## Beam duality, with application to generalized Bessel-Gaussian, and Hermite- and Laguerre-Gaussian beams

Optics Express, Vol. 17, Issue 5, pp. 3690-3697 (2009)

http://dx.doi.org/10.1364/OE.17.003690

Acrobat PDF (1003 KB)

### Abstract

The concept of the dual of a beam is discussed. The duals of Bessel-Gauss beams, elegant Hermite- or Laguerre-Gaussian beams and generalized Hermite- or Laguerre-Gauss beams are described. Duality is considered within the framework of hypergeometric beams in Cartesian and polar coordinates. The connection with the “modified Laguerre-Gauss” beam is discussed.

© 2009 Optical Society of America

## 1. Duality of beams

*ABCD*matrix formalism,

^{11. S. A. Collins, “Lens-system diffraction integral written in terms of matrix optics,” J. Opt. Soc. Am. A 60, 1168–1177 (1970). [CrossRef] }so that a dual can be generated by a 2-

*f*optical system, for which

*A*=

*D*= 0. The principle of duality as applied to optical experiments and theories was introduced many years ago by Lohmann.

^{22. A. W. Lohmann, “Ein neues Dualitatsprinzip in der Optik,” Optik 11, 478–488 (1954)., 33. A. W. Lohmann, “Duality in optics,” Optik 89, 93–97 (1992).}A beam whose cross-section is a self-Fourier transform

^{4–64. F. Riesz and B. Szökefalvi-Nagy, Functional Analysis (Dover, New York, 1990).}is its own dual, of which the standard Hermite-Gaussian (sHG) and Laguerre-Gaussian (sLG) beams are special cases. As the sHGs and sLGs are complete sets, any amplitude in the waist can be expanded as a series of these basis functions. The sHG and sLG beams have relative phase in the far field that depends on the mode numbers, and can take values 0,7

*π*/2,

*π*,3

*π*/2. Thus in terms of sHG or sLG expansions, the dual can be generated by introducing a factor of a power of

*i*depending on the mode numbers. One example of the dual of a beam that has been described is that of a flattened beam.

^{77. C. J. R. Sheppard and S. Saghafi, “Flattened light beams,” Opt. Comm. 132, 144–152 (1996). [CrossRef] }In the limiting case of the flattened beam, corresponding to a beam from a circular pupil, the focal plane is an Airy disc (N.B. not an Airy function), and its dual is a beam with an Airy disc pupil, with a uniform circular spot in the focal plane. To give the dual of the flattened beam alternating signs are introduced in the terms of the sLG expansion.

^{88. C. J. R. Sheppard and T. Wilson, “Gaussian-beam theory of lenses with annular aperture,” IEE J. Microwaves, Optics and Acoustics 2, 105–112 (1978). [CrossRef] }

*J*is a Bessel function of the first kind of integer order

_{m}*m*,

*b*is a constant defining the relative width of the Bessel and Gaussian functions,

*w*

_{0}and

*z*

_{0}=

*kw*

^{2}

_{0}/2 are the beam waist and confocal parameter for the Gaussian,

*Z*=

*z*/

*z*

_{0}and

*μ*= 1+

*iZ*, and a factor exp[

*i*(

*kz*-

*ωt*)] with

*k*= 2

*π*/

*λ*, is suppressed. An expression for the BG beam was given

^{88. C. J. R. Sheppard and T. Wilson, “Gaussian-beam theory of lenses with annular aperture,” IEE J. Microwaves, Optics and Acoustics 2, 105–112 (1978). [CrossRef] }ten years before the reintroduction of the BG beam by Gori

*et al*.,

^{99. F. Gori, G. Guatteri, and C. Padovani, “Bessel-Gauss beams,” Opt. Comm. 64, 491–495 (1987). [CrossRef] }and almost ten years even before the Bessel beam was proposed by Durnin

*et al*.

^{1010. J. Durnin, J. J. Miceli Jr., and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987). [CrossRef] [PubMed] }The intensity, defined as the squared modulus of the amplitude, in the focal region is illustrated in Fig.1a and 1b for the value

*b*= 10 and

*m*= 0. We can see the annular structure forming in Fig. 1b. If

*b*is complex we obtain the generalized Bessel-Gauss (gBG) beam.

^{1111. V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. Shirripa Spagnolo, “Generalized Bessel-Gauss beams,” J. Mod. Optics 43, 1155–1166 (1996).}The dual of the BG beam (dBG) is obtained by putting

*b*as

*b*=

*ib*′ (equivalent to introducing alternating signs in the LG expansion), giving

*I*is a modified Bessel function of the first kind. This is also called the modified Bessel Gauss (mBG) beam.

_{m}^{1111. V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. Shirripa Spagnolo, “Generalized Bessel-Gauss beams,” J. Mod. Optics 43, 1155–1166 (1996).}It has been written in this way so that the quantity in braces is approximately constant for

*Z*= 0 and large argument, so the amplitude is only appreciable near

*r*=

*b*′

*w*

_{0}/2, representing an annular waist. This is shown in Fig.1c for

*b*′ = 10. We see a ring structure in the waist, and subsidiary peaks on the axis either side of focus that are analogous to the Poisson spot in the diffraction pattern of a circular obstruction.

## 2. Hermite-Gaussian beams

^{1212. H. Kogelnik and T. Li, “Laser beams and resonators,” App. Opt. 5, 1550–1567 (1966). [CrossRef] }In the waist, the sHG beams exhibit a particular scaling ratio between the widths of the Hermite polynomial and the Gaussian. Wünsche

^{1313. A. Wünsche, “Analogien zwischen ausserordentlichen und ordentlichen Wellen nach nichtorthogonaler Koordinatentransformation und die parabolischen Näherungsgleichungen,” Ann. Phys. 25, 113–135 (1970). [CrossRef] , 1414. A. Wünsche, “Generalized Gaussian beam solutions of paraxial optics and their connections to a hidden symmetry,” J. Opt. Soc. Am. A 6, 1320–1329 (1989). [CrossRef] }showed that these beams can be generalized to an arbitrary scaling parameter, with the amplitude expressed in closed form for any point in space, called generalized HG (or gHG) beams, but the behaviour was not described explicitly. For the paraxial case any form can be assumed for the waist, as long its Fourier transform exists, but the amplitude might not be expressible simply in closed form for all space. For non-paraxial beams, if there are no evanescent waves, the amplitude in the waist must be band-limited.

^{1515. C. J. R. Sheppard, “High aperture beams,” Journal of the Optical Society of America A 18, 1579–1587 (2001). [CrossRef] }Siegman investigated the important particular case of the elegant HG (written eHG) beam, in which the arguments of the Gaussian and the Hermite polynomial have the same form at any propagation distance.

^{1616. A. E. Siegman, “Hermite-Gaussian functions of complex arguments as optical-beam eigenfunctions,” J. Opt. Soc. Am. 63, 1093–1094 (1973). [CrossRef] }Pratesi and Ronchi

^{1717. R. Pratesi and L. Ronchi, “Generalized Gaussian beams in free space,” J. Opt. Soc. Am. 67, 1274–1276 (1977). [CrossRef] }investigated beams with an arbitrary complex constant parameter. They chose the values of the complex parameter to satisfy a particular condition that was useful for the study of beam propagation in a quadratic index medium. However, any complex value of the parameter gives a valid solution of the paraxial wave equation. In this paper, we investigate the particular condition with a real scaling ratio

*b*. Then, we can show that the amplitude can be written in the form

*H*is a Hermite polynomial of integer order

_{n}*n*defining the mode number,

*c*=

*b*

^{2}-1, and

*ν*= (1+

*c*) + 2

*ic*Z + (1-

*c*)

*Z*

^{2}=

*μ*[(1 +

*c*)-

*i*(1-

*c*)

*Z*] . In the waist

*Z*= 0, the argument of the Hermite polynomial is real if

*c*≥ -1 (

*b*is real). In general

*c*may be complex. On examining the argument of the Hermite function, we can see that it can be made real at an arbitrary plane by taking

*c*as complex, with a phase - 2 arctan

*Z*. In this plane the scaling factor of the Hermite polynomial relative to the Gaussian is

*b*= 1/(1 +

*c*)

^{1/2}. The modulus of

*c*controls the scaling factor, and its argument the plane in which the argument of the Hermite function is real. Thus for

*Z*→ ±𢈞, corresponding to the far field, the phase of

*c*is ∓

*π*, and

*c*is real and negative. These negative values of

*c*correspond to the dual of a beam of positive ∣

*c*∣.

*z*

_{0}and

*c*are real, so that the amplitude in the waist is real. For

*c*= 0, Eq. (3) reduces to the sHG mode,

^{1212. H. Kogelnik and T. Li, “Laser beams and resonators,” App. Opt. 5, 1550–1567 (1966). [CrossRef] }whereas for

*c*= 1 they reduce to Siegman’s eHG modes.

^{1616. A. E. Siegman, “Hermite-Gaussian functions of complex arguments as optical-beam eigenfunctions,” J. Opt. Soc. Am. 63, 1093–1094 (1973). [CrossRef] }For ∣

*c*∣ large, the amplitude reduces to the form

*Z*→ ∞, we have for the general case

*c*=1), the Hermite polynomial can be evaluated using the limit

^{1414. A. Wünsche, “Generalized Gaussian beam solutions of paraxial optics and their connections to a hidden symmetry,” J. Opt. Soc. Am. A 6, 1320–1329 (1989). [CrossRef] , 1919. S. Saghafi and C. J. R. Sheppard, “Near field and far field of elegant Hermite-Gaussian and Laguerre-Gaussian modes,” J. Mod. Optics 45, 1999–2009 (1998). [CrossRef] }

*c*> 1, then although the argument of the Hermite polynomial is imaginary, in the far field

*z*= 0

*c*= -1, again we can use Eq. (6) to give

^{77. C. J. R. Sheppard and S. Saghafi, “Flattened light beams,” Opt. Comm. 132, 144–152 (1996). [CrossRef] }of Siegman’s elegant beam. We have in general for deHG

*z*

_{0}by replacing a real

*c*with -

*c*, i.e. cd = -

*c*. The sHG beam is its own dual. Fig. 2 shows the intensity (modulus squared of the amplitude) in the waist for gHG beams for

*n*= 5 for different values of the parameter

*c*. For sHG,

*c*= 0, as is well known the pattern consists of an array of six peaks, which become slightly stronger with distance from the axis. The behavior near to the axis is close to that of a sin

^{22. A. W. Lohmann, “Ein neues Dualitatsprinzip in der Optik,” Optik 11, 478–488 (1954).}function. As

*c*is increased, the relative strength of the outer peaks is reduced. For eHG (

*c*= 1) the outer two peaks are very weak. As

*c*is decreased below zero, the relative strength of the central peaks is reduced, until for deHG (

*c*= -1) only two peaks remain. For more negative values of

*c*, the width of the central minimum is reduced. As beams with the same absolute value of

*c*but of opposite sign are duals, Fig. 2 also represents the intensity in the far field if the sign of

*c*is changed. For even values of

*n*, there is a peak on the axis the strength of which depends on the value of

*c*. For deHG (

*c*= -1), the central peak vanishes, leaving only two peaks as before. The strength of the central peak increases as

*c*is increased or decreased from this value.

*n*= 5 for different values of the parameter

*c*. We see that for sHG beam, as the number of peaks does not change with

*z*, the ridges along the peaks can be easily followed. For deHG (

*c*= -1), for example, it is seen that the two peaks in the waist are not joined continuously to the peaks in the far field as the strongest peaks are the outer peaks in the waist, but the inner peaks in the far field. This behavior is also seen in Fig. 4, which shows the phase in the focal region, for which we must now choose a particular value of

*kz*

_{0}, taken as 5, which is large enough for the paraxial treatment to be approximately valid. For sHG (

*c*= 0), the phase is clearly separated by

*π*for the three adjacent peaks, and changes progressively in the direction of propagation along each ridge. The behavior is quite different for eHG (

*c*= 1). Similar effects occur for even

*n*, where the central peak in the waist for

*c*= 1 decays on propagation while two peaks increase in strength.

## 3. Laguerre-Gauss beams

*L*is an associated Laguerre polynomial where

^{m}_{n}*m*,

*n*are integer mode numbers. In the waist

*c*= -1), an annular beam is generated in the waist, which becomes a thin annulus for large

*n*. For non-zero

*m*, the beams exhibit a phase singularity on the axis. In the far field

^{1414. A. Wünsche, “Generalized Gaussian beam solutions of paraxial optics and their connections to a hidden symmetry,” J. Opt. Soc. Am. A 6, 1320–1329 (1989). [CrossRef] , 2020. M. Porras, R. Borghi, and M. Santarsiero, “Relationship between elegant Laguerre-Gauss and Bessel-Gauss beams,” J. Opt. Soc. Am. A 18, 177–184 (2001). [CrossRef] }

*c*=1), using the identity18

*n*or

*m*the far field thus represents an annular beam.

*c*= -1 in Eq. (14)

*n*or

*m*are non-zero the beam has an annular form in its waist.

## 4. Discussion

*c*can take continuous, non-integer values. It is known that for large

*n*, eHG or eLG beams are good approximations to the cosine-Gauss (CG) or BG beams, respectively. For example, for HG we have the limit for

*n*→ ∞ and

*x*not too large

20. M. Porras, R. Borghi, and M. Santarsiero, “Relationship between elegant Laguerre-Gauss and Bessel-Gauss beams,” J. Opt. Soc. Am. A **18**, 177–184 (2001). [CrossRef]

*n*must be an integer. From Eq. (19) we see that gHG can be approximated to CG for any positive value of

*c*. This has been pointed out for the LG case by Bandres and Gutiérerrez-Vega.

^{21}Thus an approximation to a CG beam based on a gHG beam can be found for an integer value of

*n*, with a value of

*c*near to unity.

^{21–2421. M. A. Bandres and J. C. Gutierrez-Vega, “Circular beams,” Opt. Lett. 33, 177–179 (2008). [CrossRef] [PubMed] }This interesting and powerful approach can suggest new beam forms and make visible the symmetry properties. But the hypergeometric beams are no more general than gLG with complex

*n*, as the Laguerre function for complex

*n*is defined in terms of the confluent hypergeometric function. However, symmetry relationships can be revealed for particular special cases of the confluent hypergeometric function. The relationships between our parameters and the complex beam parameters used by Bandres are

*q*

_{0}= -

*iz*

_{0},

*q*̃

_{0}= -

*iz*

_{0}̃ =

*iz*

_{0}(1 +

*c*)/(1-

*c*). Hence

*c*= (

*q*̃

_{0}-

*q*

_{0})/(

*q*̃

_{0}+

*q*

_{0}),

*μ*= 1 +

*z*/

*q*

_{0}and

*ν*= 2

*μ*(

*z*+

*q*̃

_{0})/(

*q*

_{0}+

*q*̃

_{0}) .

*f*optical system, we have

*A*=

*D*= 0,

*B*=

*f*,

*C*= -1/

*f*. Thus the dual of a beam has

*q*

_{d0}= -

*f*

^{2}/

*q*

_{0}= -

*q*

^{*}

_{0},

*q*̃

_{d0}= -

*f*

^{2}/

*q*̃

_{0}= -

*q*

_{0}

*q*

^{*}

_{0}/

*q*̃

_{0}if we take

*f*

^{2}=

*q*

_{0}

*q*

^{*}

_{0}. We find that as we stated earlier without proof the dual of a beam has

*c*= -

_{d}*c*even for the general case when

*c*is complex, and also

*z*

_{d0}=

*z*

^{*}

_{0},

*w*

_{d0}=

*w*

^{*}

_{0}if its real part is taken as positive,

*μ*= 1-

_{d}*z*/

*q*

^{*}

_{0}= 2 -

*μ*

^{*}and

*ν*= -2

_{d}*μ*(

_{d}*q*̃

_{0}/

*q*

^{*}

_{0})(

*z*+

*q*̃

_{d0})/(

*q*

_{0}+

*q*̃

_{0}) .

*z*

_{0}is real, the width of the underlying Gaussian is unchanged in the dual beam, and

*μ*=

_{d}*μ*,

*χ*

^{2}becomes

^{2122. V. V. Kotlyar, R. V. Skidanov, S. N. Khonina, and V. A. Soifer, “Hypergeometric modes,” Opt. Lett. 32, 742–744 (2007). [CrossRef] [PubMed] ,2424. M. A. Bandres and J. C. Gutierrez-Vega, “Cartesian beams,” Opt. Lett. 32, 3459–3461 (2007).25. [CrossRef] [PubMed] }

_{1}

*F*

_{1}is a confluent hypergeometric function, Γ is a Gamma function, and in general

*n*can be complex. Similarly for the two dimensional case

*m*is an integer but

*n*can be complex. Note that Eqs. 22, 23 are slightly different from those of Bandres

^{21, 24}, as they are written so that they converge as

*ν*→ 0 for

*n*a positive integer. The fractional frHG and frLG beams follow directly from the definition of the Hermite and Laguerre functions of complex order

*n*.

^{2525. J. C. Gutiérrez-Vega, “Fractionalization of optical beams: I. Planar analysis,” Opt. Lett. 11, 1521–1523 (2007). [CrossRef] , 2626. J. C. Gutiérrez-Vega, “Fractionalization of optical beams: II. Elegant Laguerre-Gaussian modes,” Optics Express 15, 6300–6313 (2007). [CrossRef] [PubMed] }Formally, Gutiérrez-Vega’s

^{2525. J. C. Gutiérrez-Vega, “Fractionalization of optical beams: I. Planar analysis,” Opt. Lett. 11, 1521–1523 (2007). [CrossRef] }

*P*(

_{n}*x*) is just 2

^{-n}

*π*

^{1/2}

*H*(-

_{n}*x*). (The minus sign is a result of Gutiérrez-Vega’s definition of the creation operator.)

*q*

_{0},

*q*̃

_{0}→ ∞), so deHG can be taken as (

*q*

_{0},0). This can be recognized as the one dimensional equivalent of the MLG (“modified LG”) beam of Karimi

*et al*.

^{2121. M. A. Bandres and J. C. Gutierrez-Vega, “Circular beams,” Opt. Lett. 33, 177–179 (2008). [CrossRef] [PubMed] ,2323. E. Karimi, G. Zito, B. Piccirillo, L. Marrucci, and E. Santamato, “Hypergeometric-Gaussian beams,” Opt. Lett. 32, 3053–3055 (2007). [CrossRef] [PubMed] }. In fact Eq. (21) is equivalent to Eq. (1) of Karimi, so that his MLG can be identified as the same as the deLG. Thus the dual of Karimi’s MLG is eLG.

^{2323. E. Karimi, G. Zito, B. Piccirillo, L. Marrucci, and E. Santamato, “Hypergeometric-Gaussian beams,” Opt. Lett. 32, 3053–3055 (2007). [CrossRef] [PubMed] }also investigated beams with Bessel functions with quadratic radial dependence. For beams with

*n*= -∣

*m*∣/2 for odd

*m*, the far field gives a phase singularity without the usual

*r*

^{∣m∣}dependence and in the waist a sum of modified Bessel functions of quadratic argument. For even

*m*the waist is given by a sum of hyperbolic functions. Actually, Karimi discussed the duals of these beams, with the modified Bessel functions or exponentials in the far field and a simple Gaussian with exp(

*imϕ*) behaviour in the waist, which can be generated experimentally by applying a phase modulation to a TEM

_{00}Gaussian beam.

^{2323. E. Karimi, G. Zito, B. Piccirillo, L. Marrucci, and E. Santamato, “Hypergeometric-Gaussian beams,” Opt. Lett. 32, 3053–3055 (2007). [CrossRef] [PubMed] }

*ABCD*matrix also leads to the relationship

*q*

_{d0}/

*q*̃

_{d0}=

*q*̃

_{0}/

*q*

_{0}. An important alternative is

*f*

^{2}=

*q*

_{0}

*q*̃

_{0};

*q*

_{a0}= -

*q*̃

_{0}

*q*̃

_{a0}=

*q*̃

_{0}. For real

*z*

_{0}this case corresponds to the adjoint beams (hence the subscript

*a*) that form a biorthogonal set with the generalized beams.

^{1616. A. E. Siegman, “Hermite-Gaussian functions of complex arguments as optical-beam eigenfunctions,” J. Opt. Soc. Am. 63, 1093–1094 (1973). [CrossRef] }We find that

*c*= -

_{a}*c*as for the previous dual, but now

*z*

_{a0}=

*z*

_{0}(1+

*c*)/(1 -

*c*), so that

*z*

_{a0}(1 +

*c*) =

_{a}*z*

_{0}(1 +

*c*) and the scaling of the hypergeometric function, instead of the Gaussian, is unchanged. The adjoint beams are not so suitable for our present analysis, however, as the adjoint of the elegant beam has

*z*

_{a0}→ ∞ (i.e.

*q*

_{a0}→ ∞), so that there is no Gaussian factor in Eqs. 20, 21

21. M. A. Bandres and J. C. Gutierrez-Vega, “Circular beams,” Opt. Lett. **33**, 177–179 (2008). [CrossRef] [PubMed]

## Acknowledgment

## References and Links

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

2. | A. W. Lohmann, “Ein neues Dualitatsprinzip in der Optik,” Optik |

3. | A. W. Lohmann, “Duality in optics,” Optik |

4. | F. Riesz and B. Szökefalvi-Nagy, |

5. | M. J. Caola, “Self-Fourier functions,” J. Phys. A |

6. | A. W. Lohmann and D. Mendlovic, “Self-Fourier objects and other self-transform objects,” J. Opt. Soc. Am. A |

7. | C. J. R. Sheppard and S. Saghafi, “Flattened light beams,” Opt. Comm. |

8. | C. J. R. Sheppard and T. Wilson, “Gaussian-beam theory of lenses with annular aperture,” IEE J. Microwaves, Optics and Acoustics |

9. | F. Gori, G. Guatteri, and C. Padovani, “Bessel-Gauss beams,” Opt. Comm. |

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

11. | V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. Shirripa Spagnolo, “Generalized Bessel-Gauss beams,” J. Mod. Optics |

12. | H. Kogelnik and T. Li, “Laser beams and resonators,” App. Opt. |

13. | A. Wünsche, “Analogien zwischen ausserordentlichen und ordentlichen Wellen nach nichtorthogonaler Koordinatentransformation und die parabolischen Näherungsgleichungen,” Ann. Phys. |

14. | A. Wünsche, “Generalized Gaussian beam solutions of paraxial optics and their connections to a hidden symmetry,” J. Opt. Soc. Am. A |

15. | C. J. R. Sheppard, “High aperture beams,” Journal of the Optical Society of America A |

16. | A. E. Siegman, “Hermite-Gaussian functions of complex arguments as optical-beam eigenfunctions,” J. Opt. Soc. Am. |

17. | R. Pratesi and L. Ronchi, “Generalized Gaussian beams in free space,” J. Opt. Soc. Am. |

18. | I. S. Gradshteyn and I. M. Ryzhik, |

19. | S. Saghafi and C. J. R. Sheppard, “Near field and far field of elegant Hermite-Gaussian and Laguerre-Gaussian modes,” J. Mod. Optics |

20. | M. Porras, R. Borghi, and M. Santarsiero, “Relationship between elegant Laguerre-Gauss and Bessel-Gauss beams,” J. Opt. Soc. Am. A |

21. | M. A. Bandres and J. C. Gutierrez-Vega, “Circular beams,” Opt. Lett. |

22. | V. V. Kotlyar, R. V. Skidanov, S. N. Khonina, and V. A. Soifer, “Hypergeometric modes,” Opt. Lett. |

23. | E. Karimi, G. Zito, B. Piccirillo, L. Marrucci, and E. Santamato, “Hypergeometric-Gaussian beams,” Opt. Lett. |

24. | M. A. Bandres and J. C. Gutierrez-Vega, “Cartesian beams,” Opt. Lett. |

25. | J. C. Gutiérrez-Vega, “Fractionalization of optical beams: I. Planar analysis,” Opt. Lett. |

26. | J. C. Gutiérrez-Vega, “Fractionalization of optical beams: II. Elegant Laguerre-Gaussian modes,” Optics Express |

27. | C. F. R. Caron and R. M. Potvliege, “Bessel-modulated Gaussian beams with quadratic radial dependence,” Opt. Comm. |

**OCIS Codes**

(050.1940) Diffraction and gratings : Diffraction

(070.0070) Fourier optics and signal processing : Fourier optics and signal processing

(140.3300) Lasers and laser optics : Laser beam shaping

**ToC Category:**

Physical Optics

**History**

Original Manuscript: August 13, 2008

Revised Manuscript: December 11, 2008

Manuscript Accepted: January 7, 2009

Published: February 24, 2009

**Citation**

Colin J.R. Sheppard, "Beam duality, with application to generalized
Bessel-Gaussian, and Hermite- and Laguerre-
Gaussian beams," Opt. Express **17**, 3690-3697 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-5-3690

Sort: Year | Journal | Reset

### References

- S. A. Collins, "Lens-system diffraction integral written in terms of matrix optics," J. Opt. Soc. Am. A 60, 1168-1177 (1970). [CrossRef]
- A. W. Lohmann, "Ein neues Dualitatsprinzip in der Optik," Optik 11, 478-488 (1954).
- A. W. Lohmann, "Duality in optics," Optik 89, 93-97 (1992).
- F. Riesz and B. Szökefalvi-Nagy, Functional Analysis (Dover, New York, 1990).
- M. J. Caola, "Self-Fourier functions," J. Phys. A 24, L1143-L1144 (1991). [CrossRef]
- A. W. Lohmann and D. Mendlovic, "Self-Fourier objects and other self-transform objects," J. Opt. Soc. Am. A 9, 2009-2012 (1992). [CrossRef]
- C. J. R. Sheppard and S. Saghafi, "Flattened light beams," Opt. Comm. 132, 144-152 (1996). [CrossRef]
- C. J. R. Sheppard and T. Wilson, "Gaussian-beam theory of lenses with annular aperture," IEE J. Microwaves Opt. Acoustics 2, 105-112 (1978). [CrossRef]
- F. Gori, G. Guatteri, and C. Padovani, "Bessel-Gauss beams," Opt. Comm. 64, 491-495 (1987). [CrossRef]
- J. Durnin, J. J. Miceli, Jr., and J. H. Eberly, "Diffraction-free beams," Phys. Rev. Lett. 58, 1499-1501 (1987). [CrossRef] [PubMed]
- V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. Shirripa Spagnolo, "Generalized Bessel-Gauss beams," J. Mod. Optics 43, 1155-1166 (1996).
- H. Kogelnik and T. Li, "Laser beams and resonators," App. Opt. 5, 1550-1567 (1966). [CrossRef]
- A. Wünsche, "Analogien zwischen ausserordentlichen und ordentlichen Wellen nach nichtorthogonaler Koordinatentransformation und die parabolischen Näherungsgleichungen," Ann. Phys. 25, 113-135 (1970). [CrossRef]
- A. Wünsche, "Generalized Gaussian beam solutions of paraxial optics and their connections to a hidden symmetry," J. Opt. Soc. Am. A 6, 1320-1329 (1989). [CrossRef]
- C. J. R. Sheppard, "High aperture beams," Journal of the Optical Society of America A 18, 1579-1587 (2001). [CrossRef]
- A. E. Siegman, "Hermite-Gaussian functions of complex arguments as optical-beam eigenfunctions," J. Opt. Soc. Am. 63, 1093-1094 (1973). [CrossRef]
- R. Pratesi and L. Ronchi, "Generalized Gaussian beams in free space," J. Opt. Soc. Am. 67, 1274-1276 (1977). [CrossRef]
- I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic Press, New York, 1994).
- S. Saghafi and C. J. R. Sheppard, "Near field and far field of elegant Hermite-Gaussian and Laguerre-Gaussian modes," J. Mod. Optics 45, 1999-2009 (1998). [CrossRef]
- M. Porras, R. Borghi, and M. Santarsiero, "Relationship between elegant Laguerre-Gauss and Bessel-Gauss beams," J. Opt. Soc. Am. A 18, 177-184 (2001). [CrossRef]
- M. A. Bandres and J. C. Gutierrez-Vega, "Circular beams," Opt. Lett. 33, 177-179 (2008). [CrossRef] [PubMed]
- V. V. Kotlyar, R. V. Skidanov, S. N. Khonina, and V. A. Soifer, "Hypergeometric modes," Opt. Lett. 32, 742-744 (2007). [CrossRef] [PubMed]
- E. Karimi, G. Zito, B. Piccirillo, L. Marrucci, and E. Santamato, "Hypergeometric-Gaussian beams," Opt. Lett. 32, 3053-3055 (2007). [CrossRef] [PubMed]
- M. A. Bandres and J. C. Gutierrez-Vega, "Cartesian beams," Opt. Lett. 32, 3459-3461 (2007).25. [CrossRef] [PubMed]
- J. C. Gutiérrez-Vega, "Fractionalization of optical beams: I. Planar analysis," Opt. Lett. 11, 1521-1523 (2007). [CrossRef]
- J. C. Gutiérrez-Vega, "Fractionalization of optical beams: II. Elegant Laguerre-Gaussian modes," Optics Express 15, 6300-6313 (2007). [CrossRef] [PubMed]
- C. F. R. Caron and R. M. Potvliege, "Bessel-modulated Gaussian beams with quadratic radial dependence," Opt. Comm. 164, 83-93 (1999). [CrossRef]

## 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.