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Vectorial structure of Hermite–Laguerre–Gaussian beam in the far field
ARTICLE IN PRESS
Optics & Laser Technology 40 (2008) 858–863 www.elsevier.com/locate/optlastec
Vectorial structure of Hermite–Laguerre–Gaussian beam in the far field Guoquan Zhou, Jun Zheng School of Sciences, Zhejiang Forestry University, Lin’an 311300, Zhejiang Province, China Received 4 August 2007; received in revised form 30 October 2007; accepted 14 November 2007 Available online 20 December 2007
Abstract Starting from Maxwell’s equations, a Hermite–Laguerre–Gaussian (HLG) beam is decomposed into the TE and TM terms by using the vector angular spectrum representation. By means of the method of stationary phase, the analytical TE and TM terms are presented in the far field. The energy flux distributions of the TE and TM terms are also investigated and depicted in the far field. The influences of the additional angle parameter and Gaussian waist width on the vectorial structure and energy flux pattern of HLG beam are also investigated. This research reveals the internal vectorial structure of HLG beam and may provide a new approach to the manipulation of laser beams. r 2007 Elsevier Ltd. All rights reserved. PACS: 41.85.Ew; 42.25.Bs; 42.55.Px Keywords: Hermite–Laguerre–Gaussian beam; Vectorial structure; Method of stationary phase
1. Introduction A united family of Hermite–Gaussian (HG) and Laguerre–Gaussian (LG) beams, namely the Hermite–Laguerre– Gaussian (HLG) beam, has been presented as a structurally stable solution of the parabolic equation [1]. By changing an additional angle parameter in the HLG beam, one can transform the HLG beam from an HG beam to an LG beam in a continuous way. The HLG beam is structurally stable under propagation and focusing. HLG beams can be experimentally realized by astigmatic transformation [1,2] and mode converters [3], which are based on the fractional Fourier transform. Moreover, the concept of the HLG beam has been extended to the vectorial non-paraxial regime [4]. As HLG beams play an important role in resonators and optical waveguides, as well as in optical vortices [5,6], the investigation of the properties of HLG beams will be useful to their practical application and experimental realization. For example, the design of optical traps demands the accurate and concise expressions of HLG beams. Therefore, the purpose of this paper is to Corresponding author.
E-mail address: [email protected] (G. Zhou). 0030-3992/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlastec.2007.11.003
reveal the internal vectorial structure of HLG beam from an alternative viewpoint and to present an analytically concise expression of the structure of the HLG beam. Here, the description of the HLG beam is directly derived from Maxwell’s equations using the full vector angular spectrum method. As the vector angular spectrum can be decomposed into two terms in the frequency domain, the propagating electric field of HLG beam can be decomposed into the TE and TM terms. The TE term denotes the electric field transverse to the propagation axis, and the TM term the associated magnetic field transverse to the propagation axis [7–9]. The spatial orientation relationship between the TE and TM terms is varied upon propagation. Only in the far field are the TE and TM terms orthogonal to each other. Moreover, most practical applications of laser beams are involved in the far field. Therefore, the analytical expressions of the TE and TM terms in the far field are presented by means of the method of stationary phase. The energy flux distributions of HLG beam and its TE and TM terms are studied and depicted in the far field. The influences of the additional angle parameter and Gaussian waist width on the vectorial structure and energy flux pattern of HLG beam are also investigated.
ARTICLE IN PRESS G. Zhou, J. Zheng / Optics & Laser Technology 40 (2008) 858–863
2. Analytical structure in the far field
with
In the Cartesian coordinates system, an HLG beam propagates toward the half free space zX0, and the z-axis is taken to be the propagation axis. The electric field of the HLG beam at the source plane z=0 is polarized in the x direction and takes the form as [1] ! E n;mx ðx; y; 0jaÞ jn;m ðx; yjaÞ ¼ , (1) E n;my ðx; y; 0jaÞ 0
Ax ðp; qÞ ¼
where jn;m ðx; yjaÞ ¼
nX þm
ij ðcos aÞnj ðsin aÞmj Pjðnj;mjÞ ð cos 2aÞ
j¼0
H G nþmj;j ðx; yÞ,
ð2Þ
where Pjðnj;mjÞ ð cos 2aÞ ¼ 2j
j X n l¼0
l
!
m
1
Z
1
½s Aðp; qÞ expðikr:sÞ dp dq, 1
e2 s ¼ e1 .
(10)
In this system of frequency domain, the vector angular spectrum Aðp; qÞ can be decomposed into two terms: (11)
Accordingly, the propagating electric field of HLG beam can be decomposed into the TE and TM terms:
H j is the jth order Hermite polynomial. n and m are the transverse mode numbers in the x and y directions, respectively, w0 is the Gaussian waist width and a is an angle parameter which allows to make a continuous transition from HG to LG beam. When a is equal to 0, HLG beam reduces to an HG beam. When a is equal to p=4, HLG beam transforms to be LG beam. The time-dependent factor expðiotÞ is omitted in Eq. (1) and o is the angular frequency. By using the vector angular spectrum representation of Maxwell’s [10,11], one can obtain the propagating electromagnetic field of the HLG beam at the location vector r ¼ xi þ yj þ zk: Z 1Z 1 E n;m ðrjaÞ ¼ Aðp; qÞ expðikr:sÞ dp dq, (5) 1
e1 e2 ¼ s;
Aðp; qÞ ¼ ½Aðp; qÞ:e1 e1 þ ½Aðp; qÞ:e2 e2 .
(4)
Z
2 2 1=2 where g ¼ ð1 p2 pqffiffiffi Þ . zr ¼ kw0 is the confocal parameter and f ¼ 1=ð 2kw0 Þ. The longitudinal electric field stems from the divergence theorem of electric field. Note that the electromagnetic field of an HLG beam is obtained by Fourier transformation. In the frequency domain, two unit vectors e1 and e2 can be defined as follows [7–9]: q p e1 ¼ i j, 1=2 2 2 2 ðp þ q Þ ðp þ q2 Þ1=2 pg qg iþ j ðp2 þ q2 Þ1=2 k. ð9Þ e2 ¼ 1=2 ðp2 þ q2 Þ ðp2 þ q2 Þ1=2
s e1 ¼ e2 ;
jl
is a Jacobi polynomial and x2 þ y2 x y G H nþmj;j ðx; yÞ ¼ exp H nþmj Hj . 2 w w 2w0 0 0
H n;m ðrjaÞ ¼ c
Z Z 1 1 1 E n;mx ðx; y; 0jaÞ exp½ikðpx þ qyÞ dx dy l2 1 1 ðiÞmþn ¼ jn;m ðzr p; zr qjaÞ, ð8Þ 4pf 2
Thus, the three unit vectors s, e1 and e2 form a mutually perpendicular right-handed system
!
ð cos 2a 1Þjl ð cos 2a þ 1Þl ð3Þ
1
859
(6)
1
where s ¼ pi þ qj þ gk. k ¼ 2p=l is the wave number. l is the light wavelength. The fork denotes a vector product, and the dot means scalar product. c ¼ ð0 =m0 Þ1=2 . 0 and m0 are the electric permittivity and the magnetic permeability, respectively. Aðp; qÞ is the vector angular spectrum and is given by p (7) Aðp; qÞ ¼ Ax ðp; qÞ i k , g
n;m E n;m ðrjaÞ ¼ E n;m TE ðrjaÞ þ E TM ðrjaÞ,
(12)
with E n;m TE ðrjaÞ ¼
Z
1
Z
1
½Aðp; qÞ:e1 e1 expðikr:sÞ dp dq 1 Z1 1 Z 1
qAx ðp; qÞ ðqi pjÞ expðikr:sÞ dp dq, 2 2 1 p þ q
¼ 1
ð13Þ E n;m TM ðrjaÞ
Z
1
Z
1
½Aðp; qÞ:e2 e2 expðikr:sÞ dp dq
¼ 1 Z1 1 Z 1
¼
pAx ðp; qÞ ½pgi þ qgj ðp2 þ q2 Þk 2 þ q2 Þ gðp 1 1 expðikr:sÞ dp dq. ð14Þ
Similarly, the corresponding magnetic field of HLG beam can be viewed as the superposition of the TE and TM terms: n;m H n;m ðrjaÞ ¼ H n;m TE ðrjaÞ þ H TM ðrjaÞ,
(15)
with H n;m TE ðrjaÞ
H n;m TM ðrjaÞ
Z
1
Z
1
Z
1
Z
qAx ðp; qÞ ½pgi þ qgj ðp2 þ q2 Þk 2 2 1 1 p þ q expðikr:sÞ dp dq, ð16Þ
¼c
¼ c 1
1
pAx ðp; qÞ ðqi pjÞ expðikr:sÞ dp dq. 2 þ q2 Þ gðp 1 (17)
ARTICLE IN PRESS G. Zhou, J. Zheng / Optics & Laser Technology 40 (2008) 858–863
860
Apparently, the TE and TM terms denote that the longitudinal components of the electric and magnetic fields are equal to zero, respectively. As the divergence condition of the electric field should be satisfied and the polarization direction of every plane wave component must be perpendicular to its own wave vector, the decomposition of HLG beam into the TE and TM terms is unique. The electromagnetic field of HLG beam can be decomposed into the transverse and longitudinal components in the spatial domain, while the decomposition of the electromagnetic field into the TE and TM terms is carried out in terms of the frequency domain. In the far field regime, the evanescent plane waves have completely disappeared, which means that the integrals in Eqs. (13), (14), (16) and (17) should be restrained within the range of 0pp2 þ q2 p1. As the condition kr ¼ kðx2 þ y2 þ z2 Þ1=2 ! 1 is satisfied in the far field, the analytical propagating electromagnetic fields of TE and TM terms can be presented in the far field. According to the method of stationary phase [12,13], the surface integral of Eq. (13) is shown to have the asymptotic value [14] E n;m TE ðrjaÞ ¼
j il X r j ðjtj bj d2j jÞ1=2 Mðpj ; qj Þ exp½ikrf ðpj ; qj ; x; yÞ as kr ! 1,
ð18Þ
where ðiÞmþn q jn;m ðzr p; zr qjaÞðqi pjÞ Mðp; qÞ ¼ 4pf 2 p2 þ q2
(19)
and f ðp; q; x; yÞ ¼ ðpx þ qy þ gzÞ=r.
(20)
The stationary points (pj ,qj ) are solutions of the following simultaneous equations: 8 qf ðp; q; x; yÞ > > > p¼pj ¼ 0; > qp > < q¼qj (21) qf ðp; q; x; yÞ > > > ¼ 0: > p¼pj > qq : q¼qj
Substituting Eq. (20) into (21), one can obtain p1 ¼ x=r;
q1 ¼ y=r.
(22)
There is only one stationary point. As a result, the parameters t1 , b1 , d1 and 1 turn out to be qf 2 ðp; q; x; yÞ 2 2 t1 ¼ (23) p¼p1 ¼ ð1 þ x =z Þ, qp2 q¼q1
qf 2 ðp; q; x; yÞ 2 2 b1 ¼ p¼p1 ¼ ð1 þ y =z Þ, qq2 q¼q1
qf 2 ðp; q; x; yÞ 2 d1 ¼ p¼p1 ¼ xy=z , qpqq
(25)
1 ¼ 1.
(26)
q¼q1
Therefore, the analytical electric field of TE term in the far field yields mþnþ3 zr yz E n;m j ðx1 ; y1 jaÞ TE ðrjaÞ ¼ ðiÞ r2 r2 n;m expðikrÞðyi xjÞ, ð27Þ where r ¼ ðx2 þ y2 Þ1=2 ;
x1 ¼ zr x=r;
y1 ¼ zr y=r.
(28)
Similarly, the analytical electric field of TM term in the far field turns out to be mþnþ3 zr x E n;m j ðx1 ; y1 jaÞ TM ðrjaÞ ¼ ðiÞ r2 r2 n;m expðikrÞðxzi þ yzj r2 kÞ.
ð29Þ
The corresponding analytical magnetic fields of TE and TM terms in the far field are found to be mþnþ3 czr yz H n;m j ðx1 ; y1 jaÞ TE ðrjaÞ ¼ ðiÞ r2 r3 n;m ðxzi þ yzj r2 kÞ expðikrÞ,
ð30Þ
czr x j ðx1 ; y1 jaÞ r2 r n;m ðyi xjÞ expðikrÞ.
ð31Þ
mþnþ1 H n;m TM ðrjaÞ ¼ ðiÞ
Apparently, the electromagnetic fields of TE and TM terms are orthogonal to each other in the far field. The HLG beam is given by the sum of the TE and TM terms: zr z E n;m ðrjaÞ ¼ ðiÞmþnþ3 2 jn;m ðx1 ; y1 jaÞ r x ð32Þ expðikrÞ i k . z czr j ðx1 ; y1 jaÞ r3 n;m ½xyi þ ðx2 þ z2 Þj yzk expðikrÞ.
H n;m ðrjaÞ ¼ ðiÞmþnþ1
ð33Þ
At the paraxial case, Eq. (32) reduces to zr z E n;m ðrjaÞ ¼ ðiÞmþnþ3 2 jn;m ðx1 ; y1 jaÞ expðikrÞi (34) r because the optical field is confined to the neighbor of the propagation axis. As the range of w0 is not limited, the formulae obtained here are applicable not only to the paraxial case, but also to the non-paraxial case. As the non-paraxial case is also taken into our consideration, the light intensity should be replaced by the energy flux. The energy flux distribution of the TE and TM terms are separately given by 1 n;m Re½E n;m TE ðrjaÞ H TE ðrjaÞz 2 cz2 y2 z3 ¼ r 2 5 jjn;m ðx1 ; y1 jaÞj2 , 2r r
hSz in;m TE ðrjaÞ ¼ (24)
ð35Þ
ARTICLE IN PRESS G. Zhou, J. Zheng / Optics & Laser Technology 40 (2008) 858–863
hS z in;m TM ðrjaÞ
cz2 x2 z ¼ r 2 3 jjn;m ðx1 ; y1 jaÞj2 , 2r r
(36)
where Re means taking the real part and the asterisk denotes complex conjugation. The energy flux distribution of HLG beam yields cz2r z z2 þ x2 2 hS z in;m ðrjaÞ ¼ 2 3 r jjn;m ðx1 ; y1 jaÞj2 2r r r2 n;m ¼ hS z in;m TE ðrjaÞ þ hS z iTM ðrjaÞ.
ð37Þ
The last step is satisfied because the TE and TM terms are orthogonal to each other in the far field. At the paraxial case, Eq. (37) can be simplified to be hS z in;m ðrjaÞ ¼
cz2r jj ðx1 ; y1 jaÞj2 . 2z2 n;m
(38)
When a is equal to 0, HLG beam reduces to be HG beam. At this case, Eqs. (35)–(37) reduce to hS z in;m TE ðrj0Þ
hS z in;m TM ðrj0Þ
cz2r y2 z3 r2 ¼ exp 2 2r2 r5 2f r2 x y 2 2 H n pffiffiffi H m pffiffiffi , 2fr 2fr
hS z in;m ðrj0Þ ¼
2
2
nXm;
npm:
(43) As a result, the energy flux of LG beam turns out to be 2 2 nm 8 2 2 r > 2 czr z z þ x 2 nþm > 2 ðm!Þ r > 2 2 2 r3 2 > r 2r > 2f r > > > 2 2 > 2 > > r r > > exp 2 Lnm ; m 2 2 < p > 2 2f r 2f r hSz in;m r ¼ mn > nþm 2 cz2r z z2 þ x2 2 r2 4 > > 2 ðn!Þ r > > r2 2r2 r3 > 2f 2 r2 > > > 2 > > r2 r2 > mn > > exp ; L : n 2f 2 r2 2f 2 r2
nXm;
npm:
(44) ð39Þ
cz2r x2 z r2 ¼ exp 2 2r2 r3 2f r2 x y 2 2 H n pffiffiffi H m pffiffiffi , 2fr 2fr cz2r z 2r2 r3
2 nm 8 2 2 r > 2 czr x z nþm > 2 ðm!Þ > 2 r3 > 2r > 2f 2 r2 > > > 2 2 > > > r2 r > nm > exp 2 ; L > m p < 2 2f r 2f 2 r2 n;m 2 mn hS z iTM r ¼ 2 2 > 4 r 2 czr x z > nþm > ðn!Þ 2 > 2 3 > r 2r > 2f 2 r2 > > > 2 2 > > r2 r > mn > > exp ; L : n 2f 2 r2 2f 2 r2
861
ð40Þ
2
z þx 2 r r exp 2 2 r 2f r2 x y H 2n pffiffiffi H 2m pffiffiffi . 2fr 2fr
ð41Þ
The above results correspond to those of typical HG beam TEMn;m mode. When a is equal to p=4, HLG beam transforms to be LG beam. The energy flux distribution of the TE and TM terms for LG beam are given by 2 nm 8 2 2 3 r > 2 czr y z nþm > 2 ðm!Þ > 2 2 2 r5 > 2r > 2f r > > > 2 2 > 2 > > r r > nm > ; < exp 2f 2 r2 Lm p > 2f 2 r2 ¼ hS z in;m r mn TM 2 2 3 > 4 r2 > 2nþm ðn!Þ2 2czr y z > > 2 > r2 r5 > 2f r2 > > > 2 2 > > r2 r > mn > > ; : exp 2 2 Ln 2f r 2f 2 r2
Comparing Eqs. (42)–(44) with Eqs. (14)–(16) of Ref. [15], it is found that LG beam of this paper is different from LG beam defined in other literatures, whose dependence on the angle is cos Mj or sin Mj. M is an arbitrary integer and j ¼ tan1 ðy=xÞ. Here, the LG beam is independent of angle. Accordingly, the pattern of LG beam is ringy [1,4]. Both the paraxial case and npm are considered, Eq. (44) can be rewritten as 2 m0 p Acz2r r r2 hS z in;m r ¼ exp 2 4 2z2 2f 2 r2 2f r2 2 2 r 0 Lm , n 2f 2 r2
where m0 ¼ m n and A ¼ 2nþm ðn!Þ2 . Now, there are two paraxial LG beams, whose dependences on the angle are cos m0 j and sin m0 j, respectively. Their polarization directions are parallel to the x and y directions, respectively. The energy flux distributions of the paraxial LG beam with cos m0 j, its TE and TM terms [15] are 0 hS 0z in;m TE
nXm;
0 hS 0z in;m TM
npm;
(42)
ð45Þ
2 m0 Acz2r y2 z3 r r2 ¼ exp 2 2r2 r5 2f 2 r2 2f r2 2 2 r 0 Lm cos2 m0 j, n 2f 2 r2 2 m0 Acz2r x2 z r r2 ¼ exp 2 2r2 r3 2f 2 r2 2f r2 2 2 r 0 Lm cos2 m0 j, n 2f 2 r2
ð46Þ
ð47Þ
ARTICLE IN PRESS 862
G. Zhou, J. Zheng / Optics & Laser Technology 40 (2008) 858–863
2 m0 Acz2r r r2 exp 2z2 2f 2 r2 2f 2 r2 2 2 r 0 Lm cos2 m0 j. ð48Þ n 2f 2 r2 Similarly, one can obtain the energy flux distributions of the paraxial LG beam with sin m0 j, its TE and TM terms: 2 m0 Acz2r x2 z3 r r2 00 n;m0 hS z iTM ¼ exp 2r2 r5 2f 2 r2 2f 2 r2 2 2 r 0 Lm sin2 m0 j, ð49Þ n 2f 2 r2 2 m0 0 Acz2r y2 z r r2 hS 00z in;m ¼ exp TM 2r2 r3 2f 2 r2 2f 2 r2 2 2 r 0 Lm sin2 m0 j, ð50Þ n 2f 2 r2 2 m0 Acz2r r r2 00 hS z in;m0 ¼ exp 2 2z2 2f 2 r2 2f r2 2 2 r 0 Lm sin2 m0 j. ð51Þ n 2f 2 r2 The superposition of the two paraxial LG beam results in the following equation: 2 m0 Acz2r r 0 00 hS z in;m0 þ hS z in;m0 ¼ 2 2z 2f 2 r2 2 2 r2 r 0 exp 2 Lm n 2 2f r 2f 2 r2 p ð52Þ ¼ hSz in;m r . 4 hS 0z in;m0 ¼
However, the superposition of the corresponding TE terms is not identical to the TE term of HLG beam with a ¼ p=4: 0 n;m p 00 n;m0 . (53) hS0z in;m TE þ hS z iTE ahS z iTE r 4 The conclusion is also valid for the TM term. Therefore, the paraxial HLG beam with a ¼ p=4 can be viewed as the superposition of two paraxial LG beams with cos m0 j and sin m0 j, whose polarizations are separately parallel to the x and y directions. But its internal structure is different from the superposition of the corresponding TE and TM terms. The energy flux distributions of HLG beam, its TE and TM terms at the reference plane z ¼ 1000l are depicted in Figs. 1 and 2. For simplicity, c is set to unity. n ¼ 2 and m ¼ 1. w0 is set to 5l in Fig. 1, which corresponds to the paraxial case. In Fig. 2 w0 is equal to 0:5l, which corresponds to the non-paraxial case. (a) denotes the energy flux distribution of the TE term and (b) that of the TM term. (c), the sum of (a) and (b), means the energy flux distribution of HLG beam. The angle parameter a is chosen from 0 to p=4 with a step of p=16. The process of evolution from HG beam to LG beam is revealed in Figs. 1 and 2. The intermediate beam keeps some important features of HG beam and LG beam. The angle parameter a markedly influences on the vectorial structure and energy flux pattern of HLG beam. Also, the value of Gaussian beam waist width impacts on the vectorial structure of HLG beam so that the energy flux pattern of a nonparaxial HLG beam is distinctly different from that of a paraxial HLG beam. The physical pictures of the HLG beam depicted from the vectorial structure denote the inherent composition of HLG beam. The vectorial
Fig. 1. The energy flux distribution at the reference plane z ¼ 1000l. w0 ¼ 5l, n ¼ 2 and m ¼ 1. (a) The TE term. (b) The TM term. (c) The whole beam. The scale of the x and y is from 100l to 100l.
ARTICLE IN PRESS G. Zhou, J. Zheng / Optics & Laser Technology 40 (2008) 858–863
863
Fig. 2. The energy flux distribution at the reference plane z ¼ 1000l. w0 ¼ 0:5l, n ¼ 2 and m ¼ 1. (a) The TE term. (b) The TM term. (c) The whole beam. The scale of the x and y is from 1500l to 1500l.
structure of HLG beam in the near field can be numerically calculated using a similar procedure proposed by Ref. [16]. 3. Conclusions The description of HLG beam is directly derived from Maxwell’s equations. HLG beam is decomposed into the TE and TM terms by using the vector angular spectrum representation of Maxwell’s equations. By means of the method of stationary phase, the analytical expressions of the TE and TM terms are obtained in the far field. The energy flux distributions of the TE and TM terms are investigated and depicted in the far field. As the TE and TM terms are orthogonal to each other in the far field, the energy flux of HLG beam is equal to the sum of those of the TE and TM terms. The influences of the additional angle parameter and Gaussian waist width on the vectorial structure and energy flux pattern of the HLG beam are also investigated. This research reveals the internal vectorial structure of the HLG beam and can be regarded as the supplement of the structurally stable solution of the parabolic equation. Moreover, this research is not only useful to the propagation and transformation of laser beam but also provides a new approach to manipulate laser beams. As the electromagnetic fields of the TE and TM terms are orthogonal to each other in the far field, they can be theoretically detached in the far field. But, how to isolate the TE and TM terms experimentally? Once the TE and TM terms are separated, there may be novel potential applications. All of these deserve further investigation.
Acknowledgments This work was supported by Scientific Research Fund of Zhejiang Provincial Education Department under Grant 20060677. The authors are indebted to the reviewer for valuable comments.
References [1] Abramochkin EG, Volostnikov VG. J Opt A: Pure Appl Opt 2004;6:S157. [2] Courtial J, Padgett MJ. Opt Commun 1999;159:13. [3] Malyutin AA. Quantum Electron 2004;34:165. [4] Wang BZ, Zhao ZG, Lu¨ BD, Duan KL. Chin Phys 2007;16:143. [5] Flossmann F, Schwarz UT, Maier M. Opt Commun 2005;250:218. [6] Volpe G, Petrov D. Opt Commun 2004;237:89. [7] Martı´ nez-Herrero R, Mejı´ as PM, Bosch S, Carnicer A. J Opt Soc Am A 2001;18:1678. [8] Mejı´ as PM, Martı´ nez-Herrero R, Piquero G, Movilla JM. Prog Quantum Electron 2002;26:65. [9] Guo HM, Chen JB, Zhuang SL. Opt Express 2006;14:2095. [10] Rhodes DR. Proc IEEE 1964;52:1013. [11] Chen CG, Konkola PT, Ferrera J, Heilmann RK, Schattenburg ML. J Opt Soc Am A 2002;19:404. [12] Carter WH. J Opt Soc Am 1972;62:1195. [13] Mandel L, Wolf E. Optical coherence and quantum optics. Cambridge: Cambridge University Press; 1995. [14] Born M, Wolf E. Principles of optics. 4th ed. New York: Pergamon Press; 1970. [15] Zhou GQ. Opt Lett 2006;31:2616. [16] Martı´ nez-Herrero R, Mejı´ as PM, Bosch S, Carnicer A. J Opt A: Pure Appl Opt 2006;8:524.
Optics & Laser Technology 40 (2008) 858–863 www.elsevier.com/locate/optlastec
Vectorial structure of Hermite–Laguerre–Gaussian beam in the far field Guoquan Zhou, Jun Zheng School of Sciences, Zhejiang Forestry University, Lin’an 311300, Zhejiang Province, China Received 4 August 2007; received in revised form 30 October 2007; accepted 14 November 2007 Available online 20 December 2007
Abstract Starting from Maxwell’s equations, a Hermite–Laguerre–Gaussian (HLG) beam is decomposed into the TE and TM terms by using the vector angular spectrum representation. By means of the method of stationary phase, the analytical TE and TM terms are presented in the far field. The energy flux distributions of the TE and TM terms are also investigated and depicted in the far field. The influences of the additional angle parameter and Gaussian waist width on the vectorial structure and energy flux pattern of HLG beam are also investigated. This research reveals the internal vectorial structure of HLG beam and may provide a new approach to the manipulation of laser beams. r 2007 Elsevier Ltd. All rights reserved. PACS: 41.85.Ew; 42.25.Bs; 42.55.Px Keywords: Hermite–Laguerre–Gaussian beam; Vectorial structure; Method of stationary phase
1. Introduction A united family of Hermite–Gaussian (HG) and Laguerre–Gaussian (LG) beams, namely the Hermite–Laguerre– Gaussian (HLG) beam, has been presented as a structurally stable solution of the parabolic equation [1]. By changing an additional angle parameter in the HLG beam, one can transform the HLG beam from an HG beam to an LG beam in a continuous way. The HLG beam is structurally stable under propagation and focusing. HLG beams can be experimentally realized by astigmatic transformation [1,2] and mode converters [3], which are based on the fractional Fourier transform. Moreover, the concept of the HLG beam has been extended to the vectorial non-paraxial regime [4]. As HLG beams play an important role in resonators and optical waveguides, as well as in optical vortices [5,6], the investigation of the properties of HLG beams will be useful to their practical application and experimental realization. For example, the design of optical traps demands the accurate and concise expressions of HLG beams. Therefore, the purpose of this paper is to Corresponding author.
E-mail address: [email protected] (G. Zhou). 0030-3992/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlastec.2007.11.003
reveal the internal vectorial structure of HLG beam from an alternative viewpoint and to present an analytically concise expression of the structure of the HLG beam. Here, the description of the HLG beam is directly derived from Maxwell’s equations using the full vector angular spectrum method. As the vector angular spectrum can be decomposed into two terms in the frequency domain, the propagating electric field of HLG beam can be decomposed into the TE and TM terms. The TE term denotes the electric field transverse to the propagation axis, and the TM term the associated magnetic field transverse to the propagation axis [7–9]. The spatial orientation relationship between the TE and TM terms is varied upon propagation. Only in the far field are the TE and TM terms orthogonal to each other. Moreover, most practical applications of laser beams are involved in the far field. Therefore, the analytical expressions of the TE and TM terms in the far field are presented by means of the method of stationary phase. The energy flux distributions of HLG beam and its TE and TM terms are studied and depicted in the far field. The influences of the additional angle parameter and Gaussian waist width on the vectorial structure and energy flux pattern of HLG beam are also investigated.
ARTICLE IN PRESS G. Zhou, J. Zheng / Optics & Laser Technology 40 (2008) 858–863
2. Analytical structure in the far field
with
In the Cartesian coordinates system, an HLG beam propagates toward the half free space zX0, and the z-axis is taken to be the propagation axis. The electric field of the HLG beam at the source plane z=0 is polarized in the x direction and takes the form as [1] ! E n;mx ðx; y; 0jaÞ jn;m ðx; yjaÞ ¼ , (1) E n;my ðx; y; 0jaÞ 0
Ax ðp; qÞ ¼
where jn;m ðx; yjaÞ ¼
nX þm
ij ðcos aÞnj ðsin aÞmj Pjðnj;mjÞ ð cos 2aÞ
j¼0
H G nþmj;j ðx; yÞ,
ð2Þ
where Pjðnj;mjÞ ð cos 2aÞ ¼ 2j
j X n l¼0
l
!
m
1
Z
1
½s Aðp; qÞ expðikr:sÞ dp dq, 1
e2 s ¼ e1 .
(10)
In this system of frequency domain, the vector angular spectrum Aðp; qÞ can be decomposed into two terms: (11)
Accordingly, the propagating electric field of HLG beam can be decomposed into the TE and TM terms:
H j is the jth order Hermite polynomial. n and m are the transverse mode numbers in the x and y directions, respectively, w0 is the Gaussian waist width and a is an angle parameter which allows to make a continuous transition from HG to LG beam. When a is equal to 0, HLG beam reduces to an HG beam. When a is equal to p=4, HLG beam transforms to be LG beam. The time-dependent factor expðiotÞ is omitted in Eq. (1) and o is the angular frequency. By using the vector angular spectrum representation of Maxwell’s [10,11], one can obtain the propagating electromagnetic field of the HLG beam at the location vector r ¼ xi þ yj þ zk: Z 1Z 1 E n;m ðrjaÞ ¼ Aðp; qÞ expðikr:sÞ dp dq, (5) 1
e1 e2 ¼ s;
Aðp; qÞ ¼ ½Aðp; qÞ:e1 e1 þ ½Aðp; qÞ:e2 e2 .
(4)
Z
2 2 1=2 where g ¼ ð1 p2 pqffiffiffi Þ . zr ¼ kw0 is the confocal parameter and f ¼ 1=ð 2kw0 Þ. The longitudinal electric field stems from the divergence theorem of electric field. Note that the electromagnetic field of an HLG beam is obtained by Fourier transformation. In the frequency domain, two unit vectors e1 and e2 can be defined as follows [7–9]: q p e1 ¼ i j, 1=2 2 2 2 ðp þ q Þ ðp þ q2 Þ1=2 pg qg iþ j ðp2 þ q2 Þ1=2 k. ð9Þ e2 ¼ 1=2 ðp2 þ q2 Þ ðp2 þ q2 Þ1=2
s e1 ¼ e2 ;
jl
is a Jacobi polynomial and x2 þ y2 x y G H nþmj;j ðx; yÞ ¼ exp H nþmj Hj . 2 w w 2w0 0 0
H n;m ðrjaÞ ¼ c
Z Z 1 1 1 E n;mx ðx; y; 0jaÞ exp½ikðpx þ qyÞ dx dy l2 1 1 ðiÞmþn ¼ jn;m ðzr p; zr qjaÞ, ð8Þ 4pf 2
Thus, the three unit vectors s, e1 and e2 form a mutually perpendicular right-handed system
!
ð cos 2a 1Þjl ð cos 2a þ 1Þl ð3Þ
1
859
(6)
1
where s ¼ pi þ qj þ gk. k ¼ 2p=l is the wave number. l is the light wavelength. The fork denotes a vector product, and the dot means scalar product. c ¼ ð0 =m0 Þ1=2 . 0 and m0 are the electric permittivity and the magnetic permeability, respectively. Aðp; qÞ is the vector angular spectrum and is given by p (7) Aðp; qÞ ¼ Ax ðp; qÞ i k , g
n;m E n;m ðrjaÞ ¼ E n;m TE ðrjaÞ þ E TM ðrjaÞ,
(12)
with E n;m TE ðrjaÞ ¼
Z
1
Z
1
½Aðp; qÞ:e1 e1 expðikr:sÞ dp dq 1 Z1 1 Z 1
qAx ðp; qÞ ðqi pjÞ expðikr:sÞ dp dq, 2 2 1 p þ q
¼ 1
ð13Þ E n;m TM ðrjaÞ
Z
1
Z
1
½Aðp; qÞ:e2 e2 expðikr:sÞ dp dq
¼ 1 Z1 1 Z 1
¼
pAx ðp; qÞ ½pgi þ qgj ðp2 þ q2 Þk 2 þ q2 Þ gðp 1 1 expðikr:sÞ dp dq. ð14Þ
Similarly, the corresponding magnetic field of HLG beam can be viewed as the superposition of the TE and TM terms: n;m H n;m ðrjaÞ ¼ H n;m TE ðrjaÞ þ H TM ðrjaÞ,
(15)
with H n;m TE ðrjaÞ
H n;m TM ðrjaÞ
Z
1
Z
1
Z
1
Z
qAx ðp; qÞ ½pgi þ qgj ðp2 þ q2 Þk 2 2 1 1 p þ q expðikr:sÞ dp dq, ð16Þ
¼c
¼ c 1
1
pAx ðp; qÞ ðqi pjÞ expðikr:sÞ dp dq. 2 þ q2 Þ gðp 1 (17)
ARTICLE IN PRESS G. Zhou, J. Zheng / Optics & Laser Technology 40 (2008) 858–863
860
Apparently, the TE and TM terms denote that the longitudinal components of the electric and magnetic fields are equal to zero, respectively. As the divergence condition of the electric field should be satisfied and the polarization direction of every plane wave component must be perpendicular to its own wave vector, the decomposition of HLG beam into the TE and TM terms is unique. The electromagnetic field of HLG beam can be decomposed into the transverse and longitudinal components in the spatial domain, while the decomposition of the electromagnetic field into the TE and TM terms is carried out in terms of the frequency domain. In the far field regime, the evanescent plane waves have completely disappeared, which means that the integrals in Eqs. (13), (14), (16) and (17) should be restrained within the range of 0pp2 þ q2 p1. As the condition kr ¼ kðx2 þ y2 þ z2 Þ1=2 ! 1 is satisfied in the far field, the analytical propagating electromagnetic fields of TE and TM terms can be presented in the far field. According to the method of stationary phase [12,13], the surface integral of Eq. (13) is shown to have the asymptotic value [14] E n;m TE ðrjaÞ ¼
j il X r j ðjtj bj d2j jÞ1=2 Mðpj ; qj Þ exp½ikrf ðpj ; qj ; x; yÞ as kr ! 1,
ð18Þ
where ðiÞmþn q jn;m ðzr p; zr qjaÞðqi pjÞ Mðp; qÞ ¼ 4pf 2 p2 þ q2
(19)
and f ðp; q; x; yÞ ¼ ðpx þ qy þ gzÞ=r.
(20)
The stationary points (pj ,qj ) are solutions of the following simultaneous equations: 8 qf ðp; q; x; yÞ > > > p¼pj ¼ 0; > qp > < q¼qj (21) qf ðp; q; x; yÞ > > > ¼ 0: > p¼pj > qq : q¼qj
Substituting Eq. (20) into (21), one can obtain p1 ¼ x=r;
q1 ¼ y=r.
(22)
There is only one stationary point. As a result, the parameters t1 , b1 , d1 and 1 turn out to be qf 2 ðp; q; x; yÞ 2 2 t1 ¼ (23) p¼p1 ¼ ð1 þ x =z Þ, qp2 q¼q1
qf 2 ðp; q; x; yÞ 2 2 b1 ¼ p¼p1 ¼ ð1 þ y =z Þ, qq2 q¼q1
qf 2 ðp; q; x; yÞ 2 d1 ¼ p¼p1 ¼ xy=z , qpqq
(25)
1 ¼ 1.
(26)
q¼q1
Therefore, the analytical electric field of TE term in the far field yields mþnþ3 zr yz E n;m j ðx1 ; y1 jaÞ TE ðrjaÞ ¼ ðiÞ r2 r2 n;m expðikrÞðyi xjÞ, ð27Þ where r ¼ ðx2 þ y2 Þ1=2 ;
x1 ¼ zr x=r;
y1 ¼ zr y=r.
(28)
Similarly, the analytical electric field of TM term in the far field turns out to be mþnþ3 zr x E n;m j ðx1 ; y1 jaÞ TM ðrjaÞ ¼ ðiÞ r2 r2 n;m expðikrÞðxzi þ yzj r2 kÞ.
ð29Þ
The corresponding analytical magnetic fields of TE and TM terms in the far field are found to be mþnþ3 czr yz H n;m j ðx1 ; y1 jaÞ TE ðrjaÞ ¼ ðiÞ r2 r3 n;m ðxzi þ yzj r2 kÞ expðikrÞ,
ð30Þ
czr x j ðx1 ; y1 jaÞ r2 r n;m ðyi xjÞ expðikrÞ.
ð31Þ
mþnþ1 H n;m TM ðrjaÞ ¼ ðiÞ
Apparently, the electromagnetic fields of TE and TM terms are orthogonal to each other in the far field. The HLG beam is given by the sum of the TE and TM terms: zr z E n;m ðrjaÞ ¼ ðiÞmþnþ3 2 jn;m ðx1 ; y1 jaÞ r x ð32Þ expðikrÞ i k . z czr j ðx1 ; y1 jaÞ r3 n;m ½xyi þ ðx2 þ z2 Þj yzk expðikrÞ.
H n;m ðrjaÞ ¼ ðiÞmþnþ1
ð33Þ
At the paraxial case, Eq. (32) reduces to zr z E n;m ðrjaÞ ¼ ðiÞmþnþ3 2 jn;m ðx1 ; y1 jaÞ expðikrÞi (34) r because the optical field is confined to the neighbor of the propagation axis. As the range of w0 is not limited, the formulae obtained here are applicable not only to the paraxial case, but also to the non-paraxial case. As the non-paraxial case is also taken into our consideration, the light intensity should be replaced by the energy flux. The energy flux distribution of the TE and TM terms are separately given by 1 n;m Re½E n;m TE ðrjaÞ H TE ðrjaÞz 2 cz2 y2 z3 ¼ r 2 5 jjn;m ðx1 ; y1 jaÞj2 , 2r r
hSz in;m TE ðrjaÞ ¼ (24)
ð35Þ
ARTICLE IN PRESS G. Zhou, J. Zheng / Optics & Laser Technology 40 (2008) 858–863
hS z in;m TM ðrjaÞ
cz2 x2 z ¼ r 2 3 jjn;m ðx1 ; y1 jaÞj2 , 2r r
(36)
where Re means taking the real part and the asterisk denotes complex conjugation. The energy flux distribution of HLG beam yields cz2r z z2 þ x2 2 hS z in;m ðrjaÞ ¼ 2 3 r jjn;m ðx1 ; y1 jaÞj2 2r r r2 n;m ¼ hS z in;m TE ðrjaÞ þ hS z iTM ðrjaÞ.
ð37Þ
The last step is satisfied because the TE and TM terms are orthogonal to each other in the far field. At the paraxial case, Eq. (37) can be simplified to be hS z in;m ðrjaÞ ¼
cz2r jj ðx1 ; y1 jaÞj2 . 2z2 n;m
(38)
When a is equal to 0, HLG beam reduces to be HG beam. At this case, Eqs. (35)–(37) reduce to hS z in;m TE ðrj0Þ
hS z in;m TM ðrj0Þ
cz2r y2 z3 r2 ¼ exp 2 2r2 r5 2f r2 x y 2 2 H n pffiffiffi H m pffiffiffi , 2fr 2fr
hS z in;m ðrj0Þ ¼
2
2
nXm;
npm:
(43) As a result, the energy flux of LG beam turns out to be 2 2 nm 8 2 2 r > 2 czr z z þ x 2 nþm > 2 ðm!Þ r > 2 2 2 r3 2 > r 2r > 2f r > > > 2 2 > 2 > > r r > > exp 2 Lnm ; m 2 2 < p > 2 2f r 2f r hSz in;m r ¼ mn > nþm 2 cz2r z z2 þ x2 2 r2 4 > > 2 ðn!Þ r > > r2 2r2 r3 > 2f 2 r2 > > > 2 > > r2 r2 > mn > > exp ; L : n 2f 2 r2 2f 2 r2
nXm;
npm:
(44) ð39Þ
cz2r x2 z r2 ¼ exp 2 2r2 r3 2f r2 x y 2 2 H n pffiffiffi H m pffiffiffi , 2fr 2fr cz2r z 2r2 r3
2 nm 8 2 2 r > 2 czr x z nþm > 2 ðm!Þ > 2 r3 > 2r > 2f 2 r2 > > > 2 2 > > > r2 r > nm > exp 2 ; L > m p < 2 2f r 2f 2 r2 n;m 2 mn hS z iTM r ¼ 2 2 > 4 r 2 czr x z > nþm > ðn!Þ 2 > 2 3 > r 2r > 2f 2 r2 > > > 2 2 > > r2 r > mn > > exp ; L : n 2f 2 r2 2f 2 r2
861
ð40Þ
2
z þx 2 r r exp 2 2 r 2f r2 x y H 2n pffiffiffi H 2m pffiffiffi . 2fr 2fr
ð41Þ
The above results correspond to those of typical HG beam TEMn;m mode. When a is equal to p=4, HLG beam transforms to be LG beam. The energy flux distribution of the TE and TM terms for LG beam are given by 2 nm 8 2 2 3 r > 2 czr y z nþm > 2 ðm!Þ > 2 2 2 r5 > 2r > 2f r > > > 2 2 > 2 > > r r > nm > ; < exp 2f 2 r2 Lm p > 2f 2 r2 ¼ hS z in;m r mn TM 2 2 3 > 4 r2 > 2nþm ðn!Þ2 2czr y z > > 2 > r2 r5 > 2f r2 > > > 2 2 > > r2 r > mn > > ; : exp 2 2 Ln 2f r 2f 2 r2
Comparing Eqs. (42)–(44) with Eqs. (14)–(16) of Ref. [15], it is found that LG beam of this paper is different from LG beam defined in other literatures, whose dependence on the angle is cos Mj or sin Mj. M is an arbitrary integer and j ¼ tan1 ðy=xÞ. Here, the LG beam is independent of angle. Accordingly, the pattern of LG beam is ringy [1,4]. Both the paraxial case and npm are considered, Eq. (44) can be rewritten as 2 m0 p Acz2r r r2 hS z in;m r ¼ exp 2 4 2z2 2f 2 r2 2f r2 2 2 r 0 Lm , n 2f 2 r2
where m0 ¼ m n and A ¼ 2nþm ðn!Þ2 . Now, there are two paraxial LG beams, whose dependences on the angle are cos m0 j and sin m0 j, respectively. Their polarization directions are parallel to the x and y directions, respectively. The energy flux distributions of the paraxial LG beam with cos m0 j, its TE and TM terms [15] are 0 hS 0z in;m TE
nXm;
0 hS 0z in;m TM
npm;
(42)
ð45Þ
2 m0 Acz2r y2 z3 r r2 ¼ exp 2 2r2 r5 2f 2 r2 2f r2 2 2 r 0 Lm cos2 m0 j, n 2f 2 r2 2 m0 Acz2r x2 z r r2 ¼ exp 2 2r2 r3 2f 2 r2 2f r2 2 2 r 0 Lm cos2 m0 j, n 2f 2 r2
ð46Þ
ð47Þ
ARTICLE IN PRESS 862
G. Zhou, J. Zheng / Optics & Laser Technology 40 (2008) 858–863
2 m0 Acz2r r r2 exp 2z2 2f 2 r2 2f 2 r2 2 2 r 0 Lm cos2 m0 j. ð48Þ n 2f 2 r2 Similarly, one can obtain the energy flux distributions of the paraxial LG beam with sin m0 j, its TE and TM terms: 2 m0 Acz2r x2 z3 r r2 00 n;m0 hS z iTM ¼ exp 2r2 r5 2f 2 r2 2f 2 r2 2 2 r 0 Lm sin2 m0 j, ð49Þ n 2f 2 r2 2 m0 0 Acz2r y2 z r r2 hS 00z in;m ¼ exp TM 2r2 r3 2f 2 r2 2f 2 r2 2 2 r 0 Lm sin2 m0 j, ð50Þ n 2f 2 r2 2 m0 Acz2r r r2 00 hS z in;m0 ¼ exp 2 2z2 2f 2 r2 2f r2 2 2 r 0 Lm sin2 m0 j. ð51Þ n 2f 2 r2 The superposition of the two paraxial LG beam results in the following equation: 2 m0 Acz2r r 0 00 hS z in;m0 þ hS z in;m0 ¼ 2 2z 2f 2 r2 2 2 r2 r 0 exp 2 Lm n 2 2f r 2f 2 r2 p ð52Þ ¼ hSz in;m r . 4 hS 0z in;m0 ¼
However, the superposition of the corresponding TE terms is not identical to the TE term of HLG beam with a ¼ p=4: 0 n;m p 00 n;m0 . (53) hS0z in;m TE þ hS z iTE ahS z iTE r 4 The conclusion is also valid for the TM term. Therefore, the paraxial HLG beam with a ¼ p=4 can be viewed as the superposition of two paraxial LG beams with cos m0 j and sin m0 j, whose polarizations are separately parallel to the x and y directions. But its internal structure is different from the superposition of the corresponding TE and TM terms. The energy flux distributions of HLG beam, its TE and TM terms at the reference plane z ¼ 1000l are depicted in Figs. 1 and 2. For simplicity, c is set to unity. n ¼ 2 and m ¼ 1. w0 is set to 5l in Fig. 1, which corresponds to the paraxial case. In Fig. 2 w0 is equal to 0:5l, which corresponds to the non-paraxial case. (a) denotes the energy flux distribution of the TE term and (b) that of the TM term. (c), the sum of (a) and (b), means the energy flux distribution of HLG beam. The angle parameter a is chosen from 0 to p=4 with a step of p=16. The process of evolution from HG beam to LG beam is revealed in Figs. 1 and 2. The intermediate beam keeps some important features of HG beam and LG beam. The angle parameter a markedly influences on the vectorial structure and energy flux pattern of HLG beam. Also, the value of Gaussian beam waist width impacts on the vectorial structure of HLG beam so that the energy flux pattern of a nonparaxial HLG beam is distinctly different from that of a paraxial HLG beam. The physical pictures of the HLG beam depicted from the vectorial structure denote the inherent composition of HLG beam. The vectorial
Fig. 1. The energy flux distribution at the reference plane z ¼ 1000l. w0 ¼ 5l, n ¼ 2 and m ¼ 1. (a) The TE term. (b) The TM term. (c) The whole beam. The scale of the x and y is from 100l to 100l.
ARTICLE IN PRESS G. Zhou, J. Zheng / Optics & Laser Technology 40 (2008) 858–863
863
Fig. 2. The energy flux distribution at the reference plane z ¼ 1000l. w0 ¼ 0:5l, n ¼ 2 and m ¼ 1. (a) The TE term. (b) The TM term. (c) The whole beam. The scale of the x and y is from 1500l to 1500l.
structure of HLG beam in the near field can be numerically calculated using a similar procedure proposed by Ref. [16]. 3. Conclusions The description of HLG beam is directly derived from Maxwell’s equations. HLG beam is decomposed into the TE and TM terms by using the vector angular spectrum representation of Maxwell’s equations. By means of the method of stationary phase, the analytical expressions of the TE and TM terms are obtained in the far field. The energy flux distributions of the TE and TM terms are investigated and depicted in the far field. As the TE and TM terms are orthogonal to each other in the far field, the energy flux of HLG beam is equal to the sum of those of the TE and TM terms. The influences of the additional angle parameter and Gaussian waist width on the vectorial structure and energy flux pattern of the HLG beam are also investigated. This research reveals the internal vectorial structure of the HLG beam and can be regarded as the supplement of the structurally stable solution of the parabolic equation. Moreover, this research is not only useful to the propagation and transformation of laser beam but also provides a new approach to manipulate laser beams. As the electromagnetic fields of the TE and TM terms are orthogonal to each other in the far field, they can be theoretically detached in the far field. But, how to isolate the TE and TM terms experimentally? Once the TE and TM terms are separated, there may be novel potential applications. All of these deserve further investigation.
Acknowledgments This work was supported by Scientific Research Fund of Zhejiang Provincial Education Department under Grant 20060677. The authors are indebted to the reviewer for valuable comments.
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