Phys. Rev. D 90, 122011 (2014).
[Crossref]

Vahlbruch, H.

M. Mehmet and H. Vahlbruch, Classical Quantum Gravity 36, 015014 (2019).
[Crossref]

H. Vahlbruch, M. Mehmet, K. Danzmann, and R. Schnabel, Phys. Rev. Lett. 117, 110801 (2016).
[Crossref]

H. Vahlbruch, M. Mehmet, S. Chelkowski, B. Hage, A. Franzen, N. Lastzka, S. Goßler, K. Danzmann, and R. Schnabel, Phys. Rev. Lett. 100, 033602 (2008).
[Crossref]

J. Heinze, H. Vahlbruch, and B. Willke, “Numerical analysis of LG33 second harmonic generation in comparison to the LG00 case,” arXiv:2008.10243 (2020).

Vajente, G.

A. Allocca, A. Gatto, M. Tacca, R. A. Day, M. Barsuglia, G. Pillant, C. Buy, and G. Vajente, Phys. Rev. D 92, 102002 (2015).
[Crossref]

Vinet, J.-Y.

J.-Y. Vinet, Classical Quantum Gravity 24, 3897 (2007).
[Crossref]

Willke, B.

A. Noack, C. Bogan, and B. Willke, Opt. Lett. 42, 751 (2017).
[Crossref]

L. Carbone, C. Bogan, P. Fulda, A. Freise, and B. Willke, Phys. Rev. Lett. 110, 251101 (2013).
[Crossref]

P. Kwee, F. Seifert, B. Willke, and K. Danzmann, Rev. Sci. Instrum. 78, 073103 (2007).
[Crossref]

J. Heinze, H. Vahlbruch, and B. Willke, “Numerical analysis of LG33 second harmonic generation in comparison to the LG00 case,” arXiv:2008.10243 (2020).

Yamamoto, H.

T. Hong, J. Miller, H. Yamamoto, Y. Chen, and R. Adhikari, Phys. Rev. D 84, 102001 (2011).
[Crossref]

Zhang, W.

Zhou, Z.-Y.

Appl. Phys. B (1)

T. D. Huang and T. H. Lu, Appl. Phys. B 124, 72 (2018).
[Crossref]

Classical Quantum Gravity (5)

M. Mehmet and H. Vahlbruch, Classical Quantum Gravity 36, 015014 (2019).
[Crossref]

The LIGO Scientific Collaboration, Classical Quantum Gravity 32,074001 (2015).
[Crossref]

The Virgo Collaboration, Classical Quantum Gravity 32, 024001 (2015).
[Crossref]

J.-Y. Vinet, Classical Quantum Gravity 24, 3897 (2007).
[Crossref]

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Figures (4)

Fig. 1.
Fig. 1. Simulated circulating power in an astigmatic cavity for an injected ${{\rm{LG}}_{3,3}}$ mode using ${\cal F} = 300$ , ${R_{c2,x}} = 0.5000\;{\rm{m}}$ , ${R_{c2,y}} = 0.5012\;{\rm{m}}$ , and $L = 21\;{\rm{cm}}$ . FSR is the free spectral range, ideal resonance assumes no astigmatism.
Fig. 2.
Fig. 2. The hemilithic SHG cavity is formed by an incoupling mirror and the curved crystal face. This figure allocates all relevant fields and cavity properties used in Eq. (5).
Fig. 3.
Fig. 3. Schematic of experimental setup. Not shown: FI right after the laser and the PDH locking scheme of the MC.
Fig. 4.
Fig. 4. Measured external, corrected, and effective SHG conversion efficiencies of the ${{\rm{LG}}_{0,0}}$ and ${{\rm{LG}}_{3,3}}$ modes, including the NLCS simulations for the corrected curves. The $x$ axis refers to the full input power for the external and effective efficiencies and to the estimated matched input power for the corrected and simulated efficiencies. Bottom right: CCD picture of the harmonic output field at $P_{{\rm{in}}}^{{\rm{ext}}} = 664\;{\rm{mW}}$ for the ${{\rm{LG}}_{3,3}}$ mode. A distorted ${{\rm{LG}}_{6,6}}$ intensity pattern can be identified.

Equations (6)

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L G 3 , 3 = k = 0 9 c k , 9 k H G k , 9 k w i t h k = 0 9 | c k , 9 k | 2 = 1 ,
R c 2 ( m , n ) := m R c 2 , x + n R c 2 , y σ ,
ξ m , n r t = 2 arccos ( s i g n ( g 1 ) × g 1 g 2 ( m , n ) ) ,
P r e s ( L ) = k = 0 9 P k , 9 k 1 + ( 2 F π ) 2 sin 2 ( π L λ + 5 ξ k , 9 k r t ) ,
r r e f l := P r e f l / P i n e x t = 1 γ | κ | 2 = 1 [ 1 | a r e f l ( l a , c ) | 2 ] | κ | 2 , r t r a n s := P t r a n s / P i n e x t = | a t r a n s ( l a , c ) | 2 | κ | 2 , η e x t := P o u t / P i n e x t = | b o u t ( l a , c ) | 2 | κ | 2 ,
a r e f l ( l a , c ) = ρ 1 a ( τ 1 a ) 2 ( τ A R a ) 2 ρ 2 a ( 1 l a c ) 1 ρ 1 a ρ 2 a ( τ A R a ) 2 ( 1 l a c ) , a t r a n s ( l a , c ) = τ 1 a τ 2 a τ A R a 1 l a c 1 ρ 1 a ρ 2 a ( τ A R a ) 2 ( 1 l a c ) , b o u t ( l a , c ) = τ 1 a τ A R a τ A R b c ( ( ρ 2 b ) 2 + ( 1 l a c ) ( ρ 2 a ) 2 ) 1 ρ 1 a ρ 2 a ( τ A R a ) 2 ( 1 l a c ) ,