OSA's Digital Library

Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 8 — Apr. 12, 2010
  • pp: 8251–8260
« Show journal navigation

Near-diffraction-limited annular flattop beam shaping with dual phase only liquid crystal spatial light modulators

Haotong Ma, Pu Zhou, Xiaolin Wang, Yanxing Ma, Fengjie Xi, Xiaojun Xu, and Zejin Liu  »View Author Affiliations


Optics Express, Vol. 18, Issue 8, pp. 8251-8260 (2010)
http://dx.doi.org/10.1364/OE.18.008251


View Full Text Article

Acrobat PDF (1945 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

Abstract

We demonstrate the annular flattop beam shaping technique with dual phase only liquid crystal spatial light modulators (LC-SLM) based on the refractive laser beam shaping systems. One LC-SLM redistributes the intensity distribution, and the other restores the initial underlying wave front. Differing from the conventional annular beam shaping technique, the wave front of the output beam can be maintained. The influences of deviations of beam waist and beam shape on the output beam profile are discussed in detail. Experimental results show that approximate 71% of the power is enclosed in a region with less than 7% rms intensity variation. The 4.1mm diameter near-diffraction-limited beam retains an annular flattop intensity distribution without significant diffraction peaks for a working distance of more than 24cm in the near field.

© 2010 OSA

1. Introduction

Recently, the annular beams with flattop and other intensity distributions have attracted more and more attentions in many applications. Annular beams with plane wave front show great prospect in the laser transmitting systems, such as laser communication and laser tracking [1

1. Y. Baykal, “Log-amplitude and phase fluctuations of higher-order annular laser beams in a turbulent medium,” J. Opt. Soc. Am. A 22(4), 672–679 (2005). [CrossRef]

12

12. Y. Chen, Y. Cai, H. T. Eyyuboğlu, and Y. Baykal, “Scintillation properties of dark hollow beams in a weak turbulent atmosphere,” Appl. Phys. B 90(1), 87–92 (2008). [CrossRef]

]. Many techniques are proposed to generate annular beams, such as astigmatic mode conversion, wedge prisms, holey fiber, and multimode fiber [13

13. Ya. Izdebskaya, V. Shvedov, and A. Volyar, “Focusing of wedge-generated higher-order optical vortices,” Opt. Lett. 30(19), 2530–2532 (2005). [CrossRef] [PubMed]

15

15. C. Zhao, Y. Cai, F. Wang, X. Lu, and Y. Wang, “Generation of a high-quality partially coherent dark hollow beam with a multimode fiber,” Opt. Lett. 33(12), 1389–1391 (2008). [CrossRef] [PubMed]

]. Because of their high conversion efficiency and simple design procedure, the refractive shaping systems for generation of flattop beam and Bessel beam have been widely investigated [16

16. J. A. Hoffnagle and C. M. Jefferson, “Design and performance of a refractive optical system that converts a Gaussian to a flattop beam,” Appl. Opt. 39(30), 5488–5499 (2000). [CrossRef]

18

18. M. Arif, M. M. Hossain, A. A. S. Awwal, and M. N. Islam, “Two-element refracting system for annular Gaussian-to-Bessel beam transformation,” Appl. Opt. 37(19), 4206–4209 (1998). [CrossRef]

], however, to the best of our knowledge, the experimental generation of annular flattop beam with refractive optical elements has never been reported.

The refractive shaping system and other shaping techniques mentioned above are based on the transformation of specific input and output beam profiles, so the systems can only work well for the single input-output combination [19

19. J. Liang, R. N. Kohn Jr, M. F. Becker, and D. J. Heinzen, “1.5% root-mean-square flat-intensity laser beam formed using a binary-amplitude spatial light modulator,” Appl. Opt. 48(10), 1955–1962 (2009). [CrossRef] [PubMed]

]. Diffractive optical elements have been widely used in laser beam shaping [20

20. J. H. Li, K. J. Webb, G. J. Burke, D. A. White, and C. A. Thompson, “Design of near-field irregular diffractive optical elements by use of a multiresolution direct binary search method,” Opt. Lett. 31(9), 1181–1183 (2006). [CrossRef] [PubMed]

,21

21. G. Zhou, X. Yuan, P. Dowd, Y. L. Lam, and Y. C. Chan, “Design of diffractive phase elements for beam shaping: hybrid approach,” J. Opt. Soc. Am. A 18(4), 791–800 (2001). [CrossRef]

]. As one kind of diffractive optical elements, phase only LC-SLM stands out as an ideal candidate for laser beam shaping because of their programmable controller and high conversion efficiency. D. McGloin and D. P. Rhodes report the applications of LC-SLM in atom optics [3

3. D. P. Rhodes, D. M. Gherardi, J. Livesey, D. Mcgloin, H. Melville, T. Freegarde, and K. Dholakia, “Atom guiding along high order Laguerre–Gaussian light beams formed by spatial light modulation,” J. Mod. Opt. 53(4), 547–556 (2006). [CrossRef]

,4

4. D. McGloin, G. C. Spalding, H. Melville, W. Sibbett, and K. Dholakia, “Applications of spatial light modulators in atom optics,” Opt. Express 11(2), 158–166 (2003). [CrossRef] [PubMed]

]. In their experiments, the single LC-SLM is used to redistribute the intensity. The target annular and other intensity distributions are realized near the desired location, however, the output beam has not been re-collimated, so energy density at the far field and beam quality would be prominently degraded. In this paper, we report the conversion of quasi-Gaussian beam into near-diffraction-limited annular flattop beam by dual phase only LC-SLMs. The working principle is based on the refractive shaping system. The phase distributions of the LC-SLMs can be derived from the surface distributions of the aspheric lenses. The shaping system combines the advantages of programmable controller and simple design procedure. The target annular beam is defined as the difference between two Fermi-Dirac profiles. One phase only LC-SLM redistributes the intensity and the other restores the initial wave front. The annular flattop intensity distributions can be maintained for a certain distance without significant diffraction ripples.

This paper is organized as follows. In the second section, the working principle is introduced and some numerical simulations are given to prove the validity of the technique. The third section reports the experimental results of transformation of quasi-Gaussian beam into annular flattop beam. In the forth section, the conclusions are given.

2. Working principle and numerical simulations

In this paper, the beam shaping with dual phase only LC-SLMs is based on the refractive shaping system. Figure 1
Fig. 1 Optical configurations of the refractive beam shaping system. (a) Galilean type; (b) Keplerian type.
shows an overview of the basic optical configurations of refractive laser beam shaping system. For each case, a ray parallel to optical axis enters the first aspheric lens at arbitrary radial position r, emerges from the second aspheric lens at radial position R. The output beam is also parallel to optical axis. The terms z(r) and Z(R) represent the first and second surface profiles, respectively. The system must meet two requirements: (1). energy conservation, the total energy remains constant from the input to the output; (2). equal optical path, all rays passing through the system from input plane to output plane have zero optical path difference (OPD).

The surface distributions of the aspheric lenses can be translated into phase distributions based on the paraxial approximation in Fourier optics. The phase distributions of the Galilean shaping system can be approximately expressed as
fphase1(r)=2π[zedgez(r)+nz(r)]λ
(1)
and
fphase2(R)=2π[nZedgenZ(R)+Z(R)]λ,
(2)
where z(r) and Z(R) denote the sags of the first and second aspheric lenses, respectively, which can be numerically derived according to the treatment of Hoffnagle and Kreuzer in Ref [16

16. J. A. Hoffnagle and C. M. Jefferson, “Design and performance of a refractive optical system that converts a Gaussian to a flattop beam,” Appl. Opt. 39(30), 5488–5499 (2000). [CrossRef]

]. and Ref [22

22. J. L. Kreuzer, “Coherent light optical system yielding an output beam of desired intensity distribution at a desired equiphase surface,” U.S. patent 3,476,463, 1969.

]. zedge and Zedge denote the edge sags of the first and second aspheric lenses. λ represents the wavelength and n represents the refractive index of the shaping system. Loading the phase distributions onto the input beam with phase only LC-SLMs, the desired intensity distribution with plane wave front can be realized.

The refractive shaping system is based on the awareness of the input and output beam profiles. It is important to study the shaping effect and influences of the deviations of the input beam. The beam shaping may be influenced by many factors, including changes of beam waist, beam shape and optical alignment errors, etc. In this paper, only the input beam with deviation of beam waist and beam shape is considered. In simulation, the intensity distribution of the input beam is Gaussian profile and is given as
P(r)=exp(2r2/w2),
(3)
where w is the beam waist. The target intensity distribution is defined as the difference between two coaxial Fermi-Dirac profiles and can be expressed as
Pannular(r)={1+exp[β1(rR11)]}1{1+exp[β2(rR21)]}1,
(4)
where β1 and β2 are dimensionless parameters, which determine the range of the intensity rolls off exponentially, R1and R2 are the radius at which the intensity have fallen to half of their values on axis. The Gaussian beam with w = 3mm and the annular flattop beam with β1= 20, β2= 9, R1= 3.3mm, and R2= R1/3 are chosen as the input and output beam profiles. The refractive index is n = 1.45 and the distance between two aspheric lenses is 300mm. The surface and phase distributions of the aspheric lenses can be numerically derived by varying the radial position. Based on the Kirchhoff’s theory, we calculate the intensity and phase distributions of the beam after passing through the shaping system. The results are shown in Figs. 2(a)
Fig. 2 Intensity and phase distributions: (a) output beam after passing through the shaping system and its gray scale image (inset); (b) corresponding phase distribution and its gray scale image (inset); (c) corresponding far field intensity distribution and its gray scale image (inset).
and 2(b).

It can be found that the annular flattop beam with nearly plane wave front is realized. To study the far field intensity distribution, we use the lens with focal length 1.2m to focus the output beam. The intensity distribution of the output beam at the focal plane is shown in Fig. 2(c). The far field intensity distribution of the output beam exhibits airy disk pattern. Figure 3(a)
Fig. 3 Intensity and phase distributions of the output beam for the incident beam with beam waist 2.7mm, 3mm, 3.3mm: (a) Intensity distributions; (b) corresponding phase distributions.
shows intensity distributions of the output beam corresponding to the input beam with different beam waists. It is found that the intensity variation of the output beam becomes larger with the increase of the deviation between the beam waist and the assumed value. The difference between the intensity at the outside edge and inside edge is more than 20% at the condition of deviation reaching to 0.3mm.

A Gaussian profile is usually used to represent the intensity distribution of the laser beam. Mostly, the practical intensity distribution of the input beam is not the ideal Gaussian profile. In simulation, we use the Gaussian profile combination to expand the input beam.
Pinput(r)=ieiexp(2r2wi2),
(5)
where wi and ei are the beam waist and coefficient of the Gaussian beam. Figure 4(a)
Fig. 4 Intensity and phase distributions of the output beam for the input beam with combination of Gaussian profiles from Eq. (5), e1= 0.9, e2= 0.1: (a) Intensity distributions; (b) corresponding phase distributions.
shows the intensity distribution of the output beam corresponding to the incident beam with different Gaussian profile combination. It is found that when the deviation of the intensity distribution from the assumed profile increases, the fluctuation of the intensity distribution of the output beam becomes strong.

The corresponding phase distributions of the output beam are shown in Fig. 3(b) and Fig. 4(b). It is recognized that the phase distributions of the deviation beams are maintained after passing through the shaping system. At the condition that beam waist and shape deviates from the assumed value, the intensity variation of the output beam can be smoothed to a certain extent by changing the distance between two aspheric lenses. The result is shown in Fig. 5(a)
Fig. 5 Intensity and phase distributions of the output beam by changing the distance between two aspheric lenses. (a) Intensity distributions; (b) corresponding phase distributions.
, however, the deviations make the phase distributions worse [see Fig. 5(b)].

From Figs. 35, it can be found that the shaping system constituted by aspheric lenses can only work well for the assumed input and output beam profile. To overcome the shortcomings, we used the phase only LC-SLMs as substitutes for aspheric lenses. The phase only LC-SLMs have the advantage of programmable controller and can be used to generate target beam profile according to the incident beam profile.

3. Experimental realization

We applied the design procedure described above to the problem of transforming a quasi-Gaussian beam into an annular flattop beam. The experimental setup is shown schematically in Fig. 6
Fig. 6 Experimental setup for generation of annular flattop laser beam.
. The essential instruments of this technique are the phase only LC-SLMs, which are the products of the BNS Company. In this paper, the LC-SLM1 with 512 × 512, 15 × 15 um2 pixels is used to redistribute the intensity distribution of the input beam and the LC-SLM2 with 256 × 256, 24 × 24 um2 pixels is used to restore the initial wave front.

It is recognized that intensity distribution of the input beam shown in Fig. 7
Fig. 7 Cross section of the input quasi-Gaussian beam and its gray-scale image (inset). The solid line shows the intensity distribution of the input beam and the dashed line shows the fitting result with Gaussian profile expansion from Eq. (5).
is not the ideal Gaussian profile. Nevertheless, we can expand the input beam with the combination of Gaussian profile, which is shown in Eq. (5). The result of Gaussian profile combination expansion is shown in Fig. 7. The working principle and performance of Keplerian beam shaping system is similar to Galilean shaping system. In this paper, we just study Galilean shaping system. According to the fitting line of the input beam and the design principle, surface distributions of aspheric lens 1 and aspheric lens 2 are calculated and shown in Fig. 8(a)
Fig. 8 Surface and phase distributions of the Galilean shaping system: (a) Surface distributions; (b) phase distributions.
. The target intensity distribution is annular flattop laser beam with parameters β1= 20, β2 = 9, R1= 1.8mm, and R2= R1/3. The refractive index of the shaping system is n = 1.45.

According to the relationship between surface and phase distribution in Eq. (1), we can obtain the phase distributions of the shaping system, which is shown in Fig. 8(b). LC-SLMs have a finite aperture, requiring that the design should be truncated at some point. According to the dimensions of LC-SLMs, clear apertures of radius 3.84mm and 3.07mm are chosen for input and output dimensions. The phase only LC-SLMs imprint the calculated phase profile onto wave front of the quasi-Gaussian beam. The intensity distribution of the output beam measured by CCD 1 is shown in Fig. 9
Fig. 9 Cross section of the output annular flattop beam and its gray-scale image (inset).
. The distance between CCD 1 and LC-SLM 2 is approximately 6cm, which is nearly the shortest distance that the mechanical housing of LC-SLM 2 and CCD 1 allowed.

The uniformity and efficiency are used to evaluate the output beam. We define an annular region of interest centered on the beam axis to compute the relative intensity variance and efficiency. The relative intensity variance (rms) and efficiency of the output beam are given by
SE={2r22r12r1r2[Pannular(x)2r22r12r1r2Pannular(x)xdx]2xdx}122r22r12r1r2Pannular(x)xdx
(6)
and
η=2πr1r2Pannular(x)xdxWtotal,
(7)
where r2 and r1 are the outer and inner radius of the region, SE is the relative intensity rms variation, η is the efficiency which is defined as the ratio of the power in the region with outer radius r2 and inner radius r1 to the total power. Wtotal is the total power of the beam. From the data shown in Fig. 9, we deduce the relation between variance and the efficiency by varying the region. Figure 10(a)
Fig. 10 Relative rms variation and power-in-the-bucket curves: (a) Measured dependence of the relative rms variation of the output intensity on efficiency compared with the theoretical value; (b) power-in-the-bucket curves before and after re-collimating and their gray-scale images (inset).
shows the measured intensity variation and the results of ideal annular flattop beam.

From Fig. 10(a), it is found that approximately 71% of the power is enclosed in the region with less than 7% rms intensity variation. In our experiment, the total power is chosen as the power captured by CCD1 before shaping, the output power is chosen as the power captured by the CCD1 after shaping. This definition excludes the energy loss caused by the shaping system. From Fig. 10(a), it is found that there exist errors between the output annular flattop beam and the ideal annular flattop beam. The errors are mainly introduced by the CCD noise and the deviation of the Gaussian profile combination from the real intensity distribution of the incident beam. To study the far field intensity distribution of the output beam, we use the lens to focus the output beam. As illustrated in Fig. 10(b), the far field profile of the output beam exhibits the Airy disk pattern, which is expected from an annular flattop profile shown in Fig. 2(c). The power in the bucket (PIB) curves of the far field intensity distribution before and after being re-collimated by the LC-SLM 2 are also shown in Fig. 10(b). After being re-collimated by LC-SLM 2, the energy in the airy disk area is seventeen times larger than that before being re-collimated.

We move CCD 2 along the optical axis to capture the intensity distribution of the output beam at different position. The results are shown in Figs. 11(a)
Fig. 11 Intensity distribution of the output beam after propagation in the near field: (a) at 12cm from the LC-SLM 2; (b) at 18cm from the LC-SLM 2; (c) at 24cm from the LC-SLM 2.
, 11(b) and 11(c). As illustrated in Fig. 11, the intensity profile remains a useful annular flattop shape without significant diffraction peaks for a distance of more than 24cm.

4. Conclusion

We have demonstrated a system consisting of two phase only LC-SLMs for generating a near-diffraction-limited annular flattop beam from quasi-Gaussian beam input. The working principle is based on the refractive shaping system. The shaping effect and the deviation of the input beam from the assumed profile are analyzed in detail. Experimental results show that approximately 71% of the power is enclosed in a region with less than 7% rms intensity variation. The annular flattop beam retains a useful flat-top intensity distribution without significant diffraction peaks for a working distance of more than 24cm. Compared with other beam shaping techniques, this technique does not need complex phase retrieval algorithm. This technique provides a convenient and powerful way to translate central symmetric intensity distribution to annular flattop intensity distributions, which can be used in many applications.

References and links

1.

Y. Baykal, “Log-amplitude and phase fluctuations of higher-order annular laser beams in a turbulent medium,” J. Opt. Soc. Am. A 22(4), 672–679 (2005). [CrossRef]

2.

H. T. Eyyuboglu and Y. Baykal, “Scintillations of cos-Gaussian and annular beams,” J. Opt. Soc. Am. A 24(1), 56–162 (2007).

3.

D. P. Rhodes, D. M. Gherardi, J. Livesey, D. Mcgloin, H. Melville, T. Freegarde, and K. Dholakia, “Atom guiding along high order Laguerre–Gaussian light beams formed by spatial light modulation,” J. Mod. Opt. 53(4), 547–556 (2006). [CrossRef]

4.

D. McGloin, G. C. Spalding, H. Melville, W. Sibbett, and K. Dholakia, “Applications of spatial light modulators in atom optics,” Opt. Express 11(2), 158–166 (2003). [CrossRef] [PubMed]

5.

S. Marksteiner, C. M. Savage, P. Zoller, and S. L. Rolston, “Coherent atomic waveguides from hollow optical fibers: Quantized atomic motion,” Phys. Rev. A 50(3), 2680–2690 (1994). [CrossRef] [PubMed]

6.

M. J. Renn, D. Montgomery, O. Vdovin, D. Z. Anderson, C. E. Wieman, and E. A. Cornell, “Laser-guided atoms in hollow-core optical fibers,” Phys. Rev. Lett. 75(18), 3253–3256 (1995). [CrossRef] [PubMed]

7.

Y. Cai, Z. Wang, and Q. Lin, “An alternative theoretical model for an anomalous hollow beam,” Opt. Express 16(19), 15254–15267 (2008). [CrossRef] [PubMed]

8.

Y. J. Cai and S. L. He, “Propagation of various dark hollow beams in a turbulent atmosphere,” Opt. Express 14(4), 1353–1367 (2006). [CrossRef] [PubMed]

9.

X. Chu and G. Zhou, “Power coupling of a two-Cassegrain-telescopes system in turbulent atmosphere in a slant path,” Opt. Express 15(12), 7697–7707 (2007). [CrossRef] [PubMed]

10.

P. G. Hannan, “General analysis of two-mirror relay systems,” Appl. Opt. 31(4), 513–518 (1992). [CrossRef] [PubMed]

11.

Y. Yuan, Y. Cai, J. Qu, H. T. Eyyuboğlu, Y. Baykal, and O. Korotkova, “M2-factor of coherent and partially coherent dark hollow beams propagating in turbulent atmosphere,” Opt. Express 17(20), 17344–17356 (2009). [CrossRef] [PubMed]

12.

Y. Chen, Y. Cai, H. T. Eyyuboğlu, and Y. Baykal, “Scintillation properties of dark hollow beams in a weak turbulent atmosphere,” Appl. Phys. B 90(1), 87–92 (2008). [CrossRef]

13.

Ya. Izdebskaya, V. Shvedov, and A. Volyar, “Focusing of wedge-generated higher-order optical vortices,” Opt. Lett. 30(19), 2530–2532 (2005). [CrossRef] [PubMed]

14.

M. L. Hu, C. Y. Wang, Y. J. Song, Y. F. Li, L. Chai, E. E. Serebryannikov, and A. M. Zheltikov, “A hollow beam from a holey fiber,” Opt. Express 14(9), 4128–4134 (2006). [CrossRef] [PubMed]

15.

C. Zhao, Y. Cai, F. Wang, X. Lu, and Y. Wang, “Generation of a high-quality partially coherent dark hollow beam with a multimode fiber,” Opt. Lett. 33(12), 1389–1391 (2008). [CrossRef] [PubMed]

16.

J. A. Hoffnagle and C. M. Jefferson, “Design and performance of a refractive optical system that converts a Gaussian to a flattop beam,” Appl. Opt. 39(30), 5488–5499 (2000). [CrossRef]

17.

C. Liu and S. Zhang, “Study of singular radius and surface boundary constraints in refractive beam shaper design,” Opt. Express 16(9), 6675–6682 (2008). [CrossRef] [PubMed]

18.

M. Arif, M. M. Hossain, A. A. S. Awwal, and M. N. Islam, “Two-element refracting system for annular Gaussian-to-Bessel beam transformation,” Appl. Opt. 37(19), 4206–4209 (1998). [CrossRef]

19.

J. Liang, R. N. Kohn Jr, M. F. Becker, and D. J. Heinzen, “1.5% root-mean-square flat-intensity laser beam formed using a binary-amplitude spatial light modulator,” Appl. Opt. 48(10), 1955–1962 (2009). [CrossRef] [PubMed]

20.

J. H. Li, K. J. Webb, G. J. Burke, D. A. White, and C. A. Thompson, “Design of near-field irregular diffractive optical elements by use of a multiresolution direct binary search method,” Opt. Lett. 31(9), 1181–1183 (2006). [CrossRef] [PubMed]

21.

G. Zhou, X. Yuan, P. Dowd, Y. L. Lam, and Y. C. Chan, “Design of diffractive phase elements for beam shaping: hybrid approach,” J. Opt. Soc. Am. A 18(4), 791–800 (2001). [CrossRef]

22.

J. L. Kreuzer, “Coherent light optical system yielding an output beam of desired intensity distribution at a desired equiphase surface,” U.S. patent 3,476,463, 1969.

OCIS Codes
(120.5060) Instrumentation, measurement, and metrology : Phase modulation
(140.3300) Lasers and laser optics : Laser beam shaping
(220.2740) Optical design and fabrication : Geometric optical design
(230.6120) Optical devices : Spatial light modulators

ToC Category:
Lasers and Laser Optics

History
Original Manuscript: January 27, 2010
Revised Manuscript: March 13, 2010
Manuscript Accepted: March 28, 2010
Published: April 5, 2010

Citation
Haotong Ma, Pu Zhou, Xiaolin Wang, Yanxing Ma, Fengjie Xi, Xiaojun Xu, and Zejin Liu, "Near-diffraction-limited annular flattop beam shaping with dual phase only liquid crystal spatial light modulators," Opt. Express 18, 8251-8260 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-8-8251


Sort:  Author  |  Year  |  Journal  |  Reset  

References

  1. Y. Baykal, “Log-amplitude and phase fluctuations of higher-order annular laser beams in a turbulent medium,” J. Opt. Soc. Am. A 22(4), 672–679 (2005). [CrossRef]
  2. H. T. Eyyuboglu and Y. Baykal, “Scintillations of cos-Gaussian and annular beams,” J. Opt. Soc. Am. A 24(1), 56–162 (2007).
  3. D. P. Rhodes, D. M. Gherardi, J. Livesey, D. Mcgloin, H. Melville, T. Freegarde, and K. Dholakia, “Atom guiding along high order Laguerre–Gaussian light beams formed by spatial light modulation,” J. Mod. Opt. 53(4), 547–556 (2006). [CrossRef]
  4. D. McGloin, G. C. Spalding, H. Melville, W. Sibbett, and K. Dholakia, “Applications of spatial light modulators in atom optics,” Opt. Express 11(2), 158–166 (2003). [CrossRef] [PubMed]
  5. S. Marksteiner, C. M. Savage, P. Zoller, and S. L. Rolston, “Coherent atomic waveguides from hollow optical fibers: Quantized atomic motion,” Phys. Rev. A 50(3), 2680–2690 (1994). [CrossRef] [PubMed]
  6. M. J. Renn, D. Montgomery, O. Vdovin, D. Z. Anderson, C. E. Wieman, and E. A. Cornell, “Laser-guided atoms in hollow-core optical fibers,” Phys. Rev. Lett. 75(18), 3253–3256 (1995). [CrossRef] [PubMed]
  7. Y. Cai, Z. Wang, and Q. Lin, “An alternative theoretical model for an anomalous hollow beam,” Opt. Express 16(19), 15254–15267 (2008). [CrossRef] [PubMed]
  8. Y. J. Cai and S. L. He, “Propagation of various dark hollow beams in a turbulent atmosphere,” Opt. Express 14(4), 1353–1367 (2006). [CrossRef] [PubMed]
  9. X. Chu and G. Zhou, “Power coupling of a two-Cassegrain-telescopes system in turbulent atmosphere in a slant path,” Opt. Express 15(12), 7697–7707 (2007). [CrossRef] [PubMed]
  10. P. G. Hannan, “General analysis of two-mirror relay systems,” Appl. Opt. 31(4), 513–518 (1992). [CrossRef] [PubMed]
  11. Y. Yuan, Y. Cai, J. Qu, H. T. Eyyuboğlu, Y. Baykal, and O. Korotkova, “M2-factor of coherent and partially coherent dark hollow beams propagating in turbulent atmosphere,” Opt. Express 17(20), 17344–17356 (2009). [CrossRef] [PubMed]
  12. Y. Chen, Y. Cai, H. T. Eyyuboğlu, and Y. Baykal, “Scintillation properties of dark hollow beams in a weak turbulent atmosphere,” Appl. Phys. B 90(1), 87–92 (2008). [CrossRef]
  13. Ya. Izdebskaya, V. Shvedov, and A. Volyar, “Focusing of wedge-generated higher-order optical vortices,” Opt. Lett. 30(19), 2530–2532 (2005). [CrossRef] [PubMed]
  14. M. L. Hu, C. Y. Wang, Y. J. Song, Y. F. Li, L. Chai, E. E. Serebryannikov, and A. M. Zheltikov, “A hollow beam from a holey fiber,” Opt. Express 14(9), 4128–4134 (2006). [CrossRef] [PubMed]
  15. C. Zhao, Y. Cai, F. Wang, X. Lu, and Y. Wang, “Generation of a high-quality partially coherent dark hollow beam with a multimode fiber,” Opt. Lett. 33(12), 1389–1391 (2008). [CrossRef] [PubMed]
  16. J. A. Hoffnagle and C. M. Jefferson, “Design and performance of a refractive optical system that converts a Gaussian to a flattop beam,” Appl. Opt. 39(30), 5488–5499 (2000). [CrossRef]
  17. C. Liu and S. Zhang, “Study of singular radius and surface boundary constraints in refractive beam shaper design,” Opt. Express 16(9), 6675–6682 (2008). [CrossRef] [PubMed]
  18. M. Arif, M. M. Hossain, A. A. S. Awwal, and M. N. Islam, “Two-element refracting system for annular Gaussian-to-Bessel beam transformation,” Appl. Opt. 37(19), 4206–4209 (1998). [CrossRef]
  19. J. Liang, R. N. Kohn, M. F. Becker, and D. J. Heinzen, “1.5% root-mean-square flat-intensity laser beam formed using a binary-amplitude spatial light modulator,” Appl. Opt. 48(10), 1955–1962 (2009). [CrossRef] [PubMed]
  20. J. H. Li, K. J. Webb, G. J. Burke, D. A. White, and C. A. Thompson, “Design of near-field irregular diffractive optical elements by use of a multiresolution direct binary search method,” Opt. Lett. 31(9), 1181–1183 (2006). [CrossRef] [PubMed]
  21. G. Zhou, X. Yuan, P. Dowd, Y. L. Lam, and Y. C. Chan, “Design of diffractive phase elements for beam shaping: hybrid approach,” J. Opt. Soc. Am. A 18(4), 791–800 (2001). [CrossRef]
  22. J. L. Kreuzer, “Coherent light optical system yielding an output beam of desired intensity distribution at a desired equiphase surface,” U.S. patent 3,476,463, 1969.

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.

CrossCheck Deposited