Generation of Laguerre–Gaussian LGp0 beams using binary phase diffractive optical elements Abdelhalim Bencheikh,1,2,* Michael Fromager,3 and Kamel Aït Ameur3,4 1
Département sciences et techniques, Faculté des sciences et des technologies, Université de Bordj Bou Arréridj, 34000 Bordj Bou Arréridj, Algeria
2
Laboratoire d’optique appliqué, Institut d’optique et mécanique de precision, Université de Sétif 1, 19000 Sétif, Algeria 3
Centre de Recherche sur les Ions, les Matériaux et la Photonique, UMR 6252-CEA-CNRS-ENSICAEN-Université de Caen, ENSICAEN, 6 Boulevard Maréchal Juin, F 14050 Caen, France 4 Centre de Développement des Techniques Avancées, Division Milieux Ionisés et Lasers, P.O. Box 17 Baba Hassan, 16303 Algiers, Algeria
*Corresponding author: [email protected] Received 28 January 2014; revised 19 May 2014; accepted 3 June 2014; posted 11 June 2014 (Doc. ID 205478); published 18 July 2014
In recent years, considerable attention has been devoted to laser beams with specific intensity profile, i.e., non-Gaussian. In this work, we present a novel technique to generate high-radial-order Laguerre– Gaussian beams LGp0 based on the use of a binary phase diffractive optical element (BPDOE). The latter is a phase plate made up of annular zones introducing alternatively a phase shift equal to 0 or π modeled on positions which do not coincide with the position of the zeros of the desired LGp0 beam. The LGp0 beams are obtained by transforming a fundamental Gaussian beam through an appropriate BPDOE. The design of the latter is based on the calculation of the Fresnel–Kirchhoff integral, and the diffracted intensity at the focus plane of a lens has been modeled analytically for the first time. The numerical simulations and experiment demonstrate a good beam quality transformation. Obtained LGp0 are suitable for atom trap and pumping solid state laser applications. © 2014 Optical Society of America OCIS codes: (140.3300) Laser beam shaping; (050.1970) Diffractive optics; (050.1380) Binary optics. http://dx.doi.org/10.1364/AO.53.004761
1. Introduction
Generally, it is desirable to force the oscillation of a laser in the fundamental Gaussian mode because it leads to the highest possible brightness for a given power [1]. As a consequence, high-order transverse modes are usually not taken into consideration. However, they present very promising properties which are illustrated through the three following examples. The first one concerns Laguerre–Gaussian modes LGpl with radial and azimuthal orders (respectively, p and l) which are used for high-precision 1559-128X/14/214761-07$15.00/0 © 2014 Optical Society of America
interferometry, and especially in detection of gravitational waves [2,3]. The second one allows the production of a bottle beam for p 0 [4] used to trap or guide cold atoms [5]. We recall that this beam (LG0l ) is also able to carry an orbital angular momentum, which is of big interest in atom optics and spiral phase imaging microscopy [6,7]. The third example is the use of LGp0 beams for focal volume reduction [8,9]. This technique is based on the transformation of the alternately out of phase rings of the incident LGp0 into a unified phase, which is achieved by a binary phase diffractive optical element (BPDOE) followed by a focusing lens. This operation corresponds to a kind of “rectification” of the wavefront, having the property to generate a single lobed 20 July 2014 / Vol. 53, No. 21 / APPLIED OPTICS
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2. Generating LGp0 Beam Using BPDOE
For a laser beam of power P propagating along the z axis, the magnitude of the electric field of a collimated LGpl mode (of radial order p and azimuthal order l) is given by [21] rs 2P p! −1p exp−ilϕ U pl ρ; ϕ 2 l p! πW p l 2 ρ2 2ρ 2ρ : Llp × exp − 2 W W W2
APPLIED OPTICS / Vol. 53, No. 21 / 20 July 2014
LG20 LG30
Transversal coordinate r(a.u.) Fig. 1. Radial intensity distribution of the pure LG10 , LG20 , and LG30 beams.
in the electric field and a well-defined angular momentum of lħ per photon [21]. In the case where l 0, the expression of the transverse distribution of the electrical field reduces to 2 ρ2 2ρ U p0 ρ; ϕ U 0 exp − 2 × Lp ; W W2
(2)
where U 0 is the on-axis field amplitude, W is the width of the beam waist, p is the order of the LGp0 beam, and Lp are the Laguerre polynomials. When l p 0, the beam is Gaussian. We note that high-order LGp0 beams are made up of a central peak surrounded by p rings of light, as shown in Fig. 1. A graphical representation of the first three Laguerre–Gaussian beams (LG10 , LG20 , and LG30 ) is presented in Fig. 1. Figure 2 shows the optical layout considered for the transformation of a fundamental Gaussian laser beam into a single higher-order LGp0 using simply a BPDOE. The observation plane is the focal plane of the focusing lens. The BPDOE is a phase plate made up of annular zones introducing a phase shift equal to 0 or π (Fig. 3). It is worthwhile to know that the intensity in the focal plane is very sensitive to phase discontinuity
(1)
The exp−ilϕ term identifies this beam as a vortex state with a quantized 2πl azimuthal phase change 4762
LG10
Normalized intensity
intensity profile in the focal plane. The latter is characterized by a focal volume which can be strongly reduced in comparison with the one resulting from the focusing of a fundamental Gaussian beam. It is worthwhile to recall that the focal volume is the area where the intensity can be considered as high, and lies at the very heart of the spatial resolution of the used optical beam. So the BPDOE can be used for both LGp0 beam generation (when illuminated with a fundamental Gaussian beam) and wave rectification (when entered with an LGp0 beam). While LG beams can be produced inside the laser cavity using diffractive optical elements [10,11], the use of laser diodes or monomode fiber requires external cavity conversion from fundamental Gaussian to LG beams. Previously demonstrated external cavity methods include spiral phase plates [12,13] and computergenerated holograms (CGH) [14]. Spiral phase plates are transmission optics whose thickness varies to create an azimuthally dependent phase delay. This can convert a fundamental Gaussian beam into an LGpl beam, as well as convert between any two LGpl modes [15,16]. A more common method uses computer-generated holographic gratings displayed on spatial light modulators [17–20]. These CGH are recordings of the theoretical interference pattern between the electric field of the LG mode of interest and a reference field, commonly a plane wave [14,15]. Techniques for producing LG beams are costly and need complicated diffractive optical elements. In this work, we propose a low-cost alternative technique for producing LGp0 beams. Indeed, a BPDOE needs only one level of etching, while a diffractive optics made up of continuous phase relief usually can need more than 64 etching levels. The technique consists of transforming a fundamental Gaussian beam into an LGp0 beam using a simple BPDOE, which is made up of annular zones introducing a phase shift corresponding alternatively to 0 or π. The design of the BPDOE is based on the calculation of the Fresnel–Kirchhoff integral, which is different from all techniques based on holographic reconstruction. In addition, an elegant analytical expression is derived for the transverse intensity distribution in the focal plane of the obtained LGp0 beam.
Fig. 2. Optical layout.
Er
9 8 p N P N P P > > a2m R2km1 kmi1 2m i > = −1 r > 4m W 2k m!2 k!2k2m2 2π < k0 m0 i1
i h λf > > : −1p 1 exp − a2 W 2 r2 4 2
> > ;
;
(5) Fig. 3. Designed BPDOE containing five phase discontinuities.
positions of the BPDOE, and also the number of discontinuities. One can intuitively but wrongly set the phase discontinuity positions at the same place as the zeros of the Laguerre polynomial Lp given in Table 1 [8,9]. It will be shown later that the best positions of the phase discontinuities, allowing the transformation of a fundamental Gaussian beam into an intensity distribution in the focal plane which is of the LGp0 beam, are actually distinct from the zeros of Lp. The field distribution of the input collimated fundamental Gaussian beam Ein can be written as ρ2 : Ein ρ E0 exp W2
where a 2π∕λf . Consequently, the corresponding transverse distribution of the intensity is Ir jErj2. The more explicit form for the intensity distribution takes the form Ir
2 N N p π 2m 2km1 2m 2 P P P −1kmi1 2kλf 2 Ri r W m! k!2k2m2 2π k0 m0 i1 ; h 2 i λf r2 −1p 12 exp − πW λf (6)
setting 2m
(3)
The transmittance τρ of the binary phase DOE is alternatively equal to −1 or 1, and the radial positions of the phase discontinuities are different from positions of the Laguerre polynomial zeros. There is an interesting case when the BPDOE shows only one phase discontinuity; it takes the shape of a phase aperture, which shows the interesting capabilities of laser beam shaping in the free propagation [22] and intracavity [23] cases. The field distribution in the focal plane of the lens of an input fundamental Gaussian beam given by Eq. (3) is described by the Fresnel–Kirchhoff integral [8,9] 2π Er; z f λf
Z
2π rρ ρdρ; τρEin ρJ 0 λf DOE
p 1 2 3 4 5
(4)
Roots of Laguerre Polynomials Lp ρi ∕W
Values of Ratio ρi ∕W for the Zeros of Intensity of LGp0 Mode 0.707106 0.541195 0.455946 0.401589 0.363015
1.306562 1.071046 0.934280 0.840041
1.773407 1.506090 1.340975
2.167379 1.882260
π λf
2m!2 k!k m 1
:
(7)
Finally, we reach an elegant expression of radial intensity of a fundamental Gaussian beam with an incident waist W, diffracted by an annular BPDOE and focused by the lens as 2 N P p N P R2km1 2 P 2m i gk; m; i r 2k 2π W ; (8) k0 m0 i1 Ir i h λf −1p πW 2 2 2 exp − λf r
where Ri : is the radius of the ring i. 3. Results and Discussion
where ρ and r are the coordinates in the plane of the phase element and the focus, respectively. After a tedious calculus, we finally find the analytical expression of the field at the focal plane (Appendix A) given by
Table 1.
gk; m; i −1kmi1
2.51040
The calculus presented in Section 2 shows that generating LGp0 only requires p phase discontinuities whose positions (scaled radial variable ρ∕W) are given in Table 2. By applying the novelly obtained formula in Eq. (8), and using the wavelength of the incident laser beam λ 1064 nm, the input beam width is W 1 mm and we can construct all intensity distributions in the focus of the BPDOE. Two examples of obtained Laguerre–Gaussian beams LG10 and LG40 calculated in the focus plane using Eq. (8) are shown in Fig. 4. In order to appreciate the quality of the transformation of a unilobe fundamental Gaussian beam into a multilobed Laguerre–Gaussian beam LGp0 , we also plot in Fig. 4 the fit by an ideal LGp0 beam. We can notice from Fig. 4 that the calculated LGp0 beam is very close to the pure one (ideal LGp0 beam). We express the fit quality using two quantities: 20 July 2014 / Vol. 53, No. 21 / APPLIED OPTICS
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Table 2.
Empirical Phase Discontinuity Positions
p Phase Discontinuity Positions (Scaled Radial Variable) ρi ∕W 0.66 0.41 0.37 0.28
Intensity (a.u.)
Ideal LG 10 fit
Transverse coordinate
LG10 LG20 LG30 LG40
LG50
M2
3.02 3
11,10 11
for calculated LGp0 M 2 2p 1, for ideal LGp0
0.74 0.58 0.42
Calculated LG10
LGp0
0.89 0.58
Ideal LG 40 fit
z=f
z=f+5ZR z=f+7.5ZR z=f+10ZR
4764
LG10
LG20
LG30
LG40
LG50
0.9910 0.9731 0.9798 0.9886 0.9797 0.9736 0.9671 0.9591 0.9502
0.9781 0.9852 0.8863 0.9167 0.9659 0.9808 0.9809 0.9747 0.9699
0.9563 0.9691 0.7561 0.9681 0.9703 0.9732 0.9768 0.9790 0.9797
0.9227 0.9445 0.7732 0.9711 0.9721 0.9736 0.9671 0.9591 0.9502
0.9693 0.9662 0.9830 0.8241 0.9737 0.9732 0.9675 0.9722 0.9826
APPLIED OPTICS / Vol. 53, No. 21 / 20 July 2014
Intensitiy (a.u.)
Transverse coordinate(a.u.)
z=f z=f+2.5ZR z=f+5ZR z=f+7.5ZR z=f+10ZR z=f+12.5ZR
Transverse coordinate (a.u.)
Fig. 5. Transverse distribution of the intensity for LG10 and LG40 at different z positions along the propagation axis.
After plotting the values of Table 3 (Fig. 6), we have observed a little decrease in the quality of the calculated LG beams at the position z 2ZR for all beam orders p 0; 1; 2; 3; 4; 5. Also, we have 1.00
The correlation coefficient R²
Characteristics of the Quality of the Transformation at Distance z after the Focusing Lens
R2
50 51 52 53 54 55 56 57 58
8.73 9
z=f+2.5ZR
Transverse coordinate
– A statistical parameter noted R2 , which is the coefficient of determination. The more R2 approaches 1, the higher the quality of the beam shaping. Table 3 shows numerical results of calculated R2 for different calculated LGp0 corresponding to different positions z along the propagation axis. For R2 1, the beam is an ideal Laguerre–Gaussian mode. – The beam quality factor M 2 of the reshaped beam, calculated using the second-order moments method. A M 2 close to (2p 1) ensures a good transformation [8,9] of the fundamental Gaussian beam into an LGp0 beam (Table 4). To estimate how these beams keep their shape as they propagate, we have plotted the intensity distribution for calculated LG10 and LG40 for different axial position z. The results are shown in Fig. 5. The calculated Laguerre–Gaussian beams (LG10 and LG40 ) conserve their radial intensity distributions for at least 10 mm before and 10 mm after the lens, which corresponds to a beam shaping longitudinal range [24] more than 20ZR , which is important and sufficient, for instance, to pump a solid laser or to trap and guide microparticles. Note that ZR 0.8 mm is the Rayleigh distance of the focused Gaussian beam without any BPDOE.
zmm
6.79 7
Calculated LG 40
Fig. 4. Obtained Laguerre–Gaussian beams LG10 and LG40 , respectively.
Table 3.
5 5
0.85
Intensity (a.u.)
0.50 0.33 0.17 0.19 0.13
Intensity (a.u.)
1 2 3 4 5
Table 4. Calculated M 2 of the Obtained Laguerre–Gaussian Beam Based on the Second-Order Moments of the Intensity Distribution
0.95
0.90
LG10 LG20
0.85
LG30 LG40 LG50
0.80
0.75
50
52
54
56
58
z[mm]
Fig. 6. Variation of the parameter R2 as function of propagating distance z for calculated LG beams
observed that the quality of calculated higher-order LG beams is improved when the beam propagates far from the focal point of the used focusing lens. 4. Experimental Investigation
(a)
(b)
transverse coordinate
Generated LG20 Ideal LG20 fit
transverse coordinate
Generated LG40 Ideal LG40 fit
Transverse coordinate
Fig. 9. Fits of generated LG10 , LG20 , and LG40 beams.
fundamental Gaussian beam of a width W 1 mm. The obtained generated beams were observed and detected in the focal plane and along the propagation axis using a CCD camera (Thorlabs cam BC 106-vis). Images and profiles of generated beams are shown in Fig. 8. We have examined profiles of generated beams at the focal plane of the lens L5, using ideal LGp0 fits. As an example, we showed in Fig. 9 the intensity distribution of generated LG10 and LG20 and their ideal LG fits. The quality of the generated beams is quantized using the R2 coefficient of determination; results are given in Table 5. To examine the invariance of the transversal intensity distribution during propagation, we have moved the CCD camera along the z axis; the images obtained are shown in Fig. 10. For instance, we presented the generated LG10 and LG40 beams, and we have observed a large value of the beam-shaping longitudinal range, which can be more than 10ZR . Table 5.
Quality of the Transformation of Generated LG Beams in the Focal Plane of Lens L5
LG
LG10
LG20
LG30
LG40
LG50
0.9817
0.9673
0.9563
0.9390
0.9693
2
R
Fig. 7. (a) Experimental setup for investigating the transformation of a fundamental Gaussian beam into a Laguerre–Gaussian beam. (b) BPDOE patterns. (a,b) Phase patterns of the designed BPDOE without grating used for generating LG10 and LG40 beams, respectively. (c,d) Corresponding phase patterns of the designed (BPDOE + grating).
Normalized intensity
Generated LG10 Ideal LG10 fit
Normalized Intensity distribution
Fig. 8. Generated Laguerre–Gaussian beam in the focal plane of a lens where, from left to right, we show increasing radial orders from p 1 to p 5.
Normalized intensity
To experimentally verify the results obtained in Section 3, we used the setup shown in Fig. 7. It consists of a laser source operating at 1064 nm delivering a fundamental Gaussian beam. This beam was enlarged and collimated through a telescope and propagated onto a reflective phase-only liquid crystal on silicon (SLM: Holoeye Pluto-NIR). We recall that, for the sake of flexibility, in this experiment we have used the SLM just to simulate the different BPDOE designed above. The SLM is calibrated for a 2π phase shift at λ 1064 nm, and was programmed with a grayscale image containing the information about the BPDOE functions introduced in Fig. 3. By adding a phase grating pattern onto the BPDOE pattern, the output beam is diffracted into a different direction to a zeroth-order beam that propagates in the normal direction to the SLM surface [15]. The diffracted field from the latter was transformed by two lenses (L3 and L4), forming a 4f system in the back plane of the lens L5 of focal length f 750 mm. Only the diffracted first order was selected between L3 and L4 using a spatial filter; then we have imaged the designed BPDOE at 4f distance from the first plane of the used SLM. The latter diffracts the incident
Fig. 10. How the generated Laguerre–Gaussian beams LG10 and LG40 keep their shapes for several values of ZR 20 July 2014 / Vol. 53, No. 21 / APPLIED OPTICS
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be of interest in prototyping and machining processes [8,9]. Z=5ZR
Intensity
Intensity [a.u.]
Z=5ZR
Z=4ZR Z=3ZR Z=2ZR
Z=4ZR
Appendix A
Z=3ZR
In this appendix, we give details on the solution of the integral given in Eq. (4). The latter takes the following form, in the case of only one phase discontinuity (phase aperture) at the radius position R: 9 8 2 ρ > > 2π > > > >Z exp − 2 J 0 λf rρ ρdρ = 2π < R W : Er; z f 2 > R∞ λf > ρ 0 > > 2π > > − R exp − 2 J 0 λf rρ ρdρ ; : W (A1)
Z=2ZR Z=ZR
Z=ZR
Transverse coordinate
Transverse coordinate
Fig. 11. Transverse distribution of the intensity for generated LG10 and LG40 at different z positions along the propagation axis. Table 6. Characteristics of the Quality of Generated LG10 and LG40 Beams at Distance z after the Focusing Lens
R2 z mm Zf f ZR f 2ZR f ZR f 4ZR f 5ZR
Using the equality
LG10
LG40
0.9817 0.9710 0.9715 0.9715 0.9693 0.9743
0.9390 0.9495 0.6541 0.9430 0.9442 0.9628
It is obvious that the resulting beams keep their LGp0 form. However, along the propagation axis, we have observed some exchange of the energy content between the different rings forming the beam. By plotting profiles in Fig. 11 of images presented in Fig. 10 and fitting the different intensity distributions for LG10 and LG40 , values of R2 presented in Table 6 show the good quality of the generated beams.
Z
ρ2 exp − 2 J 0 arρrdr W R Z∞ ρ2 exp − 2 J 0 arρρdρ W 0 ZR ρ2 exp − 2 J 0 arρρdρ; − W 0
∞
(A2)
the integral in Eq. (A1) becomes 9 8 R r2 R > > 2 τ r exp − arρrdr J > > EOD 0 = W 2k 2π < 0 ; Eρ;z f > λf > R∞ r2 > > ; : − 0 exp − 2k J 0 arρrdr W (A3)
5. Conclusion
In summary, we have demonstrated an easy way to produce a Laguerre–Gaussian LGp0 beam by reshaping a fundamental Gaussian beam using a simple BPDOE, which is made up of annular zones introducing a phase shift equal to 0 or π modeled on positions different from the zeros of Laguerre polynomials. The concept of designing this BPDOE is based on the calculation of the Fresnel–Kirchhoff integral. For the sake of flexibility, we have used a phase-only spatial light modulator to implement the various designed BPDOEs. We have also derived an elegant analytical expression of the resulting beam’s intensity at the focal plane of the lens that follows the BPDOE, which allows for the optimization of the quality of the obtained beams. The numerical and experimental results show that the resulting beams are very close to ideal LGp0 . This approach for generating LGp0 beams offers many benefits over existing techniques (CGH, continuous phase elements, and spiral phase elements). Due to its simplicity, it is very easy to manufacture this type of designed BPDOE and it needs only one level of etching, and so it could 4766
APPLIED OPTICS / Vol. 53, No. 21 / 20 July 2014
setting Z E1 ρ
∞
0
2 r exp − 2 J 0 arρrdr W
(A4)
and Z E2 ρ 2
R 0
r2 τEOD r exp − 2k J 0 arρrdr: W
(A5)
For solving the first integral (A4), we use the wellknown equality [25] given by 2 a exp − p ∞ 4b exp−bxJ 0 a x dx : b 0
Z
(A6)
Finally, we get 1 W2 aρ2 : E1 ρ W 2 exp − 4 2
(A7)
For the second integral (A5), and using the series expression of Bessel and exponential functions
J 0 arρ
∞ X −1m arρ∕22m 2
m!
m0
∞ X −1m ρ2m ar2m 2m
2
m0
m!
2
(A8) and 2 X ∞ r r2k −1k 2k ; exp − 2 W W k! k0
(A9)
where i, l, m, k, and N are integers, substituting Eqs. (A8) and (A9) into Eq. (A5), E2ρ is written as Z E2 ρ
R 0
τEOD r
−1k
k0
N X
×
N X
−1m
m0
r2k W 2k k!
a2m ρ2m r2m rdr: 4m m!2
(A10)
Simplifying Eq. (A10), we find the field expression for one phase discontinuity as E2 ρ; z f
N X N X
−1km
k0 m0
Z
×
R 0
a2m ρ2m 4m W 2k m!2 k!
τEOD rr2k2m1 dr:
(A11)
In the case of several rings (i discontinuities) which corresponds to the positions Ri mentioned above, E2 becomes E2total ρ
p N X N X X
−1km −1i1
k0 m0 i1
Z
×
Ri 0
a2m ρ2m 4m W 2k m!2 k!
r2k2m1 dr:
(A12)
After evaluating the integral in Eq. (A12), we find E2total ρ
p N X N X X
−1kmi1
k0 m0 i1
×
a2m ρ2m Ri2km1 : 4m W 2k m!2 k!2k 2m 2
(A13)
Equations (A7) and (A13) give the expression of the total field given in Eq. (6). References 1. R. de Saint Denis, N. Passilly, and K. Aït-Ameur, “Laser beam brightness of apertured optical resonators,” Opt. Commun. 264, 193–202 (2006). 2. L. Carbone, P. Fulda, C. Bond, F. Brueckner, D. Brown, M. Wang, D. Lodhia, R. Palmer, and A. Freise, “The generation of higher-order Laguerre–Gauss beams for high-precision interferometry,” J. Visualized Exp. 78, 50564 (2013). 3. L. Carbone, C. Bogan, P. Fulda, A. Freise, and B. Willke, “Generation of high-purity higher-order Laguerre–Gauss beams at high laser power,” Phys. Rev. Lett. 110, 251101 (2013).
4. J. Arlt and M. J. Padgett, “Generation of a beam with a dark focus surrounded by regions of higher intensity: the optical beam,” Opt. Lett. 25, 191–193 (2000). 5. C. H. Chen, P. T. Tai, and W. F. Hsieh, “Bottle beam from a bare laser for single-beam tapping,” Appl. Opt. 43, 6001–6006 (2004). 6. S. Fürhaper, A. Jesacher, S. Bernet, and M. Ritsch-Marte, “Spiral phase contrast imaging in microscopy,” Opt. Express 13, 689–694 (2005). 7. M. SeGall, V. Rotar, J. Lumeau, S. Mokhov, B. Zeldovich, and L. B. Glebov, “Binary volume phase masks in photo-thermorefractive glass,” Opt. Lett. 37, 1190–1192 (2012). 8. A. Hasnaoui, A. Bencheikh, and K. Ait-Ameur, “Tailored TEMp0 beams for large size 3-D laser prototyping,” Opt. Lasers Eng. 49, 248–251 (2011). 9. A. Hasnaoui, A. Bencheikh, M. Fromager, E. Cagniot, and K. Aït-Ameur, “Creation of a sharper focus by using a rectified TEMp0 beam,” Opt. Commun. 284, 1331–1334 (2011). 10. S. Ngcobo, K. Aït-Ameur, N. Passilly, A. Hasnaoui, and A. Forbes, “Exciting higher-order radial Laguerre–Gaussian modes in a diode-pumped solid-state laser resonator,” Appl. Opt. 52, 2093–2101 (2013). 11. A. Ishaaya, N. Davidson, G. Machavariani, E. Hasman, and A. Friesem, “Efficient selection of high-order Laguerre– Gaussian modes in a Q-switched Nd:YAG laser,” IEEE J. Quantum Electron. 39, 74–82 (2003). 12. K. Sueda, G. Miyaji, and N. Miyanaga, “Laguerre–Gaussian beam generated with multilevel spiral phase plate for high intensity laser pulses,” Opt. Express 12, 3548–3553 (2004). 13. R. Oron, N. Davidson, A. A. Friesem, E. Hasman, and E. Wolf, “Transverse Mode Shaping and Selection in Laser Resonators,” Vol. 42 of Progress in Optics Series (Elsevier, 2001), pp. 325–386. 14. S. N. Khonina, V. V. Kotlyar, R. V. Skidanov, V. A. Soifer, P. Laakonen, and J. Turunen, “Gauss–Lagh different indices in prescribed diffraction orders of a diffractive phase element,” Opt. Commun. 175, 301–308 (2000). 15. J. Arlt, K. Dholakia, L. Allen, and M. J. Padgett, “The production of multiringed Laguerre–Gaussian modes by computer generated holograms,” J. Mod. Opt. 45, 1231–1237 (1998). 16. G. A. Turnbull, D. A. Robertson, G. M. Smith, L. Allen, and M. J. Padgett, “The generation of free-space Laguerre– Gaussian modes at millimetre-wave frequencies by use of a spiral phase plate,” Opt. Commun. 127, 183–188 (1996). 17. F. Fatemi and M. Bashkansky, “Generation of hollow beams using a binary spatial light modulator,” Opt. Lett. 31, 864–866 (2006). 18. Y. Ohtake, T. Ando, N. Fuckuchi, N. Matsumoto, H. Ito, and T. Hara, “Universal generation of higher order multiringed Laguerre–Gaussian beams by using a spatial light modulator,” Opt. Lett. 32, 1411–1413 (2007). 19. T. Ando, Y. Ohtake, N. Matsumoto, T. Inoue, and N. Fukuchi, “Mode purities of Laguerre–Gaussian beams generated via complex-amplitude modulation using phase-only spatial light modulators,” Opt. Lett. 34, 34–36 (2009). 20. A. Syouji, K. Kurihara, A. Otomo, and S. Saito, “Diffractiongrating-type phase converters for conversion of Hermite– Laguerre–Gaussian mode into Gaussian mode,” Appl. Opt. 49, 1513–1517 (2010). 21. S. A. Kennedy, M. J. Szabo, H. Teslow, J. Z. Porterfield, and E. R. I. Abraham, “Creation of Laguerre–Gaussian laser modes using diffractive optics,” Phys. Rev. A 66, 043801 (2002). 22. R. Bourouis, K. Ait Ameur, and H. Ladjouze, “Optimization of the Gaussian beam flattening using a phase-plate,” J. Mod. Opt. 44, 1417–1427 (1997). 23. E. Cagniot, M. Fromager, T. Godin, N. Passilly, M. Brunel, and K. Ait-Ameur, “A variant of the method of Fox & Li dedicated to intracavity laser beam shaping,” J. Opt. Soc. Am. A 28, 489–495 (2011). 24. R. de Saint Denis, N. Passilly, M. Laroche, T. MohamedBrahim, and K. Ait-Ameur, “Beam shaping longitudinal range of a binary diffractive element,” Appl. Opt. 45, 8136–8141 (2006). 25. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1972). 20 July 2014 / Vol. 53, No. 21 / APPLIED OPTICS
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Département sciences et techniques, Faculté des sciences et des technologies, Université de Bordj Bou Arréridj, 34000 Bordj Bou Arréridj, Algeria
2
Laboratoire d’optique appliqué, Institut d’optique et mécanique de precision, Université de Sétif 1, 19000 Sétif, Algeria 3
Centre de Recherche sur les Ions, les Matériaux et la Photonique, UMR 6252-CEA-CNRS-ENSICAEN-Université de Caen, ENSICAEN, 6 Boulevard Maréchal Juin, F 14050 Caen, France 4 Centre de Développement des Techniques Avancées, Division Milieux Ionisés et Lasers, P.O. Box 17 Baba Hassan, 16303 Algiers, Algeria
*Corresponding author: [email protected] Received 28 January 2014; revised 19 May 2014; accepted 3 June 2014; posted 11 June 2014 (Doc. ID 205478); published 18 July 2014
In recent years, considerable attention has been devoted to laser beams with specific intensity profile, i.e., non-Gaussian. In this work, we present a novel technique to generate high-radial-order Laguerre– Gaussian beams LGp0 based on the use of a binary phase diffractive optical element (BPDOE). The latter is a phase plate made up of annular zones introducing alternatively a phase shift equal to 0 or π modeled on positions which do not coincide with the position of the zeros of the desired LGp0 beam. The LGp0 beams are obtained by transforming a fundamental Gaussian beam through an appropriate BPDOE. The design of the latter is based on the calculation of the Fresnel–Kirchhoff integral, and the diffracted intensity at the focus plane of a lens has been modeled analytically for the first time. The numerical simulations and experiment demonstrate a good beam quality transformation. Obtained LGp0 are suitable for atom trap and pumping solid state laser applications. © 2014 Optical Society of America OCIS codes: (140.3300) Laser beam shaping; (050.1970) Diffractive optics; (050.1380) Binary optics. http://dx.doi.org/10.1364/AO.53.004761
1. Introduction
Generally, it is desirable to force the oscillation of a laser in the fundamental Gaussian mode because it leads to the highest possible brightness for a given power [1]. As a consequence, high-order transverse modes are usually not taken into consideration. However, they present very promising properties which are illustrated through the three following examples. The first one concerns Laguerre–Gaussian modes LGpl with radial and azimuthal orders (respectively, p and l) which are used for high-precision 1559-128X/14/214761-07$15.00/0 © 2014 Optical Society of America
interferometry, and especially in detection of gravitational waves [2,3]. The second one allows the production of a bottle beam for p 0 [4] used to trap or guide cold atoms [5]. We recall that this beam (LG0l ) is also able to carry an orbital angular momentum, which is of big interest in atom optics and spiral phase imaging microscopy [6,7]. The third example is the use of LGp0 beams for focal volume reduction [8,9]. This technique is based on the transformation of the alternately out of phase rings of the incident LGp0 into a unified phase, which is achieved by a binary phase diffractive optical element (BPDOE) followed by a focusing lens. This operation corresponds to a kind of “rectification” of the wavefront, having the property to generate a single lobed 20 July 2014 / Vol. 53, No. 21 / APPLIED OPTICS
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2. Generating LGp0 Beam Using BPDOE
For a laser beam of power P propagating along the z axis, the magnitude of the electric field of a collimated LGpl mode (of radial order p and azimuthal order l) is given by [21] rs 2P p! −1p exp−ilϕ U pl ρ; ϕ 2 l p! πW p l 2 ρ2 2ρ 2ρ : Llp × exp − 2 W W W2
APPLIED OPTICS / Vol. 53, No. 21 / 20 July 2014
LG20 LG30
Transversal coordinate r(a.u.) Fig. 1. Radial intensity distribution of the pure LG10 , LG20 , and LG30 beams.
in the electric field and a well-defined angular momentum of lħ per photon [21]. In the case where l 0, the expression of the transverse distribution of the electrical field reduces to 2 ρ2 2ρ U p0 ρ; ϕ U 0 exp − 2 × Lp ; W W2
(2)
where U 0 is the on-axis field amplitude, W is the width of the beam waist, p is the order of the LGp0 beam, and Lp are the Laguerre polynomials. When l p 0, the beam is Gaussian. We note that high-order LGp0 beams are made up of a central peak surrounded by p rings of light, as shown in Fig. 1. A graphical representation of the first three Laguerre–Gaussian beams (LG10 , LG20 , and LG30 ) is presented in Fig. 1. Figure 2 shows the optical layout considered for the transformation of a fundamental Gaussian laser beam into a single higher-order LGp0 using simply a BPDOE. The observation plane is the focal plane of the focusing lens. The BPDOE is a phase plate made up of annular zones introducing a phase shift equal to 0 or π (Fig. 3). It is worthwhile to know that the intensity in the focal plane is very sensitive to phase discontinuity
(1)
The exp−ilϕ term identifies this beam as a vortex state with a quantized 2πl azimuthal phase change 4762
LG10
Normalized intensity
intensity profile in the focal plane. The latter is characterized by a focal volume which can be strongly reduced in comparison with the one resulting from the focusing of a fundamental Gaussian beam. It is worthwhile to recall that the focal volume is the area where the intensity can be considered as high, and lies at the very heart of the spatial resolution of the used optical beam. So the BPDOE can be used for both LGp0 beam generation (when illuminated with a fundamental Gaussian beam) and wave rectification (when entered with an LGp0 beam). While LG beams can be produced inside the laser cavity using diffractive optical elements [10,11], the use of laser diodes or monomode fiber requires external cavity conversion from fundamental Gaussian to LG beams. Previously demonstrated external cavity methods include spiral phase plates [12,13] and computergenerated holograms (CGH) [14]. Spiral phase plates are transmission optics whose thickness varies to create an azimuthally dependent phase delay. This can convert a fundamental Gaussian beam into an LGpl beam, as well as convert between any two LGpl modes [15,16]. A more common method uses computer-generated holographic gratings displayed on spatial light modulators [17–20]. These CGH are recordings of the theoretical interference pattern between the electric field of the LG mode of interest and a reference field, commonly a plane wave [14,15]. Techniques for producing LG beams are costly and need complicated diffractive optical elements. In this work, we propose a low-cost alternative technique for producing LGp0 beams. Indeed, a BPDOE needs only one level of etching, while a diffractive optics made up of continuous phase relief usually can need more than 64 etching levels. The technique consists of transforming a fundamental Gaussian beam into an LGp0 beam using a simple BPDOE, which is made up of annular zones introducing a phase shift corresponding alternatively to 0 or π. The design of the BPDOE is based on the calculation of the Fresnel–Kirchhoff integral, which is different from all techniques based on holographic reconstruction. In addition, an elegant analytical expression is derived for the transverse intensity distribution in the focal plane of the obtained LGp0 beam.
Fig. 2. Optical layout.
Er
9 8 p N P N P P > > a2m R2km1 kmi1 2m i > = −1 r > 4m W 2k m!2 k!2k2m2 2π < k0 m0 i1
i h λf > > : −1p 1 exp − a2 W 2 r2 4 2
> > ;
;
(5) Fig. 3. Designed BPDOE containing five phase discontinuities.
positions of the BPDOE, and also the number of discontinuities. One can intuitively but wrongly set the phase discontinuity positions at the same place as the zeros of the Laguerre polynomial Lp given in Table 1 [8,9]. It will be shown later that the best positions of the phase discontinuities, allowing the transformation of a fundamental Gaussian beam into an intensity distribution in the focal plane which is of the LGp0 beam, are actually distinct from the zeros of Lp. The field distribution of the input collimated fundamental Gaussian beam Ein can be written as ρ2 : Ein ρ E0 exp W2
where a 2π∕λf . Consequently, the corresponding transverse distribution of the intensity is Ir jErj2. The more explicit form for the intensity distribution takes the form Ir
2 N N p π 2m 2km1 2m 2 P P P −1kmi1 2kλf 2 Ri r W m! k!2k2m2 2π k0 m0 i1 ; h 2 i λf r2 −1p 12 exp − πW λf (6)
setting 2m
(3)
The transmittance τρ of the binary phase DOE is alternatively equal to −1 or 1, and the radial positions of the phase discontinuities are different from positions of the Laguerre polynomial zeros. There is an interesting case when the BPDOE shows only one phase discontinuity; it takes the shape of a phase aperture, which shows the interesting capabilities of laser beam shaping in the free propagation [22] and intracavity [23] cases. The field distribution in the focal plane of the lens of an input fundamental Gaussian beam given by Eq. (3) is described by the Fresnel–Kirchhoff integral [8,9] 2π Er; z f λf
Z
2π rρ ρdρ; τρEin ρJ 0 λf DOE
p 1 2 3 4 5
(4)
Roots of Laguerre Polynomials Lp ρi ∕W
Values of Ratio ρi ∕W for the Zeros of Intensity of LGp0 Mode 0.707106 0.541195 0.455946 0.401589 0.363015
1.306562 1.071046 0.934280 0.840041
1.773407 1.506090 1.340975
2.167379 1.882260
π λf
2m!2 k!k m 1
:
(7)
Finally, we reach an elegant expression of radial intensity of a fundamental Gaussian beam with an incident waist W, diffracted by an annular BPDOE and focused by the lens as 2 N P p N P R2km1 2 P 2m i gk; m; i r 2k 2π W ; (8) k0 m0 i1 Ir i h λf −1p πW 2 2 2 exp − λf r
where Ri : is the radius of the ring i. 3. Results and Discussion
where ρ and r are the coordinates in the plane of the phase element and the focus, respectively. After a tedious calculus, we finally find the analytical expression of the field at the focal plane (Appendix A) given by
Table 1.
gk; m; i −1kmi1
2.51040
The calculus presented in Section 2 shows that generating LGp0 only requires p phase discontinuities whose positions (scaled radial variable ρ∕W) are given in Table 2. By applying the novelly obtained formula in Eq. (8), and using the wavelength of the incident laser beam λ 1064 nm, the input beam width is W 1 mm and we can construct all intensity distributions in the focus of the BPDOE. Two examples of obtained Laguerre–Gaussian beams LG10 and LG40 calculated in the focus plane using Eq. (8) are shown in Fig. 4. In order to appreciate the quality of the transformation of a unilobe fundamental Gaussian beam into a multilobed Laguerre–Gaussian beam LGp0 , we also plot in Fig. 4 the fit by an ideal LGp0 beam. We can notice from Fig. 4 that the calculated LGp0 beam is very close to the pure one (ideal LGp0 beam). We express the fit quality using two quantities: 20 July 2014 / Vol. 53, No. 21 / APPLIED OPTICS
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Table 2.
Empirical Phase Discontinuity Positions
p Phase Discontinuity Positions (Scaled Radial Variable) ρi ∕W 0.66 0.41 0.37 0.28
Intensity (a.u.)
Ideal LG 10 fit
Transverse coordinate
LG10 LG20 LG30 LG40
LG50
M2
3.02 3
11,10 11
for calculated LGp0 M 2 2p 1, for ideal LGp0
0.74 0.58 0.42
Calculated LG10
LGp0
0.89 0.58
Ideal LG 40 fit
z=f
z=f+5ZR z=f+7.5ZR z=f+10ZR
4764
LG10
LG20
LG30
LG40
LG50
0.9910 0.9731 0.9798 0.9886 0.9797 0.9736 0.9671 0.9591 0.9502
0.9781 0.9852 0.8863 0.9167 0.9659 0.9808 0.9809 0.9747 0.9699
0.9563 0.9691 0.7561 0.9681 0.9703 0.9732 0.9768 0.9790 0.9797
0.9227 0.9445 0.7732 0.9711 0.9721 0.9736 0.9671 0.9591 0.9502
0.9693 0.9662 0.9830 0.8241 0.9737 0.9732 0.9675 0.9722 0.9826
APPLIED OPTICS / Vol. 53, No. 21 / 20 July 2014
Intensitiy (a.u.)
Transverse coordinate(a.u.)
z=f z=f+2.5ZR z=f+5ZR z=f+7.5ZR z=f+10ZR z=f+12.5ZR
Transverse coordinate (a.u.)
Fig. 5. Transverse distribution of the intensity for LG10 and LG40 at different z positions along the propagation axis.
After plotting the values of Table 3 (Fig. 6), we have observed a little decrease in the quality of the calculated LG beams at the position z 2ZR for all beam orders p 0; 1; 2; 3; 4; 5. Also, we have 1.00
The correlation coefficient R²
Characteristics of the Quality of the Transformation at Distance z after the Focusing Lens
R2
50 51 52 53 54 55 56 57 58
8.73 9
z=f+2.5ZR
Transverse coordinate
– A statistical parameter noted R2 , which is the coefficient of determination. The more R2 approaches 1, the higher the quality of the beam shaping. Table 3 shows numerical results of calculated R2 for different calculated LGp0 corresponding to different positions z along the propagation axis. For R2 1, the beam is an ideal Laguerre–Gaussian mode. – The beam quality factor M 2 of the reshaped beam, calculated using the second-order moments method. A M 2 close to (2p 1) ensures a good transformation [8,9] of the fundamental Gaussian beam into an LGp0 beam (Table 4). To estimate how these beams keep their shape as they propagate, we have plotted the intensity distribution for calculated LG10 and LG40 for different axial position z. The results are shown in Fig. 5. The calculated Laguerre–Gaussian beams (LG10 and LG40 ) conserve their radial intensity distributions for at least 10 mm before and 10 mm after the lens, which corresponds to a beam shaping longitudinal range [24] more than 20ZR , which is important and sufficient, for instance, to pump a solid laser or to trap and guide microparticles. Note that ZR 0.8 mm is the Rayleigh distance of the focused Gaussian beam without any BPDOE.
zmm
6.79 7
Calculated LG 40
Fig. 4. Obtained Laguerre–Gaussian beams LG10 and LG40 , respectively.
Table 3.
5 5
0.85
Intensity (a.u.)
0.50 0.33 0.17 0.19 0.13
Intensity (a.u.)
1 2 3 4 5
Table 4. Calculated M 2 of the Obtained Laguerre–Gaussian Beam Based on the Second-Order Moments of the Intensity Distribution
0.95
0.90
LG10 LG20
0.85
LG30 LG40 LG50
0.80
0.75
50
52
54
56
58
z[mm]
Fig. 6. Variation of the parameter R2 as function of propagating distance z for calculated LG beams
observed that the quality of calculated higher-order LG beams is improved when the beam propagates far from the focal point of the used focusing lens. 4. Experimental Investigation
(a)
(b)
transverse coordinate
Generated LG20 Ideal LG20 fit
transverse coordinate
Generated LG40 Ideal LG40 fit
Transverse coordinate
Fig. 9. Fits of generated LG10 , LG20 , and LG40 beams.
fundamental Gaussian beam of a width W 1 mm. The obtained generated beams were observed and detected in the focal plane and along the propagation axis using a CCD camera (Thorlabs cam BC 106-vis). Images and profiles of generated beams are shown in Fig. 8. We have examined profiles of generated beams at the focal plane of the lens L5, using ideal LGp0 fits. As an example, we showed in Fig. 9 the intensity distribution of generated LG10 and LG20 and their ideal LG fits. The quality of the generated beams is quantized using the R2 coefficient of determination; results are given in Table 5. To examine the invariance of the transversal intensity distribution during propagation, we have moved the CCD camera along the z axis; the images obtained are shown in Fig. 10. For instance, we presented the generated LG10 and LG40 beams, and we have observed a large value of the beam-shaping longitudinal range, which can be more than 10ZR . Table 5.
Quality of the Transformation of Generated LG Beams in the Focal Plane of Lens L5
LG
LG10
LG20
LG30
LG40
LG50
0.9817
0.9673
0.9563
0.9390
0.9693
2
R
Fig. 7. (a) Experimental setup for investigating the transformation of a fundamental Gaussian beam into a Laguerre–Gaussian beam. (b) BPDOE patterns. (a,b) Phase patterns of the designed BPDOE without grating used for generating LG10 and LG40 beams, respectively. (c,d) Corresponding phase patterns of the designed (BPDOE + grating).
Normalized intensity
Generated LG10 Ideal LG10 fit
Normalized Intensity distribution
Fig. 8. Generated Laguerre–Gaussian beam in the focal plane of a lens where, from left to right, we show increasing radial orders from p 1 to p 5.
Normalized intensity
To experimentally verify the results obtained in Section 3, we used the setup shown in Fig. 7. It consists of a laser source operating at 1064 nm delivering a fundamental Gaussian beam. This beam was enlarged and collimated through a telescope and propagated onto a reflective phase-only liquid crystal on silicon (SLM: Holoeye Pluto-NIR). We recall that, for the sake of flexibility, in this experiment we have used the SLM just to simulate the different BPDOE designed above. The SLM is calibrated for a 2π phase shift at λ 1064 nm, and was programmed with a grayscale image containing the information about the BPDOE functions introduced in Fig. 3. By adding a phase grating pattern onto the BPDOE pattern, the output beam is diffracted into a different direction to a zeroth-order beam that propagates in the normal direction to the SLM surface [15]. The diffracted field from the latter was transformed by two lenses (L3 and L4), forming a 4f system in the back plane of the lens L5 of focal length f 750 mm. Only the diffracted first order was selected between L3 and L4 using a spatial filter; then we have imaged the designed BPDOE at 4f distance from the first plane of the used SLM. The latter diffracts the incident
Fig. 10. How the generated Laguerre–Gaussian beams LG10 and LG40 keep their shapes for several values of ZR 20 July 2014 / Vol. 53, No. 21 / APPLIED OPTICS
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be of interest in prototyping and machining processes [8,9]. Z=5ZR
Intensity
Intensity [a.u.]
Z=5ZR
Z=4ZR Z=3ZR Z=2ZR
Z=4ZR
Appendix A
Z=3ZR
In this appendix, we give details on the solution of the integral given in Eq. (4). The latter takes the following form, in the case of only one phase discontinuity (phase aperture) at the radius position R: 9 8 2 ρ > > 2π > > > >Z exp − 2 J 0 λf rρ ρdρ = 2π < R W : Er; z f 2 > R∞ λf > ρ 0 > > 2π > > − R exp − 2 J 0 λf rρ ρdρ ; : W (A1)
Z=2ZR Z=ZR
Z=ZR
Transverse coordinate
Transverse coordinate
Fig. 11. Transverse distribution of the intensity for generated LG10 and LG40 at different z positions along the propagation axis. Table 6. Characteristics of the Quality of Generated LG10 and LG40 Beams at Distance z after the Focusing Lens
R2 z mm Zf f ZR f 2ZR f ZR f 4ZR f 5ZR
Using the equality
LG10
LG40
0.9817 0.9710 0.9715 0.9715 0.9693 0.9743
0.9390 0.9495 0.6541 0.9430 0.9442 0.9628
It is obvious that the resulting beams keep their LGp0 form. However, along the propagation axis, we have observed some exchange of the energy content between the different rings forming the beam. By plotting profiles in Fig. 11 of images presented in Fig. 10 and fitting the different intensity distributions for LG10 and LG40 , values of R2 presented in Table 6 show the good quality of the generated beams.
Z
ρ2 exp − 2 J 0 arρrdr W R Z∞ ρ2 exp − 2 J 0 arρρdρ W 0 ZR ρ2 exp − 2 J 0 arρρdρ; − W 0
∞
(A2)
the integral in Eq. (A1) becomes 9 8 R r2 R > > 2 τ r exp − arρrdr J > > EOD 0 = W 2k 2π < 0 ; Eρ;z f > λf > R∞ r2 > > ; : − 0 exp − 2k J 0 arρrdr W (A3)
5. Conclusion
In summary, we have demonstrated an easy way to produce a Laguerre–Gaussian LGp0 beam by reshaping a fundamental Gaussian beam using a simple BPDOE, which is made up of annular zones introducing a phase shift equal to 0 or π modeled on positions different from the zeros of Laguerre polynomials. The concept of designing this BPDOE is based on the calculation of the Fresnel–Kirchhoff integral. For the sake of flexibility, we have used a phase-only spatial light modulator to implement the various designed BPDOEs. We have also derived an elegant analytical expression of the resulting beam’s intensity at the focal plane of the lens that follows the BPDOE, which allows for the optimization of the quality of the obtained beams. The numerical and experimental results show that the resulting beams are very close to ideal LGp0 . This approach for generating LGp0 beams offers many benefits over existing techniques (CGH, continuous phase elements, and spiral phase elements). Due to its simplicity, it is very easy to manufacture this type of designed BPDOE and it needs only one level of etching, and so it could 4766
APPLIED OPTICS / Vol. 53, No. 21 / 20 July 2014
setting Z E1 ρ
∞
0
2 r exp − 2 J 0 arρrdr W
(A4)
and Z E2 ρ 2
R 0
r2 τEOD r exp − 2k J 0 arρrdr: W
(A5)
For solving the first integral (A4), we use the wellknown equality [25] given by 2 a exp − p ∞ 4b exp−bxJ 0 a x dx : b 0
Z
(A6)
Finally, we get 1 W2 aρ2 : E1 ρ W 2 exp − 4 2
(A7)
For the second integral (A5), and using the series expression of Bessel and exponential functions
J 0 arρ
∞ X −1m arρ∕22m 2
m!
m0
∞ X −1m ρ2m ar2m 2m
2
m0
m!
2
(A8) and 2 X ∞ r r2k −1k 2k ; exp − 2 W W k! k0
(A9)
where i, l, m, k, and N are integers, substituting Eqs. (A8) and (A9) into Eq. (A5), E2ρ is written as Z E2 ρ
R 0
τEOD r
−1k
k0
N X
×
N X
−1m
m0
r2k W 2k k!
a2m ρ2m r2m rdr: 4m m!2
(A10)
Simplifying Eq. (A10), we find the field expression for one phase discontinuity as E2 ρ; z f
N X N X
−1km
k0 m0
Z
×
R 0
a2m ρ2m 4m W 2k m!2 k!
τEOD rr2k2m1 dr:
(A11)
In the case of several rings (i discontinuities) which corresponds to the positions Ri mentioned above, E2 becomes E2total ρ
p N X N X X
−1km −1i1
k0 m0 i1
Z
×
Ri 0
a2m ρ2m 4m W 2k m!2 k!
r2k2m1 dr:
(A12)
After evaluating the integral in Eq. (A12), we find E2total ρ
p N X N X X
−1kmi1
k0 m0 i1
×
a2m ρ2m Ri2km1 : 4m W 2k m!2 k!2k 2m 2
(A13)
Equations (A7) and (A13) give the expression of the total field given in Eq. (6). References 1. R. de Saint Denis, N. Passilly, and K. Aït-Ameur, “Laser beam brightness of apertured optical resonators,” Opt. Commun. 264, 193–202 (2006). 2. L. Carbone, P. Fulda, C. Bond, F. Brueckner, D. Brown, M. Wang, D. Lodhia, R. Palmer, and A. Freise, “The generation of higher-order Laguerre–Gauss beams for high-precision interferometry,” J. Visualized Exp. 78, 50564 (2013). 3. L. Carbone, C. Bogan, P. Fulda, A. Freise, and B. Willke, “Generation of high-purity higher-order Laguerre–Gauss beams at high laser power,” Phys. Rev. Lett. 110, 251101 (2013).
4. J. Arlt and M. J. Padgett, “Generation of a beam with a dark focus surrounded by regions of higher intensity: the optical beam,” Opt. Lett. 25, 191–193 (2000). 5. C. H. Chen, P. T. Tai, and W. F. Hsieh, “Bottle beam from a bare laser for single-beam tapping,” Appl. Opt. 43, 6001–6006 (2004). 6. S. Fürhaper, A. Jesacher, S. Bernet, and M. Ritsch-Marte, “Spiral phase contrast imaging in microscopy,” Opt. Express 13, 689–694 (2005). 7. M. SeGall, V. Rotar, J. Lumeau, S. Mokhov, B. Zeldovich, and L. B. Glebov, “Binary volume phase masks in photo-thermorefractive glass,” Opt. Lett. 37, 1190–1192 (2012). 8. A. Hasnaoui, A. Bencheikh, and K. Ait-Ameur, “Tailored TEMp0 beams for large size 3-D laser prototyping,” Opt. Lasers Eng. 49, 248–251 (2011). 9. A. Hasnaoui, A. Bencheikh, M. Fromager, E. Cagniot, and K. Aït-Ameur, “Creation of a sharper focus by using a rectified TEMp0 beam,” Opt. Commun. 284, 1331–1334 (2011). 10. S. Ngcobo, K. Aït-Ameur, N. Passilly, A. Hasnaoui, and A. Forbes, “Exciting higher-order radial Laguerre–Gaussian modes in a diode-pumped solid-state laser resonator,” Appl. Opt. 52, 2093–2101 (2013). 11. A. Ishaaya, N. Davidson, G. Machavariani, E. Hasman, and A. Friesem, “Efficient selection of high-order Laguerre– Gaussian modes in a Q-switched Nd:YAG laser,” IEEE J. Quantum Electron. 39, 74–82 (2003). 12. K. Sueda, G. Miyaji, and N. Miyanaga, “Laguerre–Gaussian beam generated with multilevel spiral phase plate for high intensity laser pulses,” Opt. Express 12, 3548–3553 (2004). 13. R. Oron, N. Davidson, A. A. Friesem, E. Hasman, and E. Wolf, “Transverse Mode Shaping and Selection in Laser Resonators,” Vol. 42 of Progress in Optics Series (Elsevier, 2001), pp. 325–386. 14. S. N. Khonina, V. V. Kotlyar, R. V. Skidanov, V. A. Soifer, P. Laakonen, and J. Turunen, “Gauss–Lagh different indices in prescribed diffraction orders of a diffractive phase element,” Opt. Commun. 175, 301–308 (2000). 15. J. Arlt, K. Dholakia, L. Allen, and M. J. Padgett, “The production of multiringed Laguerre–Gaussian modes by computer generated holograms,” J. Mod. Opt. 45, 1231–1237 (1998). 16. G. A. Turnbull, D. A. Robertson, G. M. Smith, L. Allen, and M. J. Padgett, “The generation of free-space Laguerre– Gaussian modes at millimetre-wave frequencies by use of a spiral phase plate,” Opt. Commun. 127, 183–188 (1996). 17. F. Fatemi and M. Bashkansky, “Generation of hollow beams using a binary spatial light modulator,” Opt. Lett. 31, 864–866 (2006). 18. Y. Ohtake, T. Ando, N. Fuckuchi, N. Matsumoto, H. Ito, and T. Hara, “Universal generation of higher order multiringed Laguerre–Gaussian beams by using a spatial light modulator,” Opt. Lett. 32, 1411–1413 (2007). 19. T. Ando, Y. Ohtake, N. Matsumoto, T. Inoue, and N. Fukuchi, “Mode purities of Laguerre–Gaussian beams generated via complex-amplitude modulation using phase-only spatial light modulators,” Opt. Lett. 34, 34–36 (2009). 20. A. Syouji, K. Kurihara, A. Otomo, and S. Saito, “Diffractiongrating-type phase converters for conversion of Hermite– Laguerre–Gaussian mode into Gaussian mode,” Appl. Opt. 49, 1513–1517 (2010). 21. S. A. Kennedy, M. J. Szabo, H. Teslow, J. Z. Porterfield, and E. R. I. Abraham, “Creation of Laguerre–Gaussian laser modes using diffractive optics,” Phys. Rev. A 66, 043801 (2002). 22. R. Bourouis, K. Ait Ameur, and H. Ladjouze, “Optimization of the Gaussian beam flattening using a phase-plate,” J. Mod. Opt. 44, 1417–1427 (1997). 23. E. Cagniot, M. Fromager, T. Godin, N. Passilly, M. Brunel, and K. Ait-Ameur, “A variant of the method of Fox & Li dedicated to intracavity laser beam shaping,” J. Opt. Soc. Am. A 28, 489–495 (2011). 24. R. de Saint Denis, N. Passilly, M. Laroche, T. MohamedBrahim, and K. Ait-Ameur, “Beam shaping longitudinal range of a binary diffractive element,” Appl. Opt. 45, 8136–8141 (2006). 25. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1972). 20 July 2014 / Vol. 53, No. 21 / APPLIED OPTICS
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