## Generalizing higher-order Bessel-Gauss beams: analytical description and demonstration |

Optics Express, Vol. 20, Issue 24, pp. 26852-26867 (2012)

http://dx.doi.org/10.1364/OE.20.026852

Acrobat PDF (2932 KB)

### Abstract

We report on a novel class of higher-order Bessel-Gauss beams in which the well-known Bessel-Gauss beam is the fundamental mode and the azimuthally symmetric Laguerre-Gaussian beams are special cases. We find these higher-order Bessel-Gauss beams by superimposing decentered Hermite-Gaussian beams. We show analytically and experimentally that these higher-order Bessel-Gauss beams resemble higher-order eigenmodes of optical resonators consisting of aspheric mirrors. This work is relevant for the many applications of Bessel-Gauss beams in particular the more recently proposed high-intensity Bessel-Gauss enhancement cavities for strong-field physics applications.

© 2012 OSA

## 1. Introduction

1. C. J. R. Sheppard and T. Wilson, “Gaussian-beam theory of lenses with annular aperture,” IEEE J. Microw. Opt. Acoust. **2**, 105–112 (1978). [CrossRef]

2. F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. **64**, 491– 495 (1987). [CrossRef]

3. F. O. Fahrbach, P. Simon, and A. Rohrbach, “Microscopy with self-reconstructing beams,” Nat. Photonics **4**, 780–785 (2010). [CrossRef]

7. D. Li and K. Imasaki, “Vacuum laser-driven acceleration by two slits-truncated Bessel beams,” Appl. Phys. Lett. **87**, 091106 (2005). [CrossRef]

8. J. C. Gutiérrez-Vega, R. Rodríguez-Masegosa, and S. Chávez-Cerda, “Bessel-Gauss resonator with spherical output mirror: geometrical- and wave-optics analysis,” J. Opt. Soc. Am. A **20**, 2113–2122 (2003). [CrossRef]

10. B. Ma, F. Wu, W. Lu, and J. Pu, “Nanosecond zero-order pulsed Bessel beam generated from unstable resonator based on an axicon,” Opt. Laser Technol. **42**, 941–944 (2010). [CrossRef]

11. W. P. Putnam, D. N. Schimpf, G. Abram, and F. X. Kärtner, “Bessel-Gauss beam enhancement cavities for high-intensity applications,” Opt. Express **20**, 24429–24443 (2012). [CrossRef]

13. R. Vasilyeu, A. Dudley, N. Khilo, and A. Forbes, “Generating superpositions of higherorder Bessel beams,” Opt. Express **17**, 23389–23395 (2009). [CrossRef]

## 2. Generalizing higher-order Bessel-Gauss beams

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

15. A. R. Al-Rashed and B. E. A. Saleh, “Decentered Gaussian beams,” Appl. Opt. **34**, 6819–6825 (1995). [CrossRef] [PubMed]

### 2.1. Decentered Hermite-Gaussian beams

### 2.2. Propagation of decentered Hermite-Gaussian beams through optical systems

### 2.3. Superposing decentered Hermite-Gaussian beams to higher-order Bessel-Gauss beams

**r**and

_{d0}**are collinear with**

*ε*_{0}*r*

_{d0}= const. and

*ε*

_{0}= const. This corresponds to a superposition of beams whose mean centers are placed on a circle with radius

*r*

_{d0}in the z=0 plane and whose mean beam directions point to the apex of a single cone with a semi-aperture angle

*ε*

_{0}, as shown in Fig. 1.

*γ*with regard to a fixed reference coordinate system (

*r*,

*θ*,

*z*) so that its abscissa points in the direction of the vector

**r**. In this auxiliary system, the decentered Hermite-Gaussian beam consists solely of the

_{d0}*H*-term (i.e.

_{m}*H*

_{n=0}= 1) and the abscissa coordinate is given by

*r*·cos(

*θ*−

*γ*). The propagated decentered Hermite-Gaussian beam is expressed in the cylindrical coordinates (

*r*,

*θ*,

*z*) as

**r**and

_{d0}*ε*

_{0}, the following beam-propagation transformations hold in addition to the relations for the Gaussian beam parameters, Eqs. (5) and (6). The formulation of higher-order Bessel-Gauss beams in terms of ABCD-matrix parameters allows us to study their transformation under the impact of optical components. We will expand upon this feature in section 3.1 describing optical resonators for these beam solutions.

*γ*angle from 0 to 2

*π*: The term ℐ

*is given by where we substitute To generalize the analysis, we have introduced an azimuthal phase-variation while superposing the component beams. This phase is characterized by the index*

_{ml}*l*. The integral ℐ

*can be solved analytically (see Appendix), and the first four expressions are given by*

_{ml}*m*= 0 and

*l*= 0, the well-known Bessel-Gauss beam is obtained [14

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

_{0l}(i.e.

*m*= 0) are identical to previously defined higher-order Bessel-Gauss beams exhibiting an azimuthal phase-variation [12, 13

13. R. Vasilyeu, A. Dudley, N. Khilo, and A. Forbes, “Generating superpositions of higherorder Bessel beams,” Opt. Express **17**, 23389–23395 (2009). [CrossRef]

*m*> 0. In Fig. 2 we illustrate the effect of the radial index by plotting the azimuthally symmetric (

*l*= 0) higher-order Bessel-Gauss beams for different orders m in the near and far-field. The parameters are

*w*

_{0}= 200

*μm*,

*r*

_{d0}= 0,

*ε*= 0.3°, and

*λ*= 1040

*nm*. In the far-field (here, calculated at z= 4 ·

*z*, where

_{R}*l*> 0 these solutions exhibit azimuthal phase-variations, as displayed in Fig. 3. To highlight the impact of the azimuthal index l on the solution, we consider the superposition (

*u*+

_{m,l}*u*

_{m,−l}) resulting in a cosine-variation over the angular coordinate.

*r*

_{d}_{0}and

*ε*? - A classification is known for the fundamental Bessel-Gauss beam [12]:

*Generalized*Bessel-Gauss beam,

*ordinary*Bessel-Gauss beam and

*modified*Bessel-Gauss beam denote the cases (

*r*

_{d0}≠ 0,

*ε*

_{0}≠ 0); (

*r*

_{d0}= 0,

*ε*

_{0}≠ 0), and (

*r*

_{d0}≠ 0,

*ε*

_{0}= 0), respectively. We extend this terminology to the higher-order Bessel-Gauss beams. For the example of a Bessel-Gauss beam with m=2, we plot the different cases in Fig. 4. On the left side, the radial profile is plotted over the propagation distance. On the right side, we display the transverse beam pattern at the position of minimum Gaussian waist (here at z=0). Figures 4(a) and 4(b) show the absolute value of the amplitude for a generalized higher-order (m=2) Bessel-Gauss beam for

*w*

_{0}= 200

*μm*,

*r*

_{d0}= 0.8

*mm*,

*ε*= 0.3°, and

*λ*= 1040

*nm*. The vertical blue line highlights the (z=0) position. For this beam type the intersection with the symmetry axis is shifted from the position where the minimum Gaussian waist occurs. Figures 4(c) and 4(d) displays the ordinary beam type. The intersection coincidences with the position of minimum waist. Figures 4(e) and 4(f) picture the modified beam type. Furthermore, if both

*r*

_{d0}and

*ε*

_{0}are zero, a azimuthally symmetric Laguerre-Gaussian beam is recovered, as displayed in Fig. 4(g) and 4(h).

### 2.4. Collapse of the solution to Laguerre-Gaussian beams for ε = 0 and r_{d0} = 0

*ε*= 0 and

*r*

_{d0}= 0. For uneven

*m*the integral vanishes due to symmetry, and for even

*m*(= 2 ·

*p*) we find: This integration is performed by substituting

*t*= cos(

*θ*−

*γ*) and applying [18]: which holds for Im[

*α*] > −1/2.

## 3. Comparing the Bessel-Gauss beam solutions to modes of optical resonators

### 3.1. Design of the optical resonator

*ε*.

*r*

_{d0},

*ε*) and (

*r*

_{d0}, −

*ε*) before and after one roundtrip, respectively. Thus, the resonator length is given by Gaussian beam optics determines the Gaussian beam parameter

*q*, which must repeat after one round trip. At the axicon mirror, one finds The resonator is stable for

*L*<

*R*. The minimum waist for the Gaussian component of the Bessel-Gauss beam occurs at the axicon mirror and can be determined from

*q*= −

*iz*.

_{R}*w*

_{0}of the Gaussian component of the Bessel-Gauss beam solution are fully governed by the radius of curvature of the spherical resonator mirror, the base angle of the axicon mirror, as well as the radius of the ring of the generalized Bessel-Gauss beam on the axicon,

*r*

_{d0}. For the following calculations, we chose a radius of curvature R = 250 mm,

*ε*= 0.5° and

*r*

_{d0}= 1.5 mm, which result in

*L*≈ 78 mm and

*w*

_{0}≈ 196

*μ*m for a light wavelength of 1040 nm. These particular parameters were selected since they approximate the experimental configuration which, in turn, was determined by the availability of base angles for the axicon mirror and radii of curvatures for the spherical mirror. It results in a ring radius of the Bessel-Gauss beam that avoids the round tip on the axicon mirror and results in a not too large ring radius on the curved mirror. This situation is advantageous as it reduces effects of surface variations on the resonator mode.

*ε*′ ≈

*n*·

_{m}*ε*, where

*n*is the refractive index of the mirror substrate, and a ring radius

_{m}*r*′

*≈*

_{d}*r*+ Δ ·

_{d}*ε*, where the ring-radius at the front surface is given by

*r*=

_{d}*ε*·

*R*and Δ is the geometrical thickness of the mirror, which is given by Δ = Δ

*−*

_{out}*s*(

*r*), where Δ

_{out}*is the mirror thickness at the outer radius*

_{out}*r*and the surface function is given by

_{out}### 3.2. Numerical mode solver

19. H. F. Johnson, “An improved method for computing a discrete hankel transform,” Comp. Phys. Comm. **43**, 181–202 (1987). [CrossRef]

21. M. Guizar-Sicairos and J. C. Gutiérrez-Vega, “Computation of quasi-discrete Hankel transforms of integer order for propagating optical wave fields,” J. Opt. Soc. Am. A **21**, 53–58 (2004). [CrossRef]

### 3.3. Comparison of the numerical mode-solution to the Bessel-Gauss beam solutions

*R*−

*L*) via an intermediate position −(0.9 ·

*R*−

*L*). In Fig. 5, these positions are denoted as P1, P2 and P3. Figure 6 shows a comparison between numerical results and the analytical solutions. It can be seen that the azimuthally symmetric higher-order Bessel-Gauss modes represent an accurate analytical model for higher-order modes in resonators supporting the (m=0, l=0) Bessel-Gauss beam as the fundamental mode. Conversely, the numerical results validate the existence of azimuthally symmetric higher-order Bessel-Gauss beams, and constitute a reference for the hitherto unknown analytical beam solution.

### 3.4. Experimental setup

*μm*). The laser is operated at a wavelength of 1040 nm and it emits up to 30 mW of power at the fiber output with a line width of 100 kHz. The laser frequency is varied by tuning the grating angle of the external cavity with a piezoelectric element. This allows us to lock the laser frequency to the resonance of the enhancement cavity by dither locking. For this, the light that is reflected from the in-coupling mirror SM is detected at the side port of an optical isolator with a photodiode and is used as a feedback signal for the locking loop. The leakage through the axicon mirror is used to record the resonator modes with a camera, which is placed behind AM.

*n*− 1) ·

*δ*= 0.45°, where n is the refractive index of fused silica (n=1.45) and

*δ*the base angle of the axicon,

*δ*= 1°. The waist of the beam illuminating the axicon is 1.5 mm. Due to the round, and thus imperfect, apex of the axicon, the resulting beam in the far field is not a pure ring-like beam but still contains unwanted light in the central region. The quality of the beam is increased by employing Fourier filtering with an opaque disk of 1 mm radius in a 4-f system (150 mm focal-length lenses L4 and L5) [24

24. O. Brzobohaty, T. Cizmar, and P. Zemanek, “High quality quasi-Bessel beam generated by round-tip axicon,” Opt. Express **16**, 12688–12700 (2008). [CrossRef] [PubMed]

*μm*from the ideal conic shape at the tip. As a consequence, we iteratively improve the distances between L5 and L6, L6 and L7, as well as L7 and SM for best excitation. The resulting values are given in parentheses in Fig. 8, and are used as a starting point for the experiment, in which we also adjust the resonator mirrors and set the final positions of L6 and L7 so that the photodiode signal shows the most pronounced resonance feature.

### 3.5. Experimental results: comparison of the experimental modes to the beam solutions

11. W. P. Putnam, D. N. Schimpf, G. Abram, and F. X. Kärtner, “Bessel-Gauss beam enhancement cavities for high-intensity applications,” Opt. Express **20**, 24429–24443 (2012). [CrossRef]

## 4. Conclusion

## Appendix

*H*in their explicit form. For this we use their recursion relation:

_{m}*H*

_{0}(

*t*) = 1,

*H*

_{1}(

*t*) = 2

*t*and

*H*

_{m}_{+1}(

*t*) = 2

*tH*(

_{m}*t*) − 2

*mH*

_{m}_{−1}(

*t*). This results in an expansions in powers of cos-functions. The n-th power of the cos-functions can be written in terms of the harmonics of its argument by using De Moivre’s formula, e.g.: Then, we make use of the following relation which can be easily shown by writing the cosine function in terms of exponentials and applying the definition of the Bessel-function, e.g. as given in [12]: Moreover, we use the relation

*J*

_{−k}(

*t*) = (−1)

*·*

^{k}*J*(

_{k}*t*).

## Acknowledgments

## References and links

1. | C. J. R. Sheppard and T. Wilson, “Gaussian-beam theory of lenses with annular aperture,” IEEE J. Microw. Opt. Acoust. |

2. | F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. |

3. | F. O. Fahrbach, P. Simon, and A. Rohrbach, “Microscopy with self-reconstructing beams,” Nat. Photonics |

4. | T. A Planchon, L. Gao, D. E Milkie, M. W Davidson, J. A Galbraith, C. G Galbraith, and Eric Betzig, “Rapid three-dimensional isotropic imaging of living cells using Bessel beam plane illumination,” Nat. Methods |

5. | M. Duocastella and C. B. Arnold, “Bessel and annular beams for materials processing,” Laser Photon. Rev. |

6. | V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature |

7. | D. Li and K. Imasaki, “Vacuum laser-driven acceleration by two slits-truncated Bessel beams,” Appl. Phys. Lett. |

8. | J. C. Gutiérrez-Vega, R. Rodríguez-Masegosa, and S. Chávez-Cerda, “Bessel-Gauss resonator with spherical output mirror: geometrical- and wave-optics analysis,” J. Opt. Soc. Am. A |

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

10. | B. Ma, F. Wu, W. Lu, and J. Pu, “Nanosecond zero-order pulsed Bessel beam generated from unstable resonator based on an axicon,” Opt. Laser Technol. |

11. | W. P. Putnam, D. N. Schimpf, G. Abram, and F. X. Kärtner, “Bessel-Gauss beam enhancement cavities for high-intensity applications,” Opt. Express |

12. | V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. Spagnolo Schirripa, “Generalized Bessel-Gauss beams,” J. Mod. Opt. |

13. | R. Vasilyeu, A. Dudley, N. Khilo, and A. Forbes, “Generating superpositions of higherorder Bessel beams,” Opt. Express |

14. | C. Palma, “Decentered Gaussian beams, ray bundles, and Bessel-Gauss beams,” Appl. Opt. |

15. | A. R. Al-Rashed and B. E. A. Saleh, “Decentered Gaussian beams,” Appl. Opt. |

16. | A. R. Al-Rashed, “Spatial and temporal modes of resonators with dispersive phase-conjugate mirrors,” PhD Thesis (1997). |

17. | S. A. Collins Jr., “Lens-System Diffraction Integral Written in Terms of Matrix Optics,” J. Opt. Soc. Am. |

18. | G. Ryshik, |

19. | H. F. Johnson, “An improved method for computing a discrete hankel transform,” Comp. Phys. Comm. |

20. | L. Yu, M. Huang, M. Chen, W. Chen, W. Huang, and Z. Zhu, “Quasi-discrete Hankel transform,” Opt. Lett. |

21. | M. Guizar-Sicairos and J. C. Gutiérrez-Vega, “Computation of quasi-discrete Hankel transforms of integer order for propagating optical wave fields,” J. Opt. Soc. Am. A |

22. | A. Fox and T. Li, “Computation of optical resonator modes by the method of resonance excitation,” IEEE J. Quantum Electron. |

23. | A. E. Siegman, |

24. | O. Brzobohaty, T. Cizmar, and P. Zemanek, “High quality quasi-Bessel beam generated by round-tip axicon,” Opt. Express |

**OCIS Codes**

(070.2580) Fourier optics and signal processing : Paraxial wave optics

(140.3300) Lasers and laser optics : Laser beam shaping

(140.3410) Lasers and laser optics : Laser resonators

**ToC Category:**

Physical Optics

**History**

Original Manuscript: October 1, 2012

Revised Manuscript: November 4, 2012

Manuscript Accepted: November 7, 2012

Published: November 14, 2012

**Citation**

Damian N. Schimpf, Jan Schulte, William P. Putnam, and Franz X. Kärtner, "Generalizing higher-order Bessel-Gauss beams: analytical description and demonstration," Opt. Express **20**, 26852-26867 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-24-26852

Sort: Year | Journal | Reset

### References

- C. J. R. Sheppard and T. Wilson, “Gaussian-beam theory of lenses with annular aperture,” IEEE J. Microw. Opt. Acoust.2, 105–112 (1978). [CrossRef]
- F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun.64, 491– 495 (1987). [CrossRef]
- F. O. Fahrbach, P. Simon, and A. Rohrbach, “Microscopy with self-reconstructing beams,” Nat. Photonics4, 780–785 (2010). [CrossRef]
- T. A Planchon, L. Gao, D. E Milkie, M. W Davidson, J. A Galbraith, C. G Galbraith, and Eric Betzig, “Rapid three-dimensional isotropic imaging of living cells using Bessel beam plane illumination,” Nat. Methods8, 417–423 (2011). [CrossRef] [PubMed]
- M. Duocastella and C. B. Arnold, “Bessel and annular beams for materials processing,” Laser Photon. Rev.6, 607–621 (2012). [CrossRef]
- V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature419, 145–147 (2002). [CrossRef] [PubMed]
- D. Li and K. Imasaki, “Vacuum laser-driven acceleration by two slits-truncated Bessel beams,” Appl. Phys. Lett.87, 091106 (2005). [CrossRef]
- J. C. Gutiérrez-Vega, R. Rodríguez-Masegosa, and S. Chávez-Cerda, “Bessel-Gauss resonator with spherical output mirror: geometrical- and wave-optics analysis,” J. Opt. Soc. Am. A20, 2113–2122 (2003). [CrossRef]
- A. N. Khilo, E. G. Katranji, and A. A. Ryzhevich, “Axicon-based Bessel resonator: analytical description and experiment,” J. Opt. Soc. Am. A18, 1986–1992 (2001). [CrossRef]
- B. Ma, F. Wu, W. Lu, and J. Pu, “Nanosecond zero-order pulsed Bessel beam generated from unstable resonator based on an axicon,” Opt. Laser Technol.42, 941–944 (2010). [CrossRef]
- W. P. Putnam, D. N. Schimpf, G. Abram, and F. X. Kärtner, “Bessel-Gauss beam enhancement cavities for high-intensity applications,” Opt. Express20, 24429–24443 (2012). [CrossRef]
- V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. Spagnolo Schirripa, “Generalized Bessel-Gauss beams,” J. Mod. Opt.43(6), 1155–1166 (1996).
- R. Vasilyeu, A. Dudley, N. Khilo, and A. Forbes, “Generating superpositions of higherorder Bessel beams,” Opt. Express17, 23389–23395 (2009). [CrossRef]
- C. Palma, “Decentered Gaussian beams, ray bundles, and Bessel-Gauss beams,” Appl. Opt.36, 1116–1120 (1997). [CrossRef] [PubMed]
- A. R. Al-Rashed and B. E. A. Saleh, “Decentered Gaussian beams,” Appl. Opt.34, 6819–6825 (1995). [CrossRef] [PubMed]
- A. R. Al-Rashed, “Spatial and temporal modes of resonators with dispersive phase-conjugate mirrors,” PhD Thesis (1997).
- S. A. Collins, “Lens-System Diffraction Integral Written in Terms of Matrix Optics,” J. Opt. Soc. Am.60, 1168–1177 (1970). [CrossRef]
- G. Ryshik, Tables of Series, Products and Integrals (Verlag Harri Deutsch, 1981).
- H. F. Johnson, “An improved method for computing a discrete hankel transform,” Comp. Phys. Comm.43, 181–202 (1987). [CrossRef]
- L. Yu, M. Huang, M. Chen, W. Chen, W. Huang, and Z. Zhu, “Quasi-discrete Hankel transform,” Opt. Lett.23, 409–411 (1998). [CrossRef]
- M. Guizar-Sicairos and J. C. Gutiérrez-Vega, “Computation of quasi-discrete Hankel transforms of integer order for propagating optical wave fields,” J. Opt. Soc. Am. A21, 53–58 (2004). [CrossRef]
- A. Fox and T. Li, “Computation of optical resonator modes by the method of resonance excitation,” IEEE J. Quantum Electron.4, 460–465 (1968). [CrossRef]
- A. E. Siegman, Lasers (University Science Books, 1986).
- O. Brzobohaty, T. Cizmar, and P. Zemanek, “High quality quasi-Bessel beam generated by round-tip axicon,” Opt. Express16, 12688–12700 (2008). [CrossRef] [PubMed]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.