Abstract

The coupled-wave approach was used for the analysis of volume gratings in a medium with light gain for the sake of oscillation threshold determination in the second-order Bragg diffraction regime. The conditions for self-starting oscillations were analytically determined and found to be optimal for mixed phase-amplitude gratings. The properties of a distributed-feedback laser scheme are derived on the basis of the obtained results.

© 2009 Optical Society of America

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References

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  1. M. V. Vasnetsov, V. Y. Bazhenov, S. S. Slussarenko, O. Sakhno, J. Stumpe, and G. Abbate, “First approach to the analysis of the lasing conditions in POLIPHEM© structures,” Mol. Cryst. Liq. Cryst. 488, 135-147 (2008).
    [CrossRef]
  2. J. E. Bjorkholm and C. V. Shank, “Higher-order distributed feedback oscillators,” Appl. Phys. Lett. 20, 306-308 (1972).
    [CrossRef]
  3. W. Song, A. E. Vasdekis, Z. Li, and D. Psaltis, “Low-order distributed feedback optofluidic dye laser with reduced threshold,” Appl. Phys. Lett. 94, 051117 (2009).
    [CrossRef]
  4. Z. Li, Z. Zhang, T. Emery, A. Scherer, and D. Psaltis, “Single mode optofluidic distributed feedback dye laser,” Opt. Express 14, 696-701 (2006).
    [CrossRef] [PubMed]
  5. M. Gersborg-Hansen and A. Kriensen, “Optofluidic third order distributed feedback dye laser,” Appl. Phys. Lett. 89, 103518 (2006).
    [CrossRef]
  6. H. Kogelnik and C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327-2335 (1972).
    [CrossRef]
  7. R. S. Chu and T. Tamir, “Guided-wave theory of light diffraction by acoustic microwaves,” IEEE Trans. Microwave Theory Tech. MTT-18, 486-504 (1970).
  8. S. Wang, “Principles of distributed feedback and distributed Bragg reflector lasers,” IEEE J. Quantum Electron. 10, 413-424 (1974).
    [CrossRef]
  9. M. Akbari, S. Shahabadi, and K. Schünemann, “A rigorous two-dimensional field analysis of DFB structures,” Prog. Electromagn. Res. PIER 22, 197-212 (1999).
    [CrossRef]
  10. A. Yariv and P. Yeh, Photonics: Optical Electronics in Modern Communications, 6th ed. (Oxford U. Press, 2007).
  11. M. V. Vasnetsov, V. Y. Bazhenov, S. S. Slussarenko, and G. Abbate, “Coupled-wave analysis of second-order Bragg diffraction. I. Reflection-type phase gratings,” J. Opt. Soc. Am. B 26, 684-690 (2009).
    [CrossRef]
  12. H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2910-2947 (1969).
  13. M. V. Vasnetsov, S. S. Slussarenko, O. Sakhno, J. Stumpe, S. S. Slussarenko, Jr., and G. Abbate, “Lasing by second-order Bragg diffraction in dye-doped POLIPHEM phase gratings,” Mol. Cryst. Liq. Cryst. (to be published).

2009

W. Song, A. E. Vasdekis, Z. Li, and D. Psaltis, “Low-order distributed feedback optofluidic dye laser with reduced threshold,” Appl. Phys. Lett. 94, 051117 (2009).
[CrossRef]

M. V. Vasnetsov, V. Y. Bazhenov, S. S. Slussarenko, and G. Abbate, “Coupled-wave analysis of second-order Bragg diffraction. I. Reflection-type phase gratings,” J. Opt. Soc. Am. B 26, 684-690 (2009).
[CrossRef]

2008

M. V. Vasnetsov, V. Y. Bazhenov, S. S. Slussarenko, O. Sakhno, J. Stumpe, and G. Abbate, “First approach to the analysis of the lasing conditions in POLIPHEM© structures,” Mol. Cryst. Liq. Cryst. 488, 135-147 (2008).
[CrossRef]

2006

Z. Li, Z. Zhang, T. Emery, A. Scherer, and D. Psaltis, “Single mode optofluidic distributed feedback dye laser,” Opt. Express 14, 696-701 (2006).
[CrossRef] [PubMed]

M. Gersborg-Hansen and A. Kriensen, “Optofluidic third order distributed feedback dye laser,” Appl. Phys. Lett. 89, 103518 (2006).
[CrossRef]

1999

M. Akbari, S. Shahabadi, and K. Schünemann, “A rigorous two-dimensional field analysis of DFB structures,” Prog. Electromagn. Res. PIER 22, 197-212 (1999).
[CrossRef]

1974

S. Wang, “Principles of distributed feedback and distributed Bragg reflector lasers,” IEEE J. Quantum Electron. 10, 413-424 (1974).
[CrossRef]

1972

J. E. Bjorkholm and C. V. Shank, “Higher-order distributed feedback oscillators,” Appl. Phys. Lett. 20, 306-308 (1972).
[CrossRef]

H. Kogelnik and C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327-2335 (1972).
[CrossRef]

1970

R. S. Chu and T. Tamir, “Guided-wave theory of light diffraction by acoustic microwaves,” IEEE Trans. Microwave Theory Tech. MTT-18, 486-504 (1970).

1969

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2910-2947 (1969).

Abbate, G.

M. V. Vasnetsov, V. Y. Bazhenov, S. S. Slussarenko, and G. Abbate, “Coupled-wave analysis of second-order Bragg diffraction. I. Reflection-type phase gratings,” J. Opt. Soc. Am. B 26, 684-690 (2009).
[CrossRef]

M. V. Vasnetsov, V. Y. Bazhenov, S. S. Slussarenko, O. Sakhno, J. Stumpe, and G. Abbate, “First approach to the analysis of the lasing conditions in POLIPHEM© structures,” Mol. Cryst. Liq. Cryst. 488, 135-147 (2008).
[CrossRef]

M. V. Vasnetsov, S. S. Slussarenko, O. Sakhno, J. Stumpe, S. S. Slussarenko, Jr., and G. Abbate, “Lasing by second-order Bragg diffraction in dye-doped POLIPHEM phase gratings,” Mol. Cryst. Liq. Cryst. (to be published).

Akbari, M.

M. Akbari, S. Shahabadi, and K. Schünemann, “A rigorous two-dimensional field analysis of DFB structures,” Prog. Electromagn. Res. PIER 22, 197-212 (1999).
[CrossRef]

Bazhenov, V. Y.

M. V. Vasnetsov, V. Y. Bazhenov, S. S. Slussarenko, and G. Abbate, “Coupled-wave analysis of second-order Bragg diffraction. I. Reflection-type phase gratings,” J. Opt. Soc. Am. B 26, 684-690 (2009).
[CrossRef]

M. V. Vasnetsov, V. Y. Bazhenov, S. S. Slussarenko, O. Sakhno, J. Stumpe, and G. Abbate, “First approach to the analysis of the lasing conditions in POLIPHEM© structures,” Mol. Cryst. Liq. Cryst. 488, 135-147 (2008).
[CrossRef]

Bjorkholm, J. E.

J. E. Bjorkholm and C. V. Shank, “Higher-order distributed feedback oscillators,” Appl. Phys. Lett. 20, 306-308 (1972).
[CrossRef]

Chu, R. S.

R. S. Chu and T. Tamir, “Guided-wave theory of light diffraction by acoustic microwaves,” IEEE Trans. Microwave Theory Tech. MTT-18, 486-504 (1970).

Emery, T.

Gersborg-Hansen, M.

M. Gersborg-Hansen and A. Kriensen, “Optofluidic third order distributed feedback dye laser,” Appl. Phys. Lett. 89, 103518 (2006).
[CrossRef]

Kogelnik, H.

H. Kogelnik and C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327-2335 (1972).
[CrossRef]

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2910-2947 (1969).

Kriensen, A.

M. Gersborg-Hansen and A. Kriensen, “Optofluidic third order distributed feedback dye laser,” Appl. Phys. Lett. 89, 103518 (2006).
[CrossRef]

Li, Z.

W. Song, A. E. Vasdekis, Z. Li, and D. Psaltis, “Low-order distributed feedback optofluidic dye laser with reduced threshold,” Appl. Phys. Lett. 94, 051117 (2009).
[CrossRef]

Z. Li, Z. Zhang, T. Emery, A. Scherer, and D. Psaltis, “Single mode optofluidic distributed feedback dye laser,” Opt. Express 14, 696-701 (2006).
[CrossRef] [PubMed]

Psaltis, D.

W. Song, A. E. Vasdekis, Z. Li, and D. Psaltis, “Low-order distributed feedback optofluidic dye laser with reduced threshold,” Appl. Phys. Lett. 94, 051117 (2009).
[CrossRef]

Z. Li, Z. Zhang, T. Emery, A. Scherer, and D. Psaltis, “Single mode optofluidic distributed feedback dye laser,” Opt. Express 14, 696-701 (2006).
[CrossRef] [PubMed]

Sakhno, O.

M. V. Vasnetsov, V. Y. Bazhenov, S. S. Slussarenko, O. Sakhno, J. Stumpe, and G. Abbate, “First approach to the analysis of the lasing conditions in POLIPHEM© structures,” Mol. Cryst. Liq. Cryst. 488, 135-147 (2008).
[CrossRef]

M. V. Vasnetsov, S. S. Slussarenko, O. Sakhno, J. Stumpe, S. S. Slussarenko, Jr., and G. Abbate, “Lasing by second-order Bragg diffraction in dye-doped POLIPHEM phase gratings,” Mol. Cryst. Liq. Cryst. (to be published).

Scherer, A.

Schünemann, K.

M. Akbari, S. Shahabadi, and K. Schünemann, “A rigorous two-dimensional field analysis of DFB structures,” Prog. Electromagn. Res. PIER 22, 197-212 (1999).
[CrossRef]

Shahabadi, S.

M. Akbari, S. Shahabadi, and K. Schünemann, “A rigorous two-dimensional field analysis of DFB structures,” Prog. Electromagn. Res. PIER 22, 197-212 (1999).
[CrossRef]

Shank, C. V.

H. Kogelnik and C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327-2335 (1972).
[CrossRef]

J. E. Bjorkholm and C. V. Shank, “Higher-order distributed feedback oscillators,” Appl. Phys. Lett. 20, 306-308 (1972).
[CrossRef]

Slussarenko, S. S.

M. V. Vasnetsov, V. Y. Bazhenov, S. S. Slussarenko, and G. Abbate, “Coupled-wave analysis of second-order Bragg diffraction. I. Reflection-type phase gratings,” J. Opt. Soc. Am. B 26, 684-690 (2009).
[CrossRef]

M. V. Vasnetsov, V. Y. Bazhenov, S. S. Slussarenko, O. Sakhno, J. Stumpe, and G. Abbate, “First approach to the analysis of the lasing conditions in POLIPHEM© structures,” Mol. Cryst. Liq. Cryst. 488, 135-147 (2008).
[CrossRef]

M. V. Vasnetsov, S. S. Slussarenko, O. Sakhno, J. Stumpe, S. S. Slussarenko, Jr., and G. Abbate, “Lasing by second-order Bragg diffraction in dye-doped POLIPHEM phase gratings,” Mol. Cryst. Liq. Cryst. (to be published).

M. V. Vasnetsov, S. S. Slussarenko, O. Sakhno, J. Stumpe, S. S. Slussarenko, Jr., and G. Abbate, “Lasing by second-order Bragg diffraction in dye-doped POLIPHEM phase gratings,” Mol. Cryst. Liq. Cryst. (to be published).

Song, W.

W. Song, A. E. Vasdekis, Z. Li, and D. Psaltis, “Low-order distributed feedback optofluidic dye laser with reduced threshold,” Appl. Phys. Lett. 94, 051117 (2009).
[CrossRef]

Stumpe, J.

M. V. Vasnetsov, V. Y. Bazhenov, S. S. Slussarenko, O. Sakhno, J. Stumpe, and G. Abbate, “First approach to the analysis of the lasing conditions in POLIPHEM© structures,” Mol. Cryst. Liq. Cryst. 488, 135-147 (2008).
[CrossRef]

M. V. Vasnetsov, S. S. Slussarenko, O. Sakhno, J. Stumpe, S. S. Slussarenko, Jr., and G. Abbate, “Lasing by second-order Bragg diffraction in dye-doped POLIPHEM phase gratings,” Mol. Cryst. Liq. Cryst. (to be published).

Tamir, T.

R. S. Chu and T. Tamir, “Guided-wave theory of light diffraction by acoustic microwaves,” IEEE Trans. Microwave Theory Tech. MTT-18, 486-504 (1970).

Vasdekis, A. E.

W. Song, A. E. Vasdekis, Z. Li, and D. Psaltis, “Low-order distributed feedback optofluidic dye laser with reduced threshold,” Appl. Phys. Lett. 94, 051117 (2009).
[CrossRef]

Vasnetsov, M. V.

M. V. Vasnetsov, V. Y. Bazhenov, S. S. Slussarenko, and G. Abbate, “Coupled-wave analysis of second-order Bragg diffraction. I. Reflection-type phase gratings,” J. Opt. Soc. Am. B 26, 684-690 (2009).
[CrossRef]

M. V. Vasnetsov, V. Y. Bazhenov, S. S. Slussarenko, O. Sakhno, J. Stumpe, and G. Abbate, “First approach to the analysis of the lasing conditions in POLIPHEM© structures,” Mol. Cryst. Liq. Cryst. 488, 135-147 (2008).
[CrossRef]

M. V. Vasnetsov, S. S. Slussarenko, O. Sakhno, J. Stumpe, S. S. Slussarenko, Jr., and G. Abbate, “Lasing by second-order Bragg diffraction in dye-doped POLIPHEM phase gratings,” Mol. Cryst. Liq. Cryst. (to be published).

Wang, S.

S. Wang, “Principles of distributed feedback and distributed Bragg reflector lasers,” IEEE J. Quantum Electron. 10, 413-424 (1974).
[CrossRef]

Yariv, A.

A. Yariv and P. Yeh, Photonics: Optical Electronics in Modern Communications, 6th ed. (Oxford U. Press, 2007).

Yeh, P.

A. Yariv and P. Yeh, Photonics: Optical Electronics in Modern Communications, 6th ed. (Oxford U. Press, 2007).

Zhang, Z.

Appl. Phys. Lett.

J. E. Bjorkholm and C. V. Shank, “Higher-order distributed feedback oscillators,” Appl. Phys. Lett. 20, 306-308 (1972).
[CrossRef]

W. Song, A. E. Vasdekis, Z. Li, and D. Psaltis, “Low-order distributed feedback optofluidic dye laser with reduced threshold,” Appl. Phys. Lett. 94, 051117 (2009).
[CrossRef]

M. Gersborg-Hansen and A. Kriensen, “Optofluidic third order distributed feedback dye laser,” Appl. Phys. Lett. 89, 103518 (2006).
[CrossRef]

Bell Syst. Tech. J.

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2910-2947 (1969).

IEEE J. Quantum Electron.

S. Wang, “Principles of distributed feedback and distributed Bragg reflector lasers,” IEEE J. Quantum Electron. 10, 413-424 (1974).
[CrossRef]

IEEE Trans. Microwave Theory Tech.

R. S. Chu and T. Tamir, “Guided-wave theory of light diffraction by acoustic microwaves,” IEEE Trans. Microwave Theory Tech. MTT-18, 486-504 (1970).

J. Appl. Phys.

H. Kogelnik and C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327-2335 (1972).
[CrossRef]

J. Opt. Soc. Am. B

Mol. Cryst. Liq. Cryst.

M. V. Vasnetsov, S. S. Slussarenko, O. Sakhno, J. Stumpe, S. S. Slussarenko, Jr., and G. Abbate, “Lasing by second-order Bragg diffraction in dye-doped POLIPHEM phase gratings,” Mol. Cryst. Liq. Cryst. (to be published).

M. V. Vasnetsov, V. Y. Bazhenov, S. S. Slussarenko, O. Sakhno, J. Stumpe, and G. Abbate, “First approach to the analysis of the lasing conditions in POLIPHEM© structures,” Mol. Cryst. Liq. Cryst. 488, 135-147 (2008).
[CrossRef]

Opt. Express

Prog. Electromagn. Res.

M. Akbari, S. Shahabadi, and K. Schünemann, “A rigorous two-dimensional field analysis of DFB structures,” Prog. Electromagn. Res. PIER 22, 197-212 (1999).
[CrossRef]

Other

A. Yariv and P. Yeh, Photonics: Optical Electronics in Modern Communications, 6th ed. (Oxford U. Press, 2007).

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Figures (8)

Fig. 1
Fig. 1

Schematic representation of a grating within a slab of thickness h and refractive index n between the substrates with the refractive index n s < n . The grating period Λ is close to the wavelength in the medium λ / n . The field is represented in the form of superposition of two symmetric plane waves, forming a waveguide mode, whose profile is schematically shown.

Fig. 2
Fig. 2

Calculated according to Eq. (25) dependence of the threshold gain factor value α ( K / k ) d on the modulation term ( 2 κ n κ α / K 3 ) d . Curve labeled “in phase” is obtained for plus sign in Eq. (27); curve labeled “out of phase” is obtained for the minus sign. Open and closed circles indicate the parameters used below for the calculation of a transmitted wave amplitude.

Fig. 3
Fig. 3

Calculated contours of transmission (intensity) | U ( d ) / U | 2 . Plot 1 corresponds to pure phase modulation and zero gain α. Parameters k 2 = 1 , K 2 = 0.75 , κ n 2 = 0.013 , and d = 25 Λ are the same as used before for the diffraction efficiency calculation (Fig. 4 in Paper I [11]). Plot 2 is obtained with the addition of gain α / k = 10 3 in Eq. (A1). Dashed lines show the corresponding uniform intensity gain factors 1 (no amplification) and exp ( 2 α k d / K ) = 1.52 .

Fig. 4
Fig. 4

Calculated according to Eq. (A1) transmission contour 1 near the oscillation threshold for the mixed grating (out-of-phase matching). The parameters of computation are α K / k = 0.02 and κ α = κ n = 0.023 (other parameters are the same as used before). Contour 2 is computed with the same parameters for the in-phase matching case. The dashed line shows the level of the uniform intensity gain factor.

Fig. 5
Fig. 5

Comparison of calculated transmission contours for a mixed grating ( κ α = κ n ) in the case of the absence of uniform gain. Contour 1 corresponds to out-of-phase matching; contour 2 corresponds to in-phase matching. Dashed line shows the unity transmission for a medium without modulation.

Fig. 6
Fig. 6

Calculated transmission contours for pure amplitude grating ( κ α = 0.05 ) without uniform gain (contour 1), the same grating with uniform gain α = 0.005 (contour 2), and for mixed grating (out-of-phase matching) with the parameters α = 0.005 , κ n = 0.025 (contour 3).

Fig. 7
Fig. 7

Determination of the roots of Eq. (B2) on a map with coordinates a = Re [ σ z 1 d ] , b = Im [ σ z 1 d ] . Shadow areas correspond to the negative values Re [ sinc 2 ( a + i b ) ] < 0 , solid curves show zeros of Im [ sinc 2 ( a + i b ) ] , dashed lcurves are drawn for the particular equality Re [ sinc 2 ( a + i b ) ] = 4 .

Fig. 8
Fig. 8

Calculated values of the threshold gain γ d and detuning factors δ m d for angular-shifted oscillating modes (pure phase second-order Bragg DFB oscillator, χ d = κ n 2 d / K 3 = 1 ). The dashed vertical line corresponds to the center of the efficient resonance, δ d = χ d / 2 .

Equations (43)

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U   exp ( i k x x + α k x x k ) exp ( i k z z + α k z z k ) = U   exp ( i k r + α r ) ,
[ U   exp ( i k x x + α k x x k ) exp ( i k z z + α k z z k ) ] [ c .c . ] = U 2   exp ( 2 α r ) ,
k 0 2 ε ( z ) = p 2 k 2 2 i p k Δ α   cos ( K z ) .
E ( x , z ) = E ( z ) exp ( i p k x x )
d 2 E ( z ) d z 2 = [ p 2 k z 2 + 2 i p k Δ α   cos ( K z ) ] E ( z ) .
E ( z ) = u   exp ( i σ z z + i K z ) + w   exp ( i σ z z ) + v   exp ( i σ z z i K z ) ,
( p 2 k z 2 σ z 2 2 K σ z K 2 ) u = i p κ α w ,
( p 2 k z 2 σ z 2 ) w = i p κ α ( u + v ) ,
( p 2 k z 2 σ z 2 + 2 K σ z K 2 ) v = i p κ α w ,
( p 2 k z 2 σ z 2 ) [ ( p 2 k z 2 σ z 2 K 2 ) 2 4 K 2 σ z 2 ] = 2 p 2 κ α 2 ( p 2 k z 2 σ z 2 K 2 ) .
σ z 2 = ( p 2 k z 2 K 2 ) 2 + 2 κ α 2 K 2 ( p 2 k z 2 K 2 ) 4 K 2 .
( k 2 K 2 ) 1 / 2 < k x < ( k 2 K 2 + 2 κ α 2 K 2 ) 1 / 2 .
κ α 2 K 3 < Δ k z < 0.
σ z 2 = ( Δ k z i α K k ) 2 + κ α 2 K 3 ( Δ k z i α K k ) .
κ 2 = ( κ n ± i κ α ) 2 ,
σ z 2 = ( Δ k z i α K k ) ( Δ k z i α K k + κ α 2 κ n 2 ± 2 i κ α κ n K 3 ) .
U ( d ) = U q 1 q 2 q 1   exp ( i σ z 1 d ) q 2   exp ( i σ z 1 d ) exp ( α d k 2 K 2 k K ) ,
q 1 = Δ k z i α K k σ z 1 Δ k z i α K k + σ z 1 = ( Δ k z i α K k σ z 1 ) 2 ( Δ k z i α K k ) 2 σ z 1 2 ,
q 2 = ( Δ k z i α K k + σ z 1 ) 2 ( Δ k z i α K k ) 2 σ z 1 2 .
( Δ k z i α K k σ z 1 ) 2 exp ( i σ z 1 d ) = ( Δ k z i α K k + σ z 1 ) 2 exp ( i σ z 1 d ) .
i ( Δ k z i α K k κ 2 2 K 3 ) sin ( σ z 1 d ) = σ z 1   cos ( σ z 1 d ) .
σ z 2 = ( α K k ) 2 ( κ α 2 2 K 3 ) 2 ,
[ ( α K k ) 2 + ( κ α 2 2 K 3 ) 2 ] 1 / 2 = α K k tanh { [ ( α K k ) 2 + ( κ α 2 2 K 3 ) 2 ] 1 / 2 d } .
( α K k ) 2 = ( κ α 2 2 K 3 ) 2 cosh 2 { [ ( α K k ) 2 + ( κ α 2 2 K 3 ) 2 ] 1 / 2 d } .
σ z 2 = α K k ( α K k 2 κ n κ α K 3 ) .
[ ( α K k ) 2 2 α K k κ n κ α K 3 ] 1 / 2 = ( α K k κ n κ α K 3 ) tan { [ ( α K k ) 2 2 α K k κ n κ α K 3 ] 1 / 2 d } .
α K k d = ψ cos   ψ + 1 sin   ψ ,
κ n κ α K 3 d = ψ sin   ψ ,
[ ( α K k ) 2 2 α K k κ n κ α K 3 ] 1 / 2 = ( α K k κ n κ α K 3 ) tanh { [ ( α K k ) 2 2 α K k κ n κ α K 3 ] 1 / 2 d } .
α K k d = ψ cosh   ψ ± 1 sinh   ψ ,
κ n κ α K 3 d = ψ sinh   ψ ,
σ z 2 = ( δ i γ ) ( δ i γ χ ) ,
σ z 1 = [ ( δ i γ ) ( δ i γ χ ) ] 1 / 2 .
U ( d ) = [ u 1   exp ( σ z 1 d ) + u 2   exp ( σ z 1 d ) ] exp [ α k 2 K 2 k K d ] .
q 1 = ( δ i γ σ z 1 ) 2 ( δ i γ ) 2 σ z 1 2 ,
q 2 = ( δ i γ + σ z 1 ) 2 ( δ i γ ) 2 σ z 1 2 .
U ( d ) = U 4 ( δ i γ ) σ z 1 2 i [ ( δ i γ ) 2 + σ z 1 2 ] sin ( σ z 1 d ) 4 ( δ i γ ) σ z 1   cos ( σ z 1 d ) exp ( α k 2 K 2 k K d ) = U σ z 1 i ( δ i γ χ 2 ) sin ( σ z 1 d ) σ z 1   cos ( σ z 1 d ) exp ( α k 2 K 2 k K d ) .
V ( 0 ) = U i χ   sin ( σ z 1 d ) 2 i ( δ i γ χ 2 ) sin ( σ z 1 d ) 2 σ z 1   cos ( σ z 1 d ) exp ( α k 2 K 2 k K ) .
( δ i γ χ 2 ) 2 sin 2 ( σ z 1 d ) = σ z 1 2 cos 2 ( σ z 1 d ) .
[ sin ( σ z 1 d ) σ z 1 d ] 2 = 4 χ 2 d 2 .
σ z 1 = [ ( δ i γ + χ 2 ) ( δ i γ χ 2 ) ] 1 / 2 = [ ( δ i γ ) 2 χ 2 4 ] ,
( δ ) 2 d 2 γ 2 d 2 = a 2 + b 2 + χ 2 d 2 4 ,
δ γ d 2 = ± a b .

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