## Propagation of generalized vector Helmholtz-Gauss beams through paraxial optical systems

Optics Express, Vol. 14, Issue 20, pp. 8974-8988 (2006)

http://dx.doi.org/10.1364/OE.14.008974

Acrobat PDF (493 KB)

### Abstract

We introduce the generalized vector Helmholtz-Gauss (gVHzG) beams that constitute a general family of localized beam solutions of the Maxwell equations in the paraxial domain. The propagation of the electromagnetic components through axisymmetric ABCD optical systems is expressed elegantly in a coordinate-free and closed-form expression that is fully characterized by the transformation of two independent complex beam parameters. The transverse mathematical structure of the gVHzG beams is form-invariant under paraxial transformations. Any paraxial beam with the same waist size and transverse spatial frequency can be expressed as a superposition of gVHzG beams with the appropriate weight factors. This formalism can be straightforwardly applied to propagate vector Bessel-Gauss, Mathieu-Gauss, and Parabolic-Gauss beams, among others.

© 2006 Optical Society of America

## 1. Introduction

1. M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys, Rev. A **11**, 1365–1370 (1975). [CrossRef]

2. L. W. Davis and G. Patsakos, “TM and TE electromagnetic beams in free space,” Opt. Lett. **6**, 22–23 (1981). [CrossRef] [PubMed]

3. Z. Bouchal and M. Olivík, “Non-diffractive vector Bessel beams,” J. Mod. Opt. **42**, 1555–1566 (1995). [CrossRef]

4. D. G. Hall, “Vector-beam solutions of Maxwell’s wave equation,” Opt. Lett. **21**, 9–11 (1996). [CrossRef] [PubMed]

5. A. A. Tovar and G. H. Clark, “Concentric-circle-grating, surface-emitting laser beam propagation in complex optical systems,” J. Opt. Soc. Am. A **14**, 3333–3340 (1997). [CrossRef]

6. A. Flores-Pérez, J. Hernández-Hernández, R. Jáuregui, and K. Volke-Sepúlveda, “Experimental generation and analysis of first-order TE and TM Bessel modes in free space,” Opt. Lett. **31**, 1732–1734 (2006). [CrossRef] [PubMed]

7. K. Volke-Sepulveda and E. Ley-Koo, “General construction and connections of vector propagation invariant optical fields: TE and TM modes and polarization states,” J. Opt. A: Pure Appl. Opt. , **8**, 867–877 (2006). [CrossRef]

8. M. A. Bandres and J. C. Gutiérrez-Vega, “Vector Helmholtz-Gauss and vector Laplace-Gauss beams,” Opt. Lett. **30**, 2155–2157 (2005). [CrossRef] [PubMed]

8. M. A. Bandres and J. C. Gutiérrez-Vega, “Vector Helmholtz-Gauss and vector Laplace-Gauss beams,” Opt. Lett. **30**, 2155–2157 (2005). [CrossRef] [PubMed]

2. L. W. Davis and G. Patsakos, “TM and TE electromagnetic beams in free space,” Opt. Lett. **6**, 22–23 (1981). [CrossRef] [PubMed]

3. Z. Bouchal and M. Olivík, “Non-diffractive vector Bessel beams,” J. Mod. Opt. **42**, 1555–1566 (1995). [CrossRef]

4. D. G. Hall, “Vector-beam solutions of Maxwell’s wave equation,” Opt. Lett. **21**, 9–11 (1996). [CrossRef] [PubMed]

9. L. J. Chu, “Electromagnetic waves in elliptic hollow pipes of metal,” J. Appl. Phys. **9**, 583–591 (1938). [CrossRef]

10. R. D. Spence and C. P. Wells, “The propagation of electromagnetic waves in parabolic pipes,” Phys. Rev. **62**, 58–62 (1942). [CrossRef]

11. J. C. Gutiérrez-Vega and M. A. Bandres, “Helmholtz-Gauss waves,” J. Opt. Soc. Am. A **22**, 289–298 (2005). [CrossRef]

12. C. López-Mariscal, M. A. Bandres, and J. C. Gutiérrez-Vega, “Observation of the experimental propagation properties of Helmholtz-Gauss beams,” Opt. Eng. **45**, 068001 (2006). [CrossRef]

*generalized*vHzG (gVHzG) beams. The paraxial propagation of the gVHzG beams is studied, not only in free space, but also through more general types of paraxial optical systems characterized by complex ABCD matrices, including lenses, Gaussian apertures, cascaded paraxial systems, and systems having quadratic amplitude as well as phase variations about the axis. By following a coordinate-free approach, rather than proposing solutions in a particular coordinate system, it was possible to derive an elegant and closed-form expression for the electromagnetic field, the vector angular spectrum, and the Poynting vector at the output plane of the ABCD system. It is found that the gVHzG beams are a class of vector fields which exhibit the property of form-invariance under paraxial optical transformations. The formulation described in this paper can be useful in applications where the polarization of the fields is of major concern [5

5. A. A. Tovar and G. H. Clark, “Concentric-circle-grating, surface-emitting laser beam propagation in complex optical systems,” J. Opt. Soc. Am. A **14**, 3333–3340 (1997). [CrossRef]

6. A. Flores-Pérez, J. Hernández-Hernández, R. Jáuregui, and K. Volke-Sepúlveda, “Experimental generation and analysis of first-order TE and TM Bessel modes in free space,” Opt. Lett. **31**, 1732–1734 (2006). [CrossRef] [PubMed]

13. Q. Zhan, “Trapping metallic Rayleigh particles with radial polarization,” Opt. Express **12**, 3377–3382 (2004). [CrossRef] [PubMed]

14. V. G. Niziev and A. V. Nesterov, “Influence of beam polarization on laser cutting efficiency,” J. Phys. D: Appl. Phys. **32**, 1455–1461 (1999). [CrossRef]

15. R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. **91**, 233901 (2003). [CrossRef] [PubMed]

## 2. Propagation of the generalized vector HzG beams

*iωt*) travelling in the

*z*direction (unit vector

**ẑ**) through an ABCD axisymmetric optical system with input and output planes located at

*z=z*

_{1}and

*z=z*

_{2}, respectively. The system is characterized by an ABCD matrix with, in general, complex elements

*A, B, C*, and

*D*that satisfy the unimodular relation

*AD-BC*=1.

**R**and the wave vector

**K**as

**r**and

**k**are the positions at the transverse planes of the configuration and frequency spaces, respectively. Additionally, we will denote a general vector field as

**F**=

**f**+

*f*

_{z}

**ẑ**, where

**f**=(

*f*

_{x}

*, f*

_{y}) represents the transverse part of the field. The transverse nabla operators in the configuration and frequency spaces are denoted by ∇=

**x̂**

*∂/∂x*+

**y**

*∂/y*and ∇=

**k**

*̂*

_{x}

*∂/∂k*

_{x}+

**k̂**

_{y}

*∂/∂k*

_{y}, respectively.

*z=z*

_{1}) and output (

*z=z*

_{2}) transverse planes of the ABCD system.

### 2.1. Definition of the generalized vector HzG beams

**e**(

**r**) and magnetic

**h**(

**r**) vectors of a first-class (TM) gVHzG beam at the input plane of the ABCD system as [8

8. M. A. Bandres and J. C. Gutiérrez-Vega, “Vector Helmholtz-Gauss and vector Laplace-Gauss beams,” Opt. Lett. **30**, 2155–2157 (2005). [CrossRef] [PubMed]

*K*=|

**K**|=

*ω*(

*µε*)

^{1/2}is the wave number.

*q*

_{1}=

*𝓡*and

*𝓘*denote the real and imaginary parts of a complex quantity, respectively. For simplicity in dealing with the ABCD system, the parameter

*q*

_{1}is used throughout the paper. The physical meaning of

*q*

_{1}is contained in the known relation 1/

*q*

_{1}=1/

*ℜ*

_{1}+

*i*2/

_{1}is the radius of curvature of the phase fronts, and

*w*

_{1}is the 1/

*e*amplitude spot size of the Gaussian modulation. In assuming a complex

*q*1 we are allowing for the possibility that the Gaussian apodization has an initial converging (

*q*

^{𝓘}1<0 must be fulfilled in order to satisfy the physical requirement that the field amplitude vanishes as

*r*becomes arbitrarily large.

_{1}

*W*(

**r**

_{1};

*κ*

_{1}) in Eq. (2) provides the vector nature of the transverse electric field. The scalar function

*W*(

**r**

_{1};

*k*

_{1}) is a solution of the two-dimensional Helmholtz equation [

*W*=0, and physically describes the transverse shape of an ideal scalar nondiffracting beam [16

16. Z. Bouchal, “Nondiffracting optical beams: physical properties, experiments, and applications” Czech. J. Phys. **53**, 537–578 (2003). [CrossRef]

22. J. C. Gutiérrez-Vega, M.D. Iturbe-Castillo, and S. Chávez-Cerda, “Alternative formulation for invariant optical fields: Mathieu beams,” Opt. Lett. **25**, 1493–1495 (2000). [CrossRef]

23. M. A. Bandres, J. C. Gutiérrez-Vega, and S. Chávez-Cerda, “Parabolic nondiffracting optical wave fields,” Opt. Lett. **29**, 44–46 (2004). [CrossRef] [PubMed]

**e**

_{1}(

**r**

_{1}) is determined from a scalar function

*W*(

**r**

_{1};

*κ*

_{1}), throughout the paper we will refer to the latter as the

*seed*function. The function

*W*can be formally expanded in terms of plane waves as

*κ*

_{1}and

*g*(

*ϕ*) are the transverse wave number and the angular spectrum of the ideal scalar nondiffracting beam, respectively. Since

*g*(

*ϕ*) is arbitrary, an infinite number of profiles can be obtained, see section 2.6 for important special cases.

*et al*. [1

1. M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys, Rev. A **11**, 1365–1370 (1975). [CrossRef]

**TE polarized gVHzG beams**: For the sake of space, throughout the paper we shall deal with the explicit expressions for the TM beams. However second-class (TE) beams can be readily obtained from Eqs (2) by applying the duality property, i.e. replacing

**E**with (

*µ/ε*)

^{1/2}

**H**and (

*β/ε*)

^{1/2}

**H**with -

**E**, namely

### 2.2. Classification of the generalized vector HzG beams

*κ*

_{1}in Eq. (3) is customarily assumed to be real and positive [16

16. Z. Bouchal, “Nondiffracting optical beams: physical properties, experiments, and applications” Czech. J. Phys. **53**, 537–578 (2003). [CrossRef]

22. J. C. Gutiérrez-Vega, M.D. Iturbe-Castillo, and S. Chávez-Cerda, “Alternative formulation for invariant optical fields: Mathieu beams,” Opt. Lett. **25**, 1493–1495 (2000). [CrossRef]

23. M. A. Bandres, J. C. Gutiérrez-Vega, and S. Chávez-Cerda, “Parabolic nondiffracting optical wave fields,” Opt. Lett. **29**, 44–46 (2004). [CrossRef] [PubMed]

*κ*

_{1}=

*Ordinary*VHzG (oVHzG) beams correspond to purely real

*κ*

_{1}=

*W*(

**r**

_{1};

*κ*

_{1}) is a two-dimensional purely oscillatory (or standing wave) function. The physical meaning of

*W*in the transverse direction. Far from the

*z*axis, the spatial period of the field oscillations tends monotonically to 2

*q*

_{1}=-

**30**, 2155–2157 (2005). [CrossRef] [PubMed]

*e*amplitude spot size of the Gaussian apodization.

*Modified*VHzG (mVHzG) beams correspond to purely imaginary

*κ*

_{1}=

*W*(

**r**

_{1},

^{2}]

*W*=0. Two concrete examples of seed functions of the modified kind are provided by the cosh-Gaussian beams [17

17. L. W. Casperson, D. G. Hall, and A. A. Tovar, “Sinusoidal-Gaussian beams in complex optical systems,” J. Opt. Soc. Am. A **14**, 3341–3348 (1997). [CrossRef]

18. S. Ruschin, “Modified Bessel nondiffracting beams,” J. Opt. Soc. Am. A **11**, 3224–3228 (1994). [CrossRef]

19. M. Santarsiero, “Propagation of generalized Bessel-Gauss beams through ABCD optical systems,” Opt. Commun. **132**, 1–7 (1996). [CrossRef]

*Generalized*VHzG (gVHzG) beams correspond to the general case when

*k*

_{1}is complex. As we will see, the gVHzG can be interpreted as intermediate solutions between oVHzG and mVHzG beams.

*K*≫1/

*w*

_{1}, i.e. the Gaussian spot width is many wavelengths wide, and additionally that

*K*≫|

*κ*

_{1}|, i.e. the spatial transverse beam oscillations must be many wavelengths wide. Two limiting cases of the gVHzG beams are of particular interest: (a) pure vector nondiffracting beams are obtained when |

*κ*

_{1}=0 leads to the generalized vector Laplace-Gauss beams, for which the seed function

*W*is now a solution of the 2D Laplace equation [8

**30**, 2155–2157 (2005). [CrossRef] [PubMed]

### 2.3. Propagation of the electromagnetic field through the ABCD system

**e**

_{1}(

**r**

_{1}) through the ABCD system can be performed by solving the Huygens diffraction integral [20

20. S. A. Collins, “Lens-system diffraction integral written in terms of matrix optics,” J. Opt. Soc. Am. **60**, 1168–1177 (1970). [CrossRef]

**e**

_{2}(

**r**

_{2}) is the output transverse electric field, and

*L*

_{0}is the optical path length from the input to the output plane of the ABCD system measured along the optical axis. The vector integral in Eq. (6) can be treated as a pair of independent scalar integrals by decomposing the transverse vector

**e**

_{1}(

**r**

_{1}) into orthogonal linearly polarized parts. After substituting each Cartesian component of Eq. (2) and the expansion given by Eq. (3) into Eq. (6), the integration can be performed applying the changes of variables

*x*

_{j}

*=u*

_{j}cos

*ϕ-v*

_{j}sinϕ, and

*y*

_{j}

*=u*

_{j}sin

*ϕ+v*

_{j}cosϕ, for

*j*=1,2. Upon returning to the original variables and regrouping the Cartesian components into a vector function, we obtain (see the detailed derivation in the appendix A)

*q*

_{1}travelling axially through the ABCD system, and the transformation laws for the parameters

*q*

_{1}and

*κ*

_{1}from the plane

*z*

_{1}to the plane

*z*

_{2}are

*G*(

**r**

_{2},

*q*

_{2}) in Eq. (7a) provides the confinement mechanism which ensures the transverse intensity distribution vanishes at large values of

*r*, and that the beam is square integrable. As expected, Eqs. (7) reduces to Eq. (2) when the ABCD matrix becomes the identity matrix.

*e*

_{z},2 and the magnetic

*h*

_{z},2 fields at the output plane can be readily obtained by applying the operator [(

*i/K*)∇

_{2}·] over the corresponding transverse components

**e**

_{2}(

**r**

_{2}) and

**h**

_{2}(

**r**

_{2}), respectively.

### 2.4. Poynting vector of the generalized vector HzG beams

**S**〉=Re(

**E**×

**H***)/2. It can be decomposed as 〈

**S**〉=〈

*s*

_{z}

**〉**ẑ+〈

**s**〉, where 〈

*s*

_{z}〉 is the longitudinal part which determines the flow of energy in the direction of propagation

*z*, and 〈

**s**〉 is the transverse part which determines the flow of energy perpendicular to this direction. The Poynting vector can be calculated at the input (

*j*=1) and output (

*j*=2) planes using the corresponding expressions for the electric and magnetic fields, we have

*f*

_{1}=exp(

*/2q*

_{1}), and

*f*

_{2}=(

*κ*

_{1}

*/κ*

_{2})exp(-

*iκ*

_{1}

*κ*

_{2}

*B/2K*)

*G*(

**r**

_{2},

*q*

_{2}).

*W*and

*q*

_{j}are, in general, complex, the beam exhibits a transverse flow of energy whose radial part is a manifestation of diffraction. For a paraxial beam it is expected that the longitudinal part of the energy flux be much more significant than the transverse part, in fact a simple analysis of orders of magnitude in Eq. (10) reveals that the longitudinal flow is at least

*Kw*

_{1}times the transverse one. Finally, for a lossless medium, the light power carried by the beam in longitudinal direction ∬〈

**S**

_{j}〉

**·**ẑd

^{2}

**r**

_{j}remains constant for any

*z*plane.

### 2.5. Propagation of the vector angular spectrum

**e**(

**r**) of the gVHzG beam at either the input or output planes of the ABCD system admits the plane wave expansion of the form

**e(r)**=(1/2

*π*)∬∞-∞

**ẽ(k)**exp(

*i*

**k·r**)d

^{2}

**k**, where

**ẽ(k)**is the vector angular spectrum whose functional form is obtained by Fourier inversion

**k**

*̂*

_{x}

*∂/∂k*

_{x}+

**k̂**

_{y}

*∂/∂k*

_{y}is the transverse nabla operator in the

**K**space, and

### 2.6. Remarks on the coordinates systems and polarization basis

11. J. C. Gutiérrez-Vega and M. A. Bandres, “Helmholtz-Gauss waves,” J. Opt. Soc. Am. A **22**, 289–298 (2005). [CrossRef]

11. J. C. Gutiérrez-Vega and M. A. Bandres, “Helmholtz-Gauss waves,” J. Opt. Soc. Am. A **22**, 289–298 (2005). [CrossRef]

*x,y*), generalized vector Gaussian beams can be constructed from superpositions of fundamental plane waves of the form

*W*=exp(

*i*κ⃗

_{1}·

**r**

_{1})=exp[

*iκ*

_{1}(

*x*cos

*κ+*ysin

*κ*)], see, for example, the cosine-Gauss beams studied in Ref. [11

**22**, 289–298 (2005). [CrossRef]

*r,θ*) corresponds to eigenfunctions

*W*=Je

_{m}(

*κr*)exp(

*imθ*) for which gVHzG beams reduce to the

*m*th-order generalized vector Bessel-Gauss beams [4

4. D. G. Hall, “Vector-beam solutions of Maxwell’s wave equation,” Opt. Lett. **21**, 9–11 (1996). [CrossRef] [PubMed]

*ξ,η*), generalized vector even Mathieu-Gauss beams of

*m*th-order and ellipticity parameter

*ε*can be constructed from the eigenfunctions

*W*=Je

_{m}(

*ξ,ε*)ce

_{m}(

*η,ε*), where Je(·) and ce(·) are the radial and angular even Mathieu functions of

*m*th-order and parameter

*ε*, respectively [22

22. J. C. Gutiérrez-Vega, M.D. Iturbe-Castillo, and S. Chávez-Cerda, “Alternative formulation for invariant optical fields: Mathieu beams,” Opt. Lett. **25**, 1493–1495 (2000). [CrossRef]

*u,v*), generalized vector even parabolic-Gauss beams can be constructed from the eigenfunctions

*p*and even parity [23

23. M. A. Bandres, J. C. Gutiérrez-Vega, and S. Chávez-Cerda, “Parabolic nondiffracting optical wave fields,” Opt. Lett. **29**, 44–46 (2004). [CrossRef] [PubMed]

**e**

_{2}(

**r**

**2**) at any point r2 into two orthogonal polarized transverse parts. For instance, in polar coordinates, the transverse vector fields can be split into radial and azimuthal polarized components, or in elliptic coordinates, into elliptic and hyperbolic polarized components. Explicit expressions for these polarization basis associated to particular coordinate systems can be found elsewhere [24, 25].

*α*and β are arbitrary constants. By combining the different polarization basis with the different seed functions

*W*(

**r**

_{2};

*κ*

_{2}) a large variety of beam profiles with specific polarization states could be constructed through superposition. As example, consider the basis of circular polarizations

**u**û±=(

**x̂**±

*i*

**ŷ**)/√2. It is easy to verify that the gradient of

*W*in Eqs. (2) and (7a) can be expressed in the basis of vectors

**û**± as

**e**

^{±}=

**e**

^{(TM)}±ie(TE) at either the input (

*j*=1) or output (

*j*=2) planes of the ABCD system, we have explicitly

*f*

_{1}=exp(

*q*

_{1}), and

*f*2=(

*κ*

_{1}/

*κ*

_{2})exp(-

*iκ*

_{1}

*κ*

_{2}

*B*/2K)

*G*(

**r**

_{2},

*q*

_{2}). A similar approach can be applied for the vector angular spectra in Eq. (12) to derive the corresponding circular polarization states.

*W*and

**ẑ**×∇

*W*, respectively. Both vector fields Ψ

^{(1)}=∇

*W*and Ψ

^{(2)}=

**ẑ**×∇

*W*constitute two independent vector solutions of the 2D vector Helmholtz equation ∇

^{2}Ψ+

*κ*

^{2}Ψ=0 [24, 25]. Now, if we set the function

*W=W*

_{m}to be the

*m*-th eigensolution belonging to a countable set of complete and orthogonal solutions of the scalar Helmholtz equation, then, because of the linearity and the one-to-one mapping of the gradient operator, the properties of linear independence, orthogonality, and completeness exhibited by the family of scalar solutions

*W*

_{m}are transferred to the corresponding families of vector fields Ψ

^{(1)}and Ψ

^{(2)}. Additionally, the transverse fields of a TM and a TE beams are orthogonal, even when their seed functions Wm are equal. In this sense, the gVHzG beams exhibit similar polarization properties than the ideal vector nondiffracting beams [3

3. Z. Bouchal and M. Olivík, “Non-diffractive vector Bessel beams,” J. Mod. Opt. **42**, 1555–1566 (1995). [CrossRef]

7. K. Volke-Sepulveda and E. Ley-Koo, “General construction and connections of vector propagation invariant optical fields: TE and TM modes and polarization states,” J. Opt. A: Pure Appl. Opt. , **8**, 867–877 (2006). [CrossRef]

16. Z. Bouchal, “Nondiffracting optical beams: physical properties, experiments, and applications” Czech. J. Phys. **53**, 537–578 (2003). [CrossRef]

9. L. J. Chu, “Electromagnetic waves in elliptic hollow pipes of metal,” J. Appl. Phys. **9**, 583–591 (1938). [CrossRef]

10. R. D. Spence and C. P. Wells, “The propagation of electromagnetic waves in parabolic pipes,” Phys. Rev. **62**, 58–62 (1942). [CrossRef]

## 3. Physical discussion of the propagation properties

### 3.1. Free space propagation

*L=z*

_{2}-

*z*

_{1}. The input and output fields are given by Eqs. (2) and (7a) with

*A*=1,

*B=L, C*=0, and

*D*=1. From Eqs. (9) the transformation laws become

*q*

_{2}

*κ*

_{2}=

*q*

_{1}

*κ*

_{1}remains constant under free-space propagation.

**22**, 289–298 (2005). [CrossRef]

*g*(

*ϕ*). This physical picture is evident after replacing Eqs. (17) into Eq. (7a) and observing that the TM polarized gVHzG beams in vacuum can be rewritten as

*L/q*

_{1}and

_{1}=κ

_{1}(cos

*ϕ*

**x̂**+sin

*ϕ*

**ŷ**) Equation (19) represents the free space propagation along a distance

*L*of a tilted Gaussian beam with input parameter

*q*

_{1}whose mean wave vector has a projection

_{1}over the transverse plane [11

**22**, 289–298 (2005). [CrossRef]

_{1}.

*initial plane at z=z*

_{1}.

*waist plane*(

*z=z*

_{waist}) corresponds to the plane where the width of the elementary Gaussian beams is minimum, i.e. where the radial factor exp (

*q*2) becomes a real Gaussian envelope. Using this condition, from Eqs. (17) we get

*z*

_{waist}

*=z*

_{1}

*q*becomes purely imaginary

*q*

_{waist}

*κ*reduces to

*κ*

_{waist}

*=κ*

_{1}(1-

*W*and

*κ*is set to be purely real, then at the waist plane the energy flow becomes purely longitudinal.

*vertex plane*(

*z=z*

_{vertex}) corresponds to the plane where the main propagation axes of the constituent Gaussian beams intersect. As shown in Fig. 1, the pseudo-nondiffracting region delimits the region where significant interference of the constituent vector Gaussian beams occurs, and where the transverse beam profile exhibits a standing-wave behavior. The evaluation of the condition

*r*

_{gen}=0 in Eq. (20) yields

*κ*becomes purely real

*κ*

_{vertex}

*q*

_{vertex}

*=κ*

_{1}

*q*

_{1}/

*κ*

_{vertex}. At the vertex plane the extent of the pseudo-nondiffracting region is maximum, and its 1/

*e*amplitude Gaussian spot size can be calculated with

*w*

_{vertex}=[

*K*Im(1/

*q*

_{vertex})/2]

^{-1/2}.

**30**, 2155–2157 (2005). [CrossRef] [PubMed]

*z*

_{1}=0,

*q𝓡*

_{1}=0, and

*z*=0, and the cone generatrix reduces to the expected

*r*

_{gen}=(

*κ*

^{𝓡}1/

*K*)

*z*.

*r*

_{gen}=

*r*

_{0}=

*K*becomes a constant, and therefore the mVHzG beams may be viewed as a superposition of vector Gaussian beams whose axes are parallel to the

*z*axis and lie on the surface of a circular cylinder of radius

*r*

_{0}.

*i2K∂/∂z*]{

**e**

_{1},

**h**

_{1}}=0 and correspond to the purely transverse zeroth-order electric and magnetic fields of the perturbative series expansion of the Maxwell equations provided by Lax

*et al*. [1

1. M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys, Rev. A **11**, 1365–1370 (1975). [CrossRef]

### 3.2. Propagation through a GRIN medium

*n*(

*r*)=

*n*

_{0}(1-

*r*

^{2}/2

*a*

^{2}). The ABCD transfer matrix from plane

*z*

_{1}to plane

*z*

_{2}=

*z*

_{1}+

*L*is given by

*z*

_{2}is described by Eq. (7a). Substitution of the matrix elements in Eq. (22) into Eqs. (9) yields the parameter transformations:

*q*and

*κ*vary periodically with a longitudinal period 2

*πa*, therefore, the initial field distribution self-reproduces after a distance 2

*πa*.

*z*

_{1}=0 belongs to the oVHzG kind (i.e.

*q*

_{1}=

*L=L*

_{F}

*=πa*/2, the ABCD matrix Eq. (22) reduces to [0,

*a*;-1/

*a*,0] which is indeed identical to the matrix transformation from the first to the second focal plane of a converging thin lens of focal length

*a*, i.e. a Fourier transformer. At the Fourier plane

*L*=

*L*

_{F}, from Eqs. (23) we see that both parameters

*q*

_{2}=

*ia*

^{2}/

*q*

^{𝓘}1 and

*κ*

_{2}=

*a*become purely imaginary. It is now evident that if an oVHzG profile is Fourier transformed, a mVHzG profile will be obtained, and vice versa. The intermediate profiles belong to the gVHzG kind where, for the particular case of the GRIN medium, the transition between both types of beams is characterized by the continuous transformations given in Eqs. (23).

*z*

_{R}

*q*

_{2}=

*q*

_{1}=-

*ia*remains constant under propagation and that the wave number

*κ*

_{2}=

*κ*

_{1}exp(-

*iL/a*) rotates at a constant rate over the complex plane (

*balanced*propagation, and a

*non-balanced*otherwise.

*κ*

_{1}=30 mm

^{-1}through a GRIN medium with

*a*=1/√2

*π*m. The input fields are given by Eq. (16) with

*j*=1 for

*K*=2

*π*/λ and λ=632.8 nm. The animations were constructed by calculating the field at 200 transverse planes evenly spaced from the input (

*z*

_{1}=0) to the output (

*z*

_{2}=4

*L*

_{F}=2

*πa*) planes using Eq. (16) with

*j*=2 to generate the left or right circularly polarized fields as the case may be.

*W*(

**r**

_{1}) given by the superposition of

*N*plane waves of the general form

*A*

_{n}are complex amplitudes. For Fig. 2(a) we have chosen a left circularly polarized oVHzG beam in a balanced condition (

*q*

_{1}=-

*ia*=-

*i*/√2

*π*) with

*N*=3, An={1,1,1}, and

*ϕ*

_{n}={90°,-30°,-150°}. Note that the width of the constituent Gaussian beams remains constant under propagation because the beams are balanced. Following the established convention, at a given

*z*plane, the transverse components of the fields rotate anti-clockwise for left-handed circular polarization as time increases. The field at the plane

*z*

_{2}=

*L*

_{F}is shown Fig. 2(b), where we note that for the selected amplitudes

*A*

_{n}=1 the beam polarization becomes purely radial. To show the non-balanced condition, in Fig. 2(c) we propagated the same gVHzG by setting now

*q*

_{1}=0.4-

*i*0.8/√2

*π*and keeping all remaining parameters unchanged. The video shows clearly that the width of the constituent Gaussian beams change under propagation and reach a minimum at the plane where

*q*

_{2}in Eqs. (23) becomes purely imaginary (~1.22

*L*

_{F})

*N*=8 constituent Gaussian beams. By means of the amplitudes and phases of the coefficients

*A*

_{n}it is possible to adjust the polarization state of the resulting beam. In this case we set

*A*

_{n}

*=i*such that the electric field at the plane

*z*

_{2}=

*L*

_{F}now becomes purely azimuthal, as shown in Fig. 2(f). In Fig. 2(g) we set

*A*

_{n}=exp(-

*iπn*/4) such that the electric field vectors at each point on the plane

*z*

_{2}=

*L*

_{F}become parallel, as shown in Fig. 2(h).

*W*(

**r**

_{1})=

*iJ*

_{3}(

*κ*

_{1}

*r*

_{1})cos (3

*θ*

_{1}). The presence of the factor i in the seed function produces that, at the plane

*z*

_{2}=

*L*

_{F}, the VBG beam be azimuthally polarized and belongs to the modified VHzG kind. The second propagation shown in Fig. 3(c) corresponds to a non-balanced VBG beam with

*W*(

**r**

_{1})=

*J*

_{3}(

*κ*

_{1}

*r*

_{1})cos (3

*θ*

_{1}) and complex input parameter

*q*

_{1}=0.4-

*i*0.8/√2

*π*.

**22**, 289–298 (2005). [CrossRef]

**25**, 1493–1495 (2000). [CrossRef]

*W*(

**r**

_{1})=Je

_{4}(ξ,3)ce

_{4}(

*η*,3)+

*i*Jo4(ξ,3)se

_{4}(

*η*,3), where (

*ξ*,h) are the elliptic coordinates defined as

*x*=

*f*cosh ξ cos

*η*and

*y*=

*f*sinh ξ sin

*η*, with

*f*being the semifocal distance. In this case the vector beam is balanced but the initial field belongs to the gVHzG kind with

*κ*

_{1}=30+

*i*15 mm

^{-1}. The input field is given by Eq. (16) where the Cartesian partial derivatives are expressed in elliptic coordinates as follows

*q*and

*κ*vary according to Eq. (23). For this value of

*κ*

_{1}, the typical elliptic annular intensity pattern of the ordinary helical VMG beams occurs approximately at

*z*⋍0.28

*L*

_{F}, while the expected circular annular pattern of the modified helical VMG beams occurs at

*z*⋍1.28

*L*

_{F}.

**22**, 289–298 (2005). [CrossRef]

**29**, 44–46 (2004). [CrossRef] [PubMed]

*p*=2. The Cartesian derivatives are expressed in the Parabolic coordinates

*x*=(

*v*

^{2}-

*u*

^{2})/2, and

*y=uv*as follows

*κ*

_{1}=

*i*30 mm

^{-1}corresponding to a VPG of the modified kind. As expected for this initial condition, the beam now will belong to the ordinary kind at the plane

*z*2=

*L*

_{F}.

**22**, 289–298 (2005). [CrossRef]

12. C. López-Mariscal, M. A. Bandres, and J. C. Gutiérrez-Vega, “Observation of the experimental propagation properties of Helmholtz-Gauss beams,” Opt. Eng. **45**, 068001 (2006). [CrossRef]

26. M. Guizar-Sicairos and J. C. Gutiérrez-Vega, “Generalized Helmholtz-Gauss beams and its transformation by paraxial optical systems,” Opt. Lett. **31**, 2912–2914 (2006). [CrossRef] [PubMed]

## 4. Conclusions

*q*and

*κ*. The derivation of the new formulation has included the possibility of propagation in complex lenslike media having at most quadratic transverse variations of the index of refraction and the gain or loss.

*κ*

_{1}and

*κ*

_{2}are not proportional to each other through a real factor leading to different profiles of the function

*W*, and moreover because the parameters

*q*

_{1}and

*κ*

_{1}are transformed according to different laws. Gaussian apodized fields with arbitrary polarization can be built up with a suitable superposition of constituent gVHzG beams with the same Gaussian envelope and transverse spatial frequency.

## A. Appendix: Derivation of the output field e_{2}(**r**_{2}) [Eq. (7a)]

_{1}can be introduced within the angular integral and its application gives raise to two Cartesian field components, namely

**e**

_{2}(

**r**

_{2})=

*e*

_{x}

**x̂**+

*e*

_{y}

**ŷ**. Working with the

*x*component we obtain the expression

*a*

^{2}=-

*i*(

*K*/2)(

*A/B*).

*W*(

**r**

_{2};

*κ*

_{2}), then we have

*q*

_{2}and

*κ*

_{2}are given by Eq. (9),

*G*(

**r**

_{2},

*q*

_{2}) is the fundamental Gaussian beam [Eq. (8)], and the unimodular condition

*AD-BC*=1 has been applied to eliminate

*D*. The expression for

*e*

_{y}is also given by Eq. (32) with the difference that the partial derivative is now taken with respect to the

*y*coordinate. Superposing both Cartesian components and using the gradient operator definition yields

## Acknowledgments

## References and links

1. | M. Lax, W. H. Louisell, and W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys, Rev. A |

2. | L. W. Davis and G. Patsakos, “TM and TE electromagnetic beams in free space,” Opt. Lett. |

3. | Z. Bouchal and M. Olivík, “Non-diffractive vector Bessel beams,” J. Mod. Opt. |

4. | D. G. Hall, “Vector-beam solutions of Maxwell’s wave equation,” Opt. Lett. |

5. | A. A. Tovar and G. H. Clark, “Concentric-circle-grating, surface-emitting laser beam propagation in complex optical systems,” J. Opt. Soc. Am. A |

6. | A. Flores-Pérez, J. Hernández-Hernández, R. Jáuregui, and K. Volke-Sepúlveda, “Experimental generation and analysis of first-order TE and TM Bessel modes in free space,” Opt. Lett. |

7. | K. Volke-Sepulveda and E. Ley-Koo, “General construction and connections of vector propagation invariant optical fields: TE and TM modes and polarization states,” J. Opt. A: Pure Appl. Opt. , |

8. | M. A. Bandres and J. C. Gutiérrez-Vega, “Vector Helmholtz-Gauss and vector Laplace-Gauss beams,” Opt. Lett. |

9. | L. J. Chu, “Electromagnetic waves in elliptic hollow pipes of metal,” J. Appl. Phys. |

10. | R. D. Spence and C. P. Wells, “The propagation of electromagnetic waves in parabolic pipes,” Phys. Rev. |

11. | J. C. Gutiérrez-Vega and M. A. Bandres, “Helmholtz-Gauss waves,” J. Opt. Soc. Am. A |

12. | C. López-Mariscal, M. A. Bandres, and J. C. Gutiérrez-Vega, “Observation of the experimental propagation properties of Helmholtz-Gauss beams,” Opt. Eng. |

13. | Q. Zhan, “Trapping metallic Rayleigh particles with radial polarization,” Opt. Express |

14. | V. G. Niziev and A. V. Nesterov, “Influence of beam polarization on laser cutting efficiency,” J. Phys. D: Appl. Phys. |

15. | R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. |

16. | Z. Bouchal, “Nondiffracting optical beams: physical properties, experiments, and applications” Czech. J. Phys. |

17. | L. W. Casperson, D. G. Hall, and A. A. Tovar, “Sinusoidal-Gaussian beams in complex optical systems,” J. Opt. Soc. Am. A |

18. | S. Ruschin, “Modified Bessel nondiffracting beams,” J. Opt. Soc. Am. A |

19. | M. Santarsiero, “Propagation of generalized Bessel-Gauss beams through ABCD optical systems,” Opt. Commun. |

20. | S. A. Collins, “Lens-system diffraction integral written in terms of matrix optics,” J. Opt. Soc. Am. |

21. | A. E. Siegman, |

22. | J. C. Gutiérrez-Vega, M.D. Iturbe-Castillo, and S. Chávez-Cerda, “Alternative formulation for invariant optical fields: Mathieu beams,” Opt. Lett. |

23. | M. A. Bandres, J. C. Gutiérrez-Vega, and S. Chávez-Cerda, “Parabolic nondiffracting optical wave fields,” Opt. Lett. |

24. | J. A. Stratton, |

25. | P. Morse and H. Feshbach, |

26. | M. Guizar-Sicairos and J. C. Gutiérrez-Vega, “Generalized Helmholtz-Gauss beams and its transformation by paraxial optical systems,” Opt. Lett. |

**OCIS Codes**

(050.1970) Diffraction and gratings : Diffractive optics

(140.3300) Lasers and laser optics : Laser beam shaping

(260.5430) Physical optics : Polarization

(350.5500) Other areas of optics : Propagation

**ToC Category:**

Diffraction and Gratings

**History**

Original Manuscript: August 2, 2006

Revised Manuscript: September 15, 2006

Manuscript Accepted: September 15, 2006

Published: October 2, 2006

**Citation**

Raul I. Hernandez-Aranda, Julio C. Gutiérrez-Vega, Manuel Guizar-Sicairos, and Miguel A. Bandres, "Propagation of generalized vector Helmholtz-Gauss beams through paraxial optical systems," Opt. Express **14**, 8974-8988 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-20-8974

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### References

- M. Lax, W. H. Louisell, and W. B. McKnight, "From Maxwell to paraxial wave optics," Phys, Rev. A 11, 1365-1370 (1975). [CrossRef]
- L. W. Davis and G. Patsakos, "TM and TE electromagnetic beams in free space," Opt. Lett. 6, 22-23 (1981). [CrossRef] [PubMed]
- Z. Bouchal and M. Olivık, "Non-diffractive vector Bessel beams," J. Mod. Opt. 42, 1555-1566 (1995). [CrossRef]
- D. G. Hall, "Vector-beam solutions of Maxwell’s wave equation," Opt. Lett. 21, 9-11 (1996). [CrossRef] [PubMed]
- A. A. Tovar and G. H. Clark, "Concentric-circle-grating, surface-emitting laser beam propagation in complex optical systems," J. Opt. Soc. Am. A 14, 3333-3340 (1997). [CrossRef]
- A. Flores-Perez, J. Hernandez-Hernandez, R. Jauregui, and K. Volke-Sepulveda, "Experimental generation and analysis of first-order TE and TM Bessel modes in free space," Opt. Lett. 31, 1732-1734 (2006). [CrossRef] [PubMed]
- K. Volke-Sepulveda and E. Ley-Koo, "General construction and connections of vector propagation invariant optical fields: TE and TM modes and polarization states," J. Opt. A: Pure Appl. Opt., 8, 867-877 (2006). [CrossRef]
- M. A. Bandres and J. C. Gutierrez-Vega, "Vector Helmholtz-Gauss and vector Laplace-Gauss beams," Opt. Lett. 30, 2155-2157 (2005). [CrossRef] [PubMed]
- L. J. Chu, "Electromagnetic waves in elliptic hollow pipes of metal," J. Appl. Phys. 9, 583-591 (1938). [CrossRef]
- R. D. Spence and C. P. Wells, "The propagation of electromagnetic waves in parabolic pipes," Phys. Rev. 62, 58-62 (1942). [CrossRef]
- J. C. Gutierrez-Vega and M. A. Bandres, "Helmholtz-Gauss waves," J. Opt. Soc. Am. A 22, 289-298 (2005). [CrossRef]
- C. Lopez-Mariscal, M. A. Bandres, and J. C. Gutierrez-Vega, "Observation of the experimental propagation properties of Helmholtz-Gauss beams," Opt. Eng. 45, 068001 (2006). [CrossRef]
- Q. Zhan, "Trapping metallic Rayleigh particles with radial polarization," Opt. Express 12, 3377-3382 (2004). [CrossRef] [PubMed]
- V. G. Niziev and A. V. Nesterov, "Influence of beam polarization on laser cutting efficiency," J. Phys. D: Appl. Phys. 32, 1455-1461 (1999). [CrossRef]
- R. Dorn, S. Quabis, and G. Leuchs, "Sharper focus for a radially polarized light beam," Phys. Rev. Lett. 91, 233901 (2003). [CrossRef] [PubMed]
- Z. Bouchal, "Nondiffracting optical beams: physical properties, experiments, and applications" Czech. J. Phys. 53,537-578 (2003). [CrossRef]
- L. W. Casperson, D. G. Hall, and A. A. Tovar, "Sinusoidal-Gaussian beams in complex optical systems," J. Opt. Soc. Am. A 14, 3341-3348 (1997). [CrossRef]
- S. Ruschin, "Modified Bessel nondiffracting beams," J. Opt. Soc. Am. A 11, 3224-3228 (1994). [CrossRef]
- M. Santarsiero, "Propagation of generalized Bessel-Gauss beams through ABCD optical systems," Opt. Commun. 132, 1-7 (1996). [CrossRef]
- S. A. Collins, "Lens-system diffraction integral written in terms of matrix optics," J. Opt. Soc. Am. 60, 1168-1177 (1970). [CrossRef]
- A. E. Siegman, Lasers (University Science, 1986).
- J. C. Gutierrez-Vega, M.D. Iturbe-Castillo, and S. Chavez-Cerda, "Alternative formulation for invariant optical fields: Mathieu beams," Opt. Lett. 25, 1493-1495 (2000). [CrossRef]
- M. A. Bandres, J. C. Gutierrez-Vega, and S. Chavez-Cerda, "Parabolic nondiffracting optical wave fields," Opt. Lett. 29, 44-46 (2004). [CrossRef] [PubMed]
- J. A. Stratton, Electromagnetic theory (McGraw-Hill, New York, 1941)
- P. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).
- M. Guizar-Sicairos and J. C. Gutierrez-Vega, "Generalized Helmholtz-Gauss beams and its transformation by paraxial optical systems," Opt. Lett. 31, 2912-2914 (2006). [CrossRef] [PubMed]

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