Vectorial structural properties of the far-field of an apertured circular flattened Gaussian beam

Eur. Phys. J. Appl. Phys. (2012) 57: 30501DOI: 10.1051/epjap/2011110217

THE EUROPEANPHYSICAL JOURNAL

APPLIED PHYSICS

Regular Article

Vectorial structural properties of the far-fieldof an apertured circular flattened Gaussian beam

G.Q. Zhoua

School of Sciences, Zhejiang A & F University, Lin’an 311300, Zhejiang Province, P.R. China

Received: 13 May 2011 / Received in final form: 15 September 2011 / Accepted: 19 September 2011Published online: 23 February 2012 – c© EDP Sciences 2012

Abstract. Based on the method of stationary-phase and the mathematic techniques, analytical expressionsfor the TE and TM terms of an apertured circular flattened Gaussian beam (CFGB) in the far-field havebeen derived without any approximation, which allows one to calculate the energy flux distributions ofthe TE term, the TM term, and the apertured CFGB. The analytical formulae of the power of the TEterm, the TM term, and the apertured CFGB are also presented. The vectorial structural properties ofthe far-field of an apertured CFGB are demonstrated. The influences of the f-parameter, the truncationparameter, and the parameter N on the energy flux distributions of the TE term, the TM term, and theapertured CFGB are examined. Also, the effects of the f-parameter, the truncation parameter, and theparameter N on the ratios of the power of the TE and TM terms to the power of the apertured CFGB areinvestigated.

1 Introduction

The optical beam with a nearly uniform intensity distri-bution is referred to as the flattened beam. As the flat-tened beams are required in some certain applications,they have attracted a lot of attention. The propagationproperties of flattened beams in free space and in theturbulent atmosphere have been extensively investigated[1–4]. The flattened beams have been experimentally gen-erated by using different optical systems [5,6]. To de-scribe the flattened beams, many theoretical models havebeen proposed, among which a circular flattened Gaussianbeam (CFGB) is a suitable one [7,8]. The advantage of theCFGB is that the intensity distribution is varied from theflattened center to zero in a continuous and nonoscilla-tory way. The propagation of a CFGB through an opti-cal system is investigated in the turbulent atmosphere [9].The propagation of a CFGB through a misaligned parax-ial ABCD optical system has been examined [10]. Also,the propagation of a CFGB is studied from the vectorialstructure [11]. Here, the vectorial structure denotes thatan optical beam can be uniquely expressed as a sum of twoterms: the TE and TM terms. The TE term means theelectric field transverse to the propagation axis, and theTM term represents the associated magnetic field trans-verse to the propagation axis. The analytical expressionsfor the TE and TM terms of a CFGB have been derivedwithout any approximation, which are applicable to an ar-bitrary observation plane. In the practical optical systems,however, there usually exist circular apertures. Therefore,

a e-mail: [email protected]

the vectorial structure of an apertured CFGB is to beinvestigated in the remainder of this paper. Only in thefar-field, the TE and TM terms are orthogonal to eachother. Moreover, many researches and applications areconducted in the far-field. Accordingly, the vectorial struc-ture of the far-field of an apertured CFGB is examined.The utilization of the stationary-phase method combinedwith the series expansion of zeroth-order Bessel functionwill result in the accurately analytical expressions of theTE and TM terms. By using the derived formulae, theenergy flux distributions of the TE term, the TM term,and the apertured CFGB are to be demonstrated in thefar-field. Moreover, the ratios of the power of the TE andTM terms to the power of the apertured CFGB are alsoto be investigated.

2 The vectorial structural propertiesof the far-field of an apertured CFGB

In the Cartesian coordinate system, a CFGB propagatestoward half free space z ≥ 0. The z-axis is taken to bethe propagation axis. In the theoretical researches and thepractical applications, the linearly polarized state is thefamiliar and simple case. Therefore, the initial electric fieldof a CFGB in the source plane z = 0 is assumed to bepolarized in the x-direction and takes the form as [7,8]

[Ex(ρ0, 0)Ey(ρ0, 0)

]=

⎡⎣

N∑m=1

bm exp(−mρ2

0w2

0

)0

⎤⎦ , (1)

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where bm = (−1)m−1N !/Nm!(N −m)! is the weight coef-ficient, the integer N ≥ 2, and ρ0 = (x2

0 + y20)1/2. w0/

√m

is the Gaussian waist. A circular aperture with radius Rcoincides with the beam waist plane of the CFGB. TheCFGB just behind the circular aperture reads as

[Ex(ρ0, 0)Ey(ρ0, 0)

]=

⎡⎣

N∑m=1

bm exp(−mρ2

0w2

0

)0

⎤⎦ circ(ζ), (2)

where ζ = ρ0/R, and the aperture function circ(ζ) is givenby

circ(ζ) ={

1 0 ≤ ζ < 10 ζ ≥ 1 . (3)

In terms of the method of vector angular spectrum, thepropagating electric field of the apertured CFGB in thez-plane is found to be

E(ρ, z) =∫ ∞

−∞

∫ ∞

−∞Ax(p, q)

(ex − p

γez

)

× exp[ik(px + qy + γz)]dp dq, (4)

where ρ = (x2 + y2)1/2, γ = (1 − p2 − q2)1/2, andk = 2π/λ is the wave number, with λ being the opti-cal wavelength. ex and ez are the two unit vectors in thex- and z-directions, respectively. Ax(p, q) is the x com-ponent of the vector angular spectrum and given by theFourier transform of the x component of initial electricfield

Ax(p, q) =1λ2

∫ ∞

−∞

∫ ∞

−∞Ex(ρ0, 0)

× exp[−ik(px0 + qy0)]dx0 dy0

=k

λ

∫ R

0

N∑m=1

bm exp(

−mρ20

w20

)J0(kρ0b)ρ0 dρ0, (5)

where b = (p2 +q2)1/2 and J0(·) is the zeroth-order Besselfunction of the first kind. According to the theorem of thevectorial structure of an optical beam, the propagatingelectric field of the apertured CFGB can be expressed asa sum of the TE and TM terms [12–16]:

E(ρ, z) = ETE(ρ, z) + ETM (ρ, z), (6)

with ETE(ρ, z) and ETM (ρ, z) given by

ETE(ρ, z) =∫ ∞

−∞

∫ ∞

−∞

q

b2Ax(p, q)

×(qex − pey) exp[ik(px + qy + γz)]dp dq, (7)

ETM (ρ, z) =∫ ∞

−∞

∫ ∞

−∞

p

γb2Ax(p, q)

×(pγex + qγey − b2ez) exp[ik(px + qy + γz)]dp dq,

(8)

where ey is the unit vector in the y-direction. Similarly,the corresponding magnetic field of the apertured CFGB

can also be expressed as a sum of the TE and TMterms [12–16]:

H(ρ, z) = HTE(ρ, z) + HTM (ρ, z), (9)

with HTE(ρ, z) and HTM (ρ, z) given by

HTE(ρ, z) = η

∫ ∞

−∞

∫ ∞

−∞

q

b2Ax(p, q)

×(pγex + qγey−b2ez) exp[ik(px + qy + γz)]dp dq,

(10)

HTM (ρ, z) = −η

∫ ∞

−∞

∫ ∞

−∞

p

γb2Ax(p, q)

×(qex − pey) exp[ik(px + qy + γz)]dp dq, (11)

where η = (ε0/μ0)1/2. ε0 and μ0 are the electric permit-tivity and the magnetic permeability of vacuum, respec-tively. Inserting equation (5) into equation (7), the TEterm of the propagating electric field for the aperturedCFGB yields

ETE(ρ, z) =k

λ

∫ R

0

N∑m=1

bm exp(

−mρ20

w20

)U(ρ0, ρ, z)ρ0 dρ0,

(12)with U(ρ0, ρ, z) given by

U(ρ0, ρ, z) =∫ ∞

−∞

∫ ∞

−∞

q

b2J0(kρ0b)(qex − pey)

× exp[ik(px + qy + γz)]dp dq. (13)

As the condition kr = k(ρ2 + z2)1/2 → ∞ is satisfiedin the far-field regime, the method of stationary phaseis applicable. By using the method of stationaryphase [17,18], the surface integral of equation (13) is shownto have the asymptotic value [19]:

U(ρ0, ρ, z) =iλ

r

∑j

{hj(∣∣αjβj − ξ2

j

∣∣)1/2

qj(qjex − pjey)p2

j + q2j

×J0

(kρ0

√p2

j + q2j

)exp[ikrF (pj , qj , x, y)]

}

as kr → ∞, (14)

where

F (pj , qj , x, y) =pjx + qjy +

√1 − p2

j − q2j z

r. (15)

The stationary points (pj , qj) are solutions of the followingsimultaneous equations:

∂F (pj , qj , x, y)∂pj

= 0, (16)

∂F (pj , qj , x, y)∂qj

= 0. (17)

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G.Q. Zhou: Vectorial structural properties of the far-field of an apertured circular flattened Gaussian beam

Substituting equation (15) into equations (16) and (17),we can obtain

p1 = x/r, q1 = y/r. (18)

There is only one stationary point. As a result, the para-meters α1, β1, ξ1 and h1 turn out to be

α1 =∂F 2(p1, q1, x, y)

∂p21

= −1 − x2

z2, (19)

β1 =∂F 2(p1, q1, x, y)

∂q21

= −1 − y2

z2, (20)

ξ1 =∂F 2(p1, q1, x, y)

∂p1∂q1= −xy

z2, (21)

h1 = −1. (22)

Therefore, equation (14) can be analytically expressed as

U(ρ0, ρ, z) = − iλyz

ρ2r2J0

(kρ0ρ

r

)exp(ikr)(yex − xey).

(23)

To obtain the analytical expression of the TE term of thepropagating electric field, the zeroth-order Bessel functionof the first kind should be expanded as [20]:

J0

(kρρ0

r

)=

∞∑l=0

(−1)l(kρρ0)2l

22l(l!)2r2l. (24)

Substituting equations (23) and (24) into equation (12),the analytical TE term of the propagating electric field forthe apertured CFGB turns out to be

ETE(ρ, z) = iyz exp(ikr)(yex − xey)

×N∑

m=1

∞∑l=0

(−1)lbmρ2l−2[Γ (1 + l, δ2) − l!]22l+1kml+1(l!)2f2l+2r2l+2

,

(25)

where f = 1/kw0 is the f-parameter and δ = R/w0 is thetruncation parameter. Γ (1+l, δ2) is an incomplete Gammafunction. The analytical expression of the TM term of thepropagating electric field for the apertured CFGB in thefar-field reads as

ETM (ρ, z) = ix exp(ikr)(xzex + yzey − ρ2ez)

×N∑

m=1

∞∑l=0


.

(26)

The analytical expressions of the TE and TM terms of thepropagating magnetic field for the apertured CFGB in thefar-field are found to be

HTE(ρ, z) = −iηyz exp(ikr)(xzex + yzey − ρ2ez)

×N∑

m=1

∞∑l=0


, (27)

HTM (ρ, z) = iηx exp(ikr)(yex − xey)

×N∑

m=1

∞∑l=0


. (28)

The energy fluxes of the TE and TM terms for the aper-tured CFGB in the far-field plane are separately given by

〈Sz〉TE =12Re[ETE(ρ, z) × H∗

TE(ρ, z)]z

=ηy2z3

2k2r5

{N∑

m=1

∞∑l=0

(−1)lbmρ2l−1[Γ (1 + l, δ2) − l!]22l+1ml+1(l!)2f2l+2r2l

}2

, (29)

〈Sz〉TM =12Re[ETM (ρ, z) × H∗

TM (ρ, z)]z

=ηx2z

2k2r3

{N∑

m=1

∞∑l=0

(−1)lbmρ2l−1[Γ (1 + l, δ2) − l!]22l+1ml+1(l!)2f2l+2r2l

}2

,

(30)

where the angle brackets indicate an average with respectto the time. Re denotes taking the real part, and the as-terisk means the complex conjugation. As the electromag-netic fields of the TE and TM terms are orthogonal toeach other in the far-field plane, the energy flux of theapertured CFGB yields

〈Sz〉 = 〈Sz〉TE + 〈Sz〉TM =ηz(z2 + x2)

2k2r5

×{

N∑m=1

∞∑l=0

(−1)lbmρ2l[Γ (1 + l, δ2) − l!]22l+1ml+1(l!)2f2l+2r2l

}2

. (31)

When the truncation parameter δ tends to infinity,it means that there is no aperture. In this case,equations (29)–(31) are simplified to be

〈Sz〉TE =ηz2

ry2z3

2ρ2r5

{N∑

m=1

bm

m

∞∑l=0

(−1)l

l!

(ρ2

4mf2r2

)l}2

=ηz2

ry2z3

2ρ2r5

{N∑

m=1

bm

mexp

(− ρ2

4mf2r2

)}2

, (32)

〈Sz〉TM =ηz2

rx2z

2ρ2r3

{N∑

m=1

bm

m

∞∑l=0

(−1)l

l!

(ρ2

4mf2r2

)l}2

=ηz2

rx2z

2ρ2r3

{N∑

m=1

bm

mexp

(− ρ2

4mf2r2

)}2

, (33)

〈Sz〉 =ηz2

rz

2ρ2r3

(x2 +

z2

r2y2

)

×{

N∑m=1

bm

mexp

(− ρ2

4mf2r2

)}2

, (34)

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Fig. 1. (Color online) Contour graphs of normalized energy flux distributions of the TE term, the TM term, and the aperturedCFGB in the far-field plane z = 1000λ. f = 5.0, N = 4 and δ = 0.8. (a) The TE term. (b) The TM term. (c) The aperturedCFGB.

Fig. 2. (Color online) Contour graphs of normalized energy flux distributions of the TE term, the TM term, and the aperturedCFGB in the far-field plane z = 1000λ, f = 0.2, N = 4 and δ = 0.8. (a) The TE term. (b) The TM term. (c) The aperturedCFGB.

where zr = kw20/2. The above equations are consistent

with equations (61)–(63) of reference [11].The power of the TE term of the apertured CFGB in

the far-field plane is given by:

see equation (35) in the next page,

where ρ2/z2 has been replaced by τ . Using the followingintegral formula [20]:

∫ ∞

0

τn−1

(1 + τ)mdτ =

Γ (n)Γ (m − n)Γ (m)

, (36)

where Γ (·) is a Gamma function, the power of the TEterm in the far-field plane can be analytically expressedas:

see equation (37) in the next page.

Similarly, the power of the TM term in the far-fieldplane turns out to be:


The ratio of the power of the TE term to that of theapertured CFGB is described by:


CTE is determined by the f-parameter, the trunca-tion parameter δ, and the parameter N. The ratio of the

power of the TM term to that of the apertured CFGByields

CTM =PTM

PTE + PTM= 1 − CTE . (40)

3 Numerical calculations and analysis

The vectorial structural properties of the far-field ofan apertured CFGB depend on the f-parameter, the trun-cation parameter δ, and the parameter N. Now weinvestigate the influences of the f-parameter, the trun-cation parameter δ, and the parameter N on the energyflux distributions of the TE term, the TM term, and theapertured CFGB. For simplicity, η is set to be unity andthe far-field reference plane is chosen as z = 1000λ. Theenergy flux distribution is normalized according tothe maximum energy flux of the apertured CFGB.Firstly, we examine the influence of the f-parameter. InFigures 1–3, N = 4, δ = 0.8 and f = 5, 0.2 and 0.05, re-spectively. When f = 5, the beam spot pattern of the TEterm is parallel to the y-axis, and that of the TM term thex-axis. The pattern size of the TM term is far larger thanthat of the TE term. Moreover, the maximum energy fluxof the TM term is also larger than that of the TE term.The beam profile of the apertured CFGB is elliptical, andthe long axis is located at the x-axis. When f = 0.2, the

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PTE =

� ∞

−∞

� ∞

−∞〈Sz〉TEdx dy =

ηπ

k2

� ∞

0

N�m=1

N�n=1

∞�l=0

∞�s=0

(−1)l+sbmbn[Γ (1 + l, δ2) − l!][Γ (1 + s, δ2) − s!]τ l+s

22(l+s+2)ml+1ns+1(l!)2(s!)2f2(l+s+2)(1 + τ)l+s+5/2dτ (35)

PTE =ηπ

k2

N�m=1

N�n=1

∞�l=0

∞�s=0

(−1)l+sbmbn(l + s)![Γ (1 + l, δ2) − l!][Γ (1 + s, δ2) − s!]

2l+s+3ml+1ns+1(l!)2(s!)2(2l + 2s + 3)!!f2(l+s+2)(37)

PTM =ηπ

k2

N�m=1

N�n=1

∞�l=0

∞�s=0


2l+s+3ml+lns+1(l!)2(s!)2(2l + 2s + 1)!!f2(l+s+2)(38)

CT E =PT E

PT E + PT M

=

N�m=1

N�n=1

∞�l=0

∞�s=0


2l+sml+1ns+1(l!)2(s!)2(2l + 2s + 3)!!f2(l+s)

×� N�

m=1

N�n=1

∞�l=0

∞�s=0


2l+sml+1ns+1(l!)2(s!)2f2(l+s)

×�

1

(2l + 2s + 1)!!+

1

(2l + 2s + 3)!!

��−1

(39)



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pattern size of the TM term is slightly larger than thatof the TE term, and the beam profile of the aperturedCFGB is approximately circular. With further decreasingthe f-parameter, a pair of side lobes, whose shape is sim-ilar to a crescent, emerges in the beam spot pattern ofthe TM term. As a result, a pair of side lobes emerges inthe beam profile of the apertured CFGB. Therefore, thef-parameter affects the beam spot shape. Also, the largef-parameter will bring about the large pattern size of theTE term, the TM term, and the apertured CFGB.

Secondly, we investigate the influence of the trunca-tion parameter δ on the energy flux distribution. δ = 1.2

in Figure 4 and δ = 0.5 in Figure 5. Other parametersin Figures 4 and 5 are same as those in Figure 3. Com-bining Figure 3 with Figures 4–5, we can find that thetruncation parameter δ not only affects the size of the pat-tern, but also influences the beam spot shape. The smalltruncation parameter will result in the large pattern sizeof the beam spot. When the truncation parameter is largesuch as δ = 1.2, the side lobes disappear in the beampattern of the TM term and the apertured CFGB.Finally, the effect of the parameter N on the energy fluxdistribution is considered. N = 8 in Figure 6 and N = 15 inFigure 7. The rest of the parameters are same as those in

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Fig. 8. (Color online) CTE as a function of the truncation parameter δ. (a) N = 4. (b) N = 10.

Fig. 9. (Color online) CTE as a function of the parameter N. (a) δ = 0.25. (b) δ = 1.0.

Figure 3. When N = 8, a pair of side lobes emerges in thebeam spot pattern of the TE term. Accordingly, a brightouter ring is detected in the beam profile of the aperturedCFGB. When N = 15, two pairs of side lobes emerge inthe beam spot pattern of the TE and TM terms, whichresults in two bright outer rings in the beam profile of theapertured CFGB.

The contributions of the power of the TE and TMterms to the power of an apertured CFGB are alsoexamined. As CTM = 1 − CTE , the influences of thef-parameter, the truncation parameter δ, and the parame-ter N on the CTM are opposite to those of the f-parameter,the truncation parameter δ, and the parameter Non the CTE , respectively. Therefore, here only CTE istaken into account. Figure 8 represents CTE as a functionof the truncation parameter δ. When f is small such asf = 0.25, there exists a fluctuation in the curves of CTE

versus δ. Except the only one fluctuation, CTE firstincreases and then tends to a saturated value with increas-ing the truncation parameter δ. When f is large such asf = 3.0, CTE is approximately equal to 0.25 and nearlykeeps invariant with varying the truncation parameterδ. From Figure 8, we can also evaluate the effect of the

f-parameter on the CTE . With increasing thef-parameter, CTE decreases. When f is large enough, CTE

approaches the minimum value of 0.25. In this case, CTE

is nearly independent of the truncation parameter δ andthe parameter N. The influence of the parameter N onCTE is shown in Figure 9. When the truncation parameterδ is equal to 0.25, CTE augments with increasing the pa-rameter N. When δ = 1.0 and f = 0.25, CTE decreaseswith increasing the parameter N. In Figure 8, the curvesof f = 0.25 have a value in the neighborhood of δ = 1.0.Moreover, the value deepens with increasing the parame-ter N. As a result, CTE decreases with increasing the para-meter N. When δ is still 1.0 and f increases such as f = 0.5,CTE increases with increasing the parameter N.

4 Conclusions

Based on the method of stationary phase and the seriesexpansion of zeroth-order Bessel function, the analyticalexpressions for the TE and TM terms of an aperturedCFGB in the far-field have been derived without anyapproximation. The formulae for the energy flux of the

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TE term, the TM term, and the apertured CFGBbeam are presented. The powers of the TE term, the TMterm, and the apertured CFGB beam are also given, whichallows one to evaluate the contributions of the powersof the TE and TM terms to the power of the aperturedCFGB. By using the formulae derived, the far-field struc-tural property of an apertured CFGB is illustrated andanalyzed with numerical examples. This research unfoldsthe far-field characteristics of an apertured CFGB fromthe vectorial structure.

This work was supported by National Natural Science Foun-dation of China (Grant No. 61178016). The author is indebtedto the referees for valuable comments.

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Documents

Vectorial structural properties of the far-field of an apertured circular flattened Gaussian beam