Abstract

We develop a higher-order method for non-paraxial beam propagation based on the wide-angle split-step spectral (WASSS) method previously reported [Clark and Thomas, Opt. Quantum. Electron., 41, 849 (2010)]. The higher-order WASSS (HOWASSS) method approximates the Helmholtz equation by keeping terms up to third-order in the propagation step size, in the Magnus expansion. A symmetric exponential operator splitting technique is used to simplify the resulting exponential operators. The HOWASSS method is applied to the problem of waveguide propagation, where an analytical solution is known, to demonstrate the performance and accuracy of the method. The performance enhancement gained by implementing the HOWASSS method on a graphics processing unit (GPU) is demonstrated. When highly accurate results are required the HOWASSS method is shown to be substantially faster than the WASSS method.

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References

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  1. G. R. Hadley, “Multistep method for wide-angle beam propagation,” Opt. Lett.17, 1743–1745 (1992).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  3. Y. Y. Lu and P. L. Ho, “Beam propagation method using a [(p−1)/p] Padé approximant of the propagator,” Opt. Lett.27, 683–685 (2002).
    [CrossRef]
  4. A. Sharma and A. Agrawal, “New method for nonparaxial beam propagation,” J. Opt. Soc. Am. B21, 1082–1087 (2004).
    [CrossRef]
  5. A. Sharma and A. Agrawal, “Non-paraxial split-step finite-difference method for beam propagation,” Opt. Quantum. Electron.38, 19–34 (2006).
    [CrossRef]
  6. C. D. Clark and R. Thomas, “Wide-angle split-step spectral method for 2D or 3D beam propagation,” Opt. Quantum. Electron.41, 849–857 (2010).
    [CrossRef]
  7. M. Guizar-Sicairos and J. C. Gutiérrez-Vega, “Computation of quasi-discrete Hankel transforms of integer order for propagating optical wave fields,” J. Opt. Soc. Am. A21, 53–58 (2004).
    [CrossRef]
  8. W. Magnus, “On the exponential solution of differential equations for a linear operator,” Comm. Pure Appl. Math.7, 649–673 (1954).
    [CrossRef]
  9. M. Bauer, R. Chetrite, K. Ebrahimi-Fard, and F. Patras, “Time-ordering and a generalized Magnus expansion,” Lett. Math. Phys.103, 331–350 (2012).
    [CrossRef]
  10. M. J. Adams, An Introduction to Optical Waveguides (Wiley, 1981).

2012 (1)

M. Bauer, R. Chetrite, K. Ebrahimi-Fard, and F. Patras, “Time-ordering and a generalized Magnus expansion,” Lett. Math. Phys.103, 331–350 (2012).
[CrossRef]

2010 (1)

C. D. Clark and R. Thomas, “Wide-angle split-step spectral method for 2D or 3D beam propagation,” Opt. Quantum. Electron.41, 849–857 (2010).
[CrossRef]

2008 (1)

2006 (1)

A. Sharma and A. Agrawal, “Non-paraxial split-step finite-difference method for beam propagation,” Opt. Quantum. Electron.38, 19–34 (2006).
[CrossRef]

2004 (2)

2002 (1)

1992 (1)

1954 (1)

W. Magnus, “On the exponential solution of differential equations for a linear operator,” Comm. Pure Appl. Math.7, 649–673 (1954).
[CrossRef]

Adams, M. J.

M. J. Adams, An Introduction to Optical Waveguides (Wiley, 1981).

Agrawal, A.

A. Sharma and A. Agrawal, “Non-paraxial split-step finite-difference method for beam propagation,” Opt. Quantum. Electron.38, 19–34 (2006).
[CrossRef]

A. Sharma and A. Agrawal, “New method for nonparaxial beam propagation,” J. Opt. Soc. Am. B21, 1082–1087 (2004).
[CrossRef]

Bauer, M.

M. Bauer, R. Chetrite, K. Ebrahimi-Fard, and F. Patras, “Time-ordering and a generalized Magnus expansion,” Lett. Math. Phys.103, 331–350 (2012).
[CrossRef]

Bienstman, P.

Chetrite, R.

M. Bauer, R. Chetrite, K. Ebrahimi-Fard, and F. Patras, “Time-ordering and a generalized Magnus expansion,” Lett. Math. Phys.103, 331–350 (2012).
[CrossRef]

Clark, C. D.

C. D. Clark and R. Thomas, “Wide-angle split-step spectral method for 2D or 3D beam propagation,” Opt. Quantum. Electron.41, 849–857 (2010).
[CrossRef]

Ebrahimi-Fard, K.

M. Bauer, R. Chetrite, K. Ebrahimi-Fard, and F. Patras, “Time-ordering and a generalized Magnus expansion,” Lett. Math. Phys.103, 331–350 (2012).
[CrossRef]

Godoy-Rubio, R.

Guizar-Sicairos, M.

Gutiérrez-Vega, J. C.

Hadley, G. R.

Ho, P. L.

Le, K. Q.

Lu, Y. Y.

Magnus, W.

W. Magnus, “On the exponential solution of differential equations for a linear operator,” Comm. Pure Appl. Math.7, 649–673 (1954).
[CrossRef]

Patras, F.

M. Bauer, R. Chetrite, K. Ebrahimi-Fard, and F. Patras, “Time-ordering and a generalized Magnus expansion,” Lett. Math. Phys.103, 331–350 (2012).
[CrossRef]

Sharma, A.

A. Sharma and A. Agrawal, “Non-paraxial split-step finite-difference method for beam propagation,” Opt. Quantum. Electron.38, 19–34 (2006).
[CrossRef]

A. Sharma and A. Agrawal, “New method for nonparaxial beam propagation,” J. Opt. Soc. Am. B21, 1082–1087 (2004).
[CrossRef]

Thomas, R.

C. D. Clark and R. Thomas, “Wide-angle split-step spectral method for 2D or 3D beam propagation,” Opt. Quantum. Electron.41, 849–857 (2010).
[CrossRef]

Comm. Pure Appl. Math. (1)

W. Magnus, “On the exponential solution of differential equations for a linear operator,” Comm. Pure Appl. Math.7, 649–673 (1954).
[CrossRef]

J. Opt. Soc. Am. A (1)

J. Opt. Soc. Am. B (1)

A. Sharma and A. Agrawal, “New method for nonparaxial beam propagation,” J. Opt. Soc. Am. B21, 1082–1087 (2004).
[CrossRef]

Lett. Math. Phys. (1)

M. Bauer, R. Chetrite, K. Ebrahimi-Fard, and F. Patras, “Time-ordering and a generalized Magnus expansion,” Lett. Math. Phys.103, 331–350 (2012).
[CrossRef]

Opt. Express (1)

Opt. Lett. (2)

Opt. Quantum. Electron. (2)

A. Sharma and A. Agrawal, “Non-paraxial split-step finite-difference method for beam propagation,” Opt. Quantum. Electron.38, 19–34 (2006).
[CrossRef]

C. D. Clark and R. Thomas, “Wide-angle split-step spectral method for 2D or 3D beam propagation,” Opt. Quantum. Electron.41, 849–857 (2010).
[CrossRef]

Other (1)

M. J. Adams, An Introduction to Optical Waveguides (Wiley, 1981).

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Figures (6)

Fig. 1
Fig. 1

Error as a function of propagation distance for an aligned waveguide with Nx = 1000 for the (a) WASSS and (b) HOWASSS methods. Note that we have sampled the error every 0.5 mm, and not at every z step calculated, because the rapid oscillations would make the graph difficult to read otherwise.

Fig. 2
Fig. 2

Error as a function of propagation distance for a waveguide rotated 50 degrees with Nx = 1000 for the (a) WASSS and (b) HOWASSS methods.

Fig. 3
Fig. 3

Plot of |ψ(z, x)| with Nx = 1000, Nz = 2000, and θ = 50° for the (a) HOWASSS and (b) exact solutions for a waveguide tilted at 50 degrees.

Fig. 4
Fig. 4

(a) The maximum error obtained as a function of waveguide tilt angle showing that the HOWASSS method actually gains a small amount of accuracy at larger angles. (b) The maximum error obtained as a function of waveguide depth, Δn.

Fig. 5
Fig. 5

(a) Run times of HOWASSS and WASSS methods on the GPU and the single-core CPU for 1000 propagation steps. (b) Comparison of HOWASSS method run times using different number of cores on the CPU and using the GPU for 1000 propagation steps.

Fig. 6
Fig. 6

Comparison of the HOWASSS method and WASSS method compute time per propagation distance with Nx = 1000 and Nx = 2000. Both methods were run on the GPU.

Tables (1)

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Table 1 Parameters used in the numerical calculations

Equations (36)

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2 z 2 ψ ( z , r ) + r 2 ψ ( z , r ) + k 0 2 n ¯ 2 ψ ( z , r ) + k 0 2 ( n 2 ( z , r ) n ¯ 2 ) ψ ( z , r ) = 0 ,
ψ ( z , x ) = i a i ( z ) ϕ i ( x ) ,
x 2 ϕ i ( x ) = λ i 2 ϕ i ( x )
ϕ i * ( x ) ϕ j ( x ) d x = δ i , j .
2 z 2 a j ( z ) = ( k 0 2 n ¯ 2 λ j 2 ) a j ( z ) i a i ( z ) k 0 2 ( n 2 ( z , x ) n ¯ 2 ) ϕ j * ( x ) ϕ i ( x ) d x .
N i , j ( z ) = k 0 2 ( n 2 ( z , x i ) n ¯ 2 ) δ i , j .
M i , j = k 0 2 n ¯ 2 λ i 2 δ i , j .
2 z 2 a ( z ) = M 2 a ( z ) SN ( z ) S 1 a ( z ) .
A ( z ) = [ a ( z ) M 1 z a ( z ) ]
H ( z ) = [ 0 M M M 1 SN ( z ) S 1 0 ] ,
z A ( z ) = H ( z ) A ( z ) .
A ( z ) = exp ( Ω 1 ( z , z 0 ) + Ω 2 ( z , z 0 ) + ) A ( z 0 ) .
Ω 1 ( z , z 0 ) = z 0 z H ( t ) d t
Ω 2 ( z , z 0 ) = 1 2 z 0 z z 0 t 1 [ H ( t 1 ) , H ( t 2 ) ] d t 2 d t 1 .
H ( z ) = H 1 + H 2 ( z ) = [ 0 M M 0 ] + [ 0 0 M 1 SN ( z ) S 1 0 ] .
Ω 1 ( z l + 1 , z l ) = H 1 Δ z + ( H 2 ( z l + 1 ) + H 2 ( z l ) ) Δ z 2 + 𝒪 ( ( Δ z ) 3 ) .
[ H ( z 1 ) , H ( z 2 ) ] = [ S ( N ( z 1 ) N ( z 2 ) ) S 1 0 0 M 1 S ( N ( z 2 ) N ( z 1 ) ) S 1 M ] .
Ω 2 ( z l + 1 , z l ) = [ H ( z l + 1 ) , H ( z l ) ] ( Δ z ) 2 8 + 𝒪 ( ( Δ z ) 3 ) .
A ( z l + 1 ) exp ( H 1 Δ z + ( H 2 ( z l + 1 ) + H 2 ( z l ) ) Δ z 2 + [ H ( z l + 1 ) , H ( z l ) ] ( Δ z ) 2 8 ) A ( z l ) .
exp ( ( A + B ) Δ z ) = exp ( A Δ z 2 ) exp ( B Δ z ) exp ( A Δ z 2 ) + 𝒪 ( ( Δ z ) 3 ) .
A ( z l + 1 ) PQ ( z l + 1 ) Q ( z l ) PC ( z l + 1 , z l ) PQ ( z l + 1 ) Q ( z l ) PA ( z l ) ,
P = exp ( H 1 Δ z 4 )
Q ( z ) = exp ( H 2 ( z ) Δ z 4 )
C ( z 1 , z 2 ) = exp ( [ H ( z 1 ) , H ( z 2 ) ] ( Δ z ) 2 8 ) .
P = [ cos ( M Δ z 4 ) sin ( M Δ z 4 ) sin ( M Δ z 4 ) cos ( M Δ z 4 ) ]
Q ( z ) = [ I 0 M 1 SN ( z ) S 1 Δ z 4 I ]
C ( z 1 , z 2 ) = [ S exp ( ( N ( z 1 ) N ( z 2 ) ) ( Δ z ) 2 8 ) S 1 0 0 M 1 S exp ( ( N ( z 2 ) N ( z 1 ) ) ( Δ z ) 2 8 ) S 1 M ] .
Q ( z l + 1 ) Q ( z l ) = [ I 0 M 1 S ( N ( z l + 1 ) + N ( z l ) ) S 1 Δ z 4 I ] .
ϕ i ( x ) = sin ( i π ( x f x 0 ) x ) .
S i , j = 2 N x + 1 sin ( π ( i + 1 ) ( j + 1 ) N x + 1 ) = S i , j 1 .
n ( z , x ) = n ¯ 2 + 2 n ¯ ( Δ n ) sech 2 ( 2 ( x ˜ cos ( θ ) z sin ( θ ) ) w ) ,
ψ ( 0 , x ) = sech W ( 2 x ˜ cos ( θ ) w ) exp ( i K 0 x ˜ sin ( θ ) ) ,
W = 1 2 ( 1 + 2 w 2 k 0 2 n ¯ Δ n 1 )
K 0 = ( 2 W w ) 2 + ( k 0 n ¯ ) 2 .
ψ e ( z , x ) = sech W ( 2 ( x ˜ cos ( θ ) z sin ( θ ) ) w ) exp ( i K 0 ( x ˜ sin ( θ ) + z cos ( θ ) ) ) .
Error ( z ) = | 1 ( x 0 x f ψ * ( z , x ) ψ ( z , x ) d x ) 2 ( x 0 x f ψ e * ( z , x ) ψ e ( z , x ) d x ) 2 | ,

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