## Intra–cavity generation of Bessel–like beams with longitudinally dependent cone angles

Optics Express, Vol. 18, Issue 5, pp. 4701-4708 (2010)

http://dx.doi.org/10.1364/OE.18.004701

Acrobat PDF (689 KB)

### Abstract

We report on two resonator systems for producing Bessel–like beams with longitudinally dependent cone angles (LDBLBs). Such beams have extended propagation distances as compared to conventional Bessel–Gauss beams, with a far field pattern that is also Bessel–like in structure (i.e. not an annular ring). The first resonator system is based on a lens doublet with spherical aberration, while the second resonator system makes use of intra–cavity axicons and lens. In both cases we show that the LDBLB is the lowest loss fundamental mode of the cavity, and show theoretically the extended propagation distance expected from such beams.

© 2010 OSA

## 1. Introduction

*J*

_{0}) as a mathematical construction was firstly introduced by Durnin [1

1. J. Durnin, “Exact solutions for nondiffracting beams. The scalar theory,” J. Opt. Soc. Am. B **4**(4), 651 (1987). [CrossRef]

2. J. Durnin, J. J. Miceli Jr, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. **58**(15), 1499–1501 (1987). [CrossRef] [PubMed]

3. A. Vasara, J. Turunen, and A. T. Friberg, “Realization of general nondiffracting beams with computer-generated holograms,” J. Opt. Soc. Am. A **6**(11), 1748–1754 (1989). [CrossRef] [PubMed]

4. R. M. Herman and T. A. Wiggins, “Production and uses of diffractionless beams,” J. Opt. Soc. Am. A **8**(6), 932–942 (1991). [CrossRef]

6. A. N. Khilo, E. G. Katranji, and A. A. Ryzhevich, “Axicon-based. Bessel resonator: Analytical description and experiment,” J. Opt. Soc. Am. A **18**(8), 1986 (2001). [CrossRef]

9. V. N. Belyi, N. S. Kasak, and N. A. Khilo, “Properties of parametric frequency conversion with Bessel light beams,” Opt. Commun. **162**(1-3), 169–176 (1999). [CrossRef]

10. N. A. Khilo, E. S. Petrova, and A. A. Ryzhevich, “Transformation of the order of Bessel beams in uniaxial crystals,” Quantum Electron. **31**(1), 85–89 (2001). [CrossRef]

11. V. N. Belyi, N. S. Kazak, and N. A. Khilo, “Frequency conversion of Bessel light beams in nonlinear crystals,” Quantum Electron. **30**(9), 753–766 (2000). [CrossRef]

12. Z. Bouchal, J. Wagner, and M. Chlup, “Self-reconstruction of a distorted nondiffracting beam,” Opt. Commun. **151**(4-6), 207–211 (1998). [CrossRef]

13. I. A. Litvin, M. G. McLaren, and A. Forbes, “A conical wave approach to calculating Bessel-Gauss beam reconstruction after complex obstacles,” Opt. Commun. **282**(6), 1078–1082 (2009). [CrossRef]

*γ*, as the nondiffracting beam length is inversely proportional to

*γ*.

*γ*during beam propagation. In so doing, if at

*z*→∞ the limiting value of the angle

*γ*(

*z*) is zero, then such beams will have the advantages of both Bessel and Gaussian beams. In what follows such beams will be referred to as longitudinal dependent Bessel–like beams (LDBLBs).

13. I. A. Litvin, M. G. McLaren, and A. Forbes, “A conical wave approach to calculating Bessel-Gauss beam reconstruction after complex obstacles,” Opt. Commun. **282**(6), 1078–1082 (2009). [CrossRef]

18. T. Tanaka and S. Yamamoto, “Comparison of aberration between axicon and lens,” Opt. Commun. **184**(1-4), 113–118 (2000). [CrossRef]

19. P. A. Bélanger and C. Paré, “Optical resonators using graded-phase mirrors,” Opt. Lett. **16**(14), 1057–1059 (1991). [CrossRef] [PubMed]

22. I. A. Litvin and A. Forbes, “Intra-cavity flat-top beam generation,” Opt. Express **17**(18), 15891–15903 (2009). [CrossRef] [PubMed]

23. T. Aruga, “Generation of long-range nondiffracting narrow light beams,” Appl. Opt. **36**(16), 3762–3768 (1997). [CrossRef] [PubMed]

25. V. Belyi, A. Forbes, N. Kazak, N. Khilo, and P. Ropot, “Bessel–like beams with z–dependent cone angles,” Opt. Express **18**(3), 1966–1973 (2010). [CrossRef] [PubMed]

23. T. Aruga, “Generation of long-range nondiffracting narrow light beams,” Appl. Opt. **36**(16), 3762–3768 (1997). [CrossRef] [PubMed]

25. V. Belyi, A. Forbes, N. Kazak, N. Khilo, and P. Ropot, “Bessel–like beams with z–dependent cone angles,” Opt. Express **18**(3), 1966–1973 (2010). [CrossRef] [PubMed]

## 2. Resonator concept 1: intra–cavity aberrated lenses

23. T. Aruga, “Generation of long-range nondiffracting narrow light beams,” Appl. Opt. **36**(16), 3762–3768 (1997). [CrossRef] [PubMed]

*M2*mimics lens

*L2*and mirror

*M1*mimics lens

*L1*, but with an additional phase term to produce the conjugate field after reflection. In this case the desired field of extra–cavity system satisfies the criteria that its wavefront matches the phase of each mirror in the cavity and is the

*TEM*

_{00}mode of the given resonator system [19

19. P. A. Bélanger and C. Paré, “Optical resonators using graded-phase mirrors,” Opt. Lett. **16**(14), 1057–1059 (1991). [CrossRef] [PubMed]

22. I. A. Litvin and A. Forbes, “Intra-cavity flat-top beam generation,” Opt. Express **17**(18), 15891–15903 (2009). [CrossRef] [PubMed]

**36**(16), 3762–3768 (1997). [CrossRef] [PubMed]

**36**(16), 3762–3768 (1997). [CrossRef] [PubMed]

*M1*is almost twice the size than on the diverging one,

*M2*. Therefore for the converging mirror we can use almost a 16 times weaker coefficient of the spherical aberration. Furthermore, in this case, we have to change the output mirror into a diverging mirror as well [see Fig. 1 (b)].

*M1*, since the shape of mirror

*M2*(as well as the resonator length) can be taken directly from the extra–cavity design. The shape of mirror

*M1*must be such that after reflection the reverse propagating field should be equivalent to the phase conjugate of the field after propagation through the extra–cavity telescope system, taking into account the spherical aberration of lens

*L1*. This problem can be solved analytically, as we now show. Consider the extra–cavity analogy of a Gaussian beam passing through the telescope, with the waist on

*L2*with half–width

*w*

_{0}. Lens

*L2*is a non–aberrated thin lens with focal length

*f*

_{1}. Lens

*L1*is a converging lens of focal length

*f*, together with some degree of spherical aberration (

*β*). If the two lenses are separated by a distance

*z*

_{1}, then the curvature of the wavefront at a plane just in front of

*L1*may be found from the Fresnel diffraction integral:where

*k*

_{0}

*= 2π/λ*,

*λ*is the wavelength of the light, and

*R*

_{0}is the radius of the mirror. It is useful to solve Eq. (1) by applying the method of stationary phase, to give:where

*L1*the field will be the product of

*a*

_{1}and the transmission function of lens

*L1*with spherical aberration, namely:

*M1*is then found from [19

19. P. A. Bélanger and C. Paré, “Optical resonators using graded-phase mirrors,” Opt. Lett. **16**(14), 1057–1059 (1991). [CrossRef] [PubMed]

22. I. A. Litvin and A. Forbes, “Intra-cavity flat-top beam generation,” Opt. Express **17**(18), 15891–15903 (2009). [CrossRef] [PubMed]

**17**(18), 15891–15903 (2009). [CrossRef] [PubMed]

*λ*= 632 nm,

*z*

_{1}

*=*0.65 m,

*w*

_{0}

*=*0.5 mm,

*f*

_{1}

*=*0.4 m,

*f*

_{2}

*=*0.7 m, and aberration coefficient

*β =*16 × 10

^{4}m

^{–3}. The mirror radii were chosen as

*R*

_{0}= 5 mm, corresponding to a Fresnel number of

*N*= 60. With these parameters the phase of mirror

_{f}*M1*was calculated directly from Eq. (4) and shown in Fig. 2 (a (blue)).

*M1*the resonator lies on the boundary between stable and unstable, and with the inclusion of spherical aberration the losses increase due to a shift into an unstable resonator configuration with the overall losses around 4 percent per trip.

*M1*and

*M2*after stabilization of the Fox–Li algorithm. We can see the required fields are obtained successfully, namely Gaussian transverse intensity profile on mirror

*M2*(see Eq. (1) and the field corresponding to Eq. (2) on mirror

*M1*.

*z*. As is seen, it is a one-peaked curve typical for Bessel beams. When

*z*increases up to several meters or more, there occurs a slow monotonic decrease in the peak on-axis intensity with propagation distance as is typical of Gaussian beams. We also note that the field widens during propagation [see Fig. 3(b) and 3(d)], as well as picks up the ring structure of Bessel beams. Thus the output beam from the resonator has the characteristics of a Bessel–like beam outlined by others [23

**36**(16), 3762–3768 (1997). [CrossRef] [PubMed]

^{1}) red dashed graphs), the first central peak due to the Gaussian nature of the beam, and the second off-center peak (with some oscillations) a characteristic of the annular ring spectrum of Bessel–Gauss beams [12

12. Z. Bouchal, J. Wagner, and M. Chlup, “Self-reconstruction of a distorted nondiffracting beam,” Opt. Commun. **151**(4-6), 207–211 (1998). [CrossRef]

**36**(16), 3762–3768 (1997). [CrossRef] [PubMed]

## 3. Resonator concept 2: intra–cavity axicons

25. V. Belyi, A. Forbes, N. Kazak, N. Khilo, and P. Ropot, “Bessel–like beams with z–dependent cone angles,” Opt. Express **18**(3), 1966–1973 (2010). [CrossRef] [PubMed]

*A1*and

*A2*, and a thin lens

*L*with focal length

*F*. The Bessel beams generated by these axicons are characterized by cone angles of γ

_{1}and γ

_{2}respectively.

*m*= 1,

*p*= 0) but with a flat phase (the beam waist fits the output plane) as the initial beam to avoid maximum intensity on apexes of axicons [see Fig. 5 (a (red))].

*R*– is the radius of the axicon

_{a2}*A2*,

*w*1.5 mm – the half-width of the input Laguerre-Gaussian beam.

_{0}=*M1*to be given by:with

*M1*the field will be the product of

*a*

_{1}and the transmission function of the axicon

*A1*, namely:

**17**(18), 15891–15903 (2009). [CrossRef] [PubMed]

*λ*= 632 nm,

*z*

_{1}

*=*0.65 m,

*w*

_{0}

*=*1.5 mm,

*γ*

_{1}

*=*0.3 degree,

*γ*

_{2}

*=*0.1 degree,

*F =*0.65 z

_{1}. The mirror radii were chosen as

*R*

_{a1,2}= 5 mm, corresponding to a Fresnel number of

*N*= 60. With these parameters the phase of mirror

_{f}*M2*was calculated directly from Eq. (7) and shown in Fig. 5 (a (blue)).

**18**(3), 1966–1973 (2010). [CrossRef] [PubMed]

4. R. M. Herman and T. A. Wiggins, “Production and uses of diffractionless beams,” J. Opt. Soc. Am. A **8**(6), 932–942 (1991). [CrossRef]

*w*

_{0}, and axicon cone angle, γ

_{1}, we find that such a Bessel–Gauss beam would have a nondiffracting length of approximately 0.3 m. Thus while the LDBLBs slowly diverge during propagation, the enveloping function remains Bessel-like. Clearly there are applications where this shape invariance during propagation would be desirable.

12. Z. Bouchal, J. Wagner, and M. Chlup, “Self-reconstruction of a distorted nondiffracting beam,” Opt. Commun. **151**(4-6), 207–211 (1998). [CrossRef]

**151**(4-6), 207–211 (1998). [CrossRef]

## 4. Conclusion

*TEM*

_{00}mode. The first resonator system is based on a doublet of the diverging and converging lenses with spherical aberration [23

**36**(16), 3762–3768 (1997). [CrossRef] [PubMed]

**18**(3), 1966–1973 (2010). [CrossRef] [PubMed]

## Acknowledgements

## References and links

1. | J. Durnin, “Exact solutions for nondiffracting beams. The scalar theory,” J. Opt. Soc. Am. B |

2. | J. Durnin, J. J. Miceli Jr, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. |

3. | A. Vasara, J. Turunen, and A. T. Friberg, “Realization of general nondiffracting beams with computer-generated holograms,” J. Opt. Soc. Am. A |

4. | R. M. Herman and T. A. Wiggins, “Production and uses of diffractionless beams,” J. Opt. Soc. Am. A |

5. | Z. Jaroszewicz, Axicons: Design and Propagation Properties, Vol. 5 of Research and Development Treatise (SPIE Polish Chapter, Warsaw, 1997). |

6. | A. N. Khilo, E. G. Katranji, and A. A. Ryzhevich, “Axicon-based. Bessel resonator: Analytical description and experiment,” J. Opt. Soc. Am. A |

7. | J. Rogel-Salazar, G. H. C. New, and S. Chavez-Cerda, “Bessel-Gauss beam optical resonator,” Opt. Commun. |

8. | I. A. Litvin and A. Forbes, “Bessel–Gauss Resonator with Internal Amplitude Filter,” Opt. Commun. |

9. | V. N. Belyi, N. S. Kasak, and N. A. Khilo, “Properties of parametric frequency conversion with Bessel light beams,” Opt. Commun. |

10. | N. A. Khilo, E. S. Petrova, and A. A. Ryzhevich, “Transformation of the order of Bessel beams in uniaxial crystals,” Quantum Electron. |

11. | V. N. Belyi, N. S. Kazak, and N. A. Khilo, “Frequency conversion of Bessel light beams in nonlinear crystals,” Quantum Electron. |

12. | Z. Bouchal, J. Wagner, and M. Chlup, “Self-reconstruction of a distorted nondiffracting beam,” Opt. Commun. |

13. | I. A. Litvin, M. G. McLaren, and A. Forbes, “A conical wave approach to calculating Bessel-Gauss beam reconstruction after complex obstacles,” Opt. Commun. |

14. | N. Davidson, A. A. Friesem, and E. Hasman, “Holographic axilens: high resolution and long focal depth,” Opt. Lett. |

15. | Z. Jaroszewicz and J. Morales, “Lens axicons: systems composed of a diverging aberrated lens and a perfect converging lens,” J. Opt. Soc. Am. A |

16. | C. Parigger, Y. Tang, D. H. Plemmons, and J. W. Lewis, “Spherical aberration effects in lens-axicon doublets: theoretical study,” Appl. Opt. |

17. | A. V. Goncharov, A. Burvall, and C. Dainty, “Systematic design of an anastigmatic lens axicon,” Appl. Opt. |

18. | T. Tanaka and S. Yamamoto, “Comparison of aberration between axicon and lens,” Opt. Commun. |

19. | P. A. Bélanger and C. Paré, “Optical resonators using graded-phase mirrors,” Opt. Lett. |

20. | C. Pare and P. A. Belanger, “Custom Laser Resonators Using Graded-Phase Mirror,” IEEE J. Quantum Electron. |

21. | P. A. Bélanger, R. L. Lachance, and C. Paré, “Super-Gaussian output from a CO(2) laser by using a graded-phase mirror resonator,” Opt. Lett. |

22. | I. A. Litvin and A. Forbes, “Intra-cavity flat-top beam generation,” Opt. Express |

23. | T. Aruga, “Generation of long-range nondiffracting narrow light beams,” Appl. Opt. |

24. | T. Aruga, S. W. Li, S. Y. Yoshikado, M. Takabe, and R. Li, “Nondiffracting narrow light beam with small atmospheric turbulence-influenced propagation,” Appl. Opt. |

25. | V. Belyi, A. Forbes, N. Kazak, N. Khilo, and P. Ropot, “Bessel–like beams with z–dependent cone angles,” Opt. Express |

**OCIS Codes**

(140.3410) Lasers and laser optics : Laser resonators

(140.3295) Lasers and laser optics : Laser beam characterization

**ToC Category:**

Lasers and Laser Optics

**History**

Original Manuscript: January 6, 2010

Revised Manuscript: February 4, 2010

Manuscript Accepted: February 4, 2010

Published: February 22, 2010

**Citation**

Igor A. Litvin, Nikolai A. Khilo, Andrew Forbes, and Vladimir N. Belyi, "Intra–cavity generation of Bessel–like beams with longitudinally dependent cone angles," Opt. Express **18**, 4701-4708 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-5-4701

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### References

- J. Durnin, “Exact solutions for nondiffracting beams. The scalar theory,” J. Opt. Soc. Am. B 4(4), 651 (1987). [CrossRef]
- J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987). [CrossRef] [PubMed]
- A. Vasara, J. Turunen, and A. T. Friberg, “Realization of general nondiffracting beams with computer-generated holograms,” J. Opt. Soc. Am. A 6(11), 1748–1754 (1989). [CrossRef] [PubMed]
- R. M. Herman and T. A. Wiggins, “Production and uses of diffractionless beams,” J. Opt. Soc. Am. A 8(6), 932–942 (1991). [CrossRef]
- Z. Jaroszewicz, Axicons: Design and Propagation Properties, Vol. 5 of Research and Development Treatise (SPIE Polish Chapter, Warsaw, 1997).
- A. N. Khilo, E. G. Katranji, and A. A. Ryzhevich, “Axicon-based. Bessel resonator: Analytical description and experiment,” J. Opt. Soc. Am. A 18(8), 1986 (2001). [CrossRef]
- J. Rogel-Salazar, G. H. C. New, and S. Chavez-Cerda, “Bessel-Gauss beam optical resonator,” Opt. Commun. 190(1-6), 117–122 (2001). [CrossRef]
- I. A. Litvin and A. Forbes, “Bessel–Gauss Resonator with Internal Amplitude Filter,” Opt. Commun. 281(9), 2385–2392 (2008). [CrossRef]
- V. N. Belyi, N. S. Kasak, and N. A. Khilo, “Properties of parametric frequency conversion with Bessel light beams,” Opt. Commun. 162(1-3), 169–176 (1999). [CrossRef]
- N. A. Khilo, E. S. Petrova, and A. A. Ryzhevich, “Transformation of the order of Bessel beams in uniaxial crystals,” Quantum Electron. 31(1), 85–89 (2001). [CrossRef]
- V. N. Belyi, N. S. Kazak, and N. A. Khilo, “Frequency conversion of Bessel light beams in nonlinear crystals,” Quantum Electron. 30(9), 753–766 (2000). [CrossRef]
- Z. Bouchal, J. Wagner, and M. Chlup, “Self-reconstruction of a distorted nondiffracting beam,” Opt. Commun. 151(4-6), 207–211 (1998). [CrossRef]
- I. A. Litvin, M. G. McLaren, and A. Forbes, “A conical wave approach to calculating Bessel-Gauss beam reconstruction after complex obstacles,” Opt. Commun. 282(6), 1078–1082 (2009). [CrossRef]
- N. Davidson, A. A. Friesem, and E. Hasman, “Holographic axilens: high resolution and long focal depth,” Opt. Lett. 16(7), 523–525 (1991). [CrossRef] [PubMed]
- Z. Jaroszewicz and J. Morales, “Lens axicons: systems composed of a diverging aberrated lens and a perfect converging lens,” J. Opt. Soc. Am. A 15(9), 2383–2390 (1998). [CrossRef]
- C. Parigger, Y. Tang, D. H. Plemmons, and J. W. Lewis, “Spherical aberration effects in lens-axicon doublets: theoretical study,” Appl. Opt. 36(31), 8214–8221 (1997). [CrossRef]
- A. V. Goncharov, A. Burvall, and C. Dainty, “Systematic design of an anastigmatic lens axicon,” Appl. Opt. 46(24), 6076–6080 (2007). [CrossRef] [PubMed]
- T. Tanaka and S. Yamamoto, “Comparison of aberration between axicon and lens,” Opt. Commun. 184(1-4), 113–118 (2000). [CrossRef]
- P. A. Bélanger and C. Paré, “Optical resonators using graded-phase mirrors,” Opt. Lett. 16(14), 1057–1059 (1991). [CrossRef] [PubMed]
- C. Pare and P. A. Belanger, “Custom Laser Resonators Using Graded-Phase Mirror,” IEEE J. Quantum Electron. 28(1), 355–362 (1992). [CrossRef]
- P. A. Bélanger, R. L. Lachance, and C. Paré, “Super-Gaussian output from a CO(2) laser by using a graded-phase mirror resonator,” Opt. Lett. 17(10), 739–741 (1992). [CrossRef] [PubMed]
- I. A. Litvin and A. Forbes, “Intra-cavity flat-top beam generation,” Opt. Express 17(18), 15891–15903 (2009). [CrossRef] [PubMed]
- T. Aruga, “Generation of long-range nondiffracting narrow light beams,” Appl. Opt. 36(16), 3762–3768 (1997). [CrossRef] [PubMed]
- T. Aruga, S. W. Li, S. Y. Yoshikado, M. Takabe, and R. Li, “Nondiffracting narrow light beam with small atmospheric turbulence-influenced propagation,” Appl. Opt. 38(15), 3152–3156 (1999). [CrossRef]
- V. Belyi, A. Forbes, N. Kazak, N. Khilo, and P. Ropot, “Bessel–like beams with z–dependent cone angles,” Opt. Express 18(3), 1966–1973 (2010). [CrossRef] [PubMed]

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