## Optical vortices with large orbital momentum: generation and interference

Optics Express, Vol. 14, Issue 7, pp. 2888-2897 (2006)

http://dx.doi.org/10.1364/OE.14.002888

Acrobat PDF (405 KB)

### Abstract

We demonstrate a method for generation of beams of light with large angular momenta. The method utilizes whispering gallery mode resonators that transform a plane electromagnetic wave into high order Bessel beams. Interference pattern among the beams as well as shadow pictures induced by the beams are observed and studied.

© 2006 Optical Society of America

## 1. Introduction

1. L. Allen, M. J. Padgett, and M. Babiker, “The orbital angular momentum of light”, Progress in Optics **39**, 291–372 (1999). [CrossRef]

2. L. Allen, “Introduction to the atoms and angular momentum of light special issue,” J. Opt. B **4**, S1–S6 (2002). [CrossRef]

4. D. McGloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys. **46**, 15–28 (2005). [CrossRef]

^{4}ħ per photon, though in principle possible, [5

5. J. Courtial, K. Dholakia, L. Allen, and M. J. Padgett, “Gaussian beams with very high orbital angular momentum,” Opt. Commun. **144**, 210–213 (1997). [CrossRef]

5. J. Courtial, K. Dholakia, L. Allen, and M. J. Padgett, “Gaussian beams with very high orbital angular momentum,” Opt. Commun. **144**, 210–213 (1997). [CrossRef]

6. J. E. Curtis, B. A. Koss, and D. G. Grier, “Dynamic holographic optical tweezers,” Opt. Commun. **207**, 169–175 (2002). [CrossRef]

7. S. Sundbeck, I. Gruzberg, and D. G. Grier, “Structure and scaling of helical modes of light,” Opt. Lett. **30**, 477–479 (2005). [CrossRef] [PubMed]

8. A. B. Matsko, A. A. Savchenkov, D. Strekalov, and L. Maleki, “Whispering gallery resonators for studying orbital angular momentum of a photon,” Phys. Rev. Lett. **95**, 143904 (2005). [CrossRef] [PubMed]

## 2. Experiment

*μ*m in diameter as an integral part of a cone. This cone starts at the WGM at diameter of about 250-280

*μ*m and expands to 1 mm in diameter over 1 cm distance (Fig. 2). Another resonator we built has a 500

*μ*m diameter, and the taper diameter changes from 450

*μ*m to 3 mm over a 3 cm distance.

*r*

_{exit}/

*r*

_{entrance}= 6 or less) and the mode order was high (> 10

^{3}), the propagation distance of Bessel beams in free space did not exceed ten millimeters. The beams spread out creating peculiar interference shapes in the far field region (Fig. 3(A)). This observation has a certain similarity with the interference pattern of zero and first order Laguerre-Gauss beams presented in [9

9. E. J. Galvez, P. R. Crawford, H. I. Sztul, M. J. Pysher, P. J. Haglin, and R. E. Williams, “Geometric phase associated with mode transformations of optical beams bearing orbital angular momentum,” Phys. Rev. Lett. **90**, 203901 (2003). [CrossRef] [PubMed]

*μ*m needle as the object. The shadow is orthogonal to the object in the center of the interference pattern, and is parallel, but displaced, far from the center. The shape of the shadow depends on the distance of the object from the surface of the taper. The closer the wire is, the longer is the region of the orthogonal shadow. We have repeated the same experiment using 532 nm laser and a 25

*μ*m thick piece of a tungsten wire, and obtained similar result. The dynamics of the behavior of the shadow with the distance change is demonstrated in Fig. 4. In what follows we explain the result of the experiment theoretically.

## 3. Theory

### 3.1. Interference

*Z*axis and, hence, different dispersion. The initially narrow angular distribution of light in the taper broadens up with the propagation distance.

*k*

_{z}for a Bessel wave can be found from equation

*z*and azimuthal index

*l*:

*k*

_{0}=

*ω*

_{0}/

*c*is the total wave vector,

*ω*

_{0}is the frequency,

*n*is the index of refraction of the material,

*k*

_{l}is the transverse propagation constant, and

*r*(

*z*) is the radius.

*l*can be expressed as

*z*from the beginning of the fiber taper (we assume that the radius changes adiabatically so that beams with different azimuthal numbers do not interact), where

*ϕ*

_{0}(

*l*) and

*A*

_{0}(

*l*,

*r*) are the phase and amplitude of the wave at

*z*= 0.

*l*

_{0}to

*l*

_{0}- Δ

*l*and the same frequency a>0 is created at the beginning of the fiber. The summation of all the waves inside the dispersion interval Δ

*l*gives:

*F*(

*r*,

*z*,

*ϕ*,

*l*

_{0},Δ

*l*) is the complex field distribution in the taper.

*n*= 1.45, as well as initial and final radii

*r*

_{0}=0.3 mm and

*r*

_{n}= 1 mm, respectively. We assumed that all the waves initially have the same phase and amplitude (

*ϕ*

_{0}(

*l*) ≡ π and

*A*

_{0}(

*l*,

*r*) ≡ 1). Fig. 6 picture shows angular dependence of [(

*F*(

*r*,

*z*,

*ϕ*(

*mod*(2

*π*)),

*l*

_{0},Δ

*l*) +

*c*.

*c*.)/2] for the given parameters and

*l*= 3000, Δ

*l*= 1000.

*z*

_{n}= 3 mm with the other parameters as used in the simulations resulting in Fig. 6. The pattern will have at least one loop for a longer taper (see, e.g., Fig. 7).

*F*(

*x*,

*y*) is given by Eq. (3) at the end of the taper crossection surface (here

*z*,

*l*and Δ

*l*are constants),

*C*is a constant,

*S*is the aperture area, and

*H*is the distance between the screen and the aperture. A direct evaluation of the integral is complicated, but we can approximate this integral by assuming that amplitudes of the particular Bessel waves (

*A*

_{0}(

*l*,

*r*)) are nonzero only in a thin belt close to the circumference of the taper crossection. The radial thickness of the belt is approximately Δ

*r*=

*r*

_{n}/

*l*

^{2/3}, as for high order WGMs. Splitting the circumference of the taper surface at small segments 1 ≫ Δ

*ϕ*≫

*λ*/(2

*π*Δ

*r*), we estimate the integral as

*r*=

*r*

_{n}- Δ

*r*/2, C

_{1}is a normalization parameter. To find the optimum step Δ

*ϕ*

_{opt}we first approximate the angle

*θ*for the region of the localization of the interference pattern and take Δ

*ϕ*

_{opt}≃

*λ*/[2

*π*(sin

*θ*)Δ

*r*]. For sin

*θ*≈ 0.22 and Δ

*r*≈ 4.3 × 10

^{-4}cm, which are the values for our taper, we get Δ

*ϕ*

_{opt}≃ 0.11. The result of the optimum step calculation is shown in Fig. 7.

### 3.2. Propagation

*L*

_{B}for the generated Bessel beams. Because the beams can be considered as a superposition of plane waves propagating at the angle

*θ*≈ arcsin(

*k*

_{l}(

*r*

_{n})/

*k*

_{0}) to the surface of the crossection of the taper, the propagation distance corresponds to

*L*

_{B}=

*r*

_{n}/arctg(

*θ*), where

*k*

_{l}(

*r*

_{n}) =

*k*

_{l}(

*r*

_{0})

*r*

_{0}/

*r*

_{n}and at the very beginning of the taper

*k*

_{l}(

*r*

_{0}) =

*k*

_{0}. Now the equations for the beam divergence

*θ*and beam propagation length

*L*

_{B}can be written as θ ≈ arcsin(

*r*

_{0}/

*r*

_{n}) and

*L*

_{B}≈

*r*

_{0},

*θ*<< 1. For parameters of our experiment

*r*

_{0}= 0.5 mm,

*r*

_{n}= 3 mm, and

*l*= 7000 we get

*L*

_{B}≈ 18 mm. At this distance the Bessel beam decays significantly, and its phase structure is very disturbed, though its momentum is still preserved. Experimental data demonstrate a Bessel beam propagation length of the same order. In order to get a distance of a meter, the exit radius of the taper should be at least seven times larger. It is difficult to reproduce such a taper experimentally using our technique of fabrication.

*r*

_{n}= 10

*μ*m as a source of light. According to Huygens the complex field amplitude can be described by the diffraction integral [11]

*f*=

*J*

_{m}(

*k*

_{l}

*r*)

*e*

^{-jlϕ},

*J*

_{m}(

*k*

_{l}

*r*) is the Bessel function of m-th order,

*C*

_{2}is a scale parameter,

*R*is the distance between point in the plane of end of the fiber and the point (

*x*,

*y*,

*z*) where diffraction pattern is obtained.

*ϕ*does not contain information on the divergence because our system is symmetrical. We can exclude it from the final consideration without any mathematical reduction. Now the diffraction integral looks like:

*ϕ*´ coordinate appeared under integral. After selection of certain angle where the optical field is to be calculated the function under the integral loose symmetry. This diffraction integral was computed numerically for

*m*= 17 and

*l*= 30 as well as for

*m*= 10 and

*l*= 15.

*B*(

*r*,

*z*)|

^{2}in the plane which is parallel to the fiber taper symmetry axis. The truncated Bessel beam does not change its shape but has radiative loss. This is seen as a straight Bessel beam and group of conically diverged beams in Fig. (8). Similar behavior was predicted for the zero-order Bessel-Gauss beam [10

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

### 3.3. Shadow

*m*≫

*l*-

*m*and consider a one dimensional rod

*AA*´ placed at the distance

*h*from the fiber and

*H*-

*h*from the screen (Fig. 9). The surfaces of the fiber crossection and the screen are parallel to the rod. The point

*A*of the rod is illuminated by the plane wave leaving point

*S*on the fiber rim. Point

*B*is the shadow of point

*A*.

*γ*(angle CFB), that determines the change of the position of the shadow (B) with respect the straight projection (C) of point A of the rod. To do this we need to know all the sides of triangle CFB.

*°*. Segment AD is equal to

*h*. Segment OS is equal to

*r*

_{n}. Angle ADS is equal to 90

*°*, and angle ASD is equal to

*α*. Therefore

*°*, and angle CBA is equal to

*α*, hence

*α*can be directly found from the formula tg

*α*=

*k*

_{z}(

*r*

_{n})/

*k*

_{l}(

*r*

_{n}). Finally, the distance from the shadow point (

*B*) to the direct projection of the rod

*AA*´ is equal to FBsiny.

*k*

_{z}(

*r*

_{n})/

*k*

_{l}(

*r*

_{n}), as is done in the experiment. Each Bessel beam projects a point of the rod to different places of the screen. The complete shadow line is shown in the inset of Fig. (9). This result directly resembles our experimental observations.

## 4. Conclusion

^{5}. Interference patterns of the multiple beams as well as the peculiar shadow pictures created with the beams are demonstrated experimentally and explained theoretically.

## Acknowledgments

## References and links

1. | L. Allen, M. J. Padgett, and M. Babiker, “The orbital angular momentum of light”, Progress in Optics |

2. | L. Allen, “Introduction to the atoms and angular momentum of light special issue,” J. Opt. B |

3. | E. Santamato, “Photon orbital angular momentum: problems and perspectives,” Progress in Physics |

4. | D. McGloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys. |

5. | J. Courtial, K. Dholakia, L. Allen, and M. J. Padgett, “Gaussian beams with very high orbital angular momentum,” Opt. Commun. |

6. | J. E. Curtis, B. A. Koss, and D. G. Grier, “Dynamic holographic optical tweezers,” Opt. Commun. |

7. | S. Sundbeck, I. Gruzberg, and D. G. Grier, “Structure and scaling of helical modes of light,” Opt. Lett. |

8. | A. B. Matsko, A. A. Savchenkov, D. Strekalov, and L. Maleki, “Whispering gallery resonators for studying orbital angular momentum of a photon,” Phys. Rev. Lett. |

9. | E. J. Galvez, P. R. Crawford, H. I. Sztul, M. J. Pysher, P. J. Haglin, and R. E. Williams, “Geometric phase associated with mode transformations of optical beams bearing orbital angular momentum,” Phys. Rev. Lett. |

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

11. | L. Landau and E. Lifshitz, |

**OCIS Codes**

(140.3300) Lasers and laser optics : Laser beam shaping

(230.5750) Optical devices : Resonators

(260.1960) Physical optics : Diffraction theory

**ToC Category:**

Optical Devices

**History**

Original Manuscript: February 6, 2006

Revised Manuscript: March 28, 2006

Manuscript Accepted: March 29, 2006

Published: April 3, 2006

**Citation**

Anatoliy A. Savchenkov, Andrey B. Matsko, Ivan Grudinin, Ekaterina A. Savchenkova, Dmitry Strekalov, and Lute Maleki, "Optical vortices with large orbital momentum: generation and interference," Opt. Express **14**, 2888-2897 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-7-2888

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### References

- L. Allen, M. J. Padgett, and M. Babiker, "The orbital angular momentum of light," Progress in Optics 39, 291-372 (1999). [CrossRef]
- L. Allen, "Introduction to the atoms and angular momentum of light special issue," J. Opt. B 4, S1-S6 (2002). [CrossRef]
- E. Santamato, "Photon orbital angular momentum: problems and perspectives," Progress in Physics 52, 1141-1153 (2004).
- D. McGloin and K. Dholakia, "Bessel beams: diffraction in a new light," Contemp. Phys. 46, 15-28 (2005). [CrossRef]
- J. Courtial, K. Dholakia, L. Allen, and M. J. Padgett, "Gaussian beams with very high orbital angular momentum," Opt. Commun. 144, 210-213 (1997). [CrossRef]
- J. E. Curtis, B. A. Koss, and D. G. Grier, "Dynamic holographic optical tweezers," Opt. Commun. 207, 169-175 (2002). [CrossRef]
- S. Sundbeck, I. Gruzberg, and D. G. Grier, "Structure and scaling of helical modes of light," Opt. Lett. 30, 477-479 (2005). [CrossRef] [PubMed]
- A. B. Matsko, A. A. Savchenkov, D. Strekalov, and L. Maleki, "Whispering gallery resonators for studying orbital angular momentum of a photon," Phys. Rev. Lett. 95, 143904 (2005). [CrossRef] [PubMed]
- E. J. Galvez, P. R. Crawford, H. I. Sztul, M. J. Pysher, P. J. Haglin, and R. E. Williams, "Geometric phase associated with mode transformations of optical beams bearing orbital angular momentum," Phys. Rev. Lett. 90, 203901 (2003). [CrossRef] [PubMed]
- F. Gori, G. Guattari, and C. Padovani, "Bessel-Gauss beams," Opt. Commun. 64, 491-495 (1987). [CrossRef]
- L. Landau and E. Lifshitz, Classical Theory of Fields (Reed International Educational and Professional Publishing, Oxford, 1980).

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