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Rosette gratings as generators of nondiverging intensity patterns
Optics Communications 300 (2013) 108–113
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Rosette gratings as generators of nondiverging intensity patterns Suzana Topuzoski n, Ljiljana Janicijevic Institute of physics, Faculty of natural sciences and mathematics, University “Ss. Cyril and Methodius”, 1000 Skopje, Republic of Macedonia
art ic l e i nf o
a b s t r a c t
Article history: Received 10 November 2012 Received in revised form 4 March 2013 Accepted 9 March 2013 Available online 30 March 2013
In this paper we present and investigate binary phase diffraction gratings whose phase reliefs have form of rosettes-the so-called rosette gratings. They are constructed by the equiphase lines of the cosineprofiled phase Siemens star axicon-a hybrid optical element consisting of an axicon and a phase optical element with azimuthal cosine profile. It is shown that the rosette gratings transform the incident Gaussian laser beam into a sum of odd-diffraction orders beams, nondiverging in defined propagation intervals. The amplitude and intensity distributions of the separate diffraction orders, as well as the interference between them are studied. Under certain conditions these gratings can act as suitable substitutes for the cosine-profiled phase Siemens star axicons in their applications. & 2013 Elsevier B.V. All rights reserved.
Keywords: Binary phase diffraction gratings Rosette gratings Cosine-profiled phase Siemens star axicon Gaussian laser beam Nondiverging beams
1. Introduction The nondiverging or so-called “non-diffracting” optical beams (optical beams whose transverse intensity profile is remarkably resistant to the diffractive spreading over a defined propagation interval), were treated in detail many years ago by Dyson [1], being generated by circular and spiral gratings. The annular aperture of small width, which creates “diffraction-free” wavefields with zeroth-order Bessel function transverse amplitude distribution along the radial coordinate, has been described by Steward [2]. The Bessel beam (BB) as a mathematical construct was first noted by Durnin [3], who discovered that particular solutions of the Helmholtz equation, being of the Bessel type, were independent on the propagation direction; since then, these nondiverging (also commonly known as Bessel beams) have become one of the topic subjects in the optics community [4]. The zeroth-order Bessel beam has a bright central core (which can be nondiverging in a defined interval) surrounded by successive dark and bright rings, and it does not possess a phase singularity. While, the higherorder Bessel beam has a dark core around its propagation axis (which can be non-spreading in a defined interval), surrounded by successive bright and dark rings; it can posses phase singularity on the optical axis and helicoidal wavefront. Due to their unique features, the Bessel beams have found many applications in alignment in some optical and mechanical engineering, atom optics, manipulation of small particles and studying of the motion
n
Corresponding author. Tel.: þ389 71 68 29 58. E-mail addresses: [email protected], [email protected]. edu.mk (S. Topuzoski). 0030-4018/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.optcom.2013.03.023
of microscopic particles in optical potentials [5]. Besides optical elements with circular symmetry having radial variation: circular diffraction gratings [1], axicons [6,7], holographic optical elements with transmission function of axicon type [8], and certain zone plates [9], for precise alignment purposes can also serve phase diffraction gratings and phase diffractive optical elements (DOEs) with azimuthal symmetrical variation [10–12]. Thus, in [11] two phase diffractive optical elements with azimuthal periodicity of the phase retardation were proposed: the first element, named the cosine-profiled phase Siemens star, possesses azimuthal cosineprofiled phase changes deposited on a plane base; while, the second element is a combination of the first element and a thin phase axicon and is named the cosine-profiled phase Siemens star axicon (CPSSA). For both of them, theoretical investigation of the problem of Fresnel diffraction of a Gaussian laser beam, which enters with its waist and passes with its axis through their centers, was performed. It was shown that the cosine profile of the phase retardation of both optical elements produces an azimuthal cosine-profiled modulation on their diffractograms; but, it destroys the vortex characteristics of their diffraction fields. Since the subject of investigation in this work is the binary phase grating representative of the second hybrid element, the CPSSA, we will recall to its transmission function defined as: T′ðr,jÞ ¼ AðrÞexp ðikα0 rÞexp −ik½δ þδcosðpjÞ :
ð1Þ
In (1) the term exp −ik½δ þ δcosðpjÞ expresses the transmission function of the transparent phase layer with azimuthal periodicity of the thickness, p is an integer showing the number of periods of the phase change when the azimuthal variable makes one cycle (0 oj o2π), δ ¼ ðc=2Þðn−1Þ, n is the phase layer
S. Topuzoski, L. Janicijevic / Optics Communications 300 (2013) 108–113
Fig. 1. a. Binary phase relief of the rosette grating with spatial frequency p ¼ 3. b. Binary phase relief of the rosette grating with spatial frequency p ¼5.
refractive index for the incident beam of wavelength λ, c is the total depth of the relief, while k ¼ 2π=λ is the wave number. Whereas, exp ðikα0 rÞ denotes the transmission function of a very thin axicon with a base radius R0 and parameter α0 ¼ ðn′−1Þγ connected to its base angle γ (for which the approximation tan γ≈sin γ≈γ is valid) and its refractive index n′. The function A (r) plays the role of a beam truncation function and is defined as: ( 1 when w0 o R0 , AðrÞ ¼ ð2Þ circðr=R0 Þ when w0 ≥R0 , where w0 is the Gaussian laser beam waist radius, and ( 1, when r≤R0 r circ ¼ 0, when r 4 R0 : R0 The total diffracted wave field consists of coaxial wave components which are nondiverging. The coaxial optical beams in the higher mth diffraction orders, positive and negative, are vortex ones (carrying topological charges þmp and –mp, respectively) and with higher-order Bessel function amplitude radial distribution. But, because of the coupling effect, when they interfere, their optical charges transform into an azimuthal-cosine characteristic of the common wave amplitude, which is not vortex. The diffraction patterns show a central dark core surrounded by 2p bright spots in a ring array, when the zeroth-diffraction-order beam is eliminated (for a special value of the relief depth or the refractive index of the cosine-profiled phase layer). They are similar to the diffraction patterns obtained by means of a composite hologram with transmittance equal to a linear combination of transmission functions of two helical axicons with opposite topological charges [13]: Tðr,θÞ ¼ exp ð−i2πr=r o Þ exp ½iðnθ þ αÞ þexp ½−iðnθ−αÞ . The helical axicon is a hybrid of an axicon and a phase spiral plate [14]. In the
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previous equation, n is the topological charge value of the helical axicons, r0 is a constant (connected to the axicon parameters), and θ is azimuthal coordinate. This transmission function was written, as a binary phase, to the magneto-optical spatial light modulator, which was then illuminated with a collimated laser beam. In [11] if the zeroth-diffraction-order beam (described by the Bessel function of zeroth order) is present, a central bright spot in the diffractogram appears. Often instead of the precise modeling of the phase optical element it is easier, using the computer techniques, to draw its corresponding grating substituent. One can achieve that by drawing the equiphase lines of the phase optical element and photoreducing this draw on a plate, which can further be etched (in order to obtain a binary phase grating with variable depth) or bleached (resulting with a binary phase grating with variable refractive index). The phase diffractive optical elements affect the incident beam phase and have higher diffraction efficiency compared to the amplitude ones. Because the binary phase DOEs impose only two levels of phase, their fabrication is significantly easer. In this article we present and investigate phase diffraction gratings obtained by using the transmission function of the CPSSA, Eq. (1), in order to draw the lines of its maximum phase values in polar coordinates; one can produce a binary phase grating by photoreducing this drawing on a transparent plate, and applying the etching or bleaching technique. We name them the rosette gratings because of their rose shapes. Theoretically, a Gaussian laser beam transformation through them is treated and analyzed, allowing to discuss the output wave field, its transverse intensity profile, its propagation distance, as well as the conditions to be satisfied for applying these gratings as suitable substitutes of the CPSSAs.
2. Binary phase rosette gratings under Gaussian laser beam illumination Let us construct the lines of maximum phase values of the CPSSA, being defined, according to Eq. (1), by the system of lines kα0 r−kδ−kδcosðpjÞ ¼ 2Nπ,
ðN ¼ 0,1,2,::Þ
or, as: r ¼ λN=α0 þ ðδ=α0 ÞcosðpjÞ þ δ=α0 ,
ð3Þ
that shows they have form of roses drawn in polar coordinates. By photoreducing the drawing of the maximum equiphase lines, Eq. (3), on a transparent plate, and applying further the etching process, a binary phase grating will be obtained having a roseshaped profile of its thickness (Fig. 1). That is why the grating is denoted as rosette grating. For the case of a rectangular profile of transmittance and transmission coefficient equal to one, the transmission function of the rosette grating is defined in the following way ∞ Tðr,jÞ ¼ AðrÞ ∑ bs exp ðiksα0 rÞexp ð−iksδÞexp½−iksδcosðpjÞ s¼1
∞
þ ∑ b−s exp ð−iksα0 rÞexp ðiksδÞexp ½iksδcosðpjÞ, s¼1
ð4Þ
where: b0 ¼ 0, b 7 s ¼ 7 ð2s0 −1Þ ¼ 2=ðπð2s′−1ÞÞ, b 7 s ¼ 7 2s′ ¼ 0 (s′¼ 1,2,3..), while the truncation function A(r) is defined as Eq. (2) taking into account that, now R0 is radius of the circular aperture which obstructs the grating. Applying the Jacoby-Anger identity for the Bessel function [15], the transmission function (4) can also be written as ∞ Tðr,jÞ ¼ AðrÞ ∑ bs exp ð−iksδÞexp ðiksα0 rÞ J 0 ðksδÞ s¼1
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∞ þ ∑ ð−iÞm J m ðksδÞ½exp ð−impjÞ þ exp ðimpjÞ
Z
m¼1 ∞
þ ∑ b−s exp ðiksδÞexp ð−iksα0 rÞ s¼1 ∞ , J 0 ðksδÞ þ ∑ im J m ðksδÞ½expð−impjÞ þ exp ðimpjÞ m¼1
ð5Þ meaning that it is decomposed in a series of transmission functions of cosine-profiled phase Siemens star axicons.
0
where Δ is the area of the diffractive grating contributing to the diffraction. After involving the expressions for the incident light beam and the grating transmission function Eq. (5), and applying the identity [15] ∞
∑ il J l ðkrρ=zÞexp ½ilðj−θÞ,
l ¼ −∞
it gets the form Uðρ,θ,zÞ ¼
ð6Þ
In the previous formula for the wave field amplitude, we have denoted
þ b−s Y −a ðrÞexp ðiksδÞ ∞
∞
s¼1
m¼1
þ exp ðimpθÞ ∞
∞
s¼1
m¼1
þ ∑ b−s exp ðiksδÞ ∑ Y −d ðrÞim imp J m ðksδÞ½exp ð−impθÞ þ exp ðimpθÞ ,
Z Y 7d ¼
∞
AðrÞexp 0
ð11Þ
0
By performing the integration over the azimuthal variable φ, the previous expressions transform into: ∞
∞
s¼1 ∞
∑ ð∓iÞm J m ðksδÞ δl,−mp þδl,mp m¼1
" # rffiffiffi ikzs2 α20 z −z2 Aðr c ÞJ mp ðksα0 ρÞexp , 2 exp k 2 w0 =sα0
in the form
∞
Y 7 2 ¼ 2π ∑ b 7 s exp ð∓iksδÞ
! 2 −r 2 r kρ r rdr: ∓sα0 r J mp exp −ik 2 z 2z w0
The upper integrals have been solved by using the stationary phase method [17,18]. Thus, for the integrals Y þ a and Y þ d the solutions are approximated around the critical point r c ¼ sα0 z, while, for the integrals Y −a and Y −d -around the critical point r′c ¼ −sα0 z. Since the waves described by the integrals Y −a and Y −d form virtual vortices, which propagate in the negative direction of the z axis, on the positive side of the z axis the diffracted wave field is calculated considering only the solutions for Y þ a and Y þ d . " # rffiffiffi ikzs2 α20 z −z2 Aðr c ÞJ 0 ðksα0 ρÞexp , Y þ a ðr c Þ ¼ sα0 z exp 2 k 2 ðw0 =sα0 Þ Y þ d ðr c Þ ¼ sα0 z
Y 7 2 ¼ ∑ b 7 s exp ð∓iksδÞ ∑ ð∓iÞm J m ðksδÞ s¼1 m¼1 (Z ! 2 ∞ −r 2 r kρ r rdr ∓sα0 r J l AðrÞexp exp −ik 2 z 2z w0 0 ) Z 2π exp ð−iljÞ½exp ð-impjÞ þ expðimpjÞdj :
Y 7 1 ¼ 2πδl,0 ∑ b 7 s exp ð∓iksδÞJ 0 ðksδÞ s¼1 ! 2 Z ∞ −r 2 r kρ r rdr, ∓sα0 r J l AðrÞexp exp −ik 2 z 2z w0 0
ð9Þ
with the following notations for the integrals over the radial variable ! 2 Z ∞ −r 2 r kρ Y 7a ¼ r rdr, ð10Þ ∓sα AðrÞexp r J exp −ik 0 0 z 2z w20 0
∞
Y 7 1 ¼ ∑ b 7 s exp ð∓iksδÞJ 0 ðksδÞ s¼1 ! 2 Z ∞ −r 2 r kρ r rdr ∓sα0 r J l AðrÞexp exp −ik 2 z 2z w0 0 Z 2π exp ð−iljÞdj, 0 ∞
ð8Þ
þ ∑ bs exp ð−iksδÞ ∑ Y d ðrÞð−iÞm imp J m ðksδÞ½exp ð−impθÞ
Normally to the grating plane Δðr,jÞ, a Gaussian beam Uðr,j,0Þ ¼ exp ð−r 2 =w20 Þ, whose waist of size w0 occurs in the grating plane, is entering, and its propagation axis is passing through the pole of the grating (where r ¼ 0). The diffracted wave field in the observation plane Πðρ,θÞ, a distance z from the plane Δðr,jÞ, is found by use of the Fresnel-Kirchhoff diffraction integral [16] ik ρ2 Uðρ,θ,zÞ ¼ exp −ik z þ 2πz 2z k r 2 2rρ cosðj−θÞ r dr dj, ∬Δ Tðr,jÞUðr,j,0Þexp -i − 2 z z
∞ ik ρ2 exp −ik z þ ∑ il exp ðilθÞ 2πz 2z l ¼ −∞ Y þ 1 þ Y −1 þ Y þ 2 þY −2 :
! 2 −r 2 r kρ r rdr, ∓sα0 r J l AðrÞexp exp −ik 2 z 2z w0
where δl,0 and δl, 7 mp are the Kronecker delta symbols. Further, the expressions (7) and (8) have been involved into Eq. (6), yielding the following equation for the total diffracted wave field: ∞ ik ρ2 Uðρ,θ,zÞ ¼ exp −ik z þ ∑ J 0 ðksδÞ½bs Y a ðrÞexp ð−iksδÞ z 2z s¼1
2.1. Diffraction of a Gaussian laser beam by rosette gratings
exp ½ðik=zÞrρcosðj−θÞ ¼
∞
sffiffiffiffiffiffiffiffiffiffiffiffiffiffi kw0 α0 ρ2 pffiffiffi exp −ik z þ Uðρ,θ,zÞ ¼ iAðr c Þ 2z 2 pffiffiffi !1=2 ∞ pffiffi ikα20 s2 z 2 ∑ bs s exp ð−iskδÞexp z ðw0 =sα0 Þ 2 s¼1 ! −z2 exp J 0 ðksδÞJ 0 ðksα0 ρÞ ðw0 =sα0 Þ2 ∞ þ 2 ∑ ð−1Þm imðp þ 1Þ J m ðksδÞJ mp ðksα0 ρÞcosðmpθÞ : ð12Þ m¼1
Now Aðr c Þ ¼
ð7Þ
(
1
when w0 o R0 ,
circðr c =R0 Þ ¼ circðsα0 z=R0 Þ
when w0 ≥R0 :
For p being an odd number, p¼ 2p′−1 (p′ is an integer) the expression in the sum over the index m in Eq. (12) becomes real and Eq. (12) gets the form sffiffiffiffiffiffiffiffiffiffiffiffiffiffi kw0 α0 ρ2 pffiffiffi exp −ik z þ Uðρ,θ,zÞ ¼ iAðr c Þ 2z 2
S. Topuzoski, L. Janicijevic / Optics Communications 300 (2013) 108–113
pffiffiffi !1=2 2 2 pffiffi ikα0 s z 2 z ∑ bs sexp ð−iskδÞexp ðw0 =sα0 Þ 2 s¼1 ! 2 −z exp J 0 ðksδÞJ 0 ðksα0 ρÞ ðw0 =sα0 Þ2 ∞
∞ þ 2 ∑ ð−1Þmðp′ þ 1Þ J m ðksδÞJ mð2p′−1Þ ðksα0 ρÞcosðmð2p0 −1ÞθÞ : m¼1
ð13Þ
Thus, the intensity distribution Iðρ,θ,zÞ of the total diffracted wave field, obtained as Iðρ,θ,zÞ∝jUðρ,θ,zÞj2 is as follows: pffiffi kw0 α0 ∞ Iðρ,θ,zÞ ¼ Aðr c Þ pffiffiffi ∑ bs sexp ð−iskδÞ 2 s¼1 pffiffiffi !1=2 2 2 ikα0 s z 2 z exp ðw0 =sα0 Þ 2 ! −z2 fJ 0 ðksδÞJ 0 ðksα0 ρÞ exp ðw0 =sα0 Þ2 ∞ 2 þ 2 ∑ ð−1Þmðp′ þ 1Þ J m ðksδÞJ mð2p0 −1Þ ðksα0 ρÞcosðmð2p′−1ÞθÞ : m¼1
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pffiffiffiffiffiffiffiffi are, respectively: L1 ¼ 3=2ðw0 =α0 Þ ¼ 38 cm, L3 ¼ L1 =3 ¼ 12:7 cm and L5 ¼ L1 =5 ¼ 7:6 cm. In Fig. 2 the transverse intensity profile of the first diffraction order (s¼1) is shown, calculated using Eq. (15) at distance z¼16 cm, showing that it is same as that in Fig. 6 in [11]. While, in Fig. 3 the intensity pattern due to the interference between the first and the third-diffraction-order beams (both bright axis), calculated using Eq. (14) at distance z¼10 cm (L1 =5 o z o L1 =3) is presented. In both graphs (having horizontal and vertical scales in millimeters, as will be the case for all forthcoming two-dimensional plots) the following parameters are used: p¼9, γ¼0.0235 rad, n¼ n′¼1.48 for incident beam wavelength λ¼1 μm, w0 ¼3.5 mm, and kδ¼13 (c¼8.62 μm). We should also mention that the first 15 terms of the sum over m were taken in calculation (this is valid for all calculated two-dimensional plots). These two plots look very similar, having the central bright core of radius 0.035 mm. The transverse intensity profile shown in Fig. 2 remains unchanged over the propagation interval from 12.7 cm to 38 cm, which characterizes this beam as nondiverging. However, our numerical results about the interference intensity profiles formed
ð14Þ From Eq. (13) we conclude that the total diffracted wave amplitude is a sum of odd-diffraction-order beams, which are nondiverging over some propagation intervals (since the argument of the zeroth and higher-order Bessel functions which define the radial amplitude distribution, does not depend on the z distance). If we take only the first term of the sum over the index s (for s ¼1), then we will get the wave field amplitude expression for the first diffraction order, same as that for the case of the CPSSA, when p ¼ 2p′−1, given by Eq. (21) in [11]. From Eq. (14) we write the intensity distribution in the sth diffraction order in the form !1=2 ! kw0 α0 2 2z2 −2z2 I s ðρ,θ,zÞ ¼ Aðr c Þ pffiffiffi sbs exp 2 ðw0 =sα0 Þ2 ðw0 =sα0 Þ2 ∞ J 0 ðksδÞJ 0 ðksα0 ρÞ þ2 ∑ ð−1Þmðp′ þ 1Þ J m ðksδÞ m¼1
J mð2p′−1Þ ðksα0 ρÞcosðmð2p′−1ÞθÞ
o2
:
ð15Þ
According to Eq. (15) it is obvious that the longitudinal intensity distribution in the sth diffraction order, determined by the multiplicators ð2z2 =ðw0 =sα0 Þ2 Þ1=2 exp ð−2z2 =ðw0 =sα0 Þ2 Þ is of Gauss-doughnut type. This factor also defines the maximum propagation distance when w0 o R0 [18] rffiffiffi rffiffiffi w0 3 w0 3 ¼ ðs′ ¼ 1,2,3,::Þ ð16Þ Lmax ¼ sα0 2 ð2s′−1Þα0 2
Fig. 2. Transverse intensity profile of the first diffraction order, at distance z¼ 16 cm, for kδ ¼ 13 (c ¼ 8.62 μm). (The central core radius is 0.035 mm.).
One can note that L1 : L3 : L5 : ::: ¼ 1 : 1=3 : 1=5 : :::, i.e. L1 ¼ 3L3 ¼ 5L5 ¼ :::Whereas, for the case when w0 ≥R0 , the maximum propagation distance of the sth-diffraction-order beam is defined by the beam truncation function Aðr c Þ as Lmax ¼
R0 R0 ¼ ðs′ ¼ 1,2,3,::Þ: sα0 ð2s′−1Þα0
ð17Þ
However, that maximum propagation distance depends on the diffraction order value s. It means that the beam in the first diffraction order, s¼1, has the biggest propagation distance and will propagate separately, non-mixed pffiffiffiffiffiffiffiffi with the higher-diffractionpffiffiffiffiffiffiffiffi order beams, in the interval: 3=2ðw0 =3α0 Þ o z o 3=2ðw0 =α0 Þ (for the case when w0 o R0 ). In that part of the z axis, the rosette grating can act as a substitute of the cosine-profiled phase Siemens star axicon in its applications. The propagation distances of the first, third and fifth diffraction orders (taking into consideration the parameters γ¼1.351¼ 0.0235 rad, n¼ n′¼1.48 for λ¼1 μm, w0 ¼3.5 mm, and for the case when w0 oR0 )
Fig. 3. Intensity pattern due to the interference between the first and the thirddiffraction-order beams (both bright axis), at distance z ¼10 cm, for kδ¼ 13 (c ¼ 8.62 μm). (The central core radius is 0.035 mm.).
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by the first and the third diffraction orders, at different z distances (varying in the interval from L1/5 ¼ 7.6 cm to L1/3 ¼12.7 cm) showed that they vary very slightly, but the central bright spot remains with same diameter because the interference effect is stronger and has impact on the patterns outward from the central part. At distances smaller than 7.6 cm the beams from the odd diffraction orders higher than the third will also interfere with the first and the third one. Since the bright axis beam described by the zeroth-order Bessel function J 0 ðksα0 ρÞ disturbs the registration of the vortex cores of the beams under the sum over m, described by the Bessel functions J mð2p′−1Þ ðksα0 ρÞ in Eq. (14), it could be eliminated for the sth diffraction order by choosing the parameter ksδ of the Bessel function J 0 ðksδÞ to have value equal to one of the zeroes of the Bessel function J 0 ðyÞ (here y ¼ ksδ). For instance, it could be done practically by manipulating the spatial frequency, depth of the relief or the refractive index of the grating. Thus, we can make the first-diffraction-order beam (s¼ 1) to have a dark axis if the parameter kδ of the grating coincides with one of the roots of the Bessel function J 0 ðkδÞ. Then from Eq. (15) we arrive to equation same as Eq. (23) in [11], and when being computed for parameters: γ ¼1.351¼ 0.0235 rad, n ¼ n′¼ 1.48 for λ¼ 1 μm, w0 ¼3.5 mm, p ¼9 and c¼ 10 μm (kδ¼15), at distance z¼ 16 cm, the transverse intensity profile will be obtained as shown in Fig. 4. As expected, the central dark spot is surrounded by 2p bright, smaller spots, situated in a circular array. Far from the transverse plane centre, bright radial lines, whose intensity is varying in azimuthal direction, are present. For this case, the calculated transverse intensity profiles for different values of the number p have been plotted, and it was concluded that the central, nondiverging, dark spot (which does not possess a phase singularity) has a bigger radius for bigger values of p [11]. The transverse intensity profile does not rotate with change of the z distance. But, the authors in [19] have generated rotating beams which are not having angular momentum, due to an interference of radial components: Tðr,θÞ ¼ Rðr,α1 Þexp ðinθÞ þ Rðr,α2 Þexp ð−inθÞ, where n is the topological charge of a single helical beam, while α1 and α2 are parameters of the radial functions. Such beams were also experimentally obtained by means of a spatial light modulator and a ring slit aperture, based on a superposition of higher-order Bessel beams [20]. While, in Fig. 5 the interference pattern formed by the first (dark axis) and the third (bright axis) diffraction-order-beams at distance z¼10 cm (L1 =5 o z oL1 =3), for the same values of the rest parameters as in Fig. 4 is shown. When being computed at different z distances, between 7.5 cm and 12.7 cm, it shows unchanged dimensions and form, and nondiverging central bright core. But, if the condition for eliminating the bright axis beam is satisfied for the third-diffraction-order beam, still it is not observed alone, and its dark core will be covered by the bright axis first-diffraction-order beam in the interval where they interfere; hence, the resulting interfering beam will have bright nondiverging core. This is shown in Fig. 6 where the interference pattern between the first-diffraction-order, bright axis beam and the third-diffraction-order, dark axis beam is presented, calculated on base on Eq. (14) at distance z ¼10 cm, for c ¼8,84 μm (kδ ¼13,3). All other parameters are same as for Fig. 4, and as previously, the scales are in millimeters. It remains almost unchanged in dimension and form along the propagation interval between 7.5 cm and 12.7 cm (only the very slight differences around the first dark ring appear), and which is of importance for alignment or guiding applications, with nondiverging central bright core. In the next Fig. 7 only the separated first diffraction order is shown at distance z¼16 cm. One can notice that the central radius core remains with same value within the intervals of the propagation
Fig. 4. Transverse intensity profile of the first-diffraction-order beam, for c¼ 10 μm (kδ ¼15), at distance z ¼ 16 cm. (The dark core radius is 0.15 mm.).
Fig. 5. Interference pattern formed by the first (dark axis) and the third (bright axis) diffraction-order-beams at distance z¼ 10 cm, for c ¼ 10 μm (kδ ¼ 15). (The central core radius is 0.01 mm.).
Fig. 6. Interference pattern between the first-diffraction-order bright axis beam and the third-diffraction-order dark axis beam, at distance z¼ 10 cm, for c¼ 8,84 μm (kδ ¼13,3). (The central core radius is 0.035 mm.).
S. Topuzoski, L. Janicijevic / Optics Communications 300 (2013) 108–113
Fig. 7. Intensity pattern for the first diffraction order at distance z ¼ 16 cm, for c ¼8,84 μm (kδ ¼ 13.3). (The central core radius is 0.035 mm.).
of the first diffraction order and of the coaxial first and third diffraction orders; thus one obtains longer nondiverging bright core with unchanged radius, compared to that of the first diffraction order only. The same conclusion is also valid for Figs. 2 and 3. The intensity distribution in the first ring array of bright spots surrounding the dark core, at given z distance and in given sth diffraction order is proportional to J 22p′−1 ðksα0 ρÞcos2 ðð2p′−1ÞθÞ since the Bessel functions with higher m-index value have their first bright rings more distant from the centre, where they interfere. Thus, the dark core radius in the sth diffraction order can be evaluated as ρs ¼ μp′,1 =ksα0 , with μp′,1 being the value of the argument of the Bessel function J 22p′−1 ðksα0 ρÞ for which the first derivative defines its first maximum. It can be calculated as a root of the equation: J 2p′ ðksα0 ρÞ ¼ J 2ðp′−1Þ ðksα0 ρÞ. While, the bright core radius in the sth diffraction order is calculated as: ρs ¼ 2:4=ksα0 .
3. Conclusion In this paper we have presented and investigated binary phase diffraction gratings whose phase profiles have form of rosettes, being constructed by the equiphase lines of the cosine-profiled phase Siemens star axicon (an optical element being proposed and investigated previously in [11]). Their diffraction characteristics when illuminated with a Gaussian laser beam, incident with its waist, were theoretically treated. The diffracted wave field was found as a sum of odd-diffraction-order coaxial nondiverging (in defined intervals) beams. Each diffraction order consists of a zeroth-order Bessel beam and a sum of higher-order Bessel beams. The bright axis of the zeroth-order BB in a given diffraction order can be eliminated by a specific choice of the parameter value ksδ; then in the transverse intensity pattern of that separate diffraction order an azimuthal variation in the first ring, in a form of a
113
nondiverging array of bright spots, surrounding the dark, nonvortex core appears. Diffraction patterns, resulting as an interference between the first and the third diffraction orders, were computer generated for different cases-when they are both with bright cores, and when one of them is with dark core. The expression for the propagation distance of the sth diffraction order was derived, showing that it decreases with the increment of the diffraction order value s. It was shown that the rosette grating can be applied as a suitable replacement of the CPSSA in its application, in the interval along the z axis where the first diffraction order exists separately, non-mixed with the higher diffraction orders. Then the wave amplitude of the first diffraction order is same as that obtained for the CPSSA. The applications of the rosette gratings are expected to be similar with those of the optical elements they represent i.e. the CPSSAs. The central bright nondiverging core is suitable for precise optical alignment, surveying, industrial inspections, optical interconnections, or trapping long thin objects, such as rods and E-coli [5]. The laser beams having transverse intensity profiles in a form of a nondiverging array of bright spots surrounding the central dark core, can be of interest for angular alignment, as well as in manipulation of micron-sized particles i.e. their trapping in dark and bright spots depending on their refractive index (compared to that of the surrounding medium). Also, studies of the escape and synchronization of a particle between two adjacent optical traps in azimuthal direction could be of interest in statistical physics research where the Bessel beams find application [5,21]. References [1] J. Dyson, Proceedings of the Royal Society 248 (1958) 93. [2] G.C. Steward, The Symmetrical Optical System, Cambridge University Press, Cambridge, 1928. [3] J. Durnin, Journal of the Optical Society of America 4 (1987) 651. [4] S. Chávez-Cerda, G.S. McDonald, G.H.C. New, Optics Communications 123 (1996) 225. [5] D. McGloin, K. Dholakia, Contemporary Physics 46 (2005) 15. [6] J.H. McLeod, Journal of the Optical Society of America 44 (1954) 592. [7] R.M. Herman, T.A. Wiggins, Journal of the Optical Society of America 8 (1991) 932. [8] J. Turunen, A. Vasara, A.T. Friberg, Applied Optics 27 (1988) 3959. [9] A.C.S. van Heel, “Modern Alignment Devices” in Vol. 1 of Progress in Optics, Springer-Verlag, New York, 1961. [10] J. Ojeda-Castañeda, P. Andrés, M. Martínez-Corral, Applied Optics 31 (1992) 4600. [11] S. Topuzoski, Lj. Janicijevic, Journal of the Optical Society of America 28 (2011) 2465. [12] S.N. Khonina, A.V. Ustinov, Advances in Optical Technologies (2012), http://dx. doi.org/10.1155/2012/918298. [13] J.A. Davis, E. Carcole, D.M. Cottrell, Applied Optics 35 (1996) 599. [14] S.N. Khonina, V.V. Kotlyar, V.A. Soifer, M.V. Shinkaryev, G.V. Uspleniev, Optics Communications 91 (1992) 158. [15] M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions, Dover publ. Inc, New York, 1964. [16] M. Born, E. Wolf, Principles of Optics, University Press, Cambridge, 1999. [17] J.A. Davis, E. Carcole, D.M. Cottrell, Applied Optics 35 (1996) 593. [18] S. Topuzoski, Lj. Janicijevic, Optics Communications 282 (2009) 3426. [19] V.V. Kotlyar, S.N. Khonina, R.V. Skidanov, V.A. Soifer, Optics Communications 274 (2007) 8. [20] R. Vasilyeu, A. Dudley, N. Khilo, A. Forbes, Optics Express 17 (2009) 23389. [21] K. Dholakia, T. Čižmár, Nature Photonics 5 (2011) 335.
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Optics Communications journal homepage: www.elsevier.com/locate/optcom
Rosette gratings as generators of nondiverging intensity patterns Suzana Topuzoski n, Ljiljana Janicijevic Institute of physics, Faculty of natural sciences and mathematics, University “Ss. Cyril and Methodius”, 1000 Skopje, Republic of Macedonia
art ic l e i nf o
a b s t r a c t
Article history: Received 10 November 2012 Received in revised form 4 March 2013 Accepted 9 March 2013 Available online 30 March 2013
In this paper we present and investigate binary phase diffraction gratings whose phase reliefs have form of rosettes-the so-called rosette gratings. They are constructed by the equiphase lines of the cosineprofiled phase Siemens star axicon-a hybrid optical element consisting of an axicon and a phase optical element with azimuthal cosine profile. It is shown that the rosette gratings transform the incident Gaussian laser beam into a sum of odd-diffraction orders beams, nondiverging in defined propagation intervals. The amplitude and intensity distributions of the separate diffraction orders, as well as the interference between them are studied. Under certain conditions these gratings can act as suitable substitutes for the cosine-profiled phase Siemens star axicons in their applications. & 2013 Elsevier B.V. All rights reserved.
Keywords: Binary phase diffraction gratings Rosette gratings Cosine-profiled phase Siemens star axicon Gaussian laser beam Nondiverging beams
1. Introduction The nondiverging or so-called “non-diffracting” optical beams (optical beams whose transverse intensity profile is remarkably resistant to the diffractive spreading over a defined propagation interval), were treated in detail many years ago by Dyson [1], being generated by circular and spiral gratings. The annular aperture of small width, which creates “diffraction-free” wavefields with zeroth-order Bessel function transverse amplitude distribution along the radial coordinate, has been described by Steward [2]. The Bessel beam (BB) as a mathematical construct was first noted by Durnin [3], who discovered that particular solutions of the Helmholtz equation, being of the Bessel type, were independent on the propagation direction; since then, these nondiverging (also commonly known as Bessel beams) have become one of the topic subjects in the optics community [4]. The zeroth-order Bessel beam has a bright central core (which can be nondiverging in a defined interval) surrounded by successive dark and bright rings, and it does not possess a phase singularity. While, the higherorder Bessel beam has a dark core around its propagation axis (which can be non-spreading in a defined interval), surrounded by successive bright and dark rings; it can posses phase singularity on the optical axis and helicoidal wavefront. Due to their unique features, the Bessel beams have found many applications in alignment in some optical and mechanical engineering, atom optics, manipulation of small particles and studying of the motion
n
Corresponding author. Tel.: þ389 71 68 29 58. E-mail addresses: [email protected], [email protected]. edu.mk (S. Topuzoski). 0030-4018/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.optcom.2013.03.023
of microscopic particles in optical potentials [5]. Besides optical elements with circular symmetry having radial variation: circular diffraction gratings [1], axicons [6,7], holographic optical elements with transmission function of axicon type [8], and certain zone plates [9], for precise alignment purposes can also serve phase diffraction gratings and phase diffractive optical elements (DOEs) with azimuthal symmetrical variation [10–12]. Thus, in [11] two phase diffractive optical elements with azimuthal periodicity of the phase retardation were proposed: the first element, named the cosine-profiled phase Siemens star, possesses azimuthal cosineprofiled phase changes deposited on a plane base; while, the second element is a combination of the first element and a thin phase axicon and is named the cosine-profiled phase Siemens star axicon (CPSSA). For both of them, theoretical investigation of the problem of Fresnel diffraction of a Gaussian laser beam, which enters with its waist and passes with its axis through their centers, was performed. It was shown that the cosine profile of the phase retardation of both optical elements produces an azimuthal cosine-profiled modulation on their diffractograms; but, it destroys the vortex characteristics of their diffraction fields. Since the subject of investigation in this work is the binary phase grating representative of the second hybrid element, the CPSSA, we will recall to its transmission function defined as: T′ðr,jÞ ¼ AðrÞexp ðikα0 rÞexp −ik½δ þδcosðpjÞ :
ð1Þ
In (1) the term exp −ik½δ þ δcosðpjÞ expresses the transmission function of the transparent phase layer with azimuthal periodicity of the thickness, p is an integer showing the number of periods of the phase change when the azimuthal variable makes one cycle (0 oj o2π), δ ¼ ðc=2Þðn−1Þ, n is the phase layer
S. Topuzoski, L. Janicijevic / Optics Communications 300 (2013) 108–113
Fig. 1. a. Binary phase relief of the rosette grating with spatial frequency p ¼ 3. b. Binary phase relief of the rosette grating with spatial frequency p ¼5.
refractive index for the incident beam of wavelength λ, c is the total depth of the relief, while k ¼ 2π=λ is the wave number. Whereas, exp ðikα0 rÞ denotes the transmission function of a very thin axicon with a base radius R0 and parameter α0 ¼ ðn′−1Þγ connected to its base angle γ (for which the approximation tan γ≈sin γ≈γ is valid) and its refractive index n′. The function A (r) plays the role of a beam truncation function and is defined as: ( 1 when w0 o R0 , AðrÞ ¼ ð2Þ circðr=R0 Þ when w0 ≥R0 , where w0 is the Gaussian laser beam waist radius, and ( 1, when r≤R0 r circ ¼ 0, when r 4 R0 : R0 The total diffracted wave field consists of coaxial wave components which are nondiverging. The coaxial optical beams in the higher mth diffraction orders, positive and negative, are vortex ones (carrying topological charges þmp and –mp, respectively) and with higher-order Bessel function amplitude radial distribution. But, because of the coupling effect, when they interfere, their optical charges transform into an azimuthal-cosine characteristic of the common wave amplitude, which is not vortex. The diffraction patterns show a central dark core surrounded by 2p bright spots in a ring array, when the zeroth-diffraction-order beam is eliminated (for a special value of the relief depth or the refractive index of the cosine-profiled phase layer). They are similar to the diffraction patterns obtained by means of a composite hologram with transmittance equal to a linear combination of transmission functions of two helical axicons with opposite topological charges [13]: Tðr,θÞ ¼ exp ð−i2πr=r o Þ exp ½iðnθ þ αÞ þexp ½−iðnθ−αÞ . The helical axicon is a hybrid of an axicon and a phase spiral plate [14]. In the
109
previous equation, n is the topological charge value of the helical axicons, r0 is a constant (connected to the axicon parameters), and θ is azimuthal coordinate. This transmission function was written, as a binary phase, to the magneto-optical spatial light modulator, which was then illuminated with a collimated laser beam. In [11] if the zeroth-diffraction-order beam (described by the Bessel function of zeroth order) is present, a central bright spot in the diffractogram appears. Often instead of the precise modeling of the phase optical element it is easier, using the computer techniques, to draw its corresponding grating substituent. One can achieve that by drawing the equiphase lines of the phase optical element and photoreducing this draw on a plate, which can further be etched (in order to obtain a binary phase grating with variable depth) or bleached (resulting with a binary phase grating with variable refractive index). The phase diffractive optical elements affect the incident beam phase and have higher diffraction efficiency compared to the amplitude ones. Because the binary phase DOEs impose only two levels of phase, their fabrication is significantly easer. In this article we present and investigate phase diffraction gratings obtained by using the transmission function of the CPSSA, Eq. (1), in order to draw the lines of its maximum phase values in polar coordinates; one can produce a binary phase grating by photoreducing this drawing on a transparent plate, and applying the etching or bleaching technique. We name them the rosette gratings because of their rose shapes. Theoretically, a Gaussian laser beam transformation through them is treated and analyzed, allowing to discuss the output wave field, its transverse intensity profile, its propagation distance, as well as the conditions to be satisfied for applying these gratings as suitable substitutes of the CPSSAs.
2. Binary phase rosette gratings under Gaussian laser beam illumination Let us construct the lines of maximum phase values of the CPSSA, being defined, according to Eq. (1), by the system of lines kα0 r−kδ−kδcosðpjÞ ¼ 2Nπ,
ðN ¼ 0,1,2,::Þ
or, as: r ¼ λN=α0 þ ðδ=α0 ÞcosðpjÞ þ δ=α0 ,
ð3Þ
that shows they have form of roses drawn in polar coordinates. By photoreducing the drawing of the maximum equiphase lines, Eq. (3), on a transparent plate, and applying further the etching process, a binary phase grating will be obtained having a roseshaped profile of its thickness (Fig. 1). That is why the grating is denoted as rosette grating. For the case of a rectangular profile of transmittance and transmission coefficient equal to one, the transmission function of the rosette grating is defined in the following way ∞ Tðr,jÞ ¼ AðrÞ ∑ bs exp ðiksα0 rÞexp ð−iksδÞexp½−iksδcosðpjÞ s¼1
∞
þ ∑ b−s exp ð−iksα0 rÞexp ðiksδÞexp ½iksδcosðpjÞ, s¼1
ð4Þ
where: b0 ¼ 0, b 7 s ¼ 7 ð2s0 −1Þ ¼ 2=ðπð2s′−1ÞÞ, b 7 s ¼ 7 2s′ ¼ 0 (s′¼ 1,2,3..), while the truncation function A(r) is defined as Eq. (2) taking into account that, now R0 is radius of the circular aperture which obstructs the grating. Applying the Jacoby-Anger identity for the Bessel function [15], the transmission function (4) can also be written as ∞ Tðr,jÞ ¼ AðrÞ ∑ bs exp ð−iksδÞexp ðiksα0 rÞ J 0 ðksδÞ s¼1
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∞ þ ∑ ð−iÞm J m ðksδÞ½exp ð−impjÞ þ exp ðimpjÞ
Z
m¼1 ∞
þ ∑ b−s exp ðiksδÞexp ð−iksα0 rÞ s¼1 ∞ , J 0 ðksδÞ þ ∑ im J m ðksδÞ½expð−impjÞ þ exp ðimpjÞ m¼1
ð5Þ meaning that it is decomposed in a series of transmission functions of cosine-profiled phase Siemens star axicons.
0
where Δ is the area of the diffractive grating contributing to the diffraction. After involving the expressions for the incident light beam and the grating transmission function Eq. (5), and applying the identity [15] ∞
∑ il J l ðkrρ=zÞexp ½ilðj−θÞ,
l ¼ −∞
it gets the form Uðρ,θ,zÞ ¼
ð6Þ
In the previous formula for the wave field amplitude, we have denoted
þ b−s Y −a ðrÞexp ðiksδÞ ∞
∞
s¼1
m¼1
þ exp ðimpθÞ ∞
∞
s¼1
m¼1
þ ∑ b−s exp ðiksδÞ ∑ Y −d ðrÞim imp J m ðksδÞ½exp ð−impθÞ þ exp ðimpθÞ ,
Z Y 7d ¼
∞
AðrÞexp 0
ð11Þ
0
By performing the integration over the azimuthal variable φ, the previous expressions transform into: ∞
∞
s¼1 ∞
∑ ð∓iÞm J m ðksδÞ δl,−mp þδl,mp m¼1
" # rffiffiffi ikzs2 α20 z −z2 Aðr c ÞJ mp ðksα0 ρÞexp , 2 exp k 2 w0 =sα0
in the form
∞
Y 7 2 ¼ 2π ∑ b 7 s exp ð∓iksδÞ
! 2 −r 2 r kρ r rdr: ∓sα0 r J mp exp −ik 2 z 2z w0
The upper integrals have been solved by using the stationary phase method [17,18]. Thus, for the integrals Y þ a and Y þ d the solutions are approximated around the critical point r c ¼ sα0 z, while, for the integrals Y −a and Y −d -around the critical point r′c ¼ −sα0 z. Since the waves described by the integrals Y −a and Y −d form virtual vortices, which propagate in the negative direction of the z axis, on the positive side of the z axis the diffracted wave field is calculated considering only the solutions for Y þ a and Y þ d . " # rffiffiffi ikzs2 α20 z −z2 Aðr c ÞJ 0 ðksα0 ρÞexp , Y þ a ðr c Þ ¼ sα0 z exp 2 k 2 ðw0 =sα0 Þ Y þ d ðr c Þ ¼ sα0 z
Y 7 2 ¼ ∑ b 7 s exp ð∓iksδÞ ∑ ð∓iÞm J m ðksδÞ s¼1 m¼1 (Z ! 2 ∞ −r 2 r kρ r rdr ∓sα0 r J l AðrÞexp exp −ik 2 z 2z w0 0 ) Z 2π exp ð−iljÞ½exp ð-impjÞ þ expðimpjÞdj :
Y 7 1 ¼ 2πδl,0 ∑ b 7 s exp ð∓iksδÞJ 0 ðksδÞ s¼1 ! 2 Z ∞ −r 2 r kρ r rdr, ∓sα0 r J l AðrÞexp exp −ik 2 z 2z w0 0
ð9Þ
with the following notations for the integrals over the radial variable ! 2 Z ∞ −r 2 r kρ Y 7a ¼ r rdr, ð10Þ ∓sα AðrÞexp r J exp −ik 0 0 z 2z w20 0
∞
Y 7 1 ¼ ∑ b 7 s exp ð∓iksδÞJ 0 ðksδÞ s¼1 ! 2 Z ∞ −r 2 r kρ r rdr ∓sα0 r J l AðrÞexp exp −ik 2 z 2z w0 0 Z 2π exp ð−iljÞdj, 0 ∞
ð8Þ
þ ∑ bs exp ð−iksδÞ ∑ Y d ðrÞð−iÞm imp J m ðksδÞ½exp ð−impθÞ
Normally to the grating plane Δðr,jÞ, a Gaussian beam Uðr,j,0Þ ¼ exp ð−r 2 =w20 Þ, whose waist of size w0 occurs in the grating plane, is entering, and its propagation axis is passing through the pole of the grating (where r ¼ 0). The diffracted wave field in the observation plane Πðρ,θÞ, a distance z from the plane Δðr,jÞ, is found by use of the Fresnel-Kirchhoff diffraction integral [16] ik ρ2 Uðρ,θ,zÞ ¼ exp −ik z þ 2πz 2z k r 2 2rρ cosðj−θÞ r dr dj, ∬Δ Tðr,jÞUðr,j,0Þexp -i − 2 z z
∞ ik ρ2 exp −ik z þ ∑ il exp ðilθÞ 2πz 2z l ¼ −∞ Y þ 1 þ Y −1 þ Y þ 2 þY −2 :
! 2 −r 2 r kρ r rdr, ∓sα0 r J l AðrÞexp exp −ik 2 z 2z w0
where δl,0 and δl, 7 mp are the Kronecker delta symbols. Further, the expressions (7) and (8) have been involved into Eq. (6), yielding the following equation for the total diffracted wave field: ∞ ik ρ2 Uðρ,θ,zÞ ¼ exp −ik z þ ∑ J 0 ðksδÞ½bs Y a ðrÞexp ð−iksδÞ z 2z s¼1
2.1. Diffraction of a Gaussian laser beam by rosette gratings
exp ½ðik=zÞrρcosðj−θÞ ¼
∞
sffiffiffiffiffiffiffiffiffiffiffiffiffiffi kw0 α0 ρ2 pffiffiffi exp −ik z þ Uðρ,θ,zÞ ¼ iAðr c Þ 2z 2 pffiffiffi !1=2 ∞ pffiffi ikα20 s2 z 2 ∑ bs s exp ð−iskδÞexp z ðw0 =sα0 Þ 2 s¼1 ! −z2 exp J 0 ðksδÞJ 0 ðksα0 ρÞ ðw0 =sα0 Þ2 ∞ þ 2 ∑ ð−1Þm imðp þ 1Þ J m ðksδÞJ mp ðksα0 ρÞcosðmpθÞ : ð12Þ m¼1
Now Aðr c Þ ¼
ð7Þ
(
1
when w0 o R0 ,
circðr c =R0 Þ ¼ circðsα0 z=R0 Þ
when w0 ≥R0 :
For p being an odd number, p¼ 2p′−1 (p′ is an integer) the expression in the sum over the index m in Eq. (12) becomes real and Eq. (12) gets the form sffiffiffiffiffiffiffiffiffiffiffiffiffiffi kw0 α0 ρ2 pffiffiffi exp −ik z þ Uðρ,θ,zÞ ¼ iAðr c Þ 2z 2
S. Topuzoski, L. Janicijevic / Optics Communications 300 (2013) 108–113
pffiffiffi !1=2 2 2 pffiffi ikα0 s z 2 z ∑ bs sexp ð−iskδÞexp ðw0 =sα0 Þ 2 s¼1 ! 2 −z exp J 0 ðksδÞJ 0 ðksα0 ρÞ ðw0 =sα0 Þ2 ∞
∞ þ 2 ∑ ð−1Þmðp′ þ 1Þ J m ðksδÞJ mð2p′−1Þ ðksα0 ρÞcosðmð2p0 −1ÞθÞ : m¼1
ð13Þ
Thus, the intensity distribution Iðρ,θ,zÞ of the total diffracted wave field, obtained as Iðρ,θ,zÞ∝jUðρ,θ,zÞj2 is as follows: pffiffi kw0 α0 ∞ Iðρ,θ,zÞ ¼ Aðr c Þ pffiffiffi ∑ bs sexp ð−iskδÞ 2 s¼1 pffiffiffi !1=2 2 2 ikα0 s z 2 z exp ðw0 =sα0 Þ 2 ! −z2 fJ 0 ðksδÞJ 0 ðksα0 ρÞ exp ðw0 =sα0 Þ2 ∞ 2 þ 2 ∑ ð−1Þmðp′ þ 1Þ J m ðksδÞJ mð2p0 −1Þ ðksα0 ρÞcosðmð2p′−1ÞθÞ : m¼1
111
pffiffiffiffiffiffiffiffi are, respectively: L1 ¼ 3=2ðw0 =α0 Þ ¼ 38 cm, L3 ¼ L1 =3 ¼ 12:7 cm and L5 ¼ L1 =5 ¼ 7:6 cm. In Fig. 2 the transverse intensity profile of the first diffraction order (s¼1) is shown, calculated using Eq. (15) at distance z¼16 cm, showing that it is same as that in Fig. 6 in [11]. While, in Fig. 3 the intensity pattern due to the interference between the first and the third-diffraction-order beams (both bright axis), calculated using Eq. (14) at distance z¼10 cm (L1 =5 o z o L1 =3) is presented. In both graphs (having horizontal and vertical scales in millimeters, as will be the case for all forthcoming two-dimensional plots) the following parameters are used: p¼9, γ¼0.0235 rad, n¼ n′¼1.48 for incident beam wavelength λ¼1 μm, w0 ¼3.5 mm, and kδ¼13 (c¼8.62 μm). We should also mention that the first 15 terms of the sum over m were taken in calculation (this is valid for all calculated two-dimensional plots). These two plots look very similar, having the central bright core of radius 0.035 mm. The transverse intensity profile shown in Fig. 2 remains unchanged over the propagation interval from 12.7 cm to 38 cm, which characterizes this beam as nondiverging. However, our numerical results about the interference intensity profiles formed
ð14Þ From Eq. (13) we conclude that the total diffracted wave amplitude is a sum of odd-diffraction-order beams, which are nondiverging over some propagation intervals (since the argument of the zeroth and higher-order Bessel functions which define the radial amplitude distribution, does not depend on the z distance). If we take only the first term of the sum over the index s (for s ¼1), then we will get the wave field amplitude expression for the first diffraction order, same as that for the case of the CPSSA, when p ¼ 2p′−1, given by Eq. (21) in [11]. From Eq. (14) we write the intensity distribution in the sth diffraction order in the form !1=2 ! kw0 α0 2 2z2 −2z2 I s ðρ,θ,zÞ ¼ Aðr c Þ pffiffiffi sbs exp 2 ðw0 =sα0 Þ2 ðw0 =sα0 Þ2 ∞ J 0 ðksδÞJ 0 ðksα0 ρÞ þ2 ∑ ð−1Þmðp′ þ 1Þ J m ðksδÞ m¼1
J mð2p′−1Þ ðksα0 ρÞcosðmð2p′−1ÞθÞ
o2
:
ð15Þ
According to Eq. (15) it is obvious that the longitudinal intensity distribution in the sth diffraction order, determined by the multiplicators ð2z2 =ðw0 =sα0 Þ2 Þ1=2 exp ð−2z2 =ðw0 =sα0 Þ2 Þ is of Gauss-doughnut type. This factor also defines the maximum propagation distance when w0 o R0 [18] rffiffiffi rffiffiffi w0 3 w0 3 ¼ ðs′ ¼ 1,2,3,::Þ ð16Þ Lmax ¼ sα0 2 ð2s′−1Þα0 2
Fig. 2. Transverse intensity profile of the first diffraction order, at distance z¼ 16 cm, for kδ ¼ 13 (c ¼ 8.62 μm). (The central core radius is 0.035 mm.).
One can note that L1 : L3 : L5 : ::: ¼ 1 : 1=3 : 1=5 : :::, i.e. L1 ¼ 3L3 ¼ 5L5 ¼ :::Whereas, for the case when w0 ≥R0 , the maximum propagation distance of the sth-diffraction-order beam is defined by the beam truncation function Aðr c Þ as Lmax ¼
R0 R0 ¼ ðs′ ¼ 1,2,3,::Þ: sα0 ð2s′−1Þα0
ð17Þ
However, that maximum propagation distance depends on the diffraction order value s. It means that the beam in the first diffraction order, s¼1, has the biggest propagation distance and will propagate separately, non-mixed pffiffiffiffiffiffiffiffi with the higher-diffractionpffiffiffiffiffiffiffiffi order beams, in the interval: 3=2ðw0 =3α0 Þ o z o 3=2ðw0 =α0 Þ (for the case when w0 o R0 ). In that part of the z axis, the rosette grating can act as a substitute of the cosine-profiled phase Siemens star axicon in its applications. The propagation distances of the first, third and fifth diffraction orders (taking into consideration the parameters γ¼1.351¼ 0.0235 rad, n¼ n′¼1.48 for λ¼1 μm, w0 ¼3.5 mm, and for the case when w0 oR0 )
Fig. 3. Intensity pattern due to the interference between the first and the thirddiffraction-order beams (both bright axis), at distance z ¼10 cm, for kδ¼ 13 (c ¼ 8.62 μm). (The central core radius is 0.035 mm.).
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by the first and the third diffraction orders, at different z distances (varying in the interval from L1/5 ¼ 7.6 cm to L1/3 ¼12.7 cm) showed that they vary very slightly, but the central bright spot remains with same diameter because the interference effect is stronger and has impact on the patterns outward from the central part. At distances smaller than 7.6 cm the beams from the odd diffraction orders higher than the third will also interfere with the first and the third one. Since the bright axis beam described by the zeroth-order Bessel function J 0 ðksα0 ρÞ disturbs the registration of the vortex cores of the beams under the sum over m, described by the Bessel functions J mð2p′−1Þ ðksα0 ρÞ in Eq. (14), it could be eliminated for the sth diffraction order by choosing the parameter ksδ of the Bessel function J 0 ðksδÞ to have value equal to one of the zeroes of the Bessel function J 0 ðyÞ (here y ¼ ksδ). For instance, it could be done practically by manipulating the spatial frequency, depth of the relief or the refractive index of the grating. Thus, we can make the first-diffraction-order beam (s¼ 1) to have a dark axis if the parameter kδ of the grating coincides with one of the roots of the Bessel function J 0 ðkδÞ. Then from Eq. (15) we arrive to equation same as Eq. (23) in [11], and when being computed for parameters: γ ¼1.351¼ 0.0235 rad, n ¼ n′¼ 1.48 for λ¼ 1 μm, w0 ¼3.5 mm, p ¼9 and c¼ 10 μm (kδ¼15), at distance z¼ 16 cm, the transverse intensity profile will be obtained as shown in Fig. 4. As expected, the central dark spot is surrounded by 2p bright, smaller spots, situated in a circular array. Far from the transverse plane centre, bright radial lines, whose intensity is varying in azimuthal direction, are present. For this case, the calculated transverse intensity profiles for different values of the number p have been plotted, and it was concluded that the central, nondiverging, dark spot (which does not possess a phase singularity) has a bigger radius for bigger values of p [11]. The transverse intensity profile does not rotate with change of the z distance. But, the authors in [19] have generated rotating beams which are not having angular momentum, due to an interference of radial components: Tðr,θÞ ¼ Rðr,α1 Þexp ðinθÞ þ Rðr,α2 Þexp ð−inθÞ, where n is the topological charge of a single helical beam, while α1 and α2 are parameters of the radial functions. Such beams were also experimentally obtained by means of a spatial light modulator and a ring slit aperture, based on a superposition of higher-order Bessel beams [20]. While, in Fig. 5 the interference pattern formed by the first (dark axis) and the third (bright axis) diffraction-order-beams at distance z¼10 cm (L1 =5 o z oL1 =3), for the same values of the rest parameters as in Fig. 4 is shown. When being computed at different z distances, between 7.5 cm and 12.7 cm, it shows unchanged dimensions and form, and nondiverging central bright core. But, if the condition for eliminating the bright axis beam is satisfied for the third-diffraction-order beam, still it is not observed alone, and its dark core will be covered by the bright axis first-diffraction-order beam in the interval where they interfere; hence, the resulting interfering beam will have bright nondiverging core. This is shown in Fig. 6 where the interference pattern between the first-diffraction-order, bright axis beam and the third-diffraction-order, dark axis beam is presented, calculated on base on Eq. (14) at distance z ¼10 cm, for c ¼8,84 μm (kδ ¼13,3). All other parameters are same as for Fig. 4, and as previously, the scales are in millimeters. It remains almost unchanged in dimension and form along the propagation interval between 7.5 cm and 12.7 cm (only the very slight differences around the first dark ring appear), and which is of importance for alignment or guiding applications, with nondiverging central bright core. In the next Fig. 7 only the separated first diffraction order is shown at distance z¼16 cm. One can notice that the central radius core remains with same value within the intervals of the propagation
Fig. 4. Transverse intensity profile of the first-diffraction-order beam, for c¼ 10 μm (kδ ¼15), at distance z ¼ 16 cm. (The dark core radius is 0.15 mm.).
Fig. 5. Interference pattern formed by the first (dark axis) and the third (bright axis) diffraction-order-beams at distance z¼ 10 cm, for c ¼ 10 μm (kδ ¼ 15). (The central core radius is 0.01 mm.).
Fig. 6. Interference pattern between the first-diffraction-order bright axis beam and the third-diffraction-order dark axis beam, at distance z¼ 10 cm, for c¼ 8,84 μm (kδ ¼13,3). (The central core radius is 0.035 mm.).
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Fig. 7. Intensity pattern for the first diffraction order at distance z ¼ 16 cm, for c ¼8,84 μm (kδ ¼ 13.3). (The central core radius is 0.035 mm.).
of the first diffraction order and of the coaxial first and third diffraction orders; thus one obtains longer nondiverging bright core with unchanged radius, compared to that of the first diffraction order only. The same conclusion is also valid for Figs. 2 and 3. The intensity distribution in the first ring array of bright spots surrounding the dark core, at given z distance and in given sth diffraction order is proportional to J 22p′−1 ðksα0 ρÞcos2 ðð2p′−1ÞθÞ since the Bessel functions with higher m-index value have their first bright rings more distant from the centre, where they interfere. Thus, the dark core radius in the sth diffraction order can be evaluated as ρs ¼ μp′,1 =ksα0 , with μp′,1 being the value of the argument of the Bessel function J 22p′−1 ðksα0 ρÞ for which the first derivative defines its first maximum. It can be calculated as a root of the equation: J 2p′ ðksα0 ρÞ ¼ J 2ðp′−1Þ ðksα0 ρÞ. While, the bright core radius in the sth diffraction order is calculated as: ρs ¼ 2:4=ksα0 .
3. Conclusion In this paper we have presented and investigated binary phase diffraction gratings whose phase profiles have form of rosettes, being constructed by the equiphase lines of the cosine-profiled phase Siemens star axicon (an optical element being proposed and investigated previously in [11]). Their diffraction characteristics when illuminated with a Gaussian laser beam, incident with its waist, were theoretically treated. The diffracted wave field was found as a sum of odd-diffraction-order coaxial nondiverging (in defined intervals) beams. Each diffraction order consists of a zeroth-order Bessel beam and a sum of higher-order Bessel beams. The bright axis of the zeroth-order BB in a given diffraction order can be eliminated by a specific choice of the parameter value ksδ; then in the transverse intensity pattern of that separate diffraction order an azimuthal variation in the first ring, in a form of a
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nondiverging array of bright spots, surrounding the dark, nonvortex core appears. Diffraction patterns, resulting as an interference between the first and the third diffraction orders, were computer generated for different cases-when they are both with bright cores, and when one of them is with dark core. The expression for the propagation distance of the sth diffraction order was derived, showing that it decreases with the increment of the diffraction order value s. It was shown that the rosette grating can be applied as a suitable replacement of the CPSSA in its application, in the interval along the z axis where the first diffraction order exists separately, non-mixed with the higher diffraction orders. Then the wave amplitude of the first diffraction order is same as that obtained for the CPSSA. The applications of the rosette gratings are expected to be similar with those of the optical elements they represent i.e. the CPSSAs. The central bright nondiverging core is suitable for precise optical alignment, surveying, industrial inspections, optical interconnections, or trapping long thin objects, such as rods and E-coli [5]. The laser beams having transverse intensity profiles in a form of a nondiverging array of bright spots surrounding the central dark core, can be of interest for angular alignment, as well as in manipulation of micron-sized particles i.e. their trapping in dark and bright spots depending on their refractive index (compared to that of the surrounding medium). Also, studies of the escape and synchronization of a particle between two adjacent optical traps in azimuthal direction could be of interest in statistical physics research where the Bessel beams find application [5,21]. References [1] J. Dyson, Proceedings of the Royal Society 248 (1958) 93. [2] G.C. Steward, The Symmetrical Optical System, Cambridge University Press, Cambridge, 1928. [3] J. Durnin, Journal of the Optical Society of America 4 (1987) 651. [4] S. Chávez-Cerda, G.S. McDonald, G.H.C. New, Optics Communications 123 (1996) 225. [5] D. McGloin, K. Dholakia, Contemporary Physics 46 (2005) 15. [6] J.H. McLeod, Journal of the Optical Society of America 44 (1954) 592. [7] R.M. Herman, T.A. Wiggins, Journal of the Optical Society of America 8 (1991) 932. [8] J. Turunen, A. Vasara, A.T. Friberg, Applied Optics 27 (1988) 3959. [9] A.C.S. van Heel, “Modern Alignment Devices” in Vol. 1 of Progress in Optics, Springer-Verlag, New York, 1961. [10] J. Ojeda-Castañeda, P. Andrés, M. Martínez-Corral, Applied Optics 31 (1992) 4600. [11] S. Topuzoski, Lj. Janicijevic, Journal of the Optical Society of America 28 (2011) 2465. [12] S.N. Khonina, A.V. Ustinov, Advances in Optical Technologies (2012), http://dx. doi.org/10.1155/2012/918298. [13] J.A. Davis, E. Carcole, D.M. Cottrell, Applied Optics 35 (1996) 599. [14] S.N. Khonina, V.V. Kotlyar, V.A. Soifer, M.V. Shinkaryev, G.V. Uspleniev, Optics Communications 91 (1992) 158. [15] M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions, Dover publ. Inc, New York, 1964. [16] M. Born, E. Wolf, Principles of Optics, University Press, Cambridge, 1999. [17] J.A. Davis, E. Carcole, D.M. Cottrell, Applied Optics 35 (1996) 593. [18] S. Topuzoski, Lj. Janicijevic, Optics Communications 282 (2009) 3426. [19] V.V. Kotlyar, S.N. Khonina, R.V. Skidanov, V.A. Soifer, Optics Communications 274 (2007) 8. [20] R. Vasilyeu, A. Dudley, N. Khilo, A. Forbes, Optics Express 17 (2009) 23389. [21] K. Dholakia, T. Čižmár, Nature Photonics 5 (2011) 335.