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Analytical vectorial structure of truncated modified Laguerre–Gaussian beam in the far field
Optics & Laser Technology 44 (2012) 1247–1255
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Analytical vectorial structure of truncated modified Laguerre–Gaussian beam in the far field Jia Li a,n, Yanru Chen b, Quanjun Cao a a b
Institute of Fiber Optic Communication & Information Engineering, College of Information Engineering, Zhejiang University of Technology, Hangzhou 310023, China Department of Optical Engineering, Nanjing University of Science & Technology, Nanjing 210094, China
a r t i c l e i n f o
abstract
Article history: Received 6 December 2011 Received in revised form 4 January 2012 Accepted 4 January 2012 Available online 24 January 2012
Based on the vectorial angular spectrum representation of Maxwell equations and the method of stationary phase, analytical expressions are derived for the TE term, TM term and whole energy flux of a hard-edged diffracted modified Laguerre–Gaussian beam (MLGBs) in the far field, respectively. It is shown that the evolutive energy flux not only depends on the beam order, initial waist width but also relates to the truncated parameter. Furthermore, influences of the truncated parameter on the discrepancies of energy flux distributions between paraxial and non-paraxial cases are also analyzed for different selections of the initial beam width. & 2012 Elsevier Ltd. All rights reserved.
Keywords: Modified Laguerre–Gaussian beam Vectorial structure Energy flux
1. Introduction Just as the paraxial solutions to the propagation of optical fields established for some earlier families of beams such as highorder mode Gaussian beams [1] or Helmholtz–Gaussian beams [2], a novel and general beam solution of the paraxial wave equation is found in Cartesian coordinates. This new solution is termed Cartesian beam which is commonly described by several complex parameters in the most general situations [3]. Hypergeometric laser beams, which can be regarded as a special class of Cartesian modes by suitably the choice of complex cofactors, have been put forward in recent reports [4–6]. The hypergeometric beam naturally contains optical vortices which may bring initial orbital angular momentum (OAM). The tight focusing properties of hypergeometric–Gaussian beams (HGBs) through aplanatic high-numerical-aperture (NA) lens system were studied by the usage of the vectorial Debye diffraction integral [7]. The nonparaxial propagation properties of vectorial HGBs in free space were investigated by the Rayleigh–Sommerfeld diffraction integral [8]. Besides, various experimental methods for the generation of HGBs by the usage of encoded binary diffractive elements [9,10] or computer-generated hologram [11] were separately proposed. One can recognize that the HGBs would exhibit plenty of features with the change of mode parameters. By suitable selections of mode parameters, solutions to paraxial wave equations of HGBs would reduce to a subfamily of modes named the modified Laguerre–Gaussian mode [5]. Propagation properties of
modified Laguerre–Gaussian beams (MLGBs) through a paraxial ABCD optical system with a hard-edged aperture are studied by the usage of the Collins integral [12]. Very recently, propagation properties of MLGBs though a fractional Fourier transform (FRFT) system are revealed in cylindrical coordinate system [13]. The purpose of the present paper is to concentrate on the internal vectorial structure of MLGBs diffracting through a hardedged circular aperture in the far field. Firstly, based on the vectorial angular spectrum representation of Maxwell equations and the method of stationary phase, analytical expressions for the TE and TM terms of the beam are derived, respectively. Second, energy flux distributions for the TE term, TM term and whole beam are revealed by numerical plots. Finally, comparisons between the paraxial and non-paraxial propagating results are made with different choices of the initial beam width, beam order and the truncated parameter, respectively.
2. Vectorial structure of truncated MLGBs in the far field As it has been reported, the propagating electric field of a polarized beam toward the half space z Z0 can be given by means of the vectorial angular spectrum representation of electromagnetic beams [14–23] Z þ1 Z þ1 E r^ ,z ¼ Aðp,qÞexp ikðpx þqy þ mzÞ dpdq, ð1Þ 1
1
with n
Corresponding author. Tel.: þ8615068725198. E-mail address: [email protected] (J. Li).
0030-3992/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlastec.2012.01.002
p Aðp,qÞ ¼ Ax ðp,qÞ e^ x e^ z , m
ð2Þ
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J. Li et al. / Optics & Laser Technology 44 (2012) 1247–1255
8 < 1p2 q2 1=2 , m¼ : i p2 þq2 1 ,
if p2 þ q2 r 1, 2
ð3Þ
2
if p þ q 41,
where r^ ¼ xe^ x þ ye^ y þze^ z is the position vector toward the output plane z. k ¼ 2p=l is the wave number. e^ x and e^ z represent unit vectors along the x and z-axis, respectively. It is assumed that the incident wave is considered of linear polarization along the x-axis. Furthermore, due to the fact that the vectorial structure of beams is evaluated in the far field region, the effects of evanescent waves are correspondingly neglected [17–20,22,23]. As a consequence, 1=2 the alternative value m ¼ 1p2 q2 in Eq. (3) is chosen for our further calculations. Aðp,qÞ denotes the vectorial angular spectrum at the initial plane z¼0. Ax ðp,qÞ represents the component of Aðp,qÞ along the x-axis, which can be written as the Fourier transform of initial electric field [16–20] Z þ1 Z þ1 1 Ax ðp,qÞ ¼ 2 Ex x0 ,y0 ,0 exp ik px0 þqy0 dx0 dy0 ,
l
1
1
ð4Þ The initial electric field distribution of a linearly polarized MLGBs can given in the cylindrical coordinate system [5,12,13] r m þ n r2 0 exp 02 þ inf circðr 0 ÞUe^ x , ð5Þ Eðr 0 , f,z ¼ 0Þ ¼ C 0
o
o
1=2 where C 0 is an arbitrary amplitude constant, r 0 ¼ x20 þy20 is the radial coordinate, o denotes the Gaussian waist width, m is the non-negative beam order, n is so-called the topological charge number of MLGBs, f ¼ arctan y0 =x0 represents the azimuthal angle. circðr 0 Þ is the so-called circ function which represents a hard-edged aperture with radius R located at the plane z ¼0, i.e., ( 1,r 0 r R, circðr 0 Þ ¼ ð6Þ 0,r 0 4R, Eq. (6) can be further expanded into a finite sum of complex Gaussian terms [24,25] B r2 Al exp l 20 , circðr 0 Þ ¼ R l¼1 N X
ð7Þ
2
h i ¼ 2pexpðinyÞJ n kðp2 þ q2 Þ1=2 r 0 ,
N X
1 0
1F1
m þv þ1 2
; v þ 1;
bv G
mþvþ1
b2
4a
,
2
e^ 1 ¼
1 F 1 ða; b þ 1; zÞ ¼
1 b! X Gða þsÞzs , GðaÞ s ¼ 0 ðbþ sÞ!s!
ð10Þ
Bl R2
12m þ jnj þ 1
,
q 1=2
ðp2 þ q2 Þ
p
e^ x
ðp2 þ q2 Þ
1=2
e^ y ,
ð11Þ
ð12Þ
s^ ¼ pe^ x þ qe^ y þ me^ z ,
2 pm qm 2 1=2 ^ ez , e^ 2 ¼ 1=2 e^ x þ 1=2 e^ y p þq 2 2 2 2 p þq p þq
ð13Þ
Correspondingly, the propagating electric field of the beam can be decomposed into the TE and TM terms [17–20,27–31] ð14Þ E r^ ,z ¼ ETE r^ ,z þETM r^ ,z , with Z
ETE r^ ,z ¼
þ1
Z
1
Z
þ1
þ1
Aðp,qÞUe^ 1 e^ 1 exp ikðpx þ qy þmzÞ dpdq,
1
Z
þ1
¼ 1
1
1
1
Ax ðp,qÞexp ikðpx þqy þ mzÞ
q2 pq ^ x ^ y dpdq, e e 2 p2 þq2 p þq2 Z þ1 Z þ1 Aðp,qÞUe^ 2 e^ 2 exp ikðpx þ qy þmzÞ dpdq, ETM r^ ,z ¼ ð15Þ Z
¼
Z
Ax ðp,qÞexp ikðpxþ qy þmzÞ p pq p ^x þ ^ y e^ z dpdq, e e 2 m p2 þq2 p þ q2 þ1
þ1
1 2
ð16Þ
similarly, the propagating magnetic field of the beam can be also decomposed into the TE and TM terms ð17Þ H r^ ,z ¼ HTE r^ ,z þ HTM r^ ,z , with HTE r^ ,z ¼
Z
Gðv þ1Þ
where J n ð:Þ denotes the Bessel function of the first kind with the order n. Gð:Þ represents the Gamma function, 1 F 1 ða; bþ 1; zÞ is the so-called Kummer function which can be expressed as the sum of power series
þ RB2l
þ
5
with e^ 1 , e^ 2 and s^ being represented by
ð9Þ
1
o2
1
o2
þ q2 Þ
s^ e^ 1 ¼ e^ 2 , e^ 1 e^ 2 ¼ s^ , e^ 2 s^ ¼ e^ 1 ,
2
mþvþ1
2v þ 1 a !
2
In the far zone, three unit vectors which describe a mutually perpendicular right-handed system are introduced [17–20,27–31]
ð8Þ
xm expðax2 ÞJ v ðbxÞdx ¼
4
Al
l¼1
¼ Z
3 2
k ðp 41 1 F 1 2 m þ jnj þ 1; jnj þ 1;
1
where N is the number of complex Gaussian terms, Al ,Bl are the coefficients of which the values can be indexed in Table 1 of [24,25]. Owing to the convergence property of exponential functions, N¼ 10 is practically large enough to provide high accuracy in numerical calculations. Let us now derive the TE and TM terms of MLGBs propagating in the far field. By substituting Eq. (5) together with (6), (7) into Eq. (4) and utilizing the following integral formulas [26] Z 2p h i exp ikðp2 þ q2 Þ1=2 r 0 cosðfyÞ þ inf 0
the component of the vectorial angular spectrum along the x-axis of the beam at the initial transversal plane yields 1=2 ijnj h C 0 G 12m þ jnj þ 1 Ax ðp,qÞ ¼ 2 exp inarctan q=p k p2 þ q2 l om þ n 2jn þ 1j Gðjnj þ 1Þ
Z
rffiffiffiffiffiffi Z
Z
Aðp,qÞUe^ 1 e^ 2 1 1 exp ikðpx þ qyþ mzÞ dpdq,
e0 m0
þ1
þ1
Ax ðp,qÞexp ikðpxþ qy þmzÞ 1 1 pqm q2 m e^ x þ 2 e^ y qe^ z dpdq, 2 2 2 p þq p þq þ1
þ1
ð18Þ
rffiffiffiffiffiffi Z þ 1 Z þ 1 e0 Aðp,qÞUe^ 2 e^ 1 HTE r^ ,z ¼ m0 1 1 exp ikðpx þ qyþ mzÞ dpdq, Z
¼
Z
Ax ðp,qÞexp ikðpxþ qy þmzÞ ! pq p2 e^ x e^ y dpdq, m p2 þq2 m p2 þq2 þ1
1
þ1
1
ð19Þ
J. Li et al. / Optics & Laser Technology 44 (2012) 1247–1255
where e0 and m0 are the electric permittivity and magnetic permeability of light propagating in vacuum, respectively. One can observe from Eqs. (14)–(19) that the TE and TM terms of the beam in the far field preserve orthogonal to each other in distributions. Subsequently, by substituting Eq. (11) into Eqs. (15) and (16) respectively and employing the method of stationary phase [32,33], after tedious but straightforward integral calculations, electric field components of the TE and TM terms of MLGBs can be obtained
rffiffiffiffiffiffi jnj2 e0 r x HTM r^ ,z ¼ i expðikrÞU ye^ x xe^ y m0 rjnj þ 1 exp inarctan y=x kðm,nÞUY r,z , ð23Þ the parametric factor kðm,nÞ and Y r,z in Eqs. (20)–(23) are represented by C G 1 m þ jnj þ 1 jn j þ 1 k kðm,nÞ ¼ m þ0 jnj 2jnj þ 1 , ð24Þ o 2 Gðjnjþ 1Þ 2
rjnj2 yz ETE r^ ,z ¼ i jnj þ 2 expðikrÞU ye^ x xe^ y exp inarctan y=x kðm,nÞUY r,z , r
ð20Þ rjnj2 x ETM r^ ,z ¼ i jnj þ 2 expðikrÞU xze^ x þ yze^ y r2 e^ z exp inarctan y=x kðm,nÞUY r,z , r
ð21Þ 2 1=2 1=2 r ¼ x þ y2 ,r ¼ r2 þ z2 :Similarly, substituting where Eq. (11) into Eqs. (18) and (19) respectively, analytical forms for the magnetic fields of the TE and TM terms in the far field yield rffiffiffiffiffiffi jnj2 e0 r yz HTE r^ ,z ¼ i expðikrÞU xze^ x þ yze^ y r2 e^ z m0 r jnj þ 3 exp inarctan y=x kðm,nÞUY r,z , ð22Þ
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N X Al Y r,z ¼ l¼1
3
41 1 F 1 2 m þ jnj þ 1; jnjþ 1; 4
1
o2
þ RB2l
12m þ n þ 1
k 2 r2 1
o2
þ
Bl R2
5 r2
,
ð25Þ
Eqs. (20)–(23) indicate that that the TE and TM terms of the propagating beam preserve orthogonal to each other in the far field. It should be noted that these analytical expressions are applicable not only for the paraxial case but also the non-paraxial case. Moreover, in order to reveal the internal vectorial structure of truncated MLGBs diffracted in the far field, the non-paraxial case is specifically studied in our following analysis. Hence here, the traditional intensity distribution of the TE and TM terms
Fig. 1. Energy flux distributions for the TE term, TM term and the whole beam for different beam orders m at the transversal plane z¼ 250l, o ¼0.5l, b ¼ 1, n¼ 1. (a)–(c): m¼ 0, (d)–(f): m ¼2, (g)–(i): m¼ 4.
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J. Li et al. / Optics & Laser Technology 44 (2012) 1247–1255
should be replaced by the z-component time-averaged Poynting vector hSz iTE ¼ 12Re ETE r^ ,z HnTE r^ ,z z , hSz iTM ¼ 12Re ETM r^ ,z HnTM r^ ,z z , hSz i ¼ hSz iTE þ hSz iTM , ð26Þ substituting Eqs. (20)–(25) into Eq. (26), respectively, the corresponding energy flux distributions for the TE term, TM term and the whole beam yield rffiffiffiffiffiffi 2jnj2 2 3 e0 r y z hSz iTE ¼ kðm,nÞY r,z 2 , ð27Þ m0 r 2jnj þ 5 hSz iTM ¼
hSz i ¼
rffiffiffiffiffiffi
e0 r2jnj2 x2 z kðm,nÞY r,z 2 , m0 r 2jnj þ 3
rffiffiffiffiffiffi
ð28Þ
r ¼ zþ
r2
z, ð30Þ 2z Correspondingly, paraxial results for the TE term, TM term and whole beam turn out to be rffiffiffiffiffiffi 2jnj2 2 e0 r y hSz ipTE ¼ kðm,nÞY p r,z 2 , ð31Þ m0 z2jnj þ 2 hSz ipTM ¼ hSz ip ¼
rffiffiffiffiffiffi
e0 r2jnj2 x2 kðm,nÞY p r,z 2 , m0 z2jnj þ 2
rffiffiffiffiffiffi
e0 r2jnj kðm,nÞY p r,z 2 , m0 z2jnj þ 2
ð33Þ
with 2
e0 r2jnj2 z 2 y2 z2 x þ 2 kðm,nÞY r,z 2 , m0 r 2jnj þ 3 r
ð32Þ
ð29Þ
When m¼n ¼0 is substituted in to above expressions, they would reduce to the analytical forms for energy flux components for the non-paraxial Gaussian beams in the far field [17,18,34,35]. In the paraxial case, the following approximation can be further made that
Yp
N X r,z ¼ Al
l¼1
3
41 1 F 1 2 m þ jnjþ 1; jnjþ 1;
Bl o þ R2 1
4
12m þ jnj þ 1
k2 r2 1
o2
þ
Bl R2
5 z2
,
ð34Þ
2
Eqs. (27)–(34) reveal the fact that the energy flux distributions of truncated MLGBs in the far field are simultaneously dependent of the beam order m (n), initial beam width (o) and truncated
Fig. 2. Energy flux distributions for the TE term, TM term and the whole beam for different topological charge numbers n at the transversal plane z ¼250l, m ¼2. (a)–(c): n¼ 0 (hollow Gauss beam), (d)–(f): n¼3, (g)–(i): n¼ 6. Other parameters are the same as those of Fig. 1.
J. Li et al. / Optics & Laser Technology 44 (2012) 1247–1255
parameter (b ¼R/o) of the aperture. When m¼n ¼0, Eqs. (31)–(34) reduce to the expressions for the energy flux distributions of the paraxial Gaussian beam in the far field.
3. Numerical results In this section, numerical simulations are performed to show the vectorial structure of MLGBs propagating through a hardedged aperture in the far field. Particularly, the influences of the beam order, initial beam width together with the truncated parameter on the evolutive energy flux distributions are separately discussed. For simplicity, e0 =m0 is set to be unit in all following numerical simulations. Fig. 1 shows the energy flux distributions for the TE term, TM term and the whole beam for different choices of the beam order m¼0, 2, 4, respectively, Other parameters are selected as o ¼ 0.5l, b ¼1, z¼250l, n¼1. It can be obviously seen from figures that the profiles for the TE term, TM term and the whole beam would spread more rapidly for larger beam order m. For example, when the beam order m¼4, the beam spot size of truncated MLGBs at the distance z¼250l (Fig. 1(g)–(i)) weighs almost four times than its initial waist width (Fig. 1(a)–(c)). Furthermore, through observing Fig. 1(h) and (i), it is found that when the beam order m approaches to a certain
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order of magnitude, i.e., m Z4, several lobes with relatively slight energy flux would emerge around the central petal. This phenomenon is similar to the numerical results shown in previous reports [36–38], which concentrated on the vectorial characteristics of Laguerre–Gaussian beams (LGB) in the far field. Fig. 2 depicts the influence of the topological charge number (n ¼0, 3, 6) of beams on the energy flux distributions at the transversal plane z¼250l. The beam order m¼2, other parameters are the same as those of Fig. 1. Fig. 2(a)–(c) corresponds to the energy flux distributions of the hollow Gaussian beam (n ¼0). They are shown in agreement with the numerical results in [19,20]. When the topological charge number increases, the Gaussian beam converts into the MLGBs of which the energy flux distribution behaves entirely in difference while comparing with those of the Gaussian beam (see Fig. 2(d)–(i)). It is shown by Fig. 2(d)–(i) that the energy flux distributions for the TE term, TM term and the whole beam would all break up into two lobes in symmetric distributions with respect to the x-axis or y-axis. Moreover, when the topological charge number subsequently increases, these two symmetric lobes would gradually detach from each other. This phenomenon can be also observed in Fig. 2(f) and (i). Fig. 3 reveals the influence of the truncated parameter b on the energy flux distributions of the TE term, TM term and whole beam
Fig. 3. Energy flux distributions for the TE term, TM term and the whole beam for different truncated parameters b at the transversal plane z ¼250l, m ¼n¼ 2, o ¼ 0.5l. (a)–(c) b-N (untruncated case), (d)–(f) b ¼ 0.5, (g)–(i) b ¼ 0.1.
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J. Li et al. / Optics & Laser Technology 44 (2012) 1247–1255
for the case o ¼0.5l. m¼n¼2, other parameters are the same as those of Fig. 1. Fig. 3(a)–(c) corresponds to energy flux distributions of MLGBs for the unapertured case (b-N). It can be observed that there exists a group of inequalities hSz ichSz iTE , hSz iTM chSz iTE which can be concluded from figures. This result is equivalent to the analytical form hSz i hSz iTM which can be observed from the comparison between Fig. 3(b) and (c). This phenomenon is induced by the fact that the energy flux distribution of the TE term shows in smaller values than that of the TM term, in general. In another point of view, this phenomenon can be also verified by the comparison between Eqs. (27) and (28). When the MLGBs are seriously truncated by the aperture, the resultant intensity would correspondingly decrease (see Fig. 3 (c), (f) and (i)). Moreover, the two symmetric petals would be gradually connected by a central tiny lobe when the truncated parameter is small enough (b ¼0.1). Fig. 4 displays the influence of the truncated parameter on energy flux distributions of MLGBs for the case o ¼5l, other parameters are the same as those of Fig. 3. By the comparison between Figs. 3 and 4, one can discover that energy flux distributions for the case o ¼0.5l are totally different from that for the case o ¼5l. This phenomenon indicates that the initial beam width has strong influences on energy flux distributions of MLGBs in the far field. For the case o ¼5l, energy flux distributions for the TM term
and the whole beam would mainly concentrate on the central lobe; correspondingly, intensities of the two symmetric lobes would gradually disappear when the truncated parameter reduces to a certain order of magnitude, i.e., b ¼0.1 (see Fig. 4(h) and (i)). The major reason which induces differences to the propagation results between Fig. 3(h, i) and Fig. 4(h, i) may attribute to the influence of the initial beam width. This is because when the beam width is large enough compared with respect to the wavelength, propagation properties of the beam in Fig. 3 can be approximately regarded as paraxial. Inversely, when the beam width is identical to or less than the wavelength, the profiles in Fig. 4 would occupy relatively high divergence angles and is naturally regarded as the non-paraxial beam [39–41]. These effects of parameters finally cause the discrepancies between Fig. 3 and 4. Figs. 5 and 6 plot differences between the paraxial and nonparaxial propagating results of MLGBs by the usage of Eqs. (27)–(34) for the truncated parameter b ¼10 and b ¼1, respectively. It should be noted that the energy flux distributions are plotted normalized with respect to their maximum values. In addition, the energy flux distribution for the TE term is plotted against the transversal line x¼0, while the TM term is plotted versus the transversal line y¼0. The propagation distance is chosen as z¼1000l, other parameters are the same as those of Fig. 3. A significant result shown in Figs. 5
Fig. 4. Energy flux distributions for the TE term, TM term and the whole beam for different truncated parameters b at the transversal plane z ¼250l, o ¼ 5l. (a)–(c) b-N (untruncated case), (d)–(f) b ¼0.5, (g)–(i) b ¼ 0.1. Other parameters are the same as those of Fig. 3.
J. Li et al. / Optics & Laser Technology 44 (2012) 1247–1255
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Fig. 5. Comparisons between the paraxial (Eqs. (27) and (28)) and non-paraxial (Eqs. (31) and (32)) results of normalized energy flux distributions for the TE term and TM term plotted against the cross section x ¼ 0 and y¼ 0, respectively. b ¼10, o ¼ 0.5l, z ¼1000l, m¼ n¼ 2. (a) the TE term, (b) the TM term.
Fig. 6. Comparisons between the paraxial (Eqs. (27) and (28)) and non-paraxial (Eqs. (31) and (32)) results of normalized energy flux distributions for the TE term and TM term plotted against the cross section x ¼ 0 and y¼ 0, respectively. b ¼1, other parameters are the same as those of Fig. 5. (a) the TE term, (b) the TM term.
Fig. 7. Comparisons between the paraxial (Eqs. (27) and (28)) and non-paraxial (Eqs. (31) and (32)) results of normalized energy flux distributions for the TE term and TM term plotted against the cross section x ¼ 0 and y¼ 0, respectively. b ¼0.1, other parameters are the same as those of Fig. 5. (a) The TE term, (b) the TM term.
and 6 is that the difference between the paraxial and non-paraxial results for the TM term shows to be more distinct than that of the TE term. This is also in agreement with the demonstrations for Figs. 3 and 4 shown above. The difference between two results mainly concentrates on the border of the patterns of energy flux distributions for the TM term (see Fig. 5(b) and Fig. 6(b)). Furthermore, one
can notice that the discrepancy between the paraxial and nonparaxial cases in Figs. 5 and 6 becomes more distinct for smaller truncated parameters b. This result is applicable for both the TE and TM terms evaluated in the far field. Figs. 7 and 8 show the difference between the normalized paraxial and non-paraxial propagating results for values o ¼0.5l
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J. Li et al. / Optics & Laser Technology 44 (2012) 1247–1255
Fig. 8. Comparisons between the paraxial (Eqs. (27) and (28)) and non-paraxial (Eqs. (31) and (32)) results of normalized energy flux distributions for the TE term, TM term plotted against the cross section x ¼0 and y¼ 0, respectively. o ¼5l, other parameters are the same as those of Fig. 7. (a) The TE term, (b) the TM term.
and o ¼5l, respectively. The truncated parameter b ¼0.1, other parameters are the same as those of Figs. 5 and 6. Similar to the phenomenon shown in Figs. 5 and 6, differences between the two propagating cases for the TM term appear to be more visible than those for the TE term. Furthermore, it can be concluded from Figs. 7–8 that the difference between the paraxial and non-paraxial results turns out to be more obvious for smaller initial beam width (o ¼0.5l) than the larger case (o ¼5l). This is because the incident beam with smaller waist width would occupy larger divergence angles upon the propagation. Generally speaking, this effect is equivalent to the slight non-paraxial propagation of laser beams, in some sense. As a consequence, the MLGBs would convert into the non-paraxial beam more rapidly for smaller value of the initial waist width. This is why the discrepancy between the two propagating cases in Fig. 8 (o ¼5l) shows to be less distinct than that in Fig. 7 (o ¼0.5l).
4. Conclusion The far-field vectorial structure of modified Laguerre–Gaussian beams propagating through a hard-edged aperture is studied based on the vectorial angular spectrum representation of Maxwell equations and the method of stationary phase. Analytical expressions are derived for the energy flux distributions for the TE term, TM term and the whole beam evaluated in the transversal output plane. It is shown by numerical plots that the energy flux distributions of the truncated beam not only depend on the beam order, topological charge number but also relate to the initial waist width and truncated parameter. Discrepancies between the paraxial and non-paraxial results normalized with respect to their maximum values are analyzed for different values of truncated parameters or initial beam widths, respectively. Furthermore, the physical mechanism which induces these discrepancies is discussed, in detail.
Acknowledgments This work is supported by the National Natural Science Foundation of China under grant 61077012, the Foundation of CAS Key Lab of Renewable Energy and Natural Gas Hydrate under grant 0907k2.
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Contents lists available at SciVerse ScienceDirect
Optics & Laser Technology journal homepage: www.elsevier.com/locate/optlastec
Analytical vectorial structure of truncated modified Laguerre–Gaussian beam in the far field Jia Li a,n, Yanru Chen b, Quanjun Cao a a b
Institute of Fiber Optic Communication & Information Engineering, College of Information Engineering, Zhejiang University of Technology, Hangzhou 310023, China Department of Optical Engineering, Nanjing University of Science & Technology, Nanjing 210094, China
a r t i c l e i n f o
abstract
Article history: Received 6 December 2011 Received in revised form 4 January 2012 Accepted 4 January 2012 Available online 24 January 2012
Based on the vectorial angular spectrum representation of Maxwell equations and the method of stationary phase, analytical expressions are derived for the TE term, TM term and whole energy flux of a hard-edged diffracted modified Laguerre–Gaussian beam (MLGBs) in the far field, respectively. It is shown that the evolutive energy flux not only depends on the beam order, initial waist width but also relates to the truncated parameter. Furthermore, influences of the truncated parameter on the discrepancies of energy flux distributions between paraxial and non-paraxial cases are also analyzed for different selections of the initial beam width. & 2012 Elsevier Ltd. All rights reserved.
Keywords: Modified Laguerre–Gaussian beam Vectorial structure Energy flux
1. Introduction Just as the paraxial solutions to the propagation of optical fields established for some earlier families of beams such as highorder mode Gaussian beams [1] or Helmholtz–Gaussian beams [2], a novel and general beam solution of the paraxial wave equation is found in Cartesian coordinates. This new solution is termed Cartesian beam which is commonly described by several complex parameters in the most general situations [3]. Hypergeometric laser beams, which can be regarded as a special class of Cartesian modes by suitably the choice of complex cofactors, have been put forward in recent reports [4–6]. The hypergeometric beam naturally contains optical vortices which may bring initial orbital angular momentum (OAM). The tight focusing properties of hypergeometric–Gaussian beams (HGBs) through aplanatic high-numerical-aperture (NA) lens system were studied by the usage of the vectorial Debye diffraction integral [7]. The nonparaxial propagation properties of vectorial HGBs in free space were investigated by the Rayleigh–Sommerfeld diffraction integral [8]. Besides, various experimental methods for the generation of HGBs by the usage of encoded binary diffractive elements [9,10] or computer-generated hologram [11] were separately proposed. One can recognize that the HGBs would exhibit plenty of features with the change of mode parameters. By suitable selections of mode parameters, solutions to paraxial wave equations of HGBs would reduce to a subfamily of modes named the modified Laguerre–Gaussian mode [5]. Propagation properties of
modified Laguerre–Gaussian beams (MLGBs) through a paraxial ABCD optical system with a hard-edged aperture are studied by the usage of the Collins integral [12]. Very recently, propagation properties of MLGBs though a fractional Fourier transform (FRFT) system are revealed in cylindrical coordinate system [13]. The purpose of the present paper is to concentrate on the internal vectorial structure of MLGBs diffracting through a hardedged circular aperture in the far field. Firstly, based on the vectorial angular spectrum representation of Maxwell equations and the method of stationary phase, analytical expressions for the TE and TM terms of the beam are derived, respectively. Second, energy flux distributions for the TE term, TM term and whole beam are revealed by numerical plots. Finally, comparisons between the paraxial and non-paraxial propagating results are made with different choices of the initial beam width, beam order and the truncated parameter, respectively.
2. Vectorial structure of truncated MLGBs in the far field As it has been reported, the propagating electric field of a polarized beam toward the half space z Z0 can be given by means of the vectorial angular spectrum representation of electromagnetic beams [14–23] Z þ1 Z þ1 E r^ ,z ¼ Aðp,qÞexp ikðpx þqy þ mzÞ dpdq, ð1Þ 1
1
with n
Corresponding author. Tel.: þ8615068725198. E-mail address: [email protected] (J. Li).
0030-3992/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlastec.2012.01.002
p Aðp,qÞ ¼ Ax ðp,qÞ e^ x e^ z , m
ð2Þ
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J. Li et al. / Optics & Laser Technology 44 (2012) 1247–1255
8 < 1p2 q2 1=2 , m¼ : i p2 þq2 1 ,
if p2 þ q2 r 1, 2
ð3Þ
2
if p þ q 41,
where r^ ¼ xe^ x þ ye^ y þze^ z is the position vector toward the output plane z. k ¼ 2p=l is the wave number. e^ x and e^ z represent unit vectors along the x and z-axis, respectively. It is assumed that the incident wave is considered of linear polarization along the x-axis. Furthermore, due to the fact that the vectorial structure of beams is evaluated in the far field region, the effects of evanescent waves are correspondingly neglected [17–20,22,23]. As a consequence, 1=2 the alternative value m ¼ 1p2 q2 in Eq. (3) is chosen for our further calculations. Aðp,qÞ denotes the vectorial angular spectrum at the initial plane z¼0. Ax ðp,qÞ represents the component of Aðp,qÞ along the x-axis, which can be written as the Fourier transform of initial electric field [16–20] Z þ1 Z þ1 1 Ax ðp,qÞ ¼ 2 Ex x0 ,y0 ,0 exp ik px0 þqy0 dx0 dy0 ,
l
1
1
ð4Þ The initial electric field distribution of a linearly polarized MLGBs can given in the cylindrical coordinate system [5,12,13] r m þ n r2 0 exp 02 þ inf circðr 0 ÞUe^ x , ð5Þ Eðr 0 , f,z ¼ 0Þ ¼ C 0
o
o
1=2 where C 0 is an arbitrary amplitude constant, r 0 ¼ x20 þy20 is the radial coordinate, o denotes the Gaussian waist width, m is the non-negative beam order, n is so-called the topological charge number of MLGBs, f ¼ arctan y0 =x0 represents the azimuthal angle. circðr 0 Þ is the so-called circ function which represents a hard-edged aperture with radius R located at the plane z ¼0, i.e., ( 1,r 0 r R, circðr 0 Þ ¼ ð6Þ 0,r 0 4R, Eq. (6) can be further expanded into a finite sum of complex Gaussian terms [24,25] B r2 Al exp l 20 , circðr 0 Þ ¼ R l¼1 N X
ð7Þ
2
h i ¼ 2pexpðinyÞJ n kðp2 þ q2 Þ1=2 r 0 ,
N X
1 0
1F1
m þv þ1 2
; v þ 1;
bv G
mþvþ1
b2
4a
,
2
e^ 1 ¼
1 F 1 ða; b þ 1; zÞ ¼
1 b! X Gða þsÞzs , GðaÞ s ¼ 0 ðbþ sÞ!s!
ð10Þ
Bl R2
12m þ jnj þ 1
,
q 1=2
ðp2 þ q2 Þ
p
e^ x
ðp2 þ q2 Þ
1=2
e^ y ,
ð11Þ
ð12Þ
s^ ¼ pe^ x þ qe^ y þ me^ z ,
2 pm qm 2 1=2 ^ ez , e^ 2 ¼ 1=2 e^ x þ 1=2 e^ y p þq 2 2 2 2 p þq p þq
ð13Þ
Correspondingly, the propagating electric field of the beam can be decomposed into the TE and TM terms [17–20,27–31] ð14Þ E r^ ,z ¼ ETE r^ ,z þETM r^ ,z , with Z
ETE r^ ,z ¼
þ1
Z
1
Z
þ1
þ1
Aðp,qÞUe^ 1 e^ 1 exp ikðpx þ qy þmzÞ dpdq,
1
Z
þ1
¼ 1
1
1
1
Ax ðp,qÞexp ikðpx þqy þ mzÞ
q2 pq ^ x ^ y dpdq, e e 2 p2 þq2 p þq2 Z þ1 Z þ1 Aðp,qÞUe^ 2 e^ 2 exp ikðpx þ qy þmzÞ dpdq, ETM r^ ,z ¼ ð15Þ Z
¼
Z
Ax ðp,qÞexp ikðpxþ qy þmzÞ p pq p ^x þ ^ y e^ z dpdq, e e 2 m p2 þq2 p þ q2 þ1
þ1
1 2
ð16Þ
similarly, the propagating magnetic field of the beam can be also decomposed into the TE and TM terms ð17Þ H r^ ,z ¼ HTE r^ ,z þ HTM r^ ,z , with HTE r^ ,z ¼
Z
Gðv þ1Þ
where J n ð:Þ denotes the Bessel function of the first kind with the order n. Gð:Þ represents the Gamma function, 1 F 1 ða; bþ 1; zÞ is the so-called Kummer function which can be expressed as the sum of power series
þ RB2l
þ
5
with e^ 1 , e^ 2 and s^ being represented by
ð9Þ
1
o2
1
o2
þ q2 Þ
s^ e^ 1 ¼ e^ 2 , e^ 1 e^ 2 ¼ s^ , e^ 2 s^ ¼ e^ 1 ,
2
mþvþ1
2v þ 1 a !
2
In the far zone, three unit vectors which describe a mutually perpendicular right-handed system are introduced [17–20,27–31]
ð8Þ
xm expðax2 ÞJ v ðbxÞdx ¼
4
Al
l¼1
¼ Z
3 2
k ðp 41 1 F 1 2 m þ jnj þ 1; jnj þ 1;
1
where N is the number of complex Gaussian terms, Al ,Bl are the coefficients of which the values can be indexed in Table 1 of [24,25]. Owing to the convergence property of exponential functions, N¼ 10 is practically large enough to provide high accuracy in numerical calculations. Let us now derive the TE and TM terms of MLGBs propagating in the far field. By substituting Eq. (5) together with (6), (7) into Eq. (4) and utilizing the following integral formulas [26] Z 2p h i exp ikðp2 þ q2 Þ1=2 r 0 cosðfyÞ þ inf 0
the component of the vectorial angular spectrum along the x-axis of the beam at the initial transversal plane yields 1=2 ijnj h C 0 G 12m þ jnj þ 1 Ax ðp,qÞ ¼ 2 exp inarctan q=p k p2 þ q2 l om þ n 2jn þ 1j Gðjnj þ 1Þ
Z
rffiffiffiffiffiffi Z
Z
Aðp,qÞUe^ 1 e^ 2 1 1 exp ikðpx þ qyþ mzÞ dpdq,
e0 m0
þ1
þ1
Ax ðp,qÞexp ikðpxþ qy þmzÞ 1 1 pqm q2 m e^ x þ 2 e^ y qe^ z dpdq, 2 2 2 p þq p þq þ1
þ1
ð18Þ
rffiffiffiffiffiffi Z þ 1 Z þ 1 e0 Aðp,qÞUe^ 2 e^ 1 HTE r^ ,z ¼ m0 1 1 exp ikðpx þ qyþ mzÞ dpdq, Z
¼
Z
Ax ðp,qÞexp ikðpxþ qy þmzÞ ! pq p2 e^ x e^ y dpdq, m p2 þq2 m p2 þq2 þ1
1
þ1
1
ð19Þ
J. Li et al. / Optics & Laser Technology 44 (2012) 1247–1255
where e0 and m0 are the electric permittivity and magnetic permeability of light propagating in vacuum, respectively. One can observe from Eqs. (14)–(19) that the TE and TM terms of the beam in the far field preserve orthogonal to each other in distributions. Subsequently, by substituting Eq. (11) into Eqs. (15) and (16) respectively and employing the method of stationary phase [32,33], after tedious but straightforward integral calculations, electric field components of the TE and TM terms of MLGBs can be obtained
rffiffiffiffiffiffi jnj2 e0 r x HTM r^ ,z ¼ i expðikrÞU ye^ x xe^ y m0 rjnj þ 1 exp inarctan y=x kðm,nÞUY r,z , ð23Þ the parametric factor kðm,nÞ and Y r,z in Eqs. (20)–(23) are represented by C G 1 m þ jnj þ 1 jn j þ 1 k kðm,nÞ ¼ m þ0 jnj 2jnj þ 1 , ð24Þ o 2 Gðjnjþ 1Þ 2
rjnj2 yz ETE r^ ,z ¼ i jnj þ 2 expðikrÞU ye^ x xe^ y exp inarctan y=x kðm,nÞUY r,z , r
ð20Þ rjnj2 x ETM r^ ,z ¼ i jnj þ 2 expðikrÞU xze^ x þ yze^ y r2 e^ z exp inarctan y=x kðm,nÞUY r,z , r
ð21Þ 2 1=2 1=2 r ¼ x þ y2 ,r ¼ r2 þ z2 :Similarly, substituting where Eq. (11) into Eqs. (18) and (19) respectively, analytical forms for the magnetic fields of the TE and TM terms in the far field yield rffiffiffiffiffiffi jnj2 e0 r yz HTE r^ ,z ¼ i expðikrÞU xze^ x þ yze^ y r2 e^ z m0 r jnj þ 3 exp inarctan y=x kðm,nÞUY r,z , ð22Þ
1249
N X Al Y r,z ¼ l¼1
3
41 1 F 1 2 m þ jnj þ 1; jnjþ 1; 4
1
o2
þ RB2l
12m þ n þ 1
k 2 r2 1
o2
þ
Bl R2
5 r2
,
ð25Þ
Eqs. (20)–(23) indicate that that the TE and TM terms of the propagating beam preserve orthogonal to each other in the far field. It should be noted that these analytical expressions are applicable not only for the paraxial case but also the non-paraxial case. Moreover, in order to reveal the internal vectorial structure of truncated MLGBs diffracted in the far field, the non-paraxial case is specifically studied in our following analysis. Hence here, the traditional intensity distribution of the TE and TM terms
Fig. 1. Energy flux distributions for the TE term, TM term and the whole beam for different beam orders m at the transversal plane z¼ 250l, o ¼0.5l, b ¼ 1, n¼ 1. (a)–(c): m¼ 0, (d)–(f): m ¼2, (g)–(i): m¼ 4.
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J. Li et al. / Optics & Laser Technology 44 (2012) 1247–1255
should be replaced by the z-component time-averaged Poynting vector hSz iTE ¼ 12Re ETE r^ ,z HnTE r^ ,z z , hSz iTM ¼ 12Re ETM r^ ,z HnTM r^ ,z z , hSz i ¼ hSz iTE þ hSz iTM , ð26Þ substituting Eqs. (20)–(25) into Eq. (26), respectively, the corresponding energy flux distributions for the TE term, TM term and the whole beam yield rffiffiffiffiffiffi 2jnj2 2 3 e0 r y z hSz iTE ¼ kðm,nÞY r,z 2 , ð27Þ m0 r 2jnj þ 5 hSz iTM ¼
hSz i ¼
rffiffiffiffiffiffi
e0 r2jnj2 x2 z kðm,nÞY r,z 2 , m0 r 2jnj þ 3
rffiffiffiffiffiffi
ð28Þ
r ¼ zþ
r2
z, ð30Þ 2z Correspondingly, paraxial results for the TE term, TM term and whole beam turn out to be rffiffiffiffiffiffi 2jnj2 2 e0 r y hSz ipTE ¼ kðm,nÞY p r,z 2 , ð31Þ m0 z2jnj þ 2 hSz ipTM ¼ hSz ip ¼
rffiffiffiffiffiffi
e0 r2jnj2 x2 kðm,nÞY p r,z 2 , m0 z2jnj þ 2
rffiffiffiffiffiffi
e0 r2jnj kðm,nÞY p r,z 2 , m0 z2jnj þ 2
ð33Þ
with 2
e0 r2jnj2 z 2 y2 z2 x þ 2 kðm,nÞY r,z 2 , m0 r 2jnj þ 3 r
ð32Þ
ð29Þ
When m¼n ¼0 is substituted in to above expressions, they would reduce to the analytical forms for energy flux components for the non-paraxial Gaussian beams in the far field [17,18,34,35]. In the paraxial case, the following approximation can be further made that
Yp
N X r,z ¼ Al
l¼1
3
41 1 F 1 2 m þ jnjþ 1; jnjþ 1;
Bl o þ R2 1
4
12m þ jnj þ 1
k2 r2 1
o2
þ
Bl R2
5 z2
,
ð34Þ
2
Eqs. (27)–(34) reveal the fact that the energy flux distributions of truncated MLGBs in the far field are simultaneously dependent of the beam order m (n), initial beam width (o) and truncated
Fig. 2. Energy flux distributions for the TE term, TM term and the whole beam for different topological charge numbers n at the transversal plane z ¼250l, m ¼2. (a)–(c): n¼ 0 (hollow Gauss beam), (d)–(f): n¼3, (g)–(i): n¼ 6. Other parameters are the same as those of Fig. 1.
J. Li et al. / Optics & Laser Technology 44 (2012) 1247–1255
parameter (b ¼R/o) of the aperture. When m¼n ¼0, Eqs. (31)–(34) reduce to the expressions for the energy flux distributions of the paraxial Gaussian beam in the far field.
3. Numerical results In this section, numerical simulations are performed to show the vectorial structure of MLGBs propagating through a hardedged aperture in the far field. Particularly, the influences of the beam order, initial beam width together with the truncated parameter on the evolutive energy flux distributions are separately discussed. For simplicity, e0 =m0 is set to be unit in all following numerical simulations. Fig. 1 shows the energy flux distributions for the TE term, TM term and the whole beam for different choices of the beam order m¼0, 2, 4, respectively, Other parameters are selected as o ¼ 0.5l, b ¼1, z¼250l, n¼1. It can be obviously seen from figures that the profiles for the TE term, TM term and the whole beam would spread more rapidly for larger beam order m. For example, when the beam order m¼4, the beam spot size of truncated MLGBs at the distance z¼250l (Fig. 1(g)–(i)) weighs almost four times than its initial waist width (Fig. 1(a)–(c)). Furthermore, through observing Fig. 1(h) and (i), it is found that when the beam order m approaches to a certain
1251
order of magnitude, i.e., m Z4, several lobes with relatively slight energy flux would emerge around the central petal. This phenomenon is similar to the numerical results shown in previous reports [36–38], which concentrated on the vectorial characteristics of Laguerre–Gaussian beams (LGB) in the far field. Fig. 2 depicts the influence of the topological charge number (n ¼0, 3, 6) of beams on the energy flux distributions at the transversal plane z¼250l. The beam order m¼2, other parameters are the same as those of Fig. 1. Fig. 2(a)–(c) corresponds to the energy flux distributions of the hollow Gaussian beam (n ¼0). They are shown in agreement with the numerical results in [19,20]. When the topological charge number increases, the Gaussian beam converts into the MLGBs of which the energy flux distribution behaves entirely in difference while comparing with those of the Gaussian beam (see Fig. 2(d)–(i)). It is shown by Fig. 2(d)–(i) that the energy flux distributions for the TE term, TM term and the whole beam would all break up into two lobes in symmetric distributions with respect to the x-axis or y-axis. Moreover, when the topological charge number subsequently increases, these two symmetric lobes would gradually detach from each other. This phenomenon can be also observed in Fig. 2(f) and (i). Fig. 3 reveals the influence of the truncated parameter b on the energy flux distributions of the TE term, TM term and whole beam
Fig. 3. Energy flux distributions for the TE term, TM term and the whole beam for different truncated parameters b at the transversal plane z ¼250l, m ¼n¼ 2, o ¼ 0.5l. (a)–(c) b-N (untruncated case), (d)–(f) b ¼ 0.5, (g)–(i) b ¼ 0.1.
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J. Li et al. / Optics & Laser Technology 44 (2012) 1247–1255
for the case o ¼0.5l. m¼n¼2, other parameters are the same as those of Fig. 1. Fig. 3(a)–(c) corresponds to energy flux distributions of MLGBs for the unapertured case (b-N). It can be observed that there exists a group of inequalities hSz ichSz iTE , hSz iTM chSz iTE which can be concluded from figures. This result is equivalent to the analytical form hSz i hSz iTM which can be observed from the comparison between Fig. 3(b) and (c). This phenomenon is induced by the fact that the energy flux distribution of the TE term shows in smaller values than that of the TM term, in general. In another point of view, this phenomenon can be also verified by the comparison between Eqs. (27) and (28). When the MLGBs are seriously truncated by the aperture, the resultant intensity would correspondingly decrease (see Fig. 3 (c), (f) and (i)). Moreover, the two symmetric petals would be gradually connected by a central tiny lobe when the truncated parameter is small enough (b ¼0.1). Fig. 4 displays the influence of the truncated parameter on energy flux distributions of MLGBs for the case o ¼5l, other parameters are the same as those of Fig. 3. By the comparison between Figs. 3 and 4, one can discover that energy flux distributions for the case o ¼0.5l are totally different from that for the case o ¼5l. This phenomenon indicates that the initial beam width has strong influences on energy flux distributions of MLGBs in the far field. For the case o ¼5l, energy flux distributions for the TM term
and the whole beam would mainly concentrate on the central lobe; correspondingly, intensities of the two symmetric lobes would gradually disappear when the truncated parameter reduces to a certain order of magnitude, i.e., b ¼0.1 (see Fig. 4(h) and (i)). The major reason which induces differences to the propagation results between Fig. 3(h, i) and Fig. 4(h, i) may attribute to the influence of the initial beam width. This is because when the beam width is large enough compared with respect to the wavelength, propagation properties of the beam in Fig. 3 can be approximately regarded as paraxial. Inversely, when the beam width is identical to or less than the wavelength, the profiles in Fig. 4 would occupy relatively high divergence angles and is naturally regarded as the non-paraxial beam [39–41]. These effects of parameters finally cause the discrepancies between Fig. 3 and 4. Figs. 5 and 6 plot differences between the paraxial and nonparaxial propagating results of MLGBs by the usage of Eqs. (27)–(34) for the truncated parameter b ¼10 and b ¼1, respectively. It should be noted that the energy flux distributions are plotted normalized with respect to their maximum values. In addition, the energy flux distribution for the TE term is plotted against the transversal line x¼0, while the TM term is plotted versus the transversal line y¼0. The propagation distance is chosen as z¼1000l, other parameters are the same as those of Fig. 3. A significant result shown in Figs. 5
Fig. 4. Energy flux distributions for the TE term, TM term and the whole beam for different truncated parameters b at the transversal plane z ¼250l, o ¼ 5l. (a)–(c) b-N (untruncated case), (d)–(f) b ¼0.5, (g)–(i) b ¼ 0.1. Other parameters are the same as those of Fig. 3.
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Fig. 5. Comparisons between the paraxial (Eqs. (27) and (28)) and non-paraxial (Eqs. (31) and (32)) results of normalized energy flux distributions for the TE term and TM term plotted against the cross section x ¼ 0 and y¼ 0, respectively. b ¼10, o ¼ 0.5l, z ¼1000l, m¼ n¼ 2. (a) the TE term, (b) the TM term.
Fig. 6. Comparisons between the paraxial (Eqs. (27) and (28)) and non-paraxial (Eqs. (31) and (32)) results of normalized energy flux distributions for the TE term and TM term plotted against the cross section x ¼ 0 and y¼ 0, respectively. b ¼1, other parameters are the same as those of Fig. 5. (a) the TE term, (b) the TM term.
Fig. 7. Comparisons between the paraxial (Eqs. (27) and (28)) and non-paraxial (Eqs. (31) and (32)) results of normalized energy flux distributions for the TE term and TM term plotted against the cross section x ¼ 0 and y¼ 0, respectively. b ¼0.1, other parameters are the same as those of Fig. 5. (a) The TE term, (b) the TM term.
and 6 is that the difference between the paraxial and non-paraxial results for the TM term shows to be more distinct than that of the TE term. This is also in agreement with the demonstrations for Figs. 3 and 4 shown above. The difference between two results mainly concentrates on the border of the patterns of energy flux distributions for the TM term (see Fig. 5(b) and Fig. 6(b)). Furthermore, one
can notice that the discrepancy between the paraxial and nonparaxial cases in Figs. 5 and 6 becomes more distinct for smaller truncated parameters b. This result is applicable for both the TE and TM terms evaluated in the far field. Figs. 7 and 8 show the difference between the normalized paraxial and non-paraxial propagating results for values o ¼0.5l
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Fig. 8. Comparisons between the paraxial (Eqs. (27) and (28)) and non-paraxial (Eqs. (31) and (32)) results of normalized energy flux distributions for the TE term, TM term plotted against the cross section x ¼0 and y¼ 0, respectively. o ¼5l, other parameters are the same as those of Fig. 7. (a) The TE term, (b) the TM term.
and o ¼5l, respectively. The truncated parameter b ¼0.1, other parameters are the same as those of Figs. 5 and 6. Similar to the phenomenon shown in Figs. 5 and 6, differences between the two propagating cases for the TM term appear to be more visible than those for the TE term. Furthermore, it can be concluded from Figs. 7–8 that the difference between the paraxial and non-paraxial results turns out to be more obvious for smaller initial beam width (o ¼0.5l) than the larger case (o ¼5l). This is because the incident beam with smaller waist width would occupy larger divergence angles upon the propagation. Generally speaking, this effect is equivalent to the slight non-paraxial propagation of laser beams, in some sense. As a consequence, the MLGBs would convert into the non-paraxial beam more rapidly for smaller value of the initial waist width. This is why the discrepancy between the two propagating cases in Fig. 8 (o ¼5l) shows to be less distinct than that in Fig. 7 (o ¼0.5l).
4. Conclusion The far-field vectorial structure of modified Laguerre–Gaussian beams propagating through a hard-edged aperture is studied based on the vectorial angular spectrum representation of Maxwell equations and the method of stationary phase. Analytical expressions are derived for the energy flux distributions for the TE term, TM term and the whole beam evaluated in the transversal output plane. It is shown by numerical plots that the energy flux distributions of the truncated beam not only depend on the beam order, topological charge number but also relate to the initial waist width and truncated parameter. Discrepancies between the paraxial and non-paraxial results normalized with respect to their maximum values are analyzed for different values of truncated parameters or initial beam widths, respectively. Furthermore, the physical mechanism which induces these discrepancies is discussed, in detail.
Acknowledgments This work is supported by the National Natural Science Foundation of China under grant 61077012, the Foundation of CAS Key Lab of Renewable Energy and Natural Gas Hydrate under grant 0907k2.
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