Abstract

The fast and accurate propagation of general optical fields in free space is still a challenging task. Most of the standard algorithms are either fast or accurate. In the paper we introduce a new algorithm for the fast propagation of non-paraxial vectorial optical fields without further physical approximations. The method is based on decomposing highly divergent (non-paraxial) fields into subfields with small divergence. These subfields can then be propagated by a new semi-analytical spectrum of plane waves (SPW) operator using fast Fourier transformations. In the target plane, all propagated subfields are added coherently. Compared to the standard SPW operator, the numerical effort is reduced drastically due to the analytical handling of linear phase terms arising after the decomposition of the fields. Numerical results are presented for two examples demonstrating the efficiency and the accuracy of the new method.

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References

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  1. F. Wyrowski and M. Kuhn, “Introduction to field tracing,” J. Mod. Opt.58(5–6), 449–466 (2011).
    [CrossRef]
  2. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995).
  3. A. Wuttig, M. Kanka, H. J. Kreuzer, and R. Riesenberg, “Packed domain Rayleigh-Sommerfeld wavefield propagation for large targets,” Opt. Express18(26), 27036–27047 (2010).
    [CrossRef]
  4. J. A. C. Veerman, J. J. Rusch, and H. P. Urbach, “Calculation of the Rayleigh–Sommerfeld diffraction integral by exact integration of the fast oscillating factor,” J. Opt. Soc. Am. A22(4), 636–646 (2005).
    [CrossRef]
  5. M. Mansuripur, “Certain computational aspects of vector diffraction problems,” J. Opt. Soc. Am. A6(5), 786–805 (1989).
    [CrossRef]
  6. J. Braat, P. Dirksen, and A. J. E. M. Janssen, “Assessment of an extended Nijboer–Zernike approach for the computation of optical point-spread functions,” J. Opt. Soc. Am. A19(5), 858–870 (2002).
    [CrossRef]
  7. P. Valtr and P. Pechac, “Domain decomposition algorithm for complex boundary modeling using the Fourier split-step parabolic equation,” IEEE Trans. Antennas Propag.6, 152–155 (2007).
  8. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).
  9. LightTrans GmbH, LightTrans VirtualLab AdvancedTM, www.lighttrans.com (2012).
  10. E. O. Brigham, The Fast Fourier Transform and its Applications (Prentice Hall, 1988).

2011 (1)

F. Wyrowski and M. Kuhn, “Introduction to field tracing,” J. Mod. Opt.58(5–6), 449–466 (2011).
[CrossRef]

2010 (1)

2007 (1)

P. Valtr and P. Pechac, “Domain decomposition algorithm for complex boundary modeling using the Fourier split-step parabolic equation,” IEEE Trans. Antennas Propag.6, 152–155 (2007).

2005 (1)

2002 (1)

1989 (1)

Braat, J.

Brigham, E. O.

E. O. Brigham, The Fast Fourier Transform and its Applications (Prentice Hall, 1988).

Dirksen, P.

Goodman, W.

W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

Janssen, A. J. E. M.

Kanka, M.

Kreuzer, H. J.

Kuhn, M.

F. Wyrowski and M. Kuhn, “Introduction to field tracing,” J. Mod. Opt.58(5–6), 449–466 (2011).
[CrossRef]

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995).

Mansuripur, M.

Pechac, P.

P. Valtr and P. Pechac, “Domain decomposition algorithm for complex boundary modeling using the Fourier split-step parabolic equation,” IEEE Trans. Antennas Propag.6, 152–155 (2007).

Riesenberg, R.

Rusch, J. J.

Urbach, H. P.

Valtr, P.

P. Valtr and P. Pechac, “Domain decomposition algorithm for complex boundary modeling using the Fourier split-step parabolic equation,” IEEE Trans. Antennas Propag.6, 152–155 (2007).

Veerman, J. A. C.

Wolf, E.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995).

Wuttig, A.

Wyrowski, F.

F. Wyrowski and M. Kuhn, “Introduction to field tracing,” J. Mod. Opt.58(5–6), 449–466 (2011).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

P. Valtr and P. Pechac, “Domain decomposition algorithm for complex boundary modeling using the Fourier split-step parabolic equation,” IEEE Trans. Antennas Propag.6, 152–155 (2007).

J. Mod. Opt. (1)

F. Wyrowski and M. Kuhn, “Introduction to field tracing,” J. Mod. Opt.58(5–6), 449–466 (2011).
[CrossRef]

J. Opt. Soc. Am. A (3)

Opt. Express (1)

Other (4)

W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

LightTrans GmbH, LightTrans VirtualLab AdvancedTM, www.lighttrans.com (2012).

E. O. Brigham, The Fast Fourier Transform and its Applications (Prentice Hall, 1988).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995).

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Figures (17)

Fig. 1
Fig. 1

Schematic illustration of the scope for common FFT-based propagation operators in field tracing. Colored areas indicate parameter spaces for which the techniques are numerically feasible. The rigorous SPW operator suffers from high numerical effort for high propagation distances or spatial frequencies. The far field and Fresnel propagation operators have a limited range of validity due to their nature of physical approximations.

Fig. 2
Fig. 2

Illustration of fundamental properties of a plane wave: Projecting the wavefront (dashed line in the left figure), which is pointing into the direction ŝ0, on the xy-plane leads to a linear phase term (central figure). According to the shift theorem of the Fourier transformation this linear phase results in a spectrum (right figure) which is located around the spatial frequency κ0.

Fig. 3
Fig. 3

2D line evaluation of the spherical phase kz(κ) (blue line) and its decomposition (magenta and red line) according to Eq. (8) along the diagonal direction ( k x , k y ) T = k ρ 2 ( 1 , 1 ) T. Left side: Normalized spherical phase function kz(κ) and linear phase function γ0, + γ1, · κ around k 0 ρ k = 0.424. Right side: Higher order phase term γ(κκ0,) for k 0 ρ k = 0.424 calculated by Eq. (12).

Fig. 4
Fig. 4

Example for a parabasal field: Amplitude (left) and phase (right) of a super Gaussian beam with 50 μm waist diameter at a wavelength of 532 nm. An additional linear phase term of 10° in x-direction and 4° in y-direction replaces the paraxial properties of the super Gaussian beam by a parabasal field behavior. The positions of the 1D cross section profiles are given by the black arrows.

Fig. 5
Fig. 5

Flowchart for the efficient propagation of non-paraxial fields using a combination of a parabasal decomposition technique (PDT) and the semi-analytical SPW operator: A general harmonic field is decomposed into a set of parabasal subfields using a PDT technique. After that each parabasal harmonic subfield is propagated by the semi-analytical SPW operator. Finally all propagated parabasal subfields are superimposed in the target plane leading to the propagated non-paraxial field.

Fig. 6
Fig. 6

Decomposition of a first order Laguerre Gaussian beam into 2 × 2 parabasal subfields in the spatial frequency domain.

Fig. 7
Fig. 7

Single parabasal subfield created by a decomposition of a Laguerre Gaussian beam of first order using a rect-basis function: The sharp edges of the rect-function in the spatial frequency domain (left) lead to an infinite extended subfield in the space domain (right).

Fig. 8
Fig. 8

Single parabasal subfield created by a decomposition of a Laguerre Gaussian beam of first order using the basis function of Eq. (19): The smooth edges of the basis function in the spatial frequency domain (left) leads to a finitely extended subfield in the space domain (right).

Fig. 9
Fig. 9

Illustration of two basic cases of non-paraxial fields. On the left-hand side a wavefront is illustrated, which possesses rough wavefront features. Then only a PDT in spatial frequency domain is reasonable. On the right hand-side a wavefront is shown, which is smooth but has a strong phase function. Adjacent positions of the wavefront contribute to almost the same spatial frequencies which enable a PDT in space domain.

Fig. 10
Fig. 10

Initial phase of a convergent spherical wave which will be propagated by z = 3.8 mm. Please note that the artefacts in the phase distribution is a Moiré pattern due to the low resolution of the picture in comparison with the very fine sampling which is required for the phase.

Fig. 11
Fig. 11

Residual amplitude (left) and phase (right) of a parabasal subfield created by a PDT in space domain of a spherical wave into 20 × 20 subfields and extraction of the linear phase term.

Fig. 12
Fig. 12

Residual amplitude (left) and phase (right) of a propagated parabasal subfield calculated by the semi-analytical SPW operator.

Fig. 13
Fig. 13

Amplitude (left) and phase (right) of the defocused non-paraxial spherical wave propagated by a combination of PDT and semi-analytical SPW operator.

Fig. 14
Fig. 14

Amplitude (left) and phase (right) of an astigmatic Gaussian beam with waist radius of 7 μm in x-direction and 10 μm in y-direction. Additional astigmatism is added by Zernike coefficients.

Fig. 15
Fig. 15

Residual amplitude (left) and phase (right) of a parabasal subfield created by a PDT in spatial frequency domain of an astigmatic Gaussian beam into 15 × 15 subfields and extraction of the local linear phase terms.

Fig. 16
Fig. 16

Residual amplitude (left) and phase (right) of a propagated parabasal subfield calculated by the semi-analytical SPW operator.

Fig. 17
Fig. 17

Amplitude of the non-paraxial astigmatic beam propagated by a combination of PDT and semi-analytical SPW operator.

Tables (3)

Tables Icon

Table 1 Accuracy and numerical effort for different propagation techniques to propagate the parabasal super Gaussian beam by 10 mm:

Tables Icon

Table 2 Accuracy and numerical effort for different propagation techniques to propagate the defocused spherical wave by 3.8 mm. In case of the PDT and semi-analytical SPW Operator the numerical effort in brackets gives the number of used parabasal subfields.

Tables Icon

Table 3 Accuracy and numerical effort for different propagation techniques to propagate the astigmatic Gaussian beam by 0.5 mm. In case of the PDT and semi-analytical SPW Operator the numerical effort in brackets gives the number of used parabasal subfields.

Equations (25)

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A ( κ , 0 ) = [ V ( ρ , 0 ) ] = 1 2 π V ( ρ , 0 ) e i κ ρ d ρ .
V ( ρ , z ) = 1 [ A ( κ , 0 ) e i k z ( κ ) z ]
k = k s ^ 0 = k ( s 0 x , s 0 y , s 0 z ) T = k ( sin θ 0 cos ϕ 0 , sin θ 0 sin ϕ 0 , cos θ 0 ) T
κ 0 = ( k 0 x , k 0 y ) T = k ( s 0 x , s 0 y ) T
V ( ρ , 0 ) = V ( ρ , 0 ) e i κ 0 , ρ
A ( κ , 0 ) = A ( κ κ 0 , , 0 )
V ( ρ , 0 ) = 1 [ A ( κ , 0 ) ] .
k z , ( κ ) = γ 0 , + γ 1 , κ + γ ( κ κ 0 , )
γ 0 , = k 0 z , + 1 k 0 z , ( k 0 x , 2 + k 0 y , 2 ) k 0 x , 2 k 0 y , 2 k 0 z , 3 ,
γ 1 , = [ k 0 x , k 0 z , ( k 0 y , 2 k 0 z , 2 1 ) , k 0 y , k 0 z , ( k 0 x , 2 k 0 z , 2 1 ) ] T
k 0 z , = k 2 k 0 x , 2 k 0 y , 2 .
γ ( κ ) = k z , ( κ + κ 0 , ) γ 0 , γ 1 , ( κ + κ 0 , )
k z , ( κ + κ 0 , ) = k 2 ( k x + k 0 x , ) 2 ( k y + k 0 y , ) 2 .
V ( ρ , z ) = V ( ρ + z γ 1 , , z ) exp [ i ( γ 0 , z + γ 1 , κ 0 , z + ρ κ 0 , ) ]
V ( ρ , z ) = 1 [ A ( κ , 0 ) e i γ ( κ ) z ]
d : = x , y | V , operator ( x , y , z ) V , reference ( x , y , z ) | 2 x , y | V , reference ( x , y , z ) | 2
η : = Complexity of SPW algorithm Complexity of semi-analytical SPW algorithm
A ( κ , 0 ) = i , j = 1 N Φ i , j ( κ ) A ( κ , 0 )
Φ ( k x ) = 0.5 [ sin ( π a ( k x a 2 ) ) + 1 ] for 0 < k x a , Φ ( k x ) = 1 for a < k x < b a , Φ ( k x ) = 1 0.5 [ sin ( π a ( k x 2 b a 2 ) ) + 1 ] for b a k x < b , and Φ ( k x ) = 0 anywhere else ,
Φ i , j ( κ ) = Φ ( k x k i , x ) Φ ( k y k j , y )
V ( ρ , 0 ) = V ^ ( ρ , 0 ) e i φ ( ρ , 0 )
φ ( ρ , 0 ) | ( ρ ρ 0 , ) = ε + κ 0 , ρ + Δ ( ρ , 0 )
ε = φ ( ρ 0 , , 0 ) + 2 φ x y | ρ 0 , ( x 0 , y 0 , ) φ x | ρ 0 , x 0 , φ y | ρ 0 , y 0 ,
κ 0 , = ( k 0 x , , k 0 y , ) T = ( φ x | ρ 0 , 2 φ x y | ρ 0 , y 0 , , φ y | ρ 0 , 2 φ x y | ρ 0 , x 0 , ) T .
V ( ρ , 0 ) = V ^ ( ρ , 0 ) exp [ i ( ε + Δ ( ρ ρ 0 , , 0 ) ] exp [ i ( ρ κ 0 , ) ]

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