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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 16, Iss. 18 — Sep. 1, 2008
  • pp: 13599–13605
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Achromatic design for the generation of optical vortices based on radial spiral phase plates

X.-C. Yuan, J. Lin, J. Bu, and R. E. Burge  »View Author Affiliations


Optics Express, Vol. 16, Issue 18, pp. 13599-13605 (2008)
http://dx.doi.org/10.1364/OE.16.013599


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Abstract

A multi-element design scheme is proposed to produce optical vortices of large spectrum width. The key component within the approach is a radially modulated spiral phase plate. Apart from a conventional spiral phase plate having an azimuthal phase function, the proposed element possesses an additional change of phase in the radial direction.

© 2008 Optical Society of America

Various unconventional spiral phase plates exhibiting radial phase variations have been used to reduce the sidelobes of an optical vortex in the far-field [7

7. J. Lin, X.-C. Yuan, S. H. Tao, and R. E. Burge, “Variable-radius focused optical vortex with suppressed sidelobes,” Opt. Lett. 31, 1600–1602 (2006). [CrossRef] [PubMed]

, 8

8. V. V. Kotlyar, A. A. Kovalev, V. A. Soifer, C. S. Tuvey, and J. A. Davis, “Sidelobe contrast reduction for optical vortex beams using a helical axicon,” Opt. Lett. 32, 921–923 (2007). [CrossRef] [PubMed]

] and create higher order Bessel beams [9

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

, 10

10. S. H. Tao, W. M. Lee, and X.-C. Yuan, “Dynamic optical manipulation using higher order fractional Bessel beam generated from a spatial light modulator,” Opt. Lett. 28, 1867–1869 (2003). [CrossRef] [PubMed]

]. Most of their phase functions take the summation of a radial function f(r)and the linear azimuthal phase term ℓθ characterizing the nature of an optical vortex. In this paper, we study another possible form, i.e. f(rθ, that will be referred to as radial spiral phase plate (RSPP). If an infinitely narrow annular aperture with the radius equal to r ο is applied to such a phase element, the transmittance within the aperture can be approximated as exp[i f(r οθ], which is very similar to that of a conventional spiral phase plate while the constant f(r ο) takes the position of topological charge . In other words, various topological charges can be achieved by the same phase element when the radius of the annular aperture is carefully selected to a series of discrete values. Meanwhile, it is reported that a pair of identical axicons transforms an incident Gaussian beam into an annular beam [11

11. M. Rioux, R. Tremblay, and P. A. Belanger, “Linear, annular, and radial focusing with axicons and applications to laser machining,” Appl. Opt. 17, 1532–1536 (1978). [CrossRef] [PubMed]

, 12

12. T. Shiina, K. Yoshida, M. Ito, and Y. Okamura, “In-line type micropulse lidar with an annular beam: theoretical approach,” Appl. Opt. 44, 7467–7474 (2005). [CrossRef] [PubMed]

]. Therefore, a combination of axicon pairs and an RSPP would enable the achromatic generation of optical vortices as proposed in Fig. 1. Compared to light with long wavelength (depicted in red), spectral component having shorter wavelength (depicted in blue) results in a larger annulus after passing through the first axicon pair. The phase delay conveyed by a thin phase element e.g. RSPP in this paper can be given by

Φ(r,θ)=2πΔn×h(r,θ)λ,
(1)

where Δn is the difference in refractive index between the phase element and its surrounding medium, h(r,θ) is the thickness of the phase element and λ is the wavelength of light in vacuum. Normally, such an element will present different phase delay function for various wavelengths. However, Fig. 1 shows that the blue ray representing light with shorter wavelength will pass through part of the phase element thinner than that experienced by the red ray, which makes it possible to have an equivalent phase delay for these two wavelengths. It is noted that this cannot be achieved by the conventional SPP as its thickness does not change in the radial direction. The second axicon pair combines the annular beams back to one laser beam.

Fig. 1. The achromatic design scheme for the generation of optical vortices under the illumination of a broadband source.

Although the RSPP is designed for more than one spectral component, for simplicity the element can be expressed by a phase-only transmittance function at a fixed wavelength. In this paper, we will focus on a straightforward example when the transmittance of the RSPP at the wavelength λο is

t(r,θ)={exp(ikrθ)0rR1else
(2)

with the constant k chosen to be ℓο/r ο. Here ℓο is the topological charge of the optical vortex generated from the infinitely narrow annular zone at r=r 0. The phase profiles of an SPP and the RSPP with the parameter k>0 are illustrated in Fig. 2 while the case of k<0 has been demonstrated in Fig. 1. It is noted that a similar phase function was first proposed by multiplying the phase function of an SPP with that of an axicon and such an element results in a spiral intensity pattern in the far-field when working with a full circular aperture [13

13. C. Alonzo, P. J. Rodrigo, and J. Glückstad, “Helico-conical optical beams: a product of helical and conical phase fronts,” Opt. Express 13, 1749–1760 (2005).

]. For the optical field produced by the RSPP illuminated by a uniform annular beam with mean radius r ο and width w, the weight of the optical vortex with topological charge n can be calculated by the mode analysis method [14

14. G. Molina-Terriza, J. P. Torres, and L. Torner, “Management of the angular momentum of light: preparation of photons in multidimensional vector states of angular momentum,” Phys. Rev. Lett. 88, 013601 (2002). [CrossRef] [PubMed]

16

16. J. Lin, X.-C. Yuan, J. Bu, B. P. S. Ahluwalia, Y. Y. Sun, and R. E. Burge, “Selective generation of high order optical vortices from a single phase wedge,” Opt. Lett. 32, 2927–2929 (2007). [CrossRef] [PubMed]

] with the thin-element approximation [17

17. J. W. Goodman, Introduction to Fourier optics, 2nd ed. (McGraw-Hill, Singapore, 1996).

]:

An(w)=r0w2r0+w202πt(r,θ)exp(inθ)dθ2rdr2π2[(r0+w2)2(r0w2)2],(w<2r0)
(3)

which can be given in an explicit form:

An(w)=r02π202w[Ci(20π2nπ0πwr0)Ci(20π2nπ+0πwr0)
+In(202n+0wr0)In(202n0wr0)+2nπSi(20π2nπ+0πwr0)
2nπSi(20π2nπ0πwr0)+80nr0w4r02(n0)202w2sin2(0πw2r0)],
(4)

where Ci(x) and Si(x) are cosine integral and sine integral, respectively. Therefore, the weight of the major topological charge can be calculated by setting n=ℓο:

A0(y)=2Si(y)ysinc2(y2)
(5)

with the dimensionless width y=πℓο w/r ο and Sinc(x)=sinx/x.

Fig. 2. (a) Phase distribution of a conventional SPP exp(iθ) with ℓ>0, (b) Phase distribution of an RSPP exp(ikrθ) with k>0.

Figure 3(a) is the plot of Eq. 5 that shows the weight of the specific optical vortex as a function of the dimensionless width. As expected, the weight reaches 100% when the width of the annular aperture is infinitely narrow and decreases with the increasing width. Since Eq. 5 is irrelevant to the value of the topological charge (as the variable y=πℓο w/r οkw), this analysis applies to other topological charges generated by the same RSPP, e.g. the topological charge 2ℓο excited from the vicinity of r=2r ο. The high concentration of power on the target topological charge is illustrated in Fig. 3(b) when a narrow annular aperture is imposed. Figure 4 shows the Fraunhofer diffraction patterns of optical vortices produced by the RSPP with different annular apertures. The topological charge of the generated optical vortex increases with the mean radius of the annular aperture when k>0. The slightly asymmetry of intensity distribution is due to the mixture of modes other than the desired one. As a result of the annular aperture, the optical vortices in the far-field are embedded in the profile of Bessel beams that exhibits concentric intensity circles [18

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

].

Fig. 3. Analysis of the RSPP with the annular aperture of mean radius r ο : (a) the weight of the major optical vortex with topological charge ℓ0 as a function of the dimensionless width y=ℓ0πw/r ο ; (b) The weight distribution over topological charges (with a logarithmic ordinate) when r ο=1.54, ℓ0=2, and w=0.25.
Fig. 4. Fraunhofer diffraction patterns calculated numerically under the condition of y=1 (specifically, r ο=1.54, ℓο=2, and w=0.25) for the mean radius of aperture (a) r=r ο ; (b) r=2r ο.

An RSPP is fabricated by the technique of direct electron beam writing in negative resist with the phase profile wrapped within the range of 0~2π [19

19. W. C. Cheong, W. M. Lee, X.-C. Yuan, L.-S. Zhang, H. Wang, and K. Dholakia, “Direct electron-beam writing of continuous spiral phase plates in negative resist with high power efficiency for optical manipulation,” Appl. Phys. Lett. 85, 5784–5786 (2004). [CrossRef]

]. The surface relief phase element with the size of 7.68mm×7.68mm is designed to support the topological charge up to 5 at the wavelength of 633nm. The maximum topological charge is limited by the size of the element and the value of k, in our case, equal to 1.3mm-1. The confinement of RSPP in the radial direction imposed by an annular aperture is replaced by an incident annular beam which is converted from a Gaussian beam by two identical axicons as shown in Fig. 5. A beam of wavelength λ=633nm from a HeNe laser is collimated and enlarged by a telescope. After that, the laser beam is truncated by a circular diaphragm that controls the width of the annular beam. One of the axicons is set to be movable so that the distance between axicons can be adjusted according to the desired size of the annular beam [11

11. M. Rioux, R. Tremblay, and P. A. Belanger, “Linear, annular, and radial focusing with axicons and applications to laser machining,” Appl. Opt. 17, 1532–1536 (1978). [CrossRef] [PubMed]

]. A central stop is placed after the axicon pair in order to remove the residual fractional intensity at the beam axis. The transverse intensity profile of the annular beam right before the RSPP is shown in Fig. 6. The optical field created after the annular beam passing through the RSPP sample is then focused into a CCD beam profiler by a lens. Optical vortices produced by the same RSPP while changing the mean radius of the annular beam are shown in Fig. 7. The corresponding topological charge of each optical vortex is measured by interfering with an oblique reference beam (See Fig. 8).

Fig. 5. Experimental setup for the dynamic switching of the topological charge of an optical vortex generated by the RSPP. Axicon AX2 can be moved along the beam propagating direction in order to change the distance between the axicons.
Fig. 6. The intensity profile of the annular beam. (a) Two-dimensional distribution, (b) The intensity plot along the horizontal line passing through the beam central in (a).
Fig. 7. The far-field intensity distribution of various optical vortices produced by the same RSPP. (a) ℓ=2, (b) ℓ=3, (c) ℓ=4, (d) ℓ=5.
Fig. 8. The interferometric measurement of topological charges of optical vortices. (a) ℓ=2, (b) ℓ=3, (c) ℓ=4, (d) ℓ=5.

In summary, we have demonstrated that an RSPP is capable of switching the topological charge of the resulting optical vortex at a fixed wavelength of light when the radius of the incident annular beam is changed. This result suggests that the RSPP can be used to generate optical vortices of an identical topological charge for various wavelengths. Based on the RSPP discussed in this paper, we propose a design scheme for the generation of an optical vortex with a wideband light source. The dispersion of the material used to construct the axicons makes the mean radius of the annular beam differ from one wavelength to another. The size change of the annular beam will be compensated by the radial variation of an RSPP in a way that the resultant optical vortex has the same topological charge within a broad range of wavelength. Although it may not be suitable for the imaging application such as the optical vortex coronagraph [20

20. G. Foo, D. M. Palacios, and G. A. Swartzlander, “Optical Vortex Coronagraph,” Opt. Lett. 30, 3308–3310 (2005). [CrossRef]

] because of the distortion of images introduced by apertures, stops and axicons, the proposed method (Fig. 1) of compensating chromatic dispersion is still able to find applications in reducing the topological dispersion of optical vortices generated by pulse laser [21

21. G. A. Swartzlander, “Broadband nulling of a vortex phase mask,” Opt. Lett. 30, 2876–2878 (2005). [CrossRef] [PubMed]

].

Acknowledgments

This work is partially supported by National Natural Science Foundation of China for grant 60778045 and National Research Foundation of Singapore under Grant No. NRF-G-CRP 2007-01 and Ministry of Education under ARC 3/06, RGM6/05 and RGM37/06. XCY acknowledges the support given by Nankai University (China) and Nanyang Technological University (Singapore). The authors would like to thank Yuyang Sun for data conversion for electron beam lithography.

References and links

1.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular-momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).

2.

M. V. Berry, “Coloured phase singularities,” New J. Phys. 4, 66 (2002). [CrossRef]

3.

M. V. Berry, “Exploring the colours of dark light,” New J. Phys. 4, 74 (2002). [CrossRef]

4.

J. Leach and M. J. Padgett, “Observation of chromatic effects near a white-light vortex,” New J. Phys. 5, 154 (2003). [CrossRef]

5.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun. 112, 321–327 (1994). [CrossRef]

6.

G. A. Swartzlander, “Achromatic optical vortex lens,” Opt. Lett. 31, 2042–2044 (2006). [CrossRef] [PubMed]

7.

J. Lin, X.-C. Yuan, S. H. Tao, and R. E. Burge, “Variable-radius focused optical vortex with suppressed sidelobes,” Opt. Lett. 31, 1600–1602 (2006). [CrossRef] [PubMed]

8.

V. V. Kotlyar, A. A. Kovalev, V. A. Soifer, C. S. Tuvey, and J. A. Davis, “Sidelobe contrast reduction for optical vortex beams using a helical axicon,” Opt. Lett. 32, 921–923 (2007). [CrossRef] [PubMed]

9.

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

10.

S. H. Tao, W. M. Lee, and X.-C. Yuan, “Dynamic optical manipulation using higher order fractional Bessel beam generated from a spatial light modulator,” Opt. Lett. 28, 1867–1869 (2003). [CrossRef] [PubMed]

11.

M. Rioux, R. Tremblay, and P. A. Belanger, “Linear, annular, and radial focusing with axicons and applications to laser machining,” Appl. Opt. 17, 1532–1536 (1978). [CrossRef] [PubMed]

12.

T. Shiina, K. Yoshida, M. Ito, and Y. Okamura, “In-line type micropulse lidar with an annular beam: theoretical approach,” Appl. Opt. 44, 7467–7474 (2005). [CrossRef] [PubMed]

13.

C. Alonzo, P. J. Rodrigo, and J. Glückstad, “Helico-conical optical beams: a product of helical and conical phase fronts,” Opt. Express 13, 1749–1760 (2005).

14.

G. Molina-Terriza, J. P. Torres, and L. Torner, “Management of the angular momentum of light: preparation of photons in multidimensional vector states of angular momentum,” Phys. Rev. Lett. 88, 013601 (2002). [CrossRef] [PubMed]

15.

J. Lin, X.-C. Yuan, J. Bu, H. L. Chen, Y. Y. Sun, and R. E. Burge, “Generalized model for orbital angular momentum states generated by parallel aligned phase wedges,” Opt. Lett. 32, 2170–2172 (2007). [CrossRef] [PubMed]

16.

J. Lin, X.-C. Yuan, J. Bu, B. P. S. Ahluwalia, Y. Y. Sun, and R. E. Burge, “Selective generation of high order optical vortices from a single phase wedge,” Opt. Lett. 32, 2927–2929 (2007). [CrossRef] [PubMed]

17.

J. W. Goodman, Introduction to Fourier optics, 2nd ed. (McGraw-Hill, Singapore, 1996).

18.

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

19.

W. C. Cheong, W. M. Lee, X.-C. Yuan, L.-S. Zhang, H. Wang, and K. Dholakia, “Direct electron-beam writing of continuous spiral phase plates in negative resist with high power efficiency for optical manipulation,” Appl. Phys. Lett. 85, 5784–5786 (2004). [CrossRef]

20.

G. Foo, D. M. Palacios, and G. A. Swartzlander, “Optical Vortex Coronagraph,” Opt. Lett. 30, 3308–3310 (2005). [CrossRef]

21.

G. A. Swartzlander, “Broadband nulling of a vortex phase mask,” Opt. Lett. 30, 2876–2878 (2005). [CrossRef] [PubMed]

OCIS Codes
(140.3300) Lasers and laser optics : Laser beam shaping
(350.5030) Other areas of optics : Phase

ToC Category:
Physical Optics

History
Original Manuscript: May 9, 2008
Revised Manuscript: June 28, 2008
Manuscript Accepted: August 5, 2008
Published: August 20, 2008

Citation
X.-C. Yuan, J. Lin, J. Bu, and R. E. Burge, "Achromatic design for the generation of optical vortices based on radial spiral phase plates," Opt. Express 16, 13599-13605 (2008)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-18-13599


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References

  1. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, "Orbital angular-momentum of light and the transformation of Laguerre-Gaussian laser modes," Phys. Rev. A 45, 8185-8189 (1992).
  2. M. V. Berry, "Coloured phase singularities," New J. Phys. 4, 66 (2002). [CrossRef]
  3. M. V. Berry, "Exploring the colours of dark light," New J. Phys. 4, 74 (2002). [CrossRef]
  4. J. Leach and M. J. Padgett, "Observation of chromatic effects near a white-light vortex," New J. Phys. 5, 154 (2003). [CrossRef]
  5. M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, "Helical-wavefront laser beams produced with a spiral phaseplate," Opt. Commun. 112, 321-327 (1994). [CrossRef]
  6. G. A. Swartzlander, "Achromatic optical vortex lens," Opt. Lett. 31, 2042-2044 (2006). [CrossRef] [PubMed]
  7. J. Lin, X.-C. Yuan, S. H. Tao, and R. E. Burge, "Variable-radius focused optical vortex with suppressed sidelobes," Opt. Lett. 31, 1600-1602 (2006). [CrossRef] [PubMed]
  8. V. V. Kotlyar, A. A. Kovalev, V. A. Soifer, C. S. Tuvey, and J. A. Davis, "Sidelobe contrast reduction for optical vortex beams using a helical axicon," Opt. Lett. 32, 921-923 (2007). [CrossRef] [PubMed]
  9. A. Vasara, J. Turunen, and A. T. Friberg, "Realization of general nondiffracting beams with computer-generated holograms," J. Opt. Soc. Am. A 6, 1748-1754 (1989). [CrossRef]
  10. S. H. Tao, W. M. Lee, and X.-C. Yuan, "Dynamic optical manipulation using higher order fractional Bessel beam generated from a spatial light modulator," Opt. Lett. 28, 1867-1869 (2003). [CrossRef] [PubMed]
  11. M. Rioux, R. Tremblay, and P. A. Belanger, "Linear, annular, and radial focusing with axicons and applications to laser machining," Appl. Opt. 17, 1532-1536 (1978). [CrossRef] [PubMed]
  12. T. Shiina, K. Yoshida, M. Ito, and Y. Okamura, "In-line type micropulse lidar with an annular beam: theoretical approach," Appl. Opt. 44, 7467-7474 (2005). [CrossRef] [PubMed]
  13. C. Alonzo, P. J. Rodrigo, and J. Glückstad, "Helico-conical optical beams: a product of helical and conical phase fronts," Opt. Express 13, 1749-1760 (2005).
  14. G. Molina-Terriza, J. P. Torres, and L. Torner, "Management of the angular momentum of light: preparation of photons in multidimensional vector states of angular momentum," Phys. Rev. Lett. 88, 013601 (2002). [CrossRef] [PubMed]
  15. J. Lin, X.-C. Yuan, J. Bu, H. L. Chen, Y. Y. Sun, and R. E. Burge, "Generalized model for orbital angular momentum states generated by parallel aligned phase wedges," Opt. Lett. 32, 2170-2172 (2007). [CrossRef] [PubMed]
  16. J. Lin, X.-C. Yuan, J. Bu, B. P. S. Ahluwalia, Y. Y. Sun, and R. E. Burge, "Selective generation of high-order optical vortices from a single phase wedge," Opt. Lett. 32, 2927-2929 (2007). [CrossRef] [PubMed]
  17. J. W. Goodman, Introduction to Fourier optics, 2nd ed. (McGraw-Hill, Singapore, 1996).
  18. J. Durnin, J. J. Miceli, and J. H. Eberly, "Diffraction-free beams," Phys. Rev. Lett. 58, 1499-1501 (1987). [CrossRef] [PubMed]
  19. W. C. Cheong, W. M. Lee, X.-C. Yuan, L.-S. Zhang, H. Wang and K. Dholakia, "Direct electron-beam writing of continuous spiral phase plates in negative resist with high power efficiency for optical manipulation," Appl. Phys. Lett. 85, 5784-5786 (2004). [CrossRef]
  20. G.  Foo, D. M.  Palacios, and G. A.  Swartzlander, "Optical Vortex Coronagraph," Opt. Lett.  30, 3308-3310 (2005). [CrossRef]
  21. G. A. Swartzlander, "Broadband nulling of a vortex phase mask," Opt. Lett. 30, 2876-2878 (2005). [CrossRef] [PubMed]

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