Modified vectorial angular spectrum formula for propagation of non-paraxial beams

Accepted Manuscript Title: Modified Vectorial Angular Spectrum Formula for Propagation of Non-paraxial Beams Authors: Chaohong Huang, Hanqin Li, Jianfeng Wu, Yupeng Yan PII: DOI: Reference:

S0030-4026(17)31284-6 https://doi.org/10.1016/j.ijleo.2017.10.071 IJLEO 59807

To appear in: Received date: Accepted date:

16-6-2017 13-10-2017

Please cite this article as: Chaohong Huang, Hanqin Li, Jianfeng Wu, Yupeng Yan, Modified Vectorial Angular Spectrum Formula for Propagation of Non-paraxial Beams, Optik - International Journal for Light and Electron Optics https://doi.org/10.1016/j.ijleo.2017.10.071 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Modified Vectorial Angular Spectrum Formula for Propagation of Non-paraxial Beams Chaohong Huang*, Hanqin Li, Jianfeng Wu, Yupeng Yan Department of Electronic Engineering, Xiamen University, Xiamen 361005, China *[email protected]

Abstract: We presented modified forms of full-vector angular spectrum formula without singular points for longitudinal spectral components. In paraxial limitation, the modified angular spectrum formula has the same form as usual angular spectrum formula. Since there are not any singular points at the angular spectrum formula, we can apply it to analyze the propagation of non-paraxial beams with subwavelengthsize waist and give results that strictly satisfy Maxwell’s equations because the calculating process includes contributions from all field components and angular spectral components. 1. Introduction The plane wave angular spectrum approach presents an effective integral technique to calculate propagation of electromagnetic waves or laser beams. For propagation of laser beams, the so-called scalar and paraxial approximations are generally adopted. In this case, the contributions from longitudinal field components and evanescent angular spectrum components can be neglected and the scalar plane wave angular spectrum formula is sufficient to describe the propagation of laser beams. However, for the non-paraxial cases where spot sizes of beams are close to or smaller than wavelength of light, the paraxial and scalar approximation of electromagnetic waves no longer holds. Recently, the longitudinal field components and evanescent waves were found to play key roles in the intrinsic dynamic properties of optical fields [1,2], such as spin-orbit coupling and transverse spin angular momentum in structured fields: evanescent waves, interference fields, and focused (non-paraxial) beams. In order to understand the novel properties that are remarkably different from those in paraxial and scalar fields, full vector properties of optical fields, including all field components and full angular spectrum components, have to be considered. Full vector angular spectrum representation of monochromatic electromagnetic fields, which is equivalent to Maxwell’s equations or Helmholtz equation, was firstly derived by Rhodes in his classic papers [3,4]. Since then, the angular spectrum method was widely applied to correct the propagation of laser beams beyond the paraxial approximation [5-8]. Nevertheless, since the traditional form of vector angular spectrum formula is singular at kz=0 for longitudinal field components [6,7,9], the contributions from homogeneous (kz is real) and inhomogeneous waves (kz is imaginary) have to be calculated separately and the spectral components with kz=0 are obliged to be omitted. In paraxial case, the neglect of singular spectral components at

kz=0 does not cause any problems since the energy of fields mainly concentrate on spectral region of kz ~k. However, for the non-paraxial cases in which spectral components near kz ~0 are dominant, the neglect is problematic and will cause nonphysical results. In addition, the existence of singular points at vectorial angular spectrum formula also results in difficulty in numerical calculation for the propagation of non-paraxial beams. In this paper, we proposed other two forms of full vector angular spectrum formula, which have not any singular points. Firstly, we described the usual vectorial angular spectrum approach and indicated the divergent problem of longitudinal spectral component in this formula. The existence of singular points results that the usual vector angular spectrum formula is not actually applicable for propagation of beams with subwavelength-size waist. Secondly, we derived full vector angular spectrum formula without any singular points and demonstrated a numerical example for propagation of non-paraxial beams using the proposed angular formula. 2. The usual vectorial angular spectrum formula For monochromatic electromagnetic fields with angular frequency ω, the complex amplitudes of electric and magnetic fields are generally introduced through separating the time harmonic factor from the complex representation of electromagnetic fields. As usual, time harmonic dependence of exp(-iωt) is assumed in this paper. The complex amplitudes of electric and magnetic fields have the relations with the real field quantities as E  r, t   Re  E(r ) exp(it )  ,    H  r, t   Re  H(r ) exp(it ) 

(1)

where E(r) and H(r) represents the complex amplitude of electric and magnetic field, respectively. E(r,t) and H(r,t) are corresponding instantaneous real electric and magnetic fields.

For laser beams in homogeneous medium, the complex amplitudes E(r) and H(r) can be expanded as the superposition of plane waves with different wave vector k. 2   1  E(r )     E(k ) exp  ik  r  dk ,   2   2  1   H ( r )     H(k ) exp  ik  r  dk   2  

(2)

where E(k ) and H(k ) are respectively the angular spectrum of complex amplitudes (E(r) and H(r)) at initial plane (z=0). The integrals at Eq.(2) are over the surface of sphere with radius of k=ωn/c in the k-space (n is the index of refraction of medium and c is the speed of light in vacuum). For convenience of calculation, Eq.(2) is generally rewritten as component forms in the Cartesian coordinate system  1  U ( x, y, z )     2 

2

+

+

-

-

 

U  kx , k y  e



i kx x  k y y kz z



dk x dk y ,

(3)

where U represents any Cartesian component of the electric or magnetic fields in real space, namely, Ex, Ey, Ez, Hx, Hy and Hz, while U denotes any initial angular spectrum component in the plane of z=0, i.e., Ex , E y , Ez , H x , H y and H z . For the case of cylindrically symmetry, Eq.(3) is usually written using cylindrical coordinate parameters  1  U (r ,  , z )     2 

2

2

+

0

0

 

U  ,   exp  i r cos(   )  i z   d d ,

(4)

where (r,φ,z) are the cylindrical coordinate parameters in real space, while (α,ϕ,β) the cylindrical coordinate parameters in k-space and obeys α2+β2=k2. Since the restriction of transverse condition

k  E(k )=0

(which is equivalent to zero

divergence condition (  E  0 ) in spatial domain), only two components for electric field angular spectrum E(k ) are independent. Conventionally, Ex and E y are selected as independent spectral components while Ez ( k x , k y ) 

The magnetic angular spectrum

E0 z

is calculated by

k kx Ex ( k x , k y )  y E y ( k x , k y ) . kz kz H(k )

is dependent on

(5) E(k )

due to a right-hand

impedance relation between E(k ) , H(k ) and k . H(k ) 

 k  E(k ) .  k

(6)

where  and  are respectively electric permittivity and magnetic permeability of medium. Eq.(3),(4),(5) and (6) represent the conventional full-vector angular spectrum formula to calculate beam propagation from two independent angular spectrum components. For a general procedure, Ex and E y are firstly calculated from spatial boundary conditions at z=0 plane (i.e., Ex (x,y,0) and Ey (x,y,0)) through 2D Fourier transformation. Further, other angular spectrum components are calculated by Eq.(5) and (6). Through substituting these angular spectrum components into Eq.(3) or (4) and setting kz  k 2   kx2  k y2   k 2   2 (for   k ) and kz  i  kx2  k y2   k 2  i  2  k 2 (for   k ), the complex amplitudes of electric and magnetic fields at semi-space of z  0

can be solved. Although the above procedure is widely used for discussing propagation of paraxial beams, it should be careful to apply the procedure to the non-paraxial case. Due to the transverse condition, the initial angular spectral components Ex and E y are restricted by the condition kx Ex (kx , k y )  k y Ey (kx , k y )  0

Otherwise,

Ez

for kx2  k y2  k 2 .

(7)

will tend to infinite at kz=0. The Eq.(7) presents a restriction in

spectral domain for settings of initial angular spectrum components or boundary

condition of fields at z=0 plane. In other words, the initial angular spectrum components or field boundary at z=0 plane are not able to be set arbitrarily. In paraxial condition, one has not to take care of the restriction since field energy hardly reach the spectral region of kz~0. However, for the non-paraxial case where field energy near kz~0 is prominent, one must to ensure initial angular spectrum to satisfy Eq.(7). For a typical example, while considering the propagation of transversely linearlypolarized Gaussian modes, the following boundary condition at z=0 plane be usually set [5-6]  x2  y 2  . Ex ( x, y )  E0 exp    02  

(8)

This electric field distribution corresponds to a Gaussian-form angular spectrum  k x2  k y2  . (9) E ( x, y )   2 E exp   2  x

0

0

 

0

4

 

Evidently, Eq.(9) does not satisfy the Eq.(7). If we still calculate

E z using

Eq.(5), the

results will tend to infinite while kz tends to zero. Figure 1 shows the x and zcomponents of angular spectra of Gaussian modes expressed by Eq.(8) with several different beam waist radii. From these figures, it can be seen that the of longitudinal angular spectrum ( Ez ) includes spectral components with infinite amplitude at kz=0. This is not an accomplishable physically result. Although, Eq.(8) still is suitable as the boundary condition in the cases of ω0>>λ if ignoring the narrow singular angular spectrum region near kz~0 (see Fig.1 (a)). The longitudinal angular spectrum components near kz~0 gradually become prominent and cannot be ignored while ω0<λ. In this case, the singular points at kz=0 for Eq.(5) causes a serious difficulty in analyze the propagation of beams with subwavelength-size waist. 3. Modified vectorial angular spectrum formula To avoid the singular points for longitudinal angular spectrum component at kz=0, we derived other two forms of angular spectrum formula that have no any singular points in this section. In the k-space (as shown in Fig. 2), E(k ) and H(k ) are always tangent to the surface of k-sphere with radius of k=ωn/c and can be orthogonally decomposed along inclined and azimuthal directions if we adopt spherical coordinates (k,θ,ϕ) E  ,    E  ,   eˆ  E  ,   eˆ   ,    E  ,   eˆ   E  ,   eˆ H  ,        

(10a)

where eˆ  and eˆ  denotes the unit vectors at inclined and azimuthal directions, respectively.

E  ,   and E  ,   are

for electric field.

the corresponding angular spectral components

Using the relation between ( eˆ  , eˆ  ) and the unit vectors ( eˆ x , eˆ y , eˆ z ) along the Cartesian coordinate axes eˆ  eˆ x cos  cos   eˆ y cos  sin   eˆ z sin  ,  eˆ  eˆ x sin   eˆ y cos 

(11)

the rectangular components of electric angular spectrum can be written as  Ex  ,    E  ,   cos  cos   E  ,   sin   .  E y  ,    E  ,   cos  sin   E  ,   cos    Ez  ,     E  ,   sin 

(12)

Since the spherical coordinates (k,θ,ϕ) have the relations with the Cartesian coordinates (kx,ky,kz)  k 2  k y2 sin   x  k  kz  cos   k

ky  sin   2 and  k x  k y2   k x2  k y2 sin   k 

(13)

we can rewritten the rectangular components of angular spectrum of electric field as the functions of kx, ky and kz.  ky kx k  E  k x , k y   Ex  k x , k y   E  k x , k y  z 2 2 2 k kx  k y k x  k y2   . ky kx kz   E  k x , k y   E y  k x , k y   E  k x , k y  k k x2  k y2 k x2  k y2    k x2  k y2 E k , k  E k , k  z x y   x y k 

(14)

Similarly, the rectangular components of magnetic angular spectrum are  ky  H x  kx , k y    E  k x , k y   2   k x  k y2   kx  E  k x , k y   H y  kx , k y   2  k x  k y2    k x2  k y2  E  k x , k y  H z  kx , k y    k 

k  E  k x , k y  z  k k  E k , k  z   x y k

kx k  k y2 2 x

ky

.

(15)

k x2  k y2

According to Eq.(14) and (15), any electromagnetic fields can be naturally decomposed as TE ( E  0 , E  0 ) and TM ( E  0 , E  0 ) modes, which is agreement with the results given by other authors [7,10-13]. The TE modes have the non-zero field components (Ex, Ey, Hx, Hy, Hz) and TM modes have the non-zero field components (Ex, Ey, Ez, Hx, Hy). Although Eq.(14) and (15) are convenient to discuss the TE or TM modes, it is not intuitive to relate the angular spectrum representation with usual linearly-polarized modes in paraxial condition. If we introduce two new functions by

ky  kx k  E  k x , k y   E1  k x , k y   E  k x , k y  2 2 2 k z k x  k y2 kx  k y   ky kx k E k , k  E k , k  E  k x , k y  2 x y   x y 2 2 2  k kx  k y k x  k y2 z 

(16)

Then we can change Eq.(14) to a simple form kz   Ex  k x , k y   E1  k x , k y  k  . kz   E y  k x , k y   E2  k x , k y  k  ky  kx  Ez  k x , k y    E1  k x , k y   E2  k x , k y  k k 

(17)

The angular spectrum formula (17) are slightly different from usually-used angular spectrum formula (Eq.(5)). Nevertheless, they give the same result in paraxial limitation (kz→k). It is noted that the angular spectra expressed by Eq.(17) show a key advantage of no singular points. Unlike Ex and E y are restricted by Eq.(7), we can arbitrarily set initial angular spectrum

E1

and

E2

. According to Eq.(17), the

electromagnetic waves can be decomposed into two modes with transverse linearlypolarized electric fields: LPx( E2  0 ) and LPy ( E1  0 ) modes. The LPx mode have the non-zero field components (Ex, Ez, Hx, Hy, Hz) and LPy modes have the non-zero field components (Ey, Ez, Hx, Hy, Hz). In paraxial limitation, these two modes corresponds to usual orthogonal linearly polarized modes in x and y directions. As an example of calculating propagation of beams with subwavelength-size spot applying Eq.(17) and Eq.(3), we set E2 (kx , k y )  0 and  k x2  k y2  . E1 (k x , k y )  02 E0 exp  02   4  

(18)

In paraxial limitation, Eq.(18) corresponds to Gaussian beams linearly-polarized along x direction, as described by Eq.(8). Figure 3 shows the angular spectra and corresponding spatial field distributions (z=0) of the non-paraxial linearly-polarized Gaussian beams with subwavelength beam waist ( 0   ). The fields demonstrated in these figures strictly obey Maxwell’s equations because no any field components or spectral components are ignored in the calculation. From these figures, it is noted that spatial distributions of the transverse field component is remarkable deviated from the Gaussian distribution while ω0<λ and the longitudinal field component becomes gradually prominent with decreasing of ω0. The results for subwavelength cases are considerably different from the paraxial Gaussian beams. Conclusions In summary, we proposed other two forms of full vector angular spectrum formula, which have the same asymptotic forms in paraxial limitation as usual angular spectrum formula. For the first form, laser fields are naturally decomposed into

transverse electric (TE) and magnetic (TM) modes. For the second form, laser fields are the superposition of two transversely linear-polarized modes (LPx and LPy). Comparing with usual forms of angular spectrum formula, our formulae have not any singular points for longitudinal angular spectral components at kz=0. As a result, the full-vector angular spectrum formulae presented by this paper are suitable to discuss analytically and calculate numerically the propagation of non-paraxial beams with subwavelength-size waist, including contributions from all field components and full angular spectrum components. References 1. K. Y. Bliokh and F. Nori, “Transverse and longitudinal angular momenta of light,” Physics Reports 592, 1-38(2015). 2. K. Y. Bliokh, A. Y. Bekshaev, and F. Nori, “Extraordinary momentum and spin in evanescent waves,” Nat. Commu. 4300, DOI:10.1038 (2014). 3. D. R. Rhodes, ‘‘On a fundamental principle in the theory of planar antennas,’’ Proc. IEEE 52, 1013–1021 (1964). 4. D. R. Rhodes, ‘‘On the stored energy of planar apertures,’’ IEEE Trans. Antennas Propag. AP-14, 676–683 (1966). 5. G. P. Agrawal and D. N. Pattanayak, ‘‘Gaussian beam propagation beyond the paraxial approximation,’’ J. Opt. Soc. Am. 69, 575–578 (1979). 6. C. G. Chen, P. T. Konkola, J. Ferrera, R. K. Heilmann, and M. L. Schattenburg, “Analyses of vector Gaussian beam propagation and the validity of paraxial and spherical approximations,” J. Opt. Soc. Am. A 19, 404–412(2002) . 7. G. Zhou, “The analytical vectorial structure of a nonparaxial Gaussian beam close to the source,” Opt. Express 16(6), 3504–3514 (2008). 8. G. Zhou, “Vectorial structures of non-paraxial linearly polarized Gaussian beam and their beam propagation factors,” Optics Communications 265, 39-46(2006) 9. 9. G. Li, S. Teng, C. Zhang, and C. Cheng, “Angular spectrum analysis of near-field diffraction of a small aperture,” Optik 124, 5478-5480(2013) 10. T. Melamed, “TE and TM beam decomposition of time-harmonic electromagnetic waves,” J. Opt. Soc. Am. A 28(3), 401-409(2011) 11. A. April, “Nonparaxial TM and TE beams in free space,” Opt. Lett. 33(14), 15631565(2008). 12. H. Guo, J. Chen, and S. Zhuang, “Vector plane wave spectrum of an arbitrary polarized electromagnetic wave,” Opt. Express 14(6), 2095–2100(2006). 13. R. Martínez-Herrero and P. M. Mejías, S. Bosch, and A. Carnicer, “Vectorial structure of nonparaxial electromagnetic beams,” J. Opt. Soc. Am. A 18(7), 1678-1680(2001)

Fig.1 The angular spectra of transversely linearly-polarized Gaussian beams with different beam waist radii. (a) ω0=1.0λ; (b) ω0=0.8λ; (c) ω0=0.6λ; (d) ω0=0.4λ.

Fig.2 The k-sphere and orthogonal decomposition of spectral components along inclined and azimuthal directions in spherical coordinate system

Fig.3 The angular spectra (the 1st column) and spatial field distributions at z=0 plane (the 2nd column) of transversely linearly-polarized Gaussian modes with different beam waist radii (ω0=1.0λ, 0.4λ and 0.2λ for the 1st, 2nd and 3rd row, respectively). The insets in figures of the 2nd column show the transverse contour distributions of field components.