## An alternative theoretical model for an anomalous hollow beam

Optics Express, Vol. 16, Issue 19, pp. 15254-15267 (2008)

http://dx.doi.org/10.1364/OE.16.015254

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### Abstract

An alternative and convenient theoretical model is proposed to describe a flexible anomalous hollow beam of elliptical symmetry with an elliptical solid core, which was observed in experiment recently (Phys. Rev. Lett, 94 (2005) 134802). In this model, the electric field of anomalous hollow beam is expressed as a finite sum of elliptical Gaussian modes. Flat-topped beams, dark hollow beams and Gaussian beams are special cases of our model. Analytical propagation formulae for coherent and partially coherent anomalous hollow beams passing through astigmatic ABCD optical systems are derived. Some numerical examples are calculated to show the propagation and focusing properties of coherent and partially coherent anomalous hollow beams.

© 2008 Optical Society of America

## 1. Introduction

1. C. Palma, “Decentered Gaussian beams, ray bundles and Bessel-Gaussian beams,” Appl. Opt. **36**, 1116–1120 (1997). [CrossRef] [PubMed]

11. C. Arpali, C. Yazicioglu, H. T. Eyyuboglu, S. A. Arpali, and Y. Baykal, “Simulator for general-type beam propagation in turbulent atmosphere,” Opt. Express **14**, 8918–8928 (2006). [CrossRef] [PubMed]

6. Y. Cai and L. Zhang, “Coherent and partially coherent dark hollow beams with rectangular symmetry and paraxial propagation properties,” J. Opt. Soc. Am. B **23**, 1398–1407 (2006). [CrossRef]

9. H. T. Eyyuboğlu, “Propagation and coherence properties of higher order partially coherent dark hollow beams in turbulence,” Opt. Laser Technol. **40**, 156–166 (2008). [CrossRef]

25. Y. Cai, H. T. Eyyuboğlu, and Y. Baykal, “Scintillation of astigmatic dark hollow beams in a weak turbulent atmosphere,” J. Opt. Soc. Am. A **25**, 1497–1503 (2008). [CrossRef]

26. Y. K. Wu, J. Li, and J. Wu, “Anomalous hollow electron beams in a storage ring,” Phys. Rev. Lett . **94**, 134802 (2005). [CrossRef] [PubMed]

26. Y. K. Wu, J. Li, and J. Wu, “Anomalous hollow electron beams in a storage ring,” Phys. Rev. Lett . **94**, 134802 (2005). [CrossRef] [PubMed]

*approximate*model was recently proposed by Cai to describe an anomalous hollow beam [27

27. Y. Cai, “Model for an anomalous hollow beam and its paraxial propagation,” Opt. Lett . **32**, 3179–3181 (2007). [CrossRef] [PubMed]

*stigmatic*(i.e., symmetric) ABCD optical systems have been derived in [27

27. Y. Cai, “Model for an anomalous hollow beam and its paraxial propagation,” Opt. Lett . **32**, 3179–3181 (2007). [CrossRef] [PubMed]

*x*) of the out elliptical ring are the same with those of the elliptical solid core, but in practical case, they can be different as shown in [26

26. Y. K. Wu, J. Li, and J. Wu, “Anomalous hollow electron beams in a storage ring,” Phys. Rev. Lett . **94**, 134802 (2005). [CrossRef] [PubMed]

*flexible*anomalous hollow beam through beam combination, which is more close to experimental results reported in [26

**94**, 134802 (2005). [CrossRef] [PubMed]

*astigmatic*(i.e., non-symmetric) ABCD optical systems are derived. Some numerical examples are given.

## 2. An alternative model for an anomalous hollow beam

**94**, 134802 (2005). [CrossRef] [PubMed]

*w*

_{0x}and

*x*

_{0y}are the beam widths along the long axis and short axis of the fundamental elliptical Gaussian mode for constructing the out ring of the elliptical anomalous hollow beam, respectively, and

*θ*is the orientation angle between the long axis of the out elliptical ring and the horizontal axis x.

*w*and

_{1x}*w*are the beam widths along the long axis and short axis of the fundamental elliptical Gaussian mode for constructing the elliptical solid core of the elliptical anomalous hollow beam, respectively, and ϕ is the orientation angle between the long axis of the elliptical solid core and the horizontal axis x. We call N the beam order of the anomalous hollow beam mainly for controlling the dark size of the anomalous hollow beam and relative peak value of the solid core. p is a parameter mainly for controlling the dark size and the relative peak value of the solid core and satisfy 0<p<1. α is a parameter mainly for controlling the relative peak value of the solid core and satisfy α > 0. β is a parameter mainly for controlling the dark size and the beam spot size of the solid core and satisfy β > 0. When α = 0 and p = 0, Eq. (1) reduces to the expression for the electric filed of an elliptical flat-topped beam [4, 5]. When α = 0, Eq. (1) reduces to the expression for the electric filed of a controllable elliptical dark hollow beam [7]. When p = 1, Eq. (1) reduces to the expression for the electric filed of an elliptical Gaussian beam. Thus, with suitable beam parameters

_{1y}*w*,

_{0x}*w*,

_{0y}*w*,

_{1x}*w*, N, p, α and β, Eq. (1) provides an alternative and convenient model for describing an anomalous hollow beam with controllable beam properties (i.e., beam spot size, orientation angle, dark size, relative peak value of the solid core and ellipticity) as shown in Fig. 1. From Eq. (1) and Fig. 1, we can find the effective beam widths of the out ring of the elliptical anomalous hollow beam are determined by N,

_{1y}*w*and

_{0x}*w*together (i.e., the first term and second term of Eq. (1) mainly determine the out ring), and the effective beam widths of the solid core are determined by β,

_{0y}*w*and

_{1x}*w*together (i.e., the third term of Eq. (1) mainly determines the solid core). For suitable value of β, we can choose the values of

_{1y}*w*and

_{1x}*w*to be larger than

_{1y}*w*and

_{0x}*w*, but of course we can’t choose the values of

_{0y}*w*and

_{1x}*w*arbitrary large as shown in Fig. 2.

_{1y}27. Y. Cai, “Model for an anomalous hollow beam and its paraxial propagation,” Opt. Lett . **32**, 3179–3181 (2007). [CrossRef] [PubMed]

*w*

_{0x}and

*w*

_{0y}, the dark size, the relative peak value and beam spot size of the solid core are fixed. By controlling the values of

*w*

_{0x},

*w*

_{0y}and

*α*, we can control the ellipticity and the orientation angle of the out elliptical ring and the elliptical solid core, but the ellipticity and the orientation angle of the out elliptical ring are the same with those of the elliptical solid core in any case. Thus the alternative model proposed in present paper is more suitable and flexible than the model in [27

**32**, 3179–3181 (2007). [CrossRef] [PubMed]

**94**, 134802 (2005). [CrossRef] [PubMed]

*k*= 2

*π*/

*λ*is the wave number,

*λ*is the wavelength of the beam,

**r**

_{1}is the position vector given by

**r**

^{T}

_{1}=(

*x*

*y*),

**Q**

^{-1}

_{1}is a 2 × 2 matrix called the complex curvature tensor for an elliptical Gaussian beam [29, 30]. In our case,

**Q**

^{-1}

_{1n},

**Q**

^{-1}

_{1np}and

**Q**

^{-1}

**1β**are given by|

## 3. Paraxial propagation of an anomalous hollow beam through ABCD optical systems

**32**, 3179–3181 (2007). [CrossRef] [PubMed]

*stigmatic*ABCD optical systems based on the proposed theoretical model. In this section, we study the propagation of a flexible anomalous hollow beam through

*astigmatic*ABCD optical systems. Within the validity of the paraxial approximation, propagation of a coherent laser beam through an

*astigmatic*ABCD optical system can be studied with the help of the following generalized Collins formula [16

16. Y. Cai and Q. Lin, “Hollow elliptical Gaussian beam and its propagation through aligned and misaligned paraxial optical systems,” J. Opt. Soc. Am. A **21**, 1058–1065 (2004). [CrossRef]

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

*E*(

*r*

_{1}, 0) and

*E*(

**ρ**1,

*l*) are the electric fields of the laser beam in the source plane (

*z*= 0) and the output plane (

*z*=

*l*), respectively.

*ρ*

^{T}

_{1}=(

*ρ*

_{1x}

*ρ*

_{1y}) with

*ρ*

_{1}being the position vectors in the output planes.

*k*= 2

*π*/

*λ*is the wave number,

*λ*is the wavelength of light.

*l*is the axial distance from the input plane to the output plane.

**A**,

**B**,

**C**and

**D**are the 2 × 2 sub-matrices of the astigmatic optical system [29, 30] and satisfy the following Luneburg relations that describe the symplecticity of an astigmatic optical system [32]

*astigmatic*ABCD optical system

**A**=

**I**,

**B**=

*z*

**I**,

**C**= 0

**I**,

**D**=

**I**with I being a 2 × 2unit matrix. Figure 3 shows the normalized 3D-intensity distribution of an anomalous hollow beam and cross line (y = 0) in free space at several different propagation distances with

*w*

_{0x}= 1

*mm*,

*w*

_{0y}= 0.5

*mm*,

*w*

_{1x}= 1

*mm*,

*w*

_{1y}= 0.5

*mm*,

*θ*=

*ϕ*= 0,

*N*= 3,

*p*= 0.8,

*α*= 0.2,

*β*= 0.5 and

*λ*= 632.8

*nm*. One sees from Fig. 3 that as the propagation distance z increases, the initial beam profile gradually disappears, i.e., the dark region disappears, the central intensity increases gradually and the beam profile becomes non-elliptical symmetry. In the far field, the anomalous hollow beam retains its elliptical symmetry and there is a small bright elliptical ring around the brightest elliptical solid beam spot (see Fig. 3 (f)), and the long axis and short axis of the elliptical beam spot in far field has interchanged their positions compared to the elliptical beam spot in near field (see Fig. 3 (a)). The interesting propagation properties of anomalous hollow beams are caused by the fact that an anomalous hollow beam is not a pure mode, but a combination of elliptical Gaussian modes, and these different modes will overlap and interfere in propagation. The propagation properties of anomalous hollow beam in free space in this paper are consistent with those in Ref. [27

**32**, 3179–3181 (2007). [CrossRef] [PubMed]

## 3. Partially coherent an anomalous hollow beam and its propagation

46. F. Wang and Y. Cai, “Experimental observation of fractional Fourier transform for a partially coherent optical beam with Gaussian statistics,” J. Opt. Soc. Am. A **24**, 1937–1944 (2007). [CrossRef]

6. Y. Cai and L. Zhang, “Coherent and partially coherent dark hollow beams with rectangular symmetry and paraxial propagation properties,” J. Opt. Soc. Am. B **23**, 1398–1407 (2006). [CrossRef]

47. X. Lü and Y. Cai, “Partially coherent circular and elliptical dark hollow beams and their paraxial propagations,” Phys. Lett. A **369**, 157–166 (2007). [CrossRef]

6. Y. Cai and L. Zhang, “Coherent and partially coherent dark hollow beams with rectangular symmetry and paraxial propagation properties,” J. Opt. Soc. Am. B **23**, 1398–1407 (2006). [CrossRef]

9. H. T. Eyyuboğlu, “Propagation and coherence properties of higher order partially coherent dark hollow beams in turbulence,” Opt. Laser Technol. **40**, 156–166 (2008). [CrossRef]

47. X. Lü and Y. Cai, “Partially coherent circular and elliptical dark hollow beams and their paraxial propagations,” Phys. Lett. A **369**, 157–166 (2007). [CrossRef]

48. C. Zhao, Y. Cai, F. Wang, X. Lu, and Y. Wang, “Generation of a high-quality partially coherent dark hollow beam with a multimode fiber,” Opt. Lett. **33**,1389–1391(2008). [CrossRef] [PubMed]

9. H. T. Eyyuboğlu, “Propagation and coherence properties of higher order partially coherent dark hollow beams in turbulence,” Opt. Laser Technol. **40**, 156–166 (2008). [CrossRef]

**32**, 3179–3181 (2007). [CrossRef] [PubMed]

*stigmatic*ABCD optical system. In this section, for the more general case, we extend the flexible model for anomalous hollow beam proposed in Section 2 to the partially coherent case, and study the propagation of a partially coherent anomalous hollow beam through paraxial

*astigmatic*ABCD optical system.

*x*

_{1},

*y*

_{1},

*x*

_{2},

*y*

_{2},

*z*)=〈

*E*(

*x*

_{1},

*y*

_{1},

*z*)

*E** (

*x*

_{2},

*y*

_{2},

*z*)〉, where 〈 〉 denotes the ensemble average and * denotes the complex conjugate. The intensity distribution of a partially coherent beam is given by

*I*(

*x*,

*y*,

*z*)=Γ(

*x*,

*y*,

*x*,

*y*,

*z*). For a partially coherent beam generated by an Schell-model source (at z = 0), the second-order correlation at z = 0 can be expressed in the following well-known form [34]

*g*(

*x*

_{1}-

*x*

_{2},

*y*

_{1}-

*y*

_{2}) is the spectral degree of coherence given by

*σ*

_{g0}is called the transverse coherence width.

*I*(

*x*,

*y*,0)=|

*E*(

*x*,

*y*,0)|2, where

*E*(

*x*,

*y*,0) is given by Eq. (1), after some operation, we can express the second-order correlation of a partially coherent anomalous hollow beam at

*z*= 0 in following tensor form:

**r**̂^{T}=(

**r**

^{T}

_{1}

**r**

_{2}

^{T})=(

*x*

_{1}

*y*

_{1}

*x*

_{2}

*y*

_{2}) with

**r**

_{1}and

**r**

_{2}being the two arbitrary position vectors in the source plane z = 0 and

**M**

^{-1}being the partially coherent complex curvature tensor [37

37. Q. Lin and Y. Cai, “Tensor *ABCD* law for partially coherent twisted anisotropic Gaussian Schell-model beams,” Opt. Lett. **27**, 216–218 (2002). [CrossRef]

*astigmatic*ABCD optical system can be studied with the help of the following generalized Collins formula [37

37. Q. Lin and Y. Cai, “Tensor *ABCD* law for partially coherent twisted anisotropic Gaussian Schell-model beams,” Opt. Lett. **27**, 216–218 (2002). [CrossRef]

*, 0) and Γ(*

**r**̂*,*

**ρ**̂*l*) are second-order correlation of a partially coherent beam in the source (z = 0) and output planes (z=

*l*),

*d*

**r**̂=

*d*

**r**

_{1}

*d*

**r**

_{2},

**ρ**

^{T}=(

**ρ**

^{T}

_{1}

**ρ**

^{T}

_{2}) and

**A**,

**B**,

**C**and

**D**are the 2 × 2 sub-matrices of the

*astigmatic*optical system, and,

**Â**,

**B̂**,

**Ĉ**and

**D**also satisfy the following Luneburg relations [37

37. Q. Lin and Y. Cai, “Tensor *ABCD* law for partially coherent twisted anisotropic Gaussian Schell-model beams,” Opt. Lett. **27**, 216–218 (2002). [CrossRef]

*ABCD* law for partially coherent twisted anisotropic Gaussian Schell-model beams,” Opt. Lett. **27**, 216–218 (2002). [CrossRef]

**A**,

**B**,

**C**and

**D**are assumed to be real quantities implying that the “*” is not needed anywhere in Eq. (14). However, for a general optical system with loss or gain (e.g. dispersive media, a Gaussian aperture, helical gas lenses, etc.)

**A**,

**B**,

**C**and

**D**take complex values and “*” is then required in Eq. (14).

*astigmatic*ABCD optical system

*ABCD* law for partially coherent twisted anisotropic Gaussian Schell-model beams,” Opt. Lett. **27**, 216–218 (2002). [CrossRef]

_{g 0}- > ∞, a partially coherent anomalous hollow beam becomes a coherent anomalous hollow beam and Eq. (16) reduces to the expression for the intensity distribution of a coherent anomalous hollow beam after propagation when

**ρ**

_{1}=

**ρ**

_{2}. But we can’t obtain the electric field (Eq. (7)) of a coherent anomalous hollow beam after propagation directly from Eq. (16). So it is necessary to describe coherent and partially coherent anomalous hollow beam separately. In some applications, coherent anomalous hollow beam are required, and it is sufficient and convenient for us to calculate the electric field of anomalous hollow beam with Eq. (7), calculation of the second-order correlation of anomalous hollow beam with Eq. (16) will make the problem more complicated. In other applications, partially coherent anomalous hollow beams are required, and we have to calculate its second-order correlation with Eq. (16).

*w*

_{0x}= 1

*mm*,

*w*

_{0y}= 0.5

*mm*,

*w*

_{1x}= 1

*mm*,

*w*

_{1y}= 0.5

*mm*,

*θ*=

*ϕ*= 0,

*N*= 3,

*p*= 0.8,

*α*= 0.2,

*β*= 0.5,

*λ*= 632.8

*nm*and

*σ*g0 = 0.5

*mm*. One sees from Figs. 4(a)–(c) that the beam profile of a partially coherent anomalous hollow beam also becomes non-elliptical symmetry at intermediate propagation distances, which is similar to that of a coherent anomalous hollow beam (see Figs. 3 (a)–(c)). In the far field, however, it is interesting to find that the partially coherent anomalous hollow beam gradually converses into a Gaussian beam (see Figs. 4 (d)–(f)), which is much different from that of a coherent anomalous hollow beam (see Figs. 3 (d)–(f)). This interesting phenomenon can be explained as follows. Partially coherent anomalous hollow beam can be regarded as a combination of a series of partially coherent modes with the same initial transverse coherence width

*σ*

_{g0}. Different modes or different points across the beam section interfere during propagation. As the initial transverse coherence width decreases, the coherence of all modes at the source plane decreases, then the interference effect between different modes on propagation decreases, which leads to the disappearance of out small ring around the main Gaussian peak in intensity distribution of the far field. Note the intensity distribution of the partially coherent anomalous hollow beam at the source plane is independent of its initial transverse coherence width. The phenomenon that decreasing the spatial coherence can lead to the disappearance of interference pattern was demonstrated in experiment recently in [48

48. C. Zhao, Y. Cai, F. Wang, X. Lu, and Y. Wang, “Generation of a high-quality partially coherent dark hollow beam with a multimode fiber,” Opt. Lett. **33**,1389–1391(2008). [CrossRef] [PubMed]

49. F. V. Dijk, G. Gbur, and T. D. Visser, “Shaping the focal intensity distribution using spatial coherence,” J. Opt. Soc. Am. A **25**, 575–581 (2008). [CrossRef]

*f*) that is located at z = 0, and the output plane is located at

*z*=

*f*(geometrical focal plane). The elements of the transfer matrix of the optical system between the source plane (z = 0) and output plane is expressed as follows

*z*=

*f*) for different values of the initial coherence width

**σ**

_{g0}. with

*w*

_{0y}1

*mm*,

*w*

_{0y}= 0.5

*mm*,

*w*

_{1x}= 1

*mm*,

*w*

_{1y}= 0.5

*mm*,

*θ*=

*ϕ*= 0,

*N*= 3,

*p*= 0.8,

*α*= 0.2,

*β*= 0.5,

*λ*= 632.8

*nm*,

*f*= 50

*mm*, and

*σ*

_{g0}= 0.5

*mm*. It is clear from Fig. 5 that the intensity distribution of an anomalous hollow beam at the geometrical focal plane is also closely controlled by its initial coherence. For a coherent anomalous hollow beam (

*σ*

_{g0}=Infinity), the focused beam profiel is of elliptical symmetry and there is a small bright elliptical ring around the brightest elliptical solid beam spot (see Fig. 5(a)), which is similar to the far field beam profile of a coherent anomalous hollow beam in free space. For a partially coherent anomalous hollow beam, the focused beam profile gradually becomes a circular Gaussian distribution as the initial coherence decreases (see Fig. 5(b)–(d)). Physical reason is the same as given for Fig. 4. One also finds from Fig. 5 (e) that the focused beam spot size decreases as the initial coherence width increases, which means that an anomalous hollow beam with higher initial coherence can be focused more tightly, which is consistent with the focusing properties of a partially coherent Gaussian beam [34]. Our results also are consistent with those in [28]. From above discussions, we find it is necessary to take the coherence of anomalous hollow beams into consideration in some practical cases.

## 5. Conclusion

## Acknowledgments

## References and links

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2. | J. D. Strohschein, H. J. J. Seguin, and C. E. Capjack, “Beam propagation constans for a radial laser array,” Appl. Opt. |

3. | N. Nishi, T. Jitsuno, K. Tsubakimoto, S. Matsuoka,, N. Miyanaga, and M. Nakatsuka, “Two-dimensional multi-lens array with circular aperture spherical lens for flat-top irradiation of inertial confinement fusion target,” Opt. Rev . |

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27. | Y. Cai, “Model for an anomalous hollow beam and its paraxial propagation,” Opt. Lett . |

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45. | Y. Cai and U. Peschel, “Second-harmonic generation by an astigmatic partially coherent beam,” Opt. Express |

46. | F. Wang and Y. Cai, “Experimental observation of fractional Fourier transform for a partially coherent optical beam with Gaussian statistics,” J. Opt. Soc. Am. A |

47. | X. Lü and Y. Cai, “Partially coherent circular and elliptical dark hollow beams and their paraxial propagations,” Phys. Lett. A |

48. | C. Zhao, Y. Cai, F. Wang, X. Lu, and Y. Wang, “Generation of a high-quality partially coherent dark hollow beam with a multimode fiber,” Opt. Lett. |

49. | F. V. Dijk, G. Gbur, and T. D. Visser, “Shaping the focal intensity distribution using spatial coherence,” J. Opt. Soc. Am. A |

50. | F. Wang and Y. Cai, “Experimental generation of a partially coherent flat-topped beam,” Opt. Lett . |

**OCIS Codes**

(030.1640) Coherence and statistical optics : Coherence

(140.3430) Lasers and laser optics : Laser theory

(350.5500) Other areas of optics : Propagation

**ToC Category:**

Physical Optics

**History**

Original Manuscript: June 17, 2008

Revised Manuscript: September 2, 2008

Manuscript Accepted: September 7, 2008

Published: September 12, 2008

**Citation**

Yangjian Cai, Zhaoying Wang, and Qiang Lin, "An alternative theoretical model for an anomalous hollow beam," Opt. Express **16**, 15254-15267 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-19-15254

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### References

- C. Palma, "Decentered Gaussian beams, ray bundles and Bessel-Gaussian beams," Appl. Opt. 36, 1116-1120 (1997). [CrossRef] [PubMed]
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