Abstract

The process of frequency conversion in a multilayer structure of different media with quadratic nonlinearity is investigated by using the constant-intensity approximation. The values of complex amplitudes of the fundamental radiation and the second harmonic at the outlet of each layer are the entrance values of the corresponding complex amplitudes of the next layer. Analytical expressions for second-harmonic conversion efficiency for the case of n layers are proposed. The factors restricting the efficiency of the process of frequency conversion are analyzed. In layered media, upon frequency conversion, from one layer to the next the optimum values of the phase mismatch between layers changes. The character of the process of frequency conversion of the fundamental radiation to the second harmonic depends not only on the value of the nonlinear coefficients of waves, the phase mismatch, but also on the signs of these parameters.

© 2008 Optical Society of America

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References

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  1. Z. H. Tagiev and R. J. Kasumova, “Phase effects at second harmonic generation of powerful laser radiation in noncentrosymmetrical media,” Opt. Commun. 268, 311-316 (2006).
    [CrossRef]
  2. N. Blombergen, Nonlinear Optics (W.A. Benjamin, 1965).
  3. V. D. Volosov and A. G. Kalintsev, “Degenerate parametric processes at three wave interactions in consecutively arranged crystals,” Tech. Phys. Lett. 2, 85-90 (1976).
  4. A. S. Chirkin and D. B. Yusupov, “Quasi-synchrone parametric interactions of optical waves at equality of group velocities,” Quantum Elektron. (Moscow) 9, 1625-1628 (1982).
  5. V. I. Kabelka, A. S. Piskarskas, A. Yu. Stabinis, and R. L. Sher, “Group matching of interacting ultrashort light pulses in the nonlinear crystals,” Quantum Elektron. (Moscow) 2, 434-436(1975).
  6. V. D. Volosov, A. G. Kalintsev, and V. N. Krylov, “On suppression of degenerate parametric processes which limit the efficiency of frequency doubling in crystals,” Quantum Elektron. (Moscow) 3, 2139-2145 (1976).
  7. B. V. Bokut, N. S. Kazak, A. G. Malashenko, and Yu. A. Sannikov, “On some peculiarities of second harmonic generation in consecutively arranged crystals,” J. Appl. Spectrosc. 37, 748 (1982).
    [CrossRef]
  8. R. B. Andreev, K. V. Vetrov, V. D. Volosov, and A. G. Kalintsev, “Experimental study of multicrystal frequency doublers,” Opt. Spectrosc. 63, 793-795 (1987).
  9. M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631-2654 (1992).
    [CrossRef]
  10. Z. H. Tagiev and R. J. Kasumova, “Phase effects at second harmonic generation in the layer media,” Opt. Commun. 281, 814-823 (2008).
    [CrossRef]
  11. N. S. Kazak, V. K. Pavlenko, and Yu. A. Sannikov, “Peculiarities second harmonic generation in consecutively arranged crystals at strong energy exchange,” J. Appl. Spectrosc. 53, 364-370 (1990).
    [CrossRef]
  12. Z. H. Tagiev, “Theory of second harmonic generation in consecutively arranged crystals,” J. Appl. Spectrosc. 53, 136-139(1990).
  13. A. V. Bokhin and V. G. Dmitriev, “Second harmonic generation in periodically poled crystals in the fixed-intensity approximation,” Quantum Elektron. (Moscow) 32, 219-222 (2002).
    [CrossRef]
  14. Z. H. Tagiev and A. S.Chirkin, “Fixed intensity approximation in the theory of nonlinear waves,” Zh. Eksp. Teor. Fiz. 73, 1271-1282 (1977) [Sov. Phys. JETP 46, 669-680 (1977)].
  15. Z. H. Tagiev, R. J. Kasumova, R. A. Salmanova, N. V. Kerimova, J. Opt. B 3, 84-87 (2001) .
    [CrossRef]
  16. V. G. Dmitriev and L. V. Tarasov, Prikladnaya Nelineynaya Optika [Applied Nonlinear Optics] (Radio i Svyaz, 1982).
  17. I. V. Kityk, “Nonlinear optical phenomena in the large-sized nanocrystallites,” J. Non-Cryst. Solids 292, 184-201 (2001).
    [CrossRef]

2008 (1)

Z. H. Tagiev and R. J. Kasumova, “Phase effects at second harmonic generation in the layer media,” Opt. Commun. 281, 814-823 (2008).
[CrossRef]

2006 (1)

Z. H. Tagiev and R. J. Kasumova, “Phase effects at second harmonic generation of powerful laser radiation in noncentrosymmetrical media,” Opt. Commun. 268, 311-316 (2006).
[CrossRef]

2002 (1)

A. V. Bokhin and V. G. Dmitriev, “Second harmonic generation in periodically poled crystals in the fixed-intensity approximation,” Quantum Elektron. (Moscow) 32, 219-222 (2002).
[CrossRef]

2001 (2)

Z. H. Tagiev, R. J. Kasumova, R. A. Salmanova, N. V. Kerimova, J. Opt. B 3, 84-87 (2001) .
[CrossRef]

I. V. Kityk, “Nonlinear optical phenomena in the large-sized nanocrystallites,” J. Non-Cryst. Solids 292, 184-201 (2001).
[CrossRef]

1992 (1)

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631-2654 (1992).
[CrossRef]

1990 (2)

N. S. Kazak, V. K. Pavlenko, and Yu. A. Sannikov, “Peculiarities second harmonic generation in consecutively arranged crystals at strong energy exchange,” J. Appl. Spectrosc. 53, 364-370 (1990).
[CrossRef]

Z. H. Tagiev, “Theory of second harmonic generation in consecutively arranged crystals,” J. Appl. Spectrosc. 53, 136-139(1990).

1987 (1)

R. B. Andreev, K. V. Vetrov, V. D. Volosov, and A. G. Kalintsev, “Experimental study of multicrystal frequency doublers,” Opt. Spectrosc. 63, 793-795 (1987).

1982 (2)

B. V. Bokut, N. S. Kazak, A. G. Malashenko, and Yu. A. Sannikov, “On some peculiarities of second harmonic generation in consecutively arranged crystals,” J. Appl. Spectrosc. 37, 748 (1982).
[CrossRef]

A. S. Chirkin and D. B. Yusupov, “Quasi-synchrone parametric interactions of optical waves at equality of group velocities,” Quantum Elektron. (Moscow) 9, 1625-1628 (1982).

1977 (1)

Z. H. Tagiev and A. S.Chirkin, “Fixed intensity approximation in the theory of nonlinear waves,” Zh. Eksp. Teor. Fiz. 73, 1271-1282 (1977) [Sov. Phys. JETP 46, 669-680 (1977)].

1976 (2)

V. D. Volosov and A. G. Kalintsev, “Degenerate parametric processes at three wave interactions in consecutively arranged crystals,” Tech. Phys. Lett. 2, 85-90 (1976).

V. D. Volosov, A. G. Kalintsev, and V. N. Krylov, “On suppression of degenerate parametric processes which limit the efficiency of frequency doubling in crystals,” Quantum Elektron. (Moscow) 3, 2139-2145 (1976).

1975 (1)

V. I. Kabelka, A. S. Piskarskas, A. Yu. Stabinis, and R. L. Sher, “Group matching of interacting ultrashort light pulses in the nonlinear crystals,” Quantum Elektron. (Moscow) 2, 434-436(1975).

Andreev, R. B.

R. B. Andreev, K. V. Vetrov, V. D. Volosov, and A. G. Kalintsev, “Experimental study of multicrystal frequency doublers,” Opt. Spectrosc. 63, 793-795 (1987).

Blombergen, N.

N. Blombergen, Nonlinear Optics (W.A. Benjamin, 1965).

Bokhin, A. V.

A. V. Bokhin and V. G. Dmitriev, “Second harmonic generation in periodically poled crystals in the fixed-intensity approximation,” Quantum Elektron. (Moscow) 32, 219-222 (2002).
[CrossRef]

Bokut, B. V.

B. V. Bokut, N. S. Kazak, A. G. Malashenko, and Yu. A. Sannikov, “On some peculiarities of second harmonic generation in consecutively arranged crystals,” J. Appl. Spectrosc. 37, 748 (1982).
[CrossRef]

Byer, R. L.

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631-2654 (1992).
[CrossRef]

Chirkin, A. S.

A. S. Chirkin and D. B. Yusupov, “Quasi-synchrone parametric interactions of optical waves at equality of group velocities,” Quantum Elektron. (Moscow) 9, 1625-1628 (1982).

Z. H. Tagiev and A. S.Chirkin, “Fixed intensity approximation in the theory of nonlinear waves,” Zh. Eksp. Teor. Fiz. 73, 1271-1282 (1977) [Sov. Phys. JETP 46, 669-680 (1977)].

Dmitriev, V. G.

A. V. Bokhin and V. G. Dmitriev, “Second harmonic generation in periodically poled crystals in the fixed-intensity approximation,” Quantum Elektron. (Moscow) 32, 219-222 (2002).
[CrossRef]

V. G. Dmitriev and L. V. Tarasov, Prikladnaya Nelineynaya Optika [Applied Nonlinear Optics] (Radio i Svyaz, 1982).

Fejer, M. M.

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631-2654 (1992).
[CrossRef]

Jundt, D. H.

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631-2654 (1992).
[CrossRef]

Kabelka, V. I.

V. I. Kabelka, A. S. Piskarskas, A. Yu. Stabinis, and R. L. Sher, “Group matching of interacting ultrashort light pulses in the nonlinear crystals,” Quantum Elektron. (Moscow) 2, 434-436(1975).

Kalintsev, A. G.

R. B. Andreev, K. V. Vetrov, V. D. Volosov, and A. G. Kalintsev, “Experimental study of multicrystal frequency doublers,” Opt. Spectrosc. 63, 793-795 (1987).

V. D. Volosov, A. G. Kalintsev, and V. N. Krylov, “On suppression of degenerate parametric processes which limit the efficiency of frequency doubling in crystals,” Quantum Elektron. (Moscow) 3, 2139-2145 (1976).

V. D. Volosov and A. G. Kalintsev, “Degenerate parametric processes at three wave interactions in consecutively arranged crystals,” Tech. Phys. Lett. 2, 85-90 (1976).

Kasumova, R. J.

Z. H. Tagiev and R. J. Kasumova, “Phase effects at second harmonic generation in the layer media,” Opt. Commun. 281, 814-823 (2008).
[CrossRef]

Z. H. Tagiev and R. J. Kasumova, “Phase effects at second harmonic generation of powerful laser radiation in noncentrosymmetrical media,” Opt. Commun. 268, 311-316 (2006).
[CrossRef]

Z. H. Tagiev, R. J. Kasumova, R. A. Salmanova, N. V. Kerimova, J. Opt. B 3, 84-87 (2001) .
[CrossRef]

Kazak, N. S.

N. S. Kazak, V. K. Pavlenko, and Yu. A. Sannikov, “Peculiarities second harmonic generation in consecutively arranged crystals at strong energy exchange,” J. Appl. Spectrosc. 53, 364-370 (1990).
[CrossRef]

B. V. Bokut, N. S. Kazak, A. G. Malashenko, and Yu. A. Sannikov, “On some peculiarities of second harmonic generation in consecutively arranged crystals,” J. Appl. Spectrosc. 37, 748 (1982).
[CrossRef]

Kerimova, N. V.

Z. H. Tagiev, R. J. Kasumova, R. A. Salmanova, N. V. Kerimova, J. Opt. B 3, 84-87 (2001) .
[CrossRef]

Kityk, I. V.

I. V. Kityk, “Nonlinear optical phenomena in the large-sized nanocrystallites,” J. Non-Cryst. Solids 292, 184-201 (2001).
[CrossRef]

Krylov, V. N.

V. D. Volosov, A. G. Kalintsev, and V. N. Krylov, “On suppression of degenerate parametric processes which limit the efficiency of frequency doubling in crystals,” Quantum Elektron. (Moscow) 3, 2139-2145 (1976).

Magel, G. A.

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631-2654 (1992).
[CrossRef]

Malashenko, A. G.

B. V. Bokut, N. S. Kazak, A. G. Malashenko, and Yu. A. Sannikov, “On some peculiarities of second harmonic generation in consecutively arranged crystals,” J. Appl. Spectrosc. 37, 748 (1982).
[CrossRef]

Pavlenko, V. K.

N. S. Kazak, V. K. Pavlenko, and Yu. A. Sannikov, “Peculiarities second harmonic generation in consecutively arranged crystals at strong energy exchange,” J. Appl. Spectrosc. 53, 364-370 (1990).
[CrossRef]

Piskarskas, A. S.

V. I. Kabelka, A. S. Piskarskas, A. Yu. Stabinis, and R. L. Sher, “Group matching of interacting ultrashort light pulses in the nonlinear crystals,” Quantum Elektron. (Moscow) 2, 434-436(1975).

Salmanova, R. A.

Z. H. Tagiev, R. J. Kasumova, R. A. Salmanova, N. V. Kerimova, J. Opt. B 3, 84-87 (2001) .
[CrossRef]

Sannikov, Yu. A.

N. S. Kazak, V. K. Pavlenko, and Yu. A. Sannikov, “Peculiarities second harmonic generation in consecutively arranged crystals at strong energy exchange,” J. Appl. Spectrosc. 53, 364-370 (1990).
[CrossRef]

B. V. Bokut, N. S. Kazak, A. G. Malashenko, and Yu. A. Sannikov, “On some peculiarities of second harmonic generation in consecutively arranged crystals,” J. Appl. Spectrosc. 37, 748 (1982).
[CrossRef]

Sher, R. L.

V. I. Kabelka, A. S. Piskarskas, A. Yu. Stabinis, and R. L. Sher, “Group matching of interacting ultrashort light pulses in the nonlinear crystals,” Quantum Elektron. (Moscow) 2, 434-436(1975).

Stabinis, A. Yu.

V. I. Kabelka, A. S. Piskarskas, A. Yu. Stabinis, and R. L. Sher, “Group matching of interacting ultrashort light pulses in the nonlinear crystals,” Quantum Elektron. (Moscow) 2, 434-436(1975).

Tagiev, Z. H.

Z. H. Tagiev and R. J. Kasumova, “Phase effects at second harmonic generation in the layer media,” Opt. Commun. 281, 814-823 (2008).
[CrossRef]

Z. H. Tagiev and R. J. Kasumova, “Phase effects at second harmonic generation of powerful laser radiation in noncentrosymmetrical media,” Opt. Commun. 268, 311-316 (2006).
[CrossRef]

Z. H. Tagiev, R. J. Kasumova, R. A. Salmanova, N. V. Kerimova, J. Opt. B 3, 84-87 (2001) .
[CrossRef]

Z. H. Tagiev, “Theory of second harmonic generation in consecutively arranged crystals,” J. Appl. Spectrosc. 53, 136-139(1990).

Z. H. Tagiev and A. S.Chirkin, “Fixed intensity approximation in the theory of nonlinear waves,” Zh. Eksp. Teor. Fiz. 73, 1271-1282 (1977) [Sov. Phys. JETP 46, 669-680 (1977)].

Tarasov, L. V.

V. G. Dmitriev and L. V. Tarasov, Prikladnaya Nelineynaya Optika [Applied Nonlinear Optics] (Radio i Svyaz, 1982).

Vetrov, K. V.

R. B. Andreev, K. V. Vetrov, V. D. Volosov, and A. G. Kalintsev, “Experimental study of multicrystal frequency doublers,” Opt. Spectrosc. 63, 793-795 (1987).

Volosov, V. D.

R. B. Andreev, K. V. Vetrov, V. D. Volosov, and A. G. Kalintsev, “Experimental study of multicrystal frequency doublers,” Opt. Spectrosc. 63, 793-795 (1987).

V. D. Volosov, A. G. Kalintsev, and V. N. Krylov, “On suppression of degenerate parametric processes which limit the efficiency of frequency doubling in crystals,” Quantum Elektron. (Moscow) 3, 2139-2145 (1976).

V. D. Volosov and A. G. Kalintsev, “Degenerate parametric processes at three wave interactions in consecutively arranged crystals,” Tech. Phys. Lett. 2, 85-90 (1976).

Yusupov, D. B.

A. S. Chirkin and D. B. Yusupov, “Quasi-synchrone parametric interactions of optical waves at equality of group velocities,” Quantum Elektron. (Moscow) 9, 1625-1628 (1982).

IEEE J. Quantum Electron. (1)

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 28, 2631-2654 (1992).
[CrossRef]

J. Appl. Spectrosc. (3)

B. V. Bokut, N. S. Kazak, A. G. Malashenko, and Yu. A. Sannikov, “On some peculiarities of second harmonic generation in consecutively arranged crystals,” J. Appl. Spectrosc. 37, 748 (1982).
[CrossRef]

N. S. Kazak, V. K. Pavlenko, and Yu. A. Sannikov, “Peculiarities second harmonic generation in consecutively arranged crystals at strong energy exchange,” J. Appl. Spectrosc. 53, 364-370 (1990).
[CrossRef]

Z. H. Tagiev, “Theory of second harmonic generation in consecutively arranged crystals,” J. Appl. Spectrosc. 53, 136-139(1990).

J. Non-Cryst. Solids (1)

I. V. Kityk, “Nonlinear optical phenomena in the large-sized nanocrystallites,” J. Non-Cryst. Solids 292, 184-201 (2001).
[CrossRef]

J. Opt. B (1)

Z. H. Tagiev, R. J. Kasumova, R. A. Salmanova, N. V. Kerimova, J. Opt. B 3, 84-87 (2001) .
[CrossRef]

Opt. Commun. (2)

Z. H. Tagiev and R. J. Kasumova, “Phase effects at second harmonic generation in the layer media,” Opt. Commun. 281, 814-823 (2008).
[CrossRef]

Z. H. Tagiev and R. J. Kasumova, “Phase effects at second harmonic generation of powerful laser radiation in noncentrosymmetrical media,” Opt. Commun. 268, 311-316 (2006).
[CrossRef]

Opt. Spectrosc. (1)

R. B. Andreev, K. V. Vetrov, V. D. Volosov, and A. G. Kalintsev, “Experimental study of multicrystal frequency doublers,” Opt. Spectrosc. 63, 793-795 (1987).

Quantum Elektron. (Moscow) (4)

A. S. Chirkin and D. B. Yusupov, “Quasi-synchrone parametric interactions of optical waves at equality of group velocities,” Quantum Elektron. (Moscow) 9, 1625-1628 (1982).

V. I. Kabelka, A. S. Piskarskas, A. Yu. Stabinis, and R. L. Sher, “Group matching of interacting ultrashort light pulses in the nonlinear crystals,” Quantum Elektron. (Moscow) 2, 434-436(1975).

V. D. Volosov, A. G. Kalintsev, and V. N. Krylov, “On suppression of degenerate parametric processes which limit the efficiency of frequency doubling in crystals,” Quantum Elektron. (Moscow) 3, 2139-2145 (1976).

A. V. Bokhin and V. G. Dmitriev, “Second harmonic generation in periodically poled crystals in the fixed-intensity approximation,” Quantum Elektron. (Moscow) 32, 219-222 (2002).
[CrossRef]

Tech. Phys. Lett. (1)

V. D. Volosov and A. G. Kalintsev, “Degenerate parametric processes at three wave interactions in consecutively arranged crystals,” Tech. Phys. Lett. 2, 85-90 (1976).

Zh. Eksp. Teor. Fiz. (1)

Z. H. Tagiev and A. S.Chirkin, “Fixed intensity approximation in the theory of nonlinear waves,” Zh. Eksp. Teor. Fiz. 73, 1271-1282 (1977) [Sov. Phys. JETP 46, 669-680 (1977)].

Other (2)

V. G. Dmitriev and L. V. Tarasov, Prikladnaya Nelineynaya Optika [Applied Nonlinear Optics] (Radio i Svyaz, 1982).

N. Blombergen, Nonlinear Optics (W.A. Benjamin, 1965).

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Figures (6)

Fig. 1
Fig. 1

Dependencies of η 2 ( 2 ) on the given phase mismatch in the first layer Δ 1 / 2 Γ 1 , at λ j j = π / 2 , δ j = 0 ( j = 1 2 ) for Δ 2 = Δ 1 (curve 2), Δ 2 = Δ 2 , opt (curve 3), and for γ 1 , 2 / γ 1 , 2 = 1 (curves 4 and 5), γ 1 , 2 / γ 1 , 2 = 1.5 (curve 3), and γ 1 , 2 / γ 1 , 2 = 1.8 (curve 2). Here for comparison the synchronism curve for one layer is given (curve 1).

Fig. 2
Fig. 2

Dependencies of η 2 ( j ) on the given lengths of layers Γ 1 j , j = 1 4 , respectively, at δ j = 0 , γ 1 , 2 = γ 1 , 2 = γ 2 = γ 2 , Δ ˜ 1 = Δ ˜ 2 = Δ ˜ 3 = Δ ˜ 4 = 3 for ˜ 1 = 0.3 , ˜ 2 = 0.4 , ˜ 3 = 0.45 (solid curves 1–4) and ˜ 1 , coh = 0.473613 , ˜ 2 , coh = 0.47778 , ˜ 3 , coh = 0.4889927 (solid curve 1 and dotted curves 5–7) and for the regular domain structure case Δ ˜ 1 = Δ ˜ 2 = Δ ˜ 3 = Δ ˜ 4 = 3 for ˜ 1 = 0.3 , ˜ 2 = 0.4 , ˜ 3 = 0.45 (solid curve 1 and dashed curves 8–10) and for ˜ 1 , coh = 0.473613 , ˜ 2 , coh = 0.47778 , ˜ 3 , coh = 0.4889927 (solid curve 1 and dotted curves 5–7). η 2 ( 1 ) , curve 1; η 2 ( 2 ) , curves 2, 5, and 8; η 2 ( 3 ) , curves 3, 6, and 9; η 2 ( 4 ) , curves 4, 7, and 10.

Fig. 3
Fig. 3

Dependencies of η 2 ( j ) on the given lengths of layers Γ 1 j , j = 1 4 , at λ j j = π / 2 , δ j = 0 , γ 1 , 2 = γ 1 , 2 = γ 1 , 2 = γ 1 , 2 , Δ ˜ 1 = Δ ˜ 2 = Δ ˜ 3 = Δ ˜ 4 for Δ ˜ 1 = 2 (dashed curves 3, 6, 9, and 12), 2.2 (solid curves 2, 5, 8, and 11) and 2.5 (dotted curves 1, 4, 7, and 10). η 2 ( 1 ) , curves 1–3; η 2 ( 2 ) , curves 4–6; η 2 ( 3 ) , curves 7–9; η 2 ( 4 ) , curves 10–12.

Fig. 4
Fig. 4

Dependencies of η 2 ( j ) on the given lengths of layers Γ 1 j , j = 1 4 , at λ j j = π / 2 , δ j = 0 ( j = 1 4 ), γ 1 , 2 = γ 1 , 2 = γ 1 , 2 = γ 1 , 2 for Δ 1 , opt = 0.8165 , Δ 2 = Δ 2 , opt = 1.2247 . η 2 ( 1 ) , curve 1; η 2 ( 2 ) , curve 2; η 2 ( 3 ) , curve 3; η 2 ( 4 ) , curve 4.

Fig. 5
Fig. 5

Dependencies of η 2 ( j ) ( j = 1 4 ) on the given pump intensity I 10 ( I 10 = Γ 1 1 ), λ j j = π / 2 , γ 1 , 2 = γ 1 , 2 = γ 1 , 2 = γ 1 , 2 , and Δ ˜ 1 = Δ ˜ 2 = Δ ˜ 3 = Δ ˜ 4 = 2.5 for δ 2 j = 2 δ 1 j = 0 (solid curves 1–4, 6) and 0.2 (dotted curve 5), calculated in the constant-intensity approximation (curves 1–5) and in the constant-field approximation (curve 6).

Fig. 6
Fig. 6

Dependencies of the gain in efficiency η = η 2 ( j ) / η 2 ( 1 ) on the given lengths of layer structure Γ 1 at λ j j = π / 2 , γ 1 , 2 = γ 1 , 2 = γ 1 , 2 = γ 1 , 2 , Δ ˜ 1 = Δ ˜ 2 = Δ ˜ 3 = Δ ˜ 4 = 3 , and δ j = 0 . The given length of structures after the first layer ( n = 1 ) is Γ 1 = Γ 1 1 , opt , after the second layer ( n = 2 ) is Γ 1 = Γ 1 1 , opt + Γ 1 2 , opt , after the third layer ( n = 3 ) is Γ 1 = Γ 1 1 , opt + Γ 1 2 , opt + Γ 1 3 , opt , after the fourth layer ( n = 4 ) is Γ 1 = Γ 1 1 , opt + Γ 1 2 , opt + Γ 1 3 , opt + Γ 1 4 , opt . Here Γ 1 1 , opt = 0.4736 , Γ 1 2 , opt = 0.4777 , Γ 1 3 , opt = 0.4889 , Γ 1 4 , opt = 0.5023 .

Equations (35)

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d A 1 d z + δ 1 A 1 = i γ 1 A 2 A 1 * exp ( i Δ 1 z ) , d A 2 d z + δ 2 A 2 = i γ 2 A 1 2 exp ( i Δ 1 z ) ,
A 2 ( 1 ) = i γ 2 A 10 2 1 t sinc λ 1 1 exp [ 2 i φ 10 ( δ 2 + 2 δ 1 i Δ 1 ) 1 / 2 ] ,
λ 1 2 = 2 Γ 1 2 ( δ 2 2 δ 1 + i Δ 1 ) 2 / 4 , Γ 1 2 = γ 1 γ 2 I 10 , sinc x = sin x / x , I j = A j A j * .
A 2 ( 2 ) = A 2 ( 1 ) { cos λ 2 2 i [ γ 2 A 1 2 ( 1 ) A 2 ( 1 ) e i [ 2 φ 1 ( 1 ) φ 2 ( 1 ) ] + δ 2 2 δ 1 + i Δ 2 2 i ] sin λ 2 2 λ 2 } exp [ i φ 2 ( 1 ) ( δ 2 + 2 δ 1 i Δ 2 ) 2 / 2 ] ,
λ 2 2 = 2 Γ 2 2 ( δ 2 2 δ 1 + i Δ 2 ) 2 / 4 , Γ 2 2 = γ 1 γ 2 I 1 ( 1 ) ,
I 1 ( 1 ) = I 10 [ ( cos λ 1 1 + δ 2 2 δ 1 2 λ 1 sin Δ 1 1 ) 2 + Δ 1 2 4 λ 1 2 sin 2 λ 1 1 ] 1 / 2 exp [ ( δ 2 + 2 δ 1 ) 1 / 2 ] ,
A 1 2 ( 1 ) A 2 ( 1 ) = i γ 2 ( λ 1 cot λ 1 1 + δ 2 2 δ 1 + i Δ 1 2 ) exp ( i Δ 1 1 ) .
A 2 ( 2 ) = A 2 ( 1 ) { cos λ 2 2 [ δ 2 2 δ 1 + i Δ 2 2 λ 2 ( λ 1 λ 2 cot λ 1 1 + δ 2 2 δ 1 + i Δ 1 2 λ 2 ) γ 2 γ 2 e i ψ ] sin λ 2 2 } exp [ i φ 2 ( 1 ) ( δ 2 + 2 δ 1 i Δ 2 ) 2 2 ] ,
I 2 ( 2 ) = I 2 ( 1 ) [ ( cos λ 2 2 ± γ 2 γ 2 λ 1 λ 2 cot λ 1 1 sin λ 2 2 ) 2 + ( Δ 2 γ 2 γ 2 Δ 1 ) 2 sin 2 λ 2 2 4 λ 2 2 ] exp ( 2 δ 2 2 ) ,
Δ ˜ 2 , opt = γ 1 γ 1 2 2 + Δ ˜ 1 2 ,
[ η 2 ( 2 ) ] max = 1 2 + Δ ˜ 1 2 ( γ 2 γ 1 + γ 2 2 γ 1 Δ ˜ 1 2 + Δ ˜ 1 2 ) .
A 2 ( 3 ) = A 2 ( 2 ) { cos λ 3 3 i ( γ 2 A 1 2 ( 2 ) A 2 ( 2 ) + Δ 3 2 ) sin λ 3 3 λ 3 } exp ( δ 2 + i Δ 3 2 ) 3 ,
λ 3 2 = 2 Γ 3 2 + Δ 3 2 / 4 , Γ 3 2 = γ 1 γ 2 I 1 ( 2 ) ;
A 1 2 ( 2 ) A 2 ( 2 ) = i γ 2 exp ( i Δ 2 2 ) c 1 cos λ 2 2 ( λ 2 2 + b Δ 2 2 ) sin λ 2 2 λ 2 + i [ ( b + Δ 2 2 ) cos λ 2 2 + c 1 Δ 2 2 sin λ 2 2 λ 2 ] cos λ 2 2 + ( c 1 + i b ) sin λ 2 2 λ 2 ,
c 1 = λ 1 γ 2 γ 2 cot λ 1 1 cos ψ + Δ 1 2 γ 2 γ 2 sin ψ ,
b = Δ 1 2 γ 2 γ 2 cos ψ λ 1 γ 2 γ 2 cot λ 1 1 sin ψ Δ 2 2 .
Δ ˜ 3 , opt = 2 ( Δ ˜ 2 Δ ˜ 1 ) ( Γ 3 / Γ 1 ) 2 2 ( Γ 2 / Γ 1 ) 2 + Δ ˜ 1 Δ ˜ 2 .
A 2 ( 4 ) = A 2 ( 3 ) { cos λ 4 4 i [ γ 2 A 1 2 ( 3 ) A 2 ( 3 ) + Δ 4 2 ] sin λ 4 4 λ 4 } exp [ ( δ 2 + i Δ 4 2 ) 4 ] ,
λ 4 2 = 2 Γ 4 2 + Δ 4 2 / 4 , Γ 4 2 = γ 1 γ 2 I 1 ( 3 ) ,
A 1 2 ( 3 ) A 2 ( 3 ) = c 14 cos Δ 3 3 + c 15 sin Δ 3 3 + i ( c 15 cos Δ 3 3 c 14 sin Δ 3 3 ) c 10 2 + c 11 2 ,
c 2 = cos λ 2 2 + c 1 sin λ 2 2 λ 2 ,
c 3 = b sin λ 2 2 λ 2 , c 4 = c 1 cos λ 2 2 λ 2 sin λ 2 2 c 3 Δ 2 2 , c 5 = c 2 Δ 2 2 + b cos λ 2 2 ,
c 6 = c 2 cos Δ 2 2 c 3 sin Δ 2 2 , c 7 = c 2 sin Δ 2 2 + c 3 cos Δ 2 2 ,
c 8 = c 4 c 6 + c 5 c 7 γ 2 ( c 2 2 + c 3 2 ) , c 9 = c 4 c 7 c 5 c 6 γ 2 ( c 2 2 + c 3 2 ) ,
c 10 = cos λ 3 3 + c 8 γ 2 sin λ 3 3 λ 3 ,
c 11 = ( c 9 γ 2 + Δ 3 2 ) sin λ 3 3 λ 3 ,
c 12 = c 8 cos λ 3 3 + [ Δ 3 2 λ 3 ( c 9 + Δ 3 2 γ 2 ) λ 3 γ 2 ] sin λ 3 3 ,
c 13 = c 9 cos λ 3 3 c 8 Δ 3 2 λ 3 sin λ 3 3 ,
c 14 = c 10 c 13 c 11 c 12 , c 15 = c 10 c 12 c 11 c 13 .
η 2 ( 4 ) = η 2 ( 3 ) [ ( cos λ 4 4 + c 19 γ 2 sin λ 4 4 λ 4 ) 2 + ( c 18 γ 2 + Δ 4 2 ) 2 sin 2 λ 4 4 λ 4 2 ] exp ( 2 δ 2 4 ) ,
c 16 = c 15 cos Δ 3 3 c 14 sin Δ 3 3 , c 17 = c 14 cos Δ 3 3 + c 15 sin Δ 3 3 ,
c 18 = c 17 c 10 2 + c 11 2 , c 19 = c 16 c 10 2 + c 11 2 .
A 2 ( n ) = A 2 ( n 1 ) { cos λ n n i [ γ 2 ( n ) A 1 2 ( n 1 ) A 2 ( n 1 ) + Δ n 2 ] sin λ n n λ n } exp [ ( δ 2 ( n ) + i Δ n 2 ) n ] ,
λ n 2 = 2 Γ n 2 + Δ n 2 / 4 , Γ n 2 = γ 1 ( n ) γ 2 ( n ) I 1 ( n 1 ) , Δ n = Δ j , j = 1 n ,
η 2 ( n ) = η 2 ( n 1 ) { ( cos λ n n + c a γ 2 ( n ) sin λ n n λ n ) 2 + ( c b γ 2 ( n ) + Δ n 2 ) 2 sin 2 λ n n λ n 2 } exp ( 2 δ 2 ( n ) n ) ,

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