Abstract

A basic mirrored counterpropagating quasi-phase-matched device is studied with a pulsed input fundamental plane wave using the method of lines and the relaxation method. Several examples are given under varying spatial pulse length to device length ratios. An approximate upper bound on the device length is established from this study for practical pulsed applications; the largest usable length is approximately the same as the spatial length of the pulse.

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References

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  1. J. A. Armstrong, N. Bloembergen, J. Ducuino, and P. S. Pershan, "Interactions between light waves in a nonlinear dielectric," Phys. Rev. 127, 1918-1939 (1962).
    [CrossRef]
  2. G. D. Miller, R. G. Batchko, W. M. Tulloch, D. R. Weise, M. M. Fejer, and R. L. Byer, "42%-efficient single- pass CW second-harmonic generation in periodically poled lithium niobate," Opt. Lett. 22, 1834-1836 (1997).
    [CrossRef]
  3. 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]
  4. M. Houe and P. D. Townsend, "An introduction to methods of periodic poling for second- harmonic generation," J. Phys. D - Appl. Phys. 28, 1747-1763 (1995).
    [CrossRef]
  5. J. Pierce and D. Lowenthal, "Periodically poled materials & devices," Lasers & Opt. 16, 25-27 (1997).
  6. Y. J. Ding and J. B. Khurgin, "Second-harmonic generation based on quasi-phase matching: a novel configuration," Opt. Lett. 21, 1445-1447 (1996).
    [CrossRef] [PubMed]
  7. G. D. Landry and T. A. Maldonado, "Second harmonic generation and cascaded second order processes in a counterpropagating quasi-phase-matched device," Appl. Opt. 37, 7809-7820 (1998).
    [CrossRef]
  8. G. D. Landry and T. A. Maldonado, "Efficient nonlinear phase shifts due to cascaded second order processes in a counter-propagating quasi-phase-matched configuration," Opt. Lett. 22, 1400-1402 (1997).
    [CrossRef]
  9. G. D. Landry and T. A. Maldonado, "Switching and second harmonic generation using counterpropagating quasi- phase-matching in a mirrorless configuration," Journal of Lightwave Technology 17, 316-327 (1999).
    [CrossRef]
  10. W. H. Press, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in C: The Art of Scientific Computing (Cambridge University Press, New York, 1992).
  11. D. Zwillinger, Handbook of Differential Equations, 2nd ed. (Academic Press, San Diego, 1992).
  12. V. G. Dmitriev, G. G. Gurzadyan, and D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals, 2nd ed. (Springer, Berlin, 1997).

Other (12)

J. A. Armstrong, N. Bloembergen, J. Ducuino, and P. S. Pershan, "Interactions between light waves in a nonlinear dielectric," Phys. Rev. 127, 1918-1939 (1962).
[CrossRef]

G. D. Miller, R. G. Batchko, W. M. Tulloch, D. R. Weise, M. M. Fejer, and R. L. Byer, "42%-efficient single- pass CW second-harmonic generation in periodically poled lithium niobate," Opt. Lett. 22, 1834-1836 (1997).
[CrossRef]

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]

M. Houe and P. D. Townsend, "An introduction to methods of periodic poling for second- harmonic generation," J. Phys. D - Appl. Phys. 28, 1747-1763 (1995).
[CrossRef]

J. Pierce and D. Lowenthal, "Periodically poled materials & devices," Lasers & Opt. 16, 25-27 (1997).

Y. J. Ding and J. B. Khurgin, "Second-harmonic generation based on quasi-phase matching: a novel configuration," Opt. Lett. 21, 1445-1447 (1996).
[CrossRef] [PubMed]

G. D. Landry and T. A. Maldonado, "Second harmonic generation and cascaded second order processes in a counterpropagating quasi-phase-matched device," Appl. Opt. 37, 7809-7820 (1998).
[CrossRef]

G. D. Landry and T. A. Maldonado, "Efficient nonlinear phase shifts due to cascaded second order processes in a counter-propagating quasi-phase-matched configuration," Opt. Lett. 22, 1400-1402 (1997).
[CrossRef]

G. D. Landry and T. A. Maldonado, "Switching and second harmonic generation using counterpropagating quasi- phase-matching in a mirrorless configuration," Journal of Lightwave Technology 17, 316-327 (1999).
[CrossRef]

W. H. Press, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in C: The Art of Scientific Computing (Cambridge University Press, New York, 1992).

D. Zwillinger, Handbook of Differential Equations, 2nd ed. (Academic Press, San Diego, 1992).

V. G. Dmitriev, G. G. Gurzadyan, and D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals, 2nd ed. (Springer, Berlin, 1997).

Supplementary Material (2)

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Figures (7)

Fig. 1.
Fig. 1.

The wavevector matching diagrams for (a) forward-QPM (f-QPM); (b) backward-QPM (b-QPM); (c) counterpropagating-QPM (c-QPM); and (d) surface-emitting-QPM (se-QPM). There are two simultaneously phase-matchable processes for cases (a), (b), and (d) while there are six for case (c). All cases except case (d) are collinear. The fundamental, second harmonic, and grating wavevectors are represented by red, blue, and black arrows, respectively.

Fig. 2.
Fig. 2.

A schematic of the counterpropagating quasi-phase-matched (c-QPM) device under study. The input/output interface is located at ρ=0. A highly reflective mirror is located at ρ=1. The domain inversion period is Λ.

Fig. 3.
Fig. 3.

The input (gray), output FF (red), and output SH (blue) pulse intensity envelopes for a 1 cm long KTP mirrored c-QPM device with an ideal mirror at perfect phase matching. The input wavelength is 1.064 µm with a 10 ns pulse (FWHM) and peak intensity of Γ2I0 =(π/4)2. The corresponding device to pulse ratio is Ξω=0.00718. The simulation parameters are Ns =30, Nt =100, and NEt =0.

Fig. 4.
Fig. 4.

The input (gray), output FF (red), and output SH (blue) pulse intensity envelopes for the same parameters of Fig. 3 except the input pulse width is 100 ps pulse (FWHM). The corresponding device to pulse ratio is Ξω=0.718. The sampling rates are (a) {Ns, Nt, NEt }={30, 100, 0} giving a numerical leakage of 8.38% and (b) {Ns, Nt, NEt }={30, 300, 0} giving a numerical leakage of 3.08%.

Fig. 5.
Fig. 5.

Normalized field profiles inside the device for the simulation parameters used in Fig. 4(b). The field magnitudes are |A(ρ)| (red), |B(ρ)| (orange), |C(ρ)| (blue), and |D(ρ)| (cyan). (363 KB)

Fig. 6.
Fig. 6.

The input (gray), output FF (red), and output SH (blue) pulse intensity envelopes for a 1 cm long KTP mirrored c-QPM device with an ideal mirror at perfect phase matching. The input wavelength is 1.064 mm with a 50 ps pulse (FWHM) and peak intensity of Γ2I0 =(π/4)2. The corresponding device to pulse ratio is Ξω=1.437. The simulation parameters are Ns =40 Nt =600, and NEt =300.

Fig. 7.
Fig. 7.

Normalized field profiles inside the device for the simulation parameters used in Fig. 6. The field magnitudes are |A(ρ)| (red), |B(ρ)| (orange), |C(ρ)| (blue), and |D(ρ)| (cyan). (1.19 MB)

Equations (28)

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E ( z , t ) = E 0 exp [ t 2 2 σ 2 ] exp ( j k z ) .
A ( ρ ) = E ω + ( ρ ) 2 Z 0 n ω , C ( ρ ) = E 2 ω + ( ρ ) 2 Z 0 n 2 ω exp ( j Δ κ ) ,
B ( ρ ) = E ω ( ρ ) 2 Z 0 n ω , D ( ρ ) = E 2 ω ( ρ ) 2 Z 0 n 2 ω exp ( + j Δ κ ) ,
Δ κ = ( Δ k ) L = ( k 2 ω K ) L = ( k 2 ω 2 π Λ ) L .
A ( ρ = 0 , t ) = I 0 exp [ t 2 2 σ 2 ] ,
σ = t FHWM 2 ln ( 2 ) ,
g = t FHWM 2 ln ( 2 ) .
t N = t g ,
ω N = ω g .
A ( ρ = 0 , t N ) = I 0 exp [ t N 2 ] .
Ξ ω = n ω L cg .
A ( ρ , t N ) ρ = + j Ξ ω 2 ω N [ 2 A ( ρ , t N ) t N 2 + j 2 ω N A ( ρ , t N ) t N ]
+ Γ ω N 2 [ 2 t N 2 { B * ( ρ , t N ) [ C ( ρ , t N ) + D ( ρ , t N ) ] } + j 2 ω N t N { B * ( ρ , t N ) [ C ( ρ , t N ) + D ( ρ , t N ) ] } ω N 2 B * ( ρ , t N ) [ C ( ρ , t N ) + D ( ρ , t N ) ] ] ,
B ( ρ , t N ) ρ = j Ξ ω 2 ω N [ 2 B ( ρ , t N ) t N 2 + j 2 ω N B ( ρ , t N ) t N ]
Γ ω N 2 [ 2 t N 2 { A * ( ρ , t N ) [ C ( ρ , t N ) + D ( ρ , t N ) ] } + j 2 ω N t N { A * ( ρ , t N ) [ C ( ρ , t N ) + D ( ρ , t N ) ] } ω N 2 { A * ( ρ , t N ) [ C ( ρ , t N ) + D ( ρ , t N ) ] } ] ,
C ( ρ , t N ) ρ = + j Ξ ω 4 ω N n 2 ω n ω [ 2 C ( ρ , t N ) t N 2 + j 4 ω N C ( ρ , t N ) t N ]
+ 2 Γ ( 2 ω N ) 2 { 2 t N 2 [ A ( ρ , t N ) B ( ρ , t N ) ] + j 4 ω N t N [ A ( ρ , t N ) B ( ρ , t N ) ] ( 2 ω N ) 2 [ A ( ρ , t N ) B ( ρ , t N ) ] } j ( Δ κ ) C ( ρ , t N ) ,
D ( ρ , t N ) ρ = j Ξ ω 4 ω N n 2 ω n ω [ 2 D ( ρ , t N ) t N 2 + j 4 ω N D ( ρ , t N ) t N ]
2 Γ ( 2 ω N ) 2 { 2 t N 2 [ A ( ρ , t N ) B ( ρ , t N ) ] + j 4 ω N t N [ A ( ρ , t N ) B ( ρ , t N ) ] ( 2 ω N ) 2 [ A ( ρ , t N ) B ( ρ , t N ) ] } + j ( Δ κ ) D ( ρ , t N ) ,
Γ = ω ( 2 π ) d 0 L c n ω 2 Z 0 n 2 ω ,
t N p = 4 + p Δ t N ,
p = 0 , 1 , , ( N t 1 ) ,
Δ t N = 8 ( N t 1 ) 2 .
f ( ρ , t N ) t N f ( ρ , t N ) t N = t N p f ( ρ , t N ) t N = t N p 1 Δ t N ,
2 f ( ρ , t N ) t N 2 f ( ρ , t N ) t N = t N p 2 f ( ρ , t N ) t N = t N p 1 + f ( ρ , t N ) t N = t N p 2 ( Δ t N ) 2 ,
ρ = q Δ ρ ,
q = 0 , 1 , , ( N s 1 ) ,
Δ ρ = 1 N s 1 .

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