Diffraction of conic and Gaussian beams by a spiral phase plate

$Page 1: Diffraction of conic and Gaussian beams by a spiral phase plate$
Diffraction of conic and Gaussian beams bya spiral phase plate

Victor V. Kotlyar, Alexey A. Kovalev, Svetlana N. Khonina, Roman V. Skidanov,Victor A. Soifer, Henna Elfstrom, Noora Tossavainen, and Jari Turunen

An analytical expression for the spatial spectrum of the conic wave diffracted by a spiral phase plate(SPP) with arbitrary integer singularity of order n is obtained. Conic wave diffraction by the SPP isequivalent to plane-wave diffraction by a helical axicon. A comparison of the conic wave and Gaussianbeam diffraction on a SPP is made. It is shown that in both cases a light ring is formed, with the intensityfunction growing in proportion to �2n at small values of radial variable � and decreasing as n2��4 at large�. By use of direct e-beam writing on the resist, a 32 level SPP of the 2nd order and diameter 5 mm ismanufactured. By use of this SPP, a He–Ne laser beam is transformed into a beam with phase singularityand ringlike intensity distribution. A four-order binary diffractive optical element (DOE) with its trans-mittance proportional to a linear superposition of four angular harmonics is also manufactured. With thisDOE, simultaneous optical trapping of several polystyrene beads of diameter 5 �m is performed. © 2006Optical Society of America

OCIS codes: 050.1940, 050.1970.

1. Introduction

A spiral phase plate (SPP), i.e., an optical elementwith its transmittance proportional to exp�in��,where � is the polar angle and n is an integer, wasmanufactured and analyzed for the first reportedtime by Khonina et al.1 Interest in the SPP has in-creased lately, especially because of new opportuni-ties for optical microparticle manipulation.2–7 A SPPwith n � 3 several millimeters in diameter and amaximal microrelief height of 5 �m was manufac-tured by use of mold technology and characterized fora wavelength of 831 nm.2 The fabrication accuracy ofthe SPP surface relief on the polymer was very high(error of �3%). With the conventional scanning-electron microscope converted for electronic lithogra-phy, a SPP with n � 1, a continuous relief profile�500 �m in diameter) and a maximal step height of

1.04 �m for a He–Ne laser of wavelength 633 nm wasrecorded by direct e-beam writing on a negativephotoresist, SU-8, as reported in Ref. 3. The SSP-produced Gaussian beam diffraction pattern wasfound to differ from the ideal doughnut shape by only10%. A SPP manufactured in this way was used forsimultaneous optical trapping of six latex micro-spheres, each of which had a diameter of 3 �m andrefractive index n� � 1.59. The same authors showedin Ref. 4 that, when the SPP center was displacedfrom the Gaussian beam axis, an off-axis vortex wasformed, with the cross-section intensity distributionrotating about the optical axis on beam propagation.

In Ref. 5 a 16 level and 16 sector SPP for an im-pulse Ti:sapphire laser for a wavelength of 789 nmwas manufactured and studied by a standard photo-lithographic technique with four binary amplitudemasks. The SPP was manufactured on a SiO2 sub-strate of 100 mm diameter and maximal step height928 nm. In Ref. 6 a SPP of 2.5 mm diameter andrelief depth 1082 nm was reported that had beenmanufactured by direct e-beam writing on a negativephotoresist for 514 nm wavelength. Besides, a theo-retical analysis of Fresnel diffraction of a plane waveand a Gaussian beam by the SPP was made in Ref. 6.In Ref. 7 a SPP with a large singularity order,n � 80, was described that was formed by use of aliquid-crystal spatial light modulator and a double-frequency neodymium laser of 532 nm wave-

V. V. Kotlyar, A. A. Kovalev, S. N. Khonina ([email protected]),R. V. Skidanov, and V. A. Soifer are with the Image ProcessingSystems Institute, Russian Academy of Sciences, Samara StateAerospace University, 151 Molodogvardejskaya, Samara 443001,Russia. H. Elfstrom, N. Tossavainen, and J. Turunen are with theDepartment of Physics, University of Joensuu, P.O. Box 111, FIN-80101 Joensuu, Finland.

Received 6 September 2005; revised 16 November 2005; accepted18 November 2005; posted 2 December 2005 (Doc. ID 64447).

0003-6935/06/122656-10$15.00/0© 2006 Optical Society of America

2656 APPLIED OPTICS � Vol. 45, No. 12 � 20 April 2006

length. Besides, in Ref. 7 analytical expressions forGaussian-beam Fraunhofer diffraction by SPPs withlarge order singularities, n �� 1, were given.

The SPP can be used as a spatial filter to imple-ment a generalized Hilbert transform1 and to in-crease contrast when phase objects are viewed undera microscope8 In Ref. 9 the results of computer sim-ulation of the vector diffraction of linearly polarizedlight by a SPP were reported. In particular, the in-tensity distribution in the focus of a spherical lens ofnear-unit numerical aperture was numerically calcu-lated. It was shown that the intensity distribution onthe ring becomes asymmetrical because of polariza-tion effects; this is indeed a general property of high-numerical-aperture focusing systems.

Alongside studies of the diffraction of a plane waveand a Gaussian beam by the SPP, the diffraction of aconic wave by a SPP was also considered. An opticalelement of transmittance proportional to the productof the transmittances of an axicon and a SPP wasmanufactured on a low-contrast photoresist by gray-scale photolithography.10 This element was called aTrochoson (i.e., an element forming a light tube) inRef. 10. Such an optical element is also referred to asa helical axicon.11 Plane-wave diffraction by such anelement is equivalent to the conic wave diffraction bythe SPP. Optical trapping and rotation (with a periodof 2 s) of yeast particles and polystyrene beads of5 �m diameter were performed with a He–Ne laserand a 16 level helical axicon of the fifth order and6 mm diameter, manufactured by direct e-beamwriting.12

In Ref. 13 the generation of helico-conical opticalbeams with helical (instead of ringlike) intensity dis-tributions by a liquid-crystal spatial light modulatorwas reported.

Reference 14 reported that a SPP for circularlypolarized radiation of 10.6 �m wavelenth had beenfabricated and studied. The element consisted of dis-crete space-variant subwavelength dielectric grat-ings. Disadvantages of this method are that theincident wave is assumed to be circularly polarizedand that the optical scheme contains extra elements(a quarter-wave plate and a polarizer), thus reducingthe diffraction efficiency. In addition, a subwave-length grating for the visible range is difficult to fab-ricate.

In this paper an analytical expression is derived thatdescribes the wave field at the focal plane of an ideal,diffraction-unlimited, positive Fourier-transform lensilluminated by a monochromatic helico-conic wave.Analysis of the derived expression shows that a far-field diffraction pattern in the form of a light ring ofinfinite intensity is formed when the plane wave isdiffracted by the helical axicon. The radial intensitydistribution near the diffraction pattern center is pro-portional to r2n, where r is the radial coordinate and nis the SPP’s singularity order. When r is large, theintensity beyond the ring decreases proportionally ton2r�4. An analytical expression for the Gaussian-beam Fraunhofer diffraction by the SPP was ob-

tained in Refs. 6 and 15. In this paper we show thatthe Gaussian beam also forms a ring in a far-fielddiffraction pattern after diffraction by the SPP. It isalso shown that when r is small (inside the ring) theintensity function changes proportionally to �wr�2n,where w is the Gaussian-beam waist radius, andwhen r is large (outside the ring) the intensity de-creases similarly to that for a helical axicon, i.e., pro-portionally to n2r�4.

Also, in this paper we describe experimental stud-ies of SPP-aided generation of a high-quality tube-shaped beam with a ringlike intensity distribution.The SPP was manufactured on a resist by directe-beam writing. The technology used, as distinct fromthat described in Ref. 3, does not impose strong re-strictions on the SPP diameter. Experimental pat-terns of Fraunhofer diffraction of a finite-radiusplane wave by a SPP with n � 2, a diffraction helicalaxicon with n � 5, and diffraction of the Gaussianbeam by a SPP with n � 2 were obtained. The calcu-lated and experimental curves for the radial intensitydistribution are in good agreement.

Also, as distinct from the DOE discussed in Ref. 3,a four-order DOE was manufactured for simulta-neous manipulation of several microparticles. ThisDOE forms four light beams simultaneously withring intensity distributions and phase singularities of�3 and �7 orders, propagating at different angleswith respect to the optical axis. The transmittance ofsuch a DOE is proportional to a linear combination offour angular harmonics, each of which is the trans-mittance of the SPP of the corresponding order.

2. Fourier Transform of an UnlimitedHelico-conic Wave

The transmittance of the helical axicon10 has the fol-lowing form:

fn�r, �� exp�i�r in��, n � 0, � 1, � 2, . . . ,(1)

where � is an axicon’s parameter, which is related toangle � at the cone’s apex, refractive index n�, andwavelength � by the following equation:

� �2�n� � 1�� tan��2�

. (2)

Equation (2) holds in the paraxial case. When theSPP is illuminated by a conic wave with apex angle2 , function (1) can also be considered the complexamplitude of light immediately after the SPP, withthe transmittance given by

gn�r, �� exp�in��, n � 0, � 1, � 2, . . . . (3)

In this case, conic wave parameter � will be given by

� �2 sin

�. (4)

20 April 2006 � Vol. 45, No. 12 � APPLIED OPTICS 2657

The spatial spectrum of an unlimited plane wavediffracted by a helical axicon or the diffraction of aconic wave by a SPP is described by the Fourier trans-form of function (1). In polar coordinates, the far-fieldcomplex light amplitude will take the form

Fn��, �� ik2f�

0

��0

2

exp�i�r in� �ikf r�

� cos�� rdrd�, (5)

where k � 2�� is the wave number and f is the focallength of the spherical lens in the rear focal plane inwhich the spatial spectrum pattern is generated.Transforming Eq. (5) yields

Fn��, �� i�n2 k

f

� exp�in��d

d� �0

�

exp�i�r�Jn�kf r�dr,

(6)

where Jn�x� is a Bessel function of nth order and thefirst kind. Using the reference integral,16 we can ob-tain the following expression:

�0

�

exp�ibx�Jn�cx�dx �

exp�in arcsin�b�c��

�c2 � b2�1�2 b � c

cnin1

�b �b2 � c2�1�2�n�b2 � c2�1�2b � c

, (7)

Substituting Eq. (7) into Eq. (6), we obtain an ex-plicit expression for the complex amplitude of thespatial spectrum of the conic wave diffracted by theSPP:

Fn��, �� kf exp�in��

��n�� in ��2 � �2�

��2 � �2�3�2�� i ��2 � �2�n� � ��

�i��n�� n �2 � ��2��2 � ��2�3�2�� 2 � ��2�n

� � ��

,

(8)

where �� k��f. From Eq. (8) it follows that, whenn � 0, the expression for the spatial spectrum ofa plane wave diffracted by a simple axicon takes

the form

Fn�� kf

�

��2 � �2�3�2 � � ��

�i�

��2 � ��2�3�2 � � ��. (9)

From Eqs. (8) and (9) it can be seen that amplitudeFn��, �� has a singularity when � � ��. This meansthat, when the distance from the Fourier plane centeris �0 � �f�k, a light ring with infinite energy densitybut finite width is being formed. The width of the ringwith infinite intensity means the radial distance be-tween two points at which the intensity equals, forexample, I�0. From Eq. (8), the intensity is given by

I�n�� Fn��, ��2

� �kf 2

�2 n2��2 � �2��2 � �2�3

� � ��,

��2n�� n �2 � �2�2

��2 � ��2�3�� 2 � ��2�2n� � ��

.

(10)

To obtain the width of the ring, we assume that theintensity equals I�0 and find from Eq. (10) (when n� 0) two numbers, �1 and �2, which are the internaland external ring radii:

�1, 2 �fk��2 �� k

I�0f2�3�1�2

. (11)

The full width of the ring with respect to the dropin intensity, I�0�� I�0, equals �1 �2 and increaseswith the growth of axicon parameter �. Let us notethat, when � � �k�f�1�2I�0

�1�4, the internal ring radiusbecomes zero or negative. This is so because on theoptical axis, where � � 0, the intensity becomeslarger than I�0 �I�0�� 0� � I�0�. Indeed, when � � 0, itfollows from Eq. (10) that

I�n�� 0� �� 0 n � 0

� k

f�22

n � 0. (12)

It can be seen from Eq. (12) that, for a helical axicon�n � 0�, a point of zero intensity, I�n�0� � 0, will befound in the center of the Fourier plane �� 0� atany �.

Let us find a function describing the intensity onthe internal side of the ring when � tends to zero.From Eq. (10) at � �� we obtain the followingexpression:

I�n�� kf 2��

�2n�n 1�2

4n�4 . (13)


It can be seen from Eq. (13) that, if both � and ��tend to zero while their ratio �� remains constant,we obtain

I�n�� → 0, � → 0� � ��4. (14)

From expression (14) it follows that near the centerof the Fourier plane, �� 0, the intensity tends toinfinity when � → 0 but in the central point itself�� 0� the intensity will remain zero, I�n�� 0� � 0,at any small value of �.

It can also be seen from Eq. (13) that the near-centerintensity in the Fourier plane will tend to infinity pro-portionally to the 2nth power of the radius:

I�n��

�2n

, �� . (15)

Furthermore, it follows from Eq. (10) that on theouter side of the ring, when � �� , the intensityfunction decreases with growing radial variable � as

I�n�� fn

k�22

. (16)

It is seen from expression (16) that, when � is large,the behavior of the intensity function is independentof �. In particular, when � � 0 and a plane waveilluminates the SPP, it follows from expression (16)that

I�n�� 0 �� 0

�kn

f��22

�� 0. (17)

The intensity distributions on the ring’s internaland external sides for some different values of indexn are illustrated in Fig. 1(a) �k�f � 1 mm�2�.

Shown in Figs. 1(b) and 1(c) are simulated patternsof Fraunhofer diffraction of a conic wave of finite

radius R by a SPP. The intensity distributions I cal-culated by use of the Fourier transform are plottedagainst radial variable ��. It is seen from Fig. 1(b)that, with increasing SPP number n, there takesplace an insignificant decrease in the maximal inten-sity value in the main ring (at �� 2 mm�1), which isdue to an insignificant widening of the ring and to aninsignificant shift of intensity maximum towardlarger values of the radial variable ��. With n chang-ing from 0 to 10, the ring radius is increased by �7%.This might be suggested by the curves in Fig. 1(a),which show that for an infinite-radius conic beam anincrease in number n also leads to a shift of the ring’sexternal intensity curves toward larger values of ��.

From Fig. 1(c) it is seen that, with n held constant,a decrease in radius R of the conic wave causes themain ring of the diffraction pattern to widen and themaximal value of the ring intensity to decrease. Notethat the ring radius, i.e., the distance of the ringmaximum to the center, �� 2 mm�1, remains un-changed. It is noteworthy that the narrowest inten-sity distribution is formed when the conic wave’sradius is infinite. Thus the analytical expressions de-rived for diffraction of an unlimited conic wave by aSPP prove to be useful for analysis of a real situationwhen the conic wave’s radius is limited.

To conclude this section, we consider in detail am-plitude [Eq. (9)] as a function of the second kind ofdiscontinuity at � � ��. Inside the ring, when �� , F0�� in Eq. (9) is a real function; outside the ring,when � � ��, the F0�� function is purely imaginary.Thus, when radius � intersects the point � � �� theF0�� function acquires a phase shift of ��2 rad.When � � ��, the F0�� function tends to infinity, andif we assume � to be equal to zero, F0�� becomes the� function. It follows from Eq. (5) that, whenn � 0 and � � 0,

F0�� ik

f �0

�

J0�r��rdr ��ik��

f�� . (18)

Fig. 1. Radial intensity distribution in the spatial spectrum plane when a conic wave is diffracted by a SPP (k�f 1 mm2). (a) Infiniteaperture, different orders of singularities n: 0 (curve A), 3 (curve B), and 5 (curve C). (b) Finite aperture, (R 35 mm), different ordersof singularities n: 0, 3, 5, 10 (curves A, B, C, and D respectively). (c) Singularity order n 3, different apertures: infinite (curve A) and R 50, 35, 20 mm (curves B, C, and D, respectively).


As the integral of the � function is equal to 1, i.e.,

��

�

��x�dx � 1,

function (18) yields

�0

�

F0��d� ��if2k . (19)

To obtain an expression analogous to Eq. (19) fromthe explicit form of the F0�� function in Eq. (9), weneed to get rid temporarily of the divergence at� � ��. This can be done by use of the well-knownintegral16

�0

�

exp��i� �r�J0��r�rdr �

i�

��2 � �2 2 2i� �3�2. (20)

It can be seen that the function on the right-handside of Eq. (20) coincides with the right-hand side ofEq. (9) when � 0. The function on the right-handside of Eq. (20) cannot have a zero in the denominatorat any real value of � and ��. Let us use this idea to

consider the following function:

F0��, � ��k

f��2 � �2 2i� �3�2. (21)

When � 0, function F0��, � 0� coincides withfunction (9). Let us take the integral of this function:

�0

�

F0��, ��d� ��if

2k�1 � 2i �1�2 . (22)

Equation (22) clearly coincides with Eq. (19) when � 0. Note that Eq. (22) is correct for any �, even for� � 0. This proves that function (21) tends to the �function when � approaches zero and when � 0.Although the integral of amplitude (22) is finite, de-spite the second-kind discontinuity at point � � �� theintegral of the light-field intensity (full energy) willbe infinite. Actually, it follows from Eq. (21) that

I�0��, � � �F0��, ��2 � ��kf 2

��2 � �2�2 4�2 2��3�2.

(23)

The integral of function (23) can be evaluated ana-lytically, and the result is

�0

�

I�0��, ��d� �1

8 2�1 �1 4 2

�2�1�2�. (24)

Fig. 2. Generation of a laser field with second-order phase singularity: (a) desired phase distribution, (b) central part of the SPPmicrorelief. Shown also are the field intensity distributions registered by a CCD camera at several distances from the SPP: (c) z 10 mm,(d) z 40 mm, (e) z 80 mm, (f) at the focal plane of the lens.


It follows from Eq. (24) that when → 0 the inte-gral of the intensity in Eq. (23) will tend to infinityindependently of �.

3. Fraunhofer Diffraction of a Gaussian Beamon a SPP

Explicit analytical expressions that describe Fresneldiffraction of the Gaussian beam by a SPP are givenin Refs. 6 and 15. The expression for the Gaussianbeam’s Fraunhofer diffraction by the SPP was ob-tained by passage of the beam to the limit from theFresnel diffraction to the far-field diffraction.6 In thissection an analytical formula for description of theGaussian beam’s Fraunhofer diffraction by a SPP,formed in the focal plane of a spherical lens, is ob-tained. Some results reported in Refs. 6 and 15 arebriefly recalled to permit a comparison of the diffrac-tion by the SPP of conical and Gaussian beams.

Let us consider the following source function, in-stead of Eq. (1):

fn��r, �� exp��r2

w2 in�, (25)

where w is the radius of the waist of the incidentGaussian beam. Then the complex amplitude of theFraunhofer diffraction of the Gaussian beam by theSPP is given by

Fn��, �� i�n1k

f exp�in��0

�

exp��r2

w2� Jn�k

f r�rdr. (26)

The following reference integral is well known16:

�0

�

exp��px2�Jn�cx�xdx �c

8p3�2 exp��c2

8p��I�n�1��2� c2

8p� I�n1��2� c2

8p�. (27)

Here I��x� is a modified Bessel function or a Besselfunction of the second kind. Considering Eq. (27), wecan rewrite expression (26) as

Fn��, �� i�n1 exp�in��kw2

4f 2x

� exp��x��I�n�1��2�x� � I�n1��2�x��,(28)

where

x �12�kw�

2f 2

.

The intensity function of the Fraunhofer diffrac-tion pattern of the Gaussian beam by the SPP takesthe form

I�n�� Fn��, ��2

� 2�kw2

4f 2

x exp��2x��I�n�1��2�x�

� I�n1��2(x)�2. (29)

It is seen from Eq. (29) that, when x � 0, theintensity equals zero �if n � 0� in the center of theFourier plane: I�n��0� � 0. The factors x exp��2x� inEq. (29) show that for the far-field diffraction a ring-like intensity distribution is formed. The radius of thering can be obtained from6

�n � 4x�I�n�1��2�x� �n 4x�I�n1��2�x� � 0. (30)

Let us find the intensity function on the external

Fig. 3. Fraunhofer diffraction patterns for (a) a conic wave (dif-fraction axicon’s parameter, � 44.5 mm1) and (b) a Gaussianbeam (waist radius, � 0.8 mm) diffracted by a SPP with numbers(a) n 5 and (b) n 2.


side of the ring when � → � �or x → ��. To do this weuse the asymptotic form of the Bessel function:

I��x� �exp�x� 2x

�1 �4�2 � 1

8x , x �� 1. (31)

So, when x → �, we obtain instead of Eq. (29) theresult that

I�n�� nf

k�22

. (32)

It is interesting that relation (32) does not depend onthe Gaussian beam’s waist and that it coincides withrelation (16). From this coincidence it can be con-cluded that, for � → �, the asymptotes of the inten-sity function are determined only by the SPP number,the focal length of the spherical lens, and the wave-length of light but do not depend on amplitude–phaseparameters � and w of the beam illuminating theSPP.

Let us note that we can obtain expression (32) fromEq. (29) by extending the radius of the Gaussian

beam’s waist to infinity �� → �� while � remainsfixed.

Let us next find the intensity function inside thering. When � tends to zero (with w fixed), Besselfunction argument x tends to zero also, and the firstmembers of the cylinder function’s decompositioninto a series can be used:

I��x� � �x2�

��1�� 1�, x �� 1, (33)

where ��x� is a gamma function. So, instead of Eq.(29), we obtain, when � → 0,

I�n�� 2�n 12 �kw2

f �kw�

4f 2n

. (34)

It is seen from relation (34) that the intensity nearthe center of the Fourier plane increases as the 2nthpower of the radial coordinate:

I�n�� w��2n, � �� 1. (35)

The form of the asymptotics [expression (35)] coin-cides with expression (15) for the diffraction of theconic wave by the SPP if w is replaced by 1��.

If � tends to zero while the Gaussian beam’s radiusw tends to infinity and their product w� remainsconstant, expression (34) suggests that the intensitynear the center of the Fourier plane will tend to in-finity proportionally to the squared waist radius:

I�n�� → 0, w → �� w2, �w � const. (36)

but at the central point itself (when � � 0) the inten-sity will be zero, I�n�� 0� � 0, at any value of w.

4. Experimental Studies of Light Diffraction by a SPP

Fabrication of SSPs and experiments with SPPs ob-tained by modern microlithography technology havebeen reported elsewhere.1–6 However, each of thesetechnologies has drawbacks of its own. For example,Ref. 3 describes a SPP fabricated by use of a conven-tional scanning-electron microscope speciallyadapted for direct writing on a resist. However, thecontrol system of the microscope’s e beam does notallow one to write on the resist high-quality patternsof diameters larger than 500 �m. The photolithogra-phy technology with use of several photomasks thatwas used for SPP fabrication by Sueda et al.15 in-volves a critical operation of photomask alignment,resulting in additional errors in microrelief fabrica-tion. Conventional e-beam lithography in the directe-beam writing mode on a resist for SPP fabrication isfree of the listed drawbacks.

In this paper we describe the generation of a lightfield with second-order singularity with a 32 levelSPP. The element size is 2.5 mm � 2.5 mm, and thepixel size is 5 �m � 5 �m. The SPPs were intended to

Fig. 4. Diffraction of (a) a conic wave and (b) a Gaussian beam bya SPP: solid curves are experimental and solid curves with circlesare theoretical intensity distributions.


operate at wavelength � � 633 nm. The microreliefdepth measured with a contact profilometer is1320 nm. The optimal depth of the 32 level microre-lief is 1341 nm if we assume that the refractive indexof the resist is nr � 1.457 (the exact value is notknown). Thus the deviation from the intended depthis only �1.5%. In Ref. 6 the fabrication of a SPP byuse of the same technology but for a wavelength of514 nm, with a wavelength of 543 nm used in exper-iments, was reported. This led to a low quality oftubular beam generation. In the present paper thesame wavelength, 633 nm of a He–Ne laser, was usedboth for designing the SPP and in experiments. Be-cause of this, the intensity distribution of the gener-ated beam really possesses a ring symmetry. Figure 2shows the experimental results of generating the la-ser field with the second-order phase singularity. Fig-ure 2(a) illustrates the theoretical phase distribution[white denotes the zero phase and black shows phase2�1 � 1�N�, where N is the number of quantizationlevels]. Figure 2(b) shows the SPP microrelief derivedby use of a Newview 5000 ZYGO interferometer�200� magnification, tilted view). Figures 2(c)–2(e)show the intensity distributions registered with aCCD camera at various distances from the SPP. Fig-ure 2(f) depicts the Fraunhofer diffraction patternobtained with use of a spherical lens of focusf � 135 mm when a plane wave is diffracted by theSPP specified above.

Figure 3 shows experimental intensity distribu-tions in the focal plane of a spherical lens of focusf � 135 mm. The annular intensity distribution inFig. 3(a) was obtained when a plane wave of radius

R � 2.5 mm was diffracted by a diffraction axiconwith parameter � � 44.5 mm�1 and a SPP with num-ber n � 5. The wavelength of light was �� 0.633 �m for a 1 mW laser beam. A second, largerring [Fig. 3(a)] appears because the maximal reliefheight fails to produce an exact phase delay of 2m.The annular intensity distribution in Fig. 3(b) wasderived when a Gaussian beam of waist radius �� 0.8 mm was diffracted by a SPP with number n� 2. Some ellipticity of the diffraction pattern is dueto inaccurate alignment of the Gaussian beam andthe SPP center. Note that there is no such problemwhen a plane wave of limited radius is diffracted bythe SPP [Fig. 2(f)].

Figure 4 compares theoretical and experimentalprofiles of the annular intensity distributions shownin Fig. 3. The ring’s radius in Fig. 4(a) approximatelyequals the estimate in Eq. (8): �0 � �f�k � 605 �m,while the ring width can be derived from �� 2�f�R � 68 �m. The radius of the ring in Fig. 4(b)can be derived from a relation in Ref. 6: �2 �0.46�f�� 45 �m. It is seen from Fig. 4 that theexperimental and theoretical curves agree fairly well.

5. Optical Micromanipulation by Use of SPPs

The use of a SPP for optical manipulation was re-ported in Ref. 3. Note that the SPP can form only onebeam with phase singularity of some order. Thebeams with different phase singularity orders havedifferent orbital angular momenta and thus interactwith microparticles in different ways. In this paperwe report that a diffractive optical element (DOE)

Fig. 5. Four-order DOE to generate laser beams with singularities of the �3rd and �7th orders.

Fig. 6. Microparticle motion in a laser beam with the 7th order angular harmonic.


working as four SPPs with different singularityorders was manufactured and applied for opticaltrapping. The transmittance of such a DOE is pro-portional to a linear combination of four angular har-monics of the form of Eq. (3). Note that the amplitudecoefficients of these harmonics and their orders arechosen such that the DOE’s transmittance is binary.

Figure 5(a) shows that the phase of a four-orderDOE simultaneously generates laser beams with sin-gularities of the �3rd and �7th orders; Fig. 5(b)shows the central part of the microrelief; and Fig. 5(c)shows the experimental intensity distribution in theFourier plane.

We conducted a number of experiments on micro-manipulation with a laser beam composed of fourorders with 3rd- (small rings) and 7th- (large rings)order angular harmonics.

Figure 6 depicts stages of motion of a polystyrenemicroparticle (5 m in diameter) trapped in a laserbeam with the 7th-order angular harmonic. The mi-croparticle is clearly seen to be drawn into the brightring, moving along it with a constant velocity. Agroup of microparticles at the right that were trappedinto the 3rd-order bright ring fails to move, havingstuck to the cell’s bottom. Figure 7 depicts variousstages of trapping of a four-microparticle group in alaser beam with the 3rd-order angular harmonic. Itcan be seen from Fig. 7 that all four microparticlesare being drawn into the bright ring.

6. Conclusions

In the research reported here the following resultswere obtained: An analytical expression for the spa-tial spectrum of the conic wave diffracted by a spiralphase plate with arbitrary integer singularity order nwas obtained. The diffraction of the conic wave by aSPP is equivalent to the diffraction of the plane waveby a helical axicon. An analytical expression for theGaussian beam’s Fraunhofer diffraction by a SPPlocated in the beam’s waist was obtained. Diffractionof the conic wave and that of the Gaussian beam bythe SPP have been compared analytically. It has beenshown that in both cases a light ring is formed, withthe intensity function increasing proportionally to �2n

at small values of the radial variable � and decreasingas n2��4 at large values of �.

Using direct electron-beam writing on a resist, wefabricated a 32 level SPP of 2nd order, of diameter5 mm. This SPP was used to convert a He–Ne laserbeam into a beam with phase singularity and ringlikeintensity distribution. Fraunhofer diffraction pat-

terns for a finite-radius plane wave diffracted by aSPP with n � 2 and a diffractive helical axicon withn � 5 and for a Gaussian beam diffracted by a SPPwith n � 2 have been derived experimentally. Calcu-lated and experimental curves of the radial intensitydistribution are in good agreement.

A four-order binary DOE with the transmittanceproportional to a linear superposition of four angularharmonics also has been manufactured. With thehelp of this DOE, simultaneous optical trapping ofseveral polystyrene beads �5 �m in diameter) hasbeen performed.

The work was financially supported by the RussianFederation Ministry of Education, Samara RegionAdministration, the American Civilian Research andDevelopment Foundation (CRDF project SA-014-02)as part of joint Russian–American program Basic Re-search and Higher Education, by Russian Federationpresidential grant MD-209.2003.01 and NSh-1964.2006.1, and by Russian Foundation for BasicResearch grants 05-01-96505 and 05-08-50298. TheFinnish authors acknowledge the support of theFinnish Network of Excellence in Micro-Optics(NEMO; www.micro-optics.org).

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Diffraction of conic and Gaussian beams by a spiral phase plate