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Propagation properties of partially coherent modified Bessel–Gauss beams
ARTICLE IN PRESS
Optik
Optics
Optik 116 (2005) 65–70 www.elsevier.de/ijleo
Propagation properties of partially coherent modified Bessel–Gauss beams Li Wanga, Xiqing Wanga, Baida Lu¨b, a
Department of Applied Physics, Southwest Jiaotong University, Chengdu 610031, PR China Institute of Laser Physics and Laser Chemistry, Sichuan University, Chengdu 610064, PR China
b
Received 18 August 2004; accepted 27 November 2004
Abstract Starting from the propagation law of partially coherent light, the analytical propagation equations of partially coherent modified Bessel–Gauss beams (MBGBs) through a paraxial optical ABCD system are derived and illustrated with typical application examples. Furthermore, by using the intensity moments method and integral transformation technique, the important characteristic parameters, including the beam width, far-field divergence angle, M 2 factor and kurtosis parameter of partially coherent MBGBs, are expressed in a closed and simple form. As a result, some basic properties of MBGBs and the dependence of the M 2 factor and kurtosis parameter on the spectral degree of coherence and beam order are illustrated both analytically and numerically. r 2005 Elsevier GmbH. All rights reserved. Keywords: Partially coherent modified Bessel–Gauss beam (MBGB); Propagation; Beam width; M2 factor; Kurtosis parameter
1. Introduction Recently, Ponomarenko [1] introduced a new class of partially coherent beams with a separable phase, that can be represented as an incoherent superposition of fully coherent Laguerre–Gauss modes. The new beams carry optical vortices and possess remarkable properties, for example, their spectral degree of coherence does not depend on the relative orientation of a pair of points at the transversal plane. In addition, the new beams propagate over large distance with little spreading, which may be useful for some applications [2]. Such a type of beam can be called the partially coherent modified Bessel–Gauss beam (MBGB). In Ref. [1], the free-space propagation properties of partially coherent MBGBs were analyzed. The present paper is aimed at studying the propagation properties of partially coherent MBGBs in the general case. In Section 2, starting from the Corresponding author. Fax: +86 28 8540 3260.
E-mail address: [email protected] (B. Lu¨). 0030-4026/$ - see front matter r 2005 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2004.11.006
propagation law of partially coherent light, the explicit propagation equations of partially coherent MBGBs through a paraxial optical ABCD system are derived and two application examples, i.e., the free-space propagation and focusing by a thin lens are analyzed. Then, by using the intensity moments method and integral transformation technique, the analytical expressions for the beam characteristic parameters, such as the beam width, farfield divergence angle, M 2 factor and kurtosis parameter of partially coherent MBGBs, are deduced and illustrated with numerical examples in Section 3. Finally, Section 4 summarizes the main results obtained in this paper.
2. Propagation of partially coherent modified Bessel–Gauss beams through a paraxial optical ABCD system According to Ref. [1], the cross-spectral density function W ðr01 ; r02 ; j01 ; j02 ; 0Þ of partially coherent
ARTICLE IN PRESS 66
L. Wang et al. / Optik 116 (2005) 65–70
MBGBs at the plane z ¼ 0 reads as [1]
Applying the integral formulae [4] Z 1 expðbx2 ÞJ m ðxyÞI m ðaxÞx dx 0 2 1 a y2 ay exp ¼ ; Jm 2b 2b 4b
W ðr01 ; r02 ; j01 ; j02 ; 0Þ xm=2 exp imðj01 j02 Þ 1x pffiffiffi 0 0 0 0 1 þ x r12 þ r22 4 x r1 r2 exp Im ; 2 1 x w0 1 x w20
¼
ð1Þ
where I m ðxÞ denotes the modified Bessel function of order m, ðr01 ; j01 Þ; ðr02 ; j02 Þ are polar coordinates of two points at the plane z ¼ 0; w0 stands for the waist width of the Gaussian part ðm ¼ 0; x ¼ 0Þ; x is the spectral degree of coherence, 0oxo1; and two limiting cases x ! 0 and x ! 1 correspond to the fully coherent and incoherent cases, respectively. Within the framework of the paraxial approximation, the propagation of the cross-spectral density function through a paraxial optical system parameterized by A B transfer matrix obeys [3] C D W ðr1 ; r2 ; j1 ; j2 ; zÞ ZZ ZZ k 2 W 0 ðr01 ; r02 ; z0 ¼ 0Þ ¼ 2pB ik 0 2 0 exp Aðr1 r22 Þ 2B 2 r1 r01 cosðj1 j01 Þ r2 r02 cosðj2 j02 Þ
0 0 0 0 2 2 þDðr1 r2 Þ r1 r2 dr1 dr2 dj01 dj02 ;
x1=2 expðex2 ÞJ m ðxf ÞJ m ðgxÞðxf Þ1=2 dx
0
¼
2 f 1=2 g þf2 gf exp Im 2e 2e 4e
ð6Þ
tedious but straightforward integral calculations yield W ðr1 ; r2 ; j1 ; j2 ; zÞ w20 xm=2 exp½imðj1 j2 Þ w2 1 x pffiffiffi 1 þ x ðr21 þ r22 Þ 4 x r1 r2 exp Im 1x w2 1 x w2 2 2 ðr r2 Þ exp ik 1 ; 2R
¼
ð7Þ
w2 ¼ w20 ½A2 þ ðB=z0 Þ2 ;
(8)
A2 þ ð4B2 =k2 w40 Þ AC þ ð4BD=k2 w40 Þ
(9)
R¼
ð2Þ
2p
exp½ix cosðj fÞ imf df 0
¼ 2pim expðimjÞJ m ðxÞ
1
where
where k is the wave number related to the wavelength l by k ¼ 2p=l: Recalling the integral formula Z
Z
ð5Þ
ð3Þ
with J m ðxÞ being the Bessel function of order m, and substituting Eq. (1) into Eq. (2), the integrals with respect to j01 and j02 lead to W ðr1 ; r2 ; j1 ; j2 ; zÞ k 2 xm=2 2 exp½imðj1 j2 Þ ¼ 4p 2pB 1 x pffiffiffi 0 0 Z 1Z 1 0 0 1 þ x r12 þ r22 4 x r1 r2 exp Im 2 1 x w0 1 x w20 0 0
ik 0 0 ½Aðr12 r22 Þ þ Dðr21 r22 Þ exp 2B k 0 k J m r1 r1 J m r2 r02 r01 r02 dr01 dr02 : ð4Þ B B
denote the beam width and radius of curvature of the corresponding Gaussian beam propagating through a paraxial optical ABCD system, and z0 ¼ pw20 =l is the Rayleigh length of the corresponding Gaussian beam. On placing r1 ¼ r2 ¼ r; j1 ¼ j2 ¼ j into Eq. (7), the optical intensity is given by pffiffiffi 2 w20 xm=2 1 þ x 2r2 4 x r Iðr; zÞ ¼ 2 exp Im : 1 x w2 1 x w2 w 1x (10) Eqs. (7) and (10) provide the general propagation equations of partially coherent MBGBs through a paraxial optical ABCD system and indicate that generally partially coherent MBGBs preserve their shape upon propagation through the paraxial optical ABCD system. Additionally, for m ¼ 0 Eq. (10) reduces to pffiffiffi 2 w2 1 1 þ x 2r2 4 x r Iðr; zÞ ¼ 02 exp I0 : 1 x w2 1 x w2 w 1x (11) On the other hand, if ma0 we have Ið0; zÞ ¼ 0
(12)
because I m ð0Þ ¼ 0: Eq. (12) indicates that there is a central shadow in the intensity profile.
ARTICLE IN PRESS L. Wang et al. / Optik 116 (2005) 65–70
Two typical application examples of Eqs. (7)–(10) are of interest.
67
1.0
0.8
For the free-space propagation the ABCD matrix reduces to A B 1 z ¼ : (13) C D 0 1 It is readily seen that the expression (7) remains unchanged, but Eqs. (8) and (9) simplify to w2 ðzÞ ¼ w20 ½1 þ ðz=z0 Þ2 ;
(14)
RðzÞ ¼ z0 ðz=z0 þ z0 =zÞ;
(15)
I/Imax
2.1. Free-space propagation 0.6
ξ=0.1 ξ=0.5 ξ=0.9
0.4
0.2
0.0 0
1
2
3
4
r/w0
(a) 1.0
Eqs. (14)–(16) are essentially consistent with the results in Ref. [1]. Fig. 1 gives the normalized intensity distributions I=I max at the plane z ¼ 0 of partially coherent MBGBs for different values of m ¼ 0; 2 and x ¼ 0:1; 0:5; 0:9; where I max denotes the maximum intensity. As can be seen, apart from m ¼ 0; where the intensity shows a Gaussian shape, there is a central dip in intensity profile for ma0: Additionally, Fig. 1b indicates that for the nearly coherent case of x ¼ 0:1; the intensity profile is fairly symmetric about its maximum. However, the symmetry vanishes and a long tail appears in the intensity profile as the spatial coherence decreases (i.e. x increases).
2.2. Focusing by a thin lens For this case the ABCD matrix reads as ! Dz z A B ¼ ; 1=f 1 C D
0.6
ξ= 0.1 ξ= 0.5 ξ= 0.9
0.4
0.2
0.0 0
1
2
(b)
3
4
r/w0
Fig. 1. Normalized intensity distributions I=I max at the plane z ¼ 0 of partially coherent MBGBs for different values of x ¼ 0:1; 0:5 and 0.9: (a) m ¼ 0 and (b) m ¼ 2:
R¼
Dz2 þ ð4z2 =k2 w40 Þ ; ðDz=f Þ þ ð4z=k2 w40 Þ
(20)
respectively. The substitution from Eq. (17) into Eq. (10) leads to Iðr; DzÞ ¼
(17)
where zf Dz ¼ f
0.8
I/Imax
respectively. The substitution from Eq. (13) into Eq. (10) yields pffiffiffi 2 w2 xm=2 1 þ x 2r2 4 x r exp Iðr; zÞ ¼ 2 0 Im : 1 x w2 ðzÞ 1 x w2 ðzÞ w ðzÞ 1 x (16)
p2 N 2 xm=2 2 p2 N 2 Dz2 þ ð1 þ DzÞ 1 x 1þx p2 N 2 2r2 exp 1 x p2 N 2 Dz2 þ ð1 þ DzÞ2 w20 pffiffiffi 4 x p2 N 2 r2 I m ; ð21Þ 1 x p2 N 2 Dz2 þ ð1 þ DzÞ2 w20
where (18)
and f is the focal length of the lens. The expression (7) remains unchanged, but Eqs. (8) and (9) reduce to ! 4z2 2 2 2 w ¼ w0 Dz þ 2 4 ¼ w20 ½Dz2 þ ðz=z0 Þ2 ; (19) k w0
N¼
w20 lf
(22)
is the Fresnel number associated with the corresponding Gaussian beam. On placing r ¼ 0 and m ¼ 0 into Eq. (21), we obtain Ið0; DzÞ ¼ I 0
1 ; ð1 þ DzÞ þ p2 N 2 Dz2 2
(23)
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L. Wang et al. / Optik 116 (2005) 65–70
where I0 ¼
p2 N 2 1x
(24)
is the intensity at the geometrical focus of Dz ¼ 0: Eq. (23) denotes the axial intensity distribution of partially coherent MBGBs with m ¼ 0 and indicates that Ið0; DzÞ depends on the Fresnel number N, spectral degree of coherence x and relative distance Dz: It is easily seen that for the case of ma0 we have Ið0; DzÞ ¼ 0: From Eq. (23) the relative focal shift, namely, the relative position of axial maximum intensity for the case of m ¼ 0 turns out to be Df ¼
zmax f 1 ¼ ; f 1 þ p2 N 2
The normalized intensity distributions I=I max at the focal plane of focused partially coherent MBGBs are plotted in Fig. 2 for m ¼ 0; 2 and x ¼ 0:1; 0:5; 0:9: A comparison of Figs. 1 and 2 shows the focusing effect of the lens, but the shape of the intensity profile of focused partially coherent MBGBs remains unchanged.
(25)
where zmax is the position of axial maximum intensity referred to the lens. Eq. (25) implies that Df is dependent only on the Fresnel number N, but independent of the spectral degree of coherence x: In addition, Df (absolute value) increases with decreasing N. Interesting is that the relative focal shift Df of focused partially coherent
1.0
0.6
ξ=0.1 ξ=0.5 ξ=0.9
0.4
3. Characteristic parameters of partially coherent modified Bessel–Gauss beams 3.1. Beam width and far-field divergence angle The second-order intensity moment in the space domain is expressed as [6] R 2p R 1 2 r Iðr; j; zÞr dr dj s2r ðzÞ ¼ R0 2p R0 1 : (27) 0 0 Iðr; j; zÞr dr dj Recalling the integral formulae [4] Z 1 expðaxÞI n ðbxÞ dx
0.8
I/Imax
MBGBs takes the same form of focused Gaussian beams [5]. On substituting from Eq. (25) into Eq. (23), the axial maximum intensity is given by 1 I max ¼ I 0 1 þ 2 2 : (26) pN
0
bn qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 b2 ða þ a2 b2 Þn
0.2
0.0 0.0
½Re n4 1 Z 0.2
0.4
0.6
0.8
0
1.0
I/Imax
0.8 ξ=0.1 ξ=0.5 ξ=0.9
0.2
(b)
0.2
0.4
0.6
0.8
1.0
Re a4jRe bj
ð29Þ
the substitution from Eq. (10) into Eq. (27) yields 1 1þx þ m ðA2 þ B2 =z20 Þ: s2r ðzÞ ¼ w20 (30) 2 1x
0.4
0.0 0.0
x expðaxÞI n ðbxÞ dx qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi bn ða þ n a2 b2 Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ða2 b2 Þ3 ða þ a2 b2 Þn ½Re n4 2
0.6
ð28Þ
1
1.0
r/w0
(a)
Re a4jRe bj ;
1.2
r/w0
Fig. 2. Normalized intensity distributions I=I max at the focal plane of partially coherent MBGBs for x ¼ 0:1; 0:5 and 0.9, N ¼ 1: (a) m ¼ 0 and (b) m ¼ 2:
It is worth noting that, differing from Ref. [1], in our treatment cumbersome combinations of hypergeometric functions were avoided, so that simple analytical expression of s2r ðzÞ is available. From Eq. (30) the beam width of partially coherent MBGBs at the z plane turns out to be 1þx þ m ðA2 þ B2 =z20 Þ: W 2 ðzÞ ¼ 2s2r ¼ w20 (31) 1x
ARTICLE IN PRESS L. Wang et al. / Optik 116 (2005) 65–70
For the case of the free-space propagation Eq. (31) reduces to 1þx þ m ð1 þ z2 =z20 Þ: W 2 ðzÞ ¼ w20 (32) 1x
which is located at the plane z ¼ 0: The far-field divergence angle y0 can be obtained from the knowledge of the second-order intensity moment in the spatial-frequency domain, or equally, sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi W ðzÞ l 2x ¼ y0 ¼ lim : (34) 1þmþ z!1 z pw0 1þx
3.2. M2 factor From Eqs. (33) and (34) the M 2 factor of partially coherent MBGBs can be derived and is given by M2 ¼
W 0 y0 2x : ¼1þmþ 1x l=p
(35)
Alternatively, the M 2 factor and far-field divergence angle can be simply obtained by a hyperbolic fit once the waist width W 0 and its position L0 are known. The wellknown hyperbolic law of the beam width reads as [6] 2 lM 2 ðz L0 Þ2 W 2 ðzÞ ¼ W 20 þ pW 0 ¼
W 20
þ
y20 ðz
2
L0 Þ :
ð36Þ
The substitution from Eq. (32) and L0 ¼ 0 into Eq. (36) leads to the same results as Eqs. (34) and (35). Eq. (35) indicates that the M 2 factor of partially coherent MBGBs depends on the spectral degree of coherence x and order m. As usual M 2 41; the limiting case of M 2 ¼ 1 corresponds to the fully coherent Gaussian beam (m ¼ 0 and x ¼ 0). For comparison with Ref. [1], the M 2 factor versus x for different values of m ¼ 0; 2; 3 and 4 is plotted in Fig. 3. It is seen that, apart from a scaling factor (there are some errors in Ref. [1], for example, sr and s1 on the left-hand of Eqs. (37) and (38) in Ref. [1] should be replaced by s2r and s21 ; respectively) Fig. 3 is similar to Fig. 5 in Ref. [1], where no analytical
12 m=0 m=2 m=3 m=4
10 8
M2
A comparison of Eqs. (31), (32) and (8), (14) shows that the beam width of partially coherent MBGBs depends on the beam parameters m, x and transfer matrix elements A and B, and differs from that of the corresponding Gaussian beam. Only for the case of m ¼ 0 and x ¼ 0 Eqs. (31) and (32) reduce to Eqs. (8) and (14), respectively. From Eq. (32) it is readily shown that the waist width of partially coherent MBGBs is expressed by sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2x W 0 ¼ w0 1 þ m þ ; (33) 1x
69
6 4 2 0 0.0
0.2
0.4
0.6
0.8
ξ
Fig. 3. The M 2 factor of partially coherent MBGBs versus spectral degree of coherence x for m ¼ 0; 2, 3 and 4.
expressions for the M 2 factor and beam width, etc. were found. Nevertheless, by means of the intensity moments method and integral transformation technique, we express them in a closed and simple form.
3.3. Kurtosis parameter The kurtosis parameter (K parameter) expressed in terms of the second- and fourth-intensity moments is used to characterize the degree of sharpness (or flatness) of a beam. In the polar coordinate system the K parameter of rotationally symmetrical beams is defined as K¼
hr4 i ; hr2 i2
(37)
where 1 p
hrn i ¼
Z
1
rn Iðr; zÞr dr;
n ¼ 2; 4
(38)
0
and Z
1
Iðr; zÞr dr:
p¼
(39)
0
Iðr; zÞ is given by Eq. (10). The direct integral of Eq. (38) will lead to complicated hypergeometric functions, which, however, can be avoided by using a suitable integral transformation. Now let us introduce Z w2 xm=2 1 2 f ða; bÞ ¼ 2 0 r expðar2 ÞI m ðbr2 Þr dr; (40) w ðzÞ 1 x 0 where a¼
b¼
2 w2 ðzÞ
1þx ; 1x
pffiffiffi 4 x 1 1 x w2 ðzÞ
(41)
(42)
ARTICLE IN PRESS 70
L. Wang et al. / Optik 116 (2005) 65–70 3.0
4. Conclusion
2.7
In this study the analytical expressions for partially coherent MBGBs propagating through a paraxial optical ABCD system have been derived, which permit us to illustrate the shape invariant property of partially coherent MBGBs in the general case. By using the intensity moments method and integral transformation technique, the important characteristic parameters, such as the beam width, far-field divergence angle, M 2 factor and K parameter of partially coherent MBGBs have been expressed in a closed and simple form. The M 2 factor of partially coherent MBGBs increases with increasing spectral degree of coherence x and beam order m, and reduce to the values of M 2 ¼ 1 for the limiting case of the fully coherent Gaussian beam (m ¼ 0 and x ¼ 0). The K parameter increases as x increases and m decreases, and K ¼ 2 for the case of m ¼ 0 and x ¼ 0:
2.4
K
m=0 m=2 m=3 m=4
2.1
1.8
1.5 0.0
0.2
0.4
0.6
0.8
1.0
ξ
Fig. 4. The K parameter of partially coherent MBGBs as a function of spectral degree of coherence x for m ¼ 0; 2, 3 and 4.
and wðzÞ is given by Eq. (8). By means of Eqs. (40)–(42), the second- and fourth-order intensity moments in Eq. (37) are expressed as hr2 i ¼
1 f ða; bÞ; p
hr4 i ¼
1 qf ða; bÞ : p qa
(43) (44)
Therefore, Eq. (37) is rewritten as K ¼ p
ðqf ða; bÞ=qaÞ : f 2 ða; bÞ
This work was supported by the Foundation of Sichuan Provincial Natural Science and the Foundation of Science and Technology Development of Southwest Jiaotong University of China.
(45)
Making use of the integral formula (29), after tedious algebra, the finial result can be arranged as 2m2 þ 1 K ¼3 2 : ð1 þ xÞ=ð1 xÞ þ m
Acknowledgments
(46)
Eq. (46) implies that the K parameter of partially coherent MBGBs propagating through a paraxial optical ABCD system is dependent on the spectral degree of coherence x and order m, and is invariant upon propagation. It means that their shape is invariant upon propagation, which agrees with the above analysis in Section 2. Fig. 4 gives the K parameter of partially coherent MBGBs as a function of spectral degree of coherence x for different values of m ¼ 0; 2, 3 and 4. From Fig. 4 we see that, K increases with increasing x and decreasing m. The limiting case of K ¼ 2 (m ¼ 0 and x ¼ 0) corresponds to the fully coherent Gaussian beam in the polar coordinate system [7]. For the case of x ! 1 K approaches the asymptotic value of K ¼ 3:
References [1] S.A. Ponomarenko, A class of partially coherent beams carrying optical vortices, J. Opt. Soc. Am. A 18 (2001) 150–156. [2] K.M. Iftekharuddin, M.A. Karim, Heterodyne detection by using diffraction-free beam: tilt and offset effects, Appl. Opt. 31 (1992) 4853–4856. [3] J. Turunen, A.T. Friberg, Matrix representation of Gaussian–Schell-model beams in optical system, Opt. Laser Technol. 18 (1986) 259–266. [4] A. Erdelyhi (Ed.), Tables of Integral Transforms, vol. 2, McGraw-Hill, New York, 1954. [5] Y. Li, E. Wolf, Focal shift formula for focused apertured, Gaussian beams, J. Mod. Opt. 42 (1982) 151–156. [6] A.E. Siegman, New developments in resonators, Proc. SPIE 1224 (1990) 6–9. [7] B. Lu¨, X. Wang, Kurtosis parameter of Bessel-modulated Gaussian beams propagating through ABCD optical systems, Opt. Commun. 204 (2002) 91–97.
Optik
Optics
Optik 116 (2005) 65–70 www.elsevier.de/ijleo
Propagation properties of partially coherent modified Bessel–Gauss beams Li Wanga, Xiqing Wanga, Baida Lu¨b, a
Department of Applied Physics, Southwest Jiaotong University, Chengdu 610031, PR China Institute of Laser Physics and Laser Chemistry, Sichuan University, Chengdu 610064, PR China
b
Received 18 August 2004; accepted 27 November 2004
Abstract Starting from the propagation law of partially coherent light, the analytical propagation equations of partially coherent modified Bessel–Gauss beams (MBGBs) through a paraxial optical ABCD system are derived and illustrated with typical application examples. Furthermore, by using the intensity moments method and integral transformation technique, the important characteristic parameters, including the beam width, far-field divergence angle, M 2 factor and kurtosis parameter of partially coherent MBGBs, are expressed in a closed and simple form. As a result, some basic properties of MBGBs and the dependence of the M 2 factor and kurtosis parameter on the spectral degree of coherence and beam order are illustrated both analytically and numerically. r 2005 Elsevier GmbH. All rights reserved. Keywords: Partially coherent modified Bessel–Gauss beam (MBGB); Propagation; Beam width; M2 factor; Kurtosis parameter
1. Introduction Recently, Ponomarenko [1] introduced a new class of partially coherent beams with a separable phase, that can be represented as an incoherent superposition of fully coherent Laguerre–Gauss modes. The new beams carry optical vortices and possess remarkable properties, for example, their spectral degree of coherence does not depend on the relative orientation of a pair of points at the transversal plane. In addition, the new beams propagate over large distance with little spreading, which may be useful for some applications [2]. Such a type of beam can be called the partially coherent modified Bessel–Gauss beam (MBGB). In Ref. [1], the free-space propagation properties of partially coherent MBGBs were analyzed. The present paper is aimed at studying the propagation properties of partially coherent MBGBs in the general case. In Section 2, starting from the Corresponding author. Fax: +86 28 8540 3260.
E-mail address: [email protected] (B. Lu¨). 0030-4026/$ - see front matter r 2005 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2004.11.006
propagation law of partially coherent light, the explicit propagation equations of partially coherent MBGBs through a paraxial optical ABCD system are derived and two application examples, i.e., the free-space propagation and focusing by a thin lens are analyzed. Then, by using the intensity moments method and integral transformation technique, the analytical expressions for the beam characteristic parameters, such as the beam width, farfield divergence angle, M 2 factor and kurtosis parameter of partially coherent MBGBs, are deduced and illustrated with numerical examples in Section 3. Finally, Section 4 summarizes the main results obtained in this paper.
2. Propagation of partially coherent modified Bessel–Gauss beams through a paraxial optical ABCD system According to Ref. [1], the cross-spectral density function W ðr01 ; r02 ; j01 ; j02 ; 0Þ of partially coherent
ARTICLE IN PRESS 66
L. Wang et al. / Optik 116 (2005) 65–70
MBGBs at the plane z ¼ 0 reads as [1]
Applying the integral formulae [4] Z 1 expðbx2 ÞJ m ðxyÞI m ðaxÞx dx 0 2 1 a y2 ay exp ¼ ; Jm 2b 2b 4b
W ðr01 ; r02 ; j01 ; j02 ; 0Þ xm=2 exp imðj01 j02 Þ 1x pffiffiffi 0 0 0 0 1 þ x r12 þ r22 4 x r1 r2 exp Im ; 2 1 x w0 1 x w20
¼
ð1Þ
where I m ðxÞ denotes the modified Bessel function of order m, ðr01 ; j01 Þ; ðr02 ; j02 Þ are polar coordinates of two points at the plane z ¼ 0; w0 stands for the waist width of the Gaussian part ðm ¼ 0; x ¼ 0Þ; x is the spectral degree of coherence, 0oxo1; and two limiting cases x ! 0 and x ! 1 correspond to the fully coherent and incoherent cases, respectively. Within the framework of the paraxial approximation, the propagation of the cross-spectral density function through a paraxial optical system parameterized by A B transfer matrix obeys [3] C D W ðr1 ; r2 ; j1 ; j2 ; zÞ ZZ ZZ k 2 W 0 ðr01 ; r02 ; z0 ¼ 0Þ ¼ 2pB ik 0 2 0 exp Aðr1 r22 Þ 2B 2 r1 r01 cosðj1 j01 Þ r2 r02 cosðj2 j02 Þ
0 0 0 0 2 2 þDðr1 r2 Þ r1 r2 dr1 dr2 dj01 dj02 ;
x1=2 expðex2 ÞJ m ðxf ÞJ m ðgxÞðxf Þ1=2 dx
0
¼
2 f 1=2 g þf2 gf exp Im 2e 2e 4e
ð6Þ
tedious but straightforward integral calculations yield W ðr1 ; r2 ; j1 ; j2 ; zÞ w20 xm=2 exp½imðj1 j2 Þ w2 1 x pffiffiffi 1 þ x ðr21 þ r22 Þ 4 x r1 r2 exp Im 1x w2 1 x w2 2 2 ðr r2 Þ exp ik 1 ; 2R
¼
ð7Þ
w2 ¼ w20 ½A2 þ ðB=z0 Þ2 ;
(8)
A2 þ ð4B2 =k2 w40 Þ AC þ ð4BD=k2 w40 Þ
(9)
R¼
ð2Þ
2p
exp½ix cosðj fÞ imf df 0
¼ 2pim expðimjÞJ m ðxÞ
1
where
where k is the wave number related to the wavelength l by k ¼ 2p=l: Recalling the integral formula Z
Z
ð5Þ
ð3Þ
with J m ðxÞ being the Bessel function of order m, and substituting Eq. (1) into Eq. (2), the integrals with respect to j01 and j02 lead to W ðr1 ; r2 ; j1 ; j2 ; zÞ k 2 xm=2 2 exp½imðj1 j2 Þ ¼ 4p 2pB 1 x pffiffiffi 0 0 Z 1Z 1 0 0 1 þ x r12 þ r22 4 x r1 r2 exp Im 2 1 x w0 1 x w20 0 0
ik 0 0 ½Aðr12 r22 Þ þ Dðr21 r22 Þ exp 2B k 0 k J m r1 r1 J m r2 r02 r01 r02 dr01 dr02 : ð4Þ B B
denote the beam width and radius of curvature of the corresponding Gaussian beam propagating through a paraxial optical ABCD system, and z0 ¼ pw20 =l is the Rayleigh length of the corresponding Gaussian beam. On placing r1 ¼ r2 ¼ r; j1 ¼ j2 ¼ j into Eq. (7), the optical intensity is given by pffiffiffi 2 w20 xm=2 1 þ x 2r2 4 x r Iðr; zÞ ¼ 2 exp Im : 1 x w2 1 x w2 w 1x (10) Eqs. (7) and (10) provide the general propagation equations of partially coherent MBGBs through a paraxial optical ABCD system and indicate that generally partially coherent MBGBs preserve their shape upon propagation through the paraxial optical ABCD system. Additionally, for m ¼ 0 Eq. (10) reduces to pffiffiffi 2 w2 1 1 þ x 2r2 4 x r Iðr; zÞ ¼ 02 exp I0 : 1 x w2 1 x w2 w 1x (11) On the other hand, if ma0 we have Ið0; zÞ ¼ 0
(12)
because I m ð0Þ ¼ 0: Eq. (12) indicates that there is a central shadow in the intensity profile.
ARTICLE IN PRESS L. Wang et al. / Optik 116 (2005) 65–70
Two typical application examples of Eqs. (7)–(10) are of interest.
67
1.0
0.8
For the free-space propagation the ABCD matrix reduces to A B 1 z ¼ : (13) C D 0 1 It is readily seen that the expression (7) remains unchanged, but Eqs. (8) and (9) simplify to w2 ðzÞ ¼ w20 ½1 þ ðz=z0 Þ2 ;
(14)
RðzÞ ¼ z0 ðz=z0 þ z0 =zÞ;
(15)
I/Imax
2.1. Free-space propagation 0.6
ξ=0.1 ξ=0.5 ξ=0.9
0.4
0.2
0.0 0
1
2
3
4
r/w0
(a) 1.0
Eqs. (14)–(16) are essentially consistent with the results in Ref. [1]. Fig. 1 gives the normalized intensity distributions I=I max at the plane z ¼ 0 of partially coherent MBGBs for different values of m ¼ 0; 2 and x ¼ 0:1; 0:5; 0:9; where I max denotes the maximum intensity. As can be seen, apart from m ¼ 0; where the intensity shows a Gaussian shape, there is a central dip in intensity profile for ma0: Additionally, Fig. 1b indicates that for the nearly coherent case of x ¼ 0:1; the intensity profile is fairly symmetric about its maximum. However, the symmetry vanishes and a long tail appears in the intensity profile as the spatial coherence decreases (i.e. x increases).
2.2. Focusing by a thin lens For this case the ABCD matrix reads as ! Dz z A B ¼ ; 1=f 1 C D
0.6
ξ= 0.1 ξ= 0.5 ξ= 0.9
0.4
0.2
0.0 0
1
2
(b)
3
4
r/w0
Fig. 1. Normalized intensity distributions I=I max at the plane z ¼ 0 of partially coherent MBGBs for different values of x ¼ 0:1; 0:5 and 0.9: (a) m ¼ 0 and (b) m ¼ 2:
R¼
Dz2 þ ð4z2 =k2 w40 Þ ; ðDz=f Þ þ ð4z=k2 w40 Þ
(20)
respectively. The substitution from Eq. (17) into Eq. (10) leads to Iðr; DzÞ ¼
(17)
where zf Dz ¼ f
0.8
I/Imax
respectively. The substitution from Eq. (13) into Eq. (10) yields pffiffiffi 2 w2 xm=2 1 þ x 2r2 4 x r exp Iðr; zÞ ¼ 2 0 Im : 1 x w2 ðzÞ 1 x w2 ðzÞ w ðzÞ 1 x (16)
p2 N 2 xm=2 2 p2 N 2 Dz2 þ ð1 þ DzÞ 1 x 1þx p2 N 2 2r2 exp 1 x p2 N 2 Dz2 þ ð1 þ DzÞ2 w20 pffiffiffi 4 x p2 N 2 r2 I m ; ð21Þ 1 x p2 N 2 Dz2 þ ð1 þ DzÞ2 w20
where (18)
and f is the focal length of the lens. The expression (7) remains unchanged, but Eqs. (8) and (9) reduce to ! 4z2 2 2 2 w ¼ w0 Dz þ 2 4 ¼ w20 ½Dz2 þ ðz=z0 Þ2 ; (19) k w0
N¼
w20 lf
(22)
is the Fresnel number associated with the corresponding Gaussian beam. On placing r ¼ 0 and m ¼ 0 into Eq. (21), we obtain Ið0; DzÞ ¼ I 0
1 ; ð1 þ DzÞ þ p2 N 2 Dz2 2
(23)
ARTICLE IN PRESS 68
L. Wang et al. / Optik 116 (2005) 65–70
where I0 ¼
p2 N 2 1x
(24)
is the intensity at the geometrical focus of Dz ¼ 0: Eq. (23) denotes the axial intensity distribution of partially coherent MBGBs with m ¼ 0 and indicates that Ið0; DzÞ depends on the Fresnel number N, spectral degree of coherence x and relative distance Dz: It is easily seen that for the case of ma0 we have Ið0; DzÞ ¼ 0: From Eq. (23) the relative focal shift, namely, the relative position of axial maximum intensity for the case of m ¼ 0 turns out to be Df ¼
zmax f 1 ¼ ; f 1 þ p2 N 2
The normalized intensity distributions I=I max at the focal plane of focused partially coherent MBGBs are plotted in Fig. 2 for m ¼ 0; 2 and x ¼ 0:1; 0:5; 0:9: A comparison of Figs. 1 and 2 shows the focusing effect of the lens, but the shape of the intensity profile of focused partially coherent MBGBs remains unchanged.
(25)
where zmax is the position of axial maximum intensity referred to the lens. Eq. (25) implies that Df is dependent only on the Fresnel number N, but independent of the spectral degree of coherence x: In addition, Df (absolute value) increases with decreasing N. Interesting is that the relative focal shift Df of focused partially coherent
1.0
0.6
ξ=0.1 ξ=0.5 ξ=0.9
0.4
3. Characteristic parameters of partially coherent modified Bessel–Gauss beams 3.1. Beam width and far-field divergence angle The second-order intensity moment in the space domain is expressed as [6] R 2p R 1 2 r Iðr; j; zÞr dr dj s2r ðzÞ ¼ R0 2p R0 1 : (27) 0 0 Iðr; j; zÞr dr dj Recalling the integral formulae [4] Z 1 expðaxÞI n ðbxÞ dx
0.8
I/Imax
MBGBs takes the same form of focused Gaussian beams [5]. On substituting from Eq. (25) into Eq. (23), the axial maximum intensity is given by 1 I max ¼ I 0 1 þ 2 2 : (26) pN
0
bn qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a2 b2 ða þ a2 b2 Þn
0.2
0.0 0.0
½Re n4 1 Z 0.2
0.4
0.6
0.8
0
1.0
I/Imax
0.8 ξ=0.1 ξ=0.5 ξ=0.9
0.2
(b)
0.2
0.4
0.6
0.8
1.0
Re a4jRe bj
ð29Þ
the substitution from Eq. (10) into Eq. (27) yields 1 1þx þ m ðA2 þ B2 =z20 Þ: s2r ðzÞ ¼ w20 (30) 2 1x
0.4
0.0 0.0
x expðaxÞI n ðbxÞ dx qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi bn ða þ n a2 b2 Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ða2 b2 Þ3 ða þ a2 b2 Þn ½Re n4 2
0.6
ð28Þ
1
1.0
r/w0
(a)
Re a4jRe bj ;
1.2
r/w0
Fig. 2. Normalized intensity distributions I=I max at the focal plane of partially coherent MBGBs for x ¼ 0:1; 0:5 and 0.9, N ¼ 1: (a) m ¼ 0 and (b) m ¼ 2:
It is worth noting that, differing from Ref. [1], in our treatment cumbersome combinations of hypergeometric functions were avoided, so that simple analytical expression of s2r ðzÞ is available. From Eq. (30) the beam width of partially coherent MBGBs at the z plane turns out to be 1þx þ m ðA2 þ B2 =z20 Þ: W 2 ðzÞ ¼ 2s2r ¼ w20 (31) 1x
ARTICLE IN PRESS L. Wang et al. / Optik 116 (2005) 65–70
For the case of the free-space propagation Eq. (31) reduces to 1þx þ m ð1 þ z2 =z20 Þ: W 2 ðzÞ ¼ w20 (32) 1x
which is located at the plane z ¼ 0: The far-field divergence angle y0 can be obtained from the knowledge of the second-order intensity moment in the spatial-frequency domain, or equally, sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi W ðzÞ l 2x ¼ y0 ¼ lim : (34) 1þmþ z!1 z pw0 1þx
3.2. M2 factor From Eqs. (33) and (34) the M 2 factor of partially coherent MBGBs can be derived and is given by M2 ¼
W 0 y0 2x : ¼1þmþ 1x l=p
(35)
Alternatively, the M 2 factor and far-field divergence angle can be simply obtained by a hyperbolic fit once the waist width W 0 and its position L0 are known. The wellknown hyperbolic law of the beam width reads as [6] 2 lM 2 ðz L0 Þ2 W 2 ðzÞ ¼ W 20 þ pW 0 ¼
W 20
þ
y20 ðz
2
L0 Þ :
ð36Þ
The substitution from Eq. (32) and L0 ¼ 0 into Eq. (36) leads to the same results as Eqs. (34) and (35). Eq. (35) indicates that the M 2 factor of partially coherent MBGBs depends on the spectral degree of coherence x and order m. As usual M 2 41; the limiting case of M 2 ¼ 1 corresponds to the fully coherent Gaussian beam (m ¼ 0 and x ¼ 0). For comparison with Ref. [1], the M 2 factor versus x for different values of m ¼ 0; 2; 3 and 4 is plotted in Fig. 3. It is seen that, apart from a scaling factor (there are some errors in Ref. [1], for example, sr and s1 on the left-hand of Eqs. (37) and (38) in Ref. [1] should be replaced by s2r and s21 ; respectively) Fig. 3 is similar to Fig. 5 in Ref. [1], where no analytical
12 m=0 m=2 m=3 m=4
10 8
M2
A comparison of Eqs. (31), (32) and (8), (14) shows that the beam width of partially coherent MBGBs depends on the beam parameters m, x and transfer matrix elements A and B, and differs from that of the corresponding Gaussian beam. Only for the case of m ¼ 0 and x ¼ 0 Eqs. (31) and (32) reduce to Eqs. (8) and (14), respectively. From Eq. (32) it is readily shown that the waist width of partially coherent MBGBs is expressed by sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2x W 0 ¼ w0 1 þ m þ ; (33) 1x
69
6 4 2 0 0.0
0.2
0.4
0.6
0.8
ξ
Fig. 3. The M 2 factor of partially coherent MBGBs versus spectral degree of coherence x for m ¼ 0; 2, 3 and 4.
expressions for the M 2 factor and beam width, etc. were found. Nevertheless, by means of the intensity moments method and integral transformation technique, we express them in a closed and simple form.
3.3. Kurtosis parameter The kurtosis parameter (K parameter) expressed in terms of the second- and fourth-intensity moments is used to characterize the degree of sharpness (or flatness) of a beam. In the polar coordinate system the K parameter of rotationally symmetrical beams is defined as K¼
hr4 i ; hr2 i2
(37)
where 1 p
hrn i ¼
Z
1
rn Iðr; zÞr dr;
n ¼ 2; 4
(38)
0
and Z
1
Iðr; zÞr dr:
p¼
(39)
0
Iðr; zÞ is given by Eq. (10). The direct integral of Eq. (38) will lead to complicated hypergeometric functions, which, however, can be avoided by using a suitable integral transformation. Now let us introduce Z w2 xm=2 1 2 f ða; bÞ ¼ 2 0 r expðar2 ÞI m ðbr2 Þr dr; (40) w ðzÞ 1 x 0 where a¼
b¼
2 w2 ðzÞ
1þx ; 1x
pffiffiffi 4 x 1 1 x w2 ðzÞ
(41)
(42)
ARTICLE IN PRESS 70
L. Wang et al. / Optik 116 (2005) 65–70 3.0
4. Conclusion
2.7
In this study the analytical expressions for partially coherent MBGBs propagating through a paraxial optical ABCD system have been derived, which permit us to illustrate the shape invariant property of partially coherent MBGBs in the general case. By using the intensity moments method and integral transformation technique, the important characteristic parameters, such as the beam width, far-field divergence angle, M 2 factor and K parameter of partially coherent MBGBs have been expressed in a closed and simple form. The M 2 factor of partially coherent MBGBs increases with increasing spectral degree of coherence x and beam order m, and reduce to the values of M 2 ¼ 1 for the limiting case of the fully coherent Gaussian beam (m ¼ 0 and x ¼ 0). The K parameter increases as x increases and m decreases, and K ¼ 2 for the case of m ¼ 0 and x ¼ 0:
2.4
K
m=0 m=2 m=3 m=4
2.1
1.8
1.5 0.0
0.2
0.4
0.6
0.8
1.0
ξ
Fig. 4. The K parameter of partially coherent MBGBs as a function of spectral degree of coherence x for m ¼ 0; 2, 3 and 4.
and wðzÞ is given by Eq. (8). By means of Eqs. (40)–(42), the second- and fourth-order intensity moments in Eq. (37) are expressed as hr2 i ¼
1 f ða; bÞ; p
hr4 i ¼
1 qf ða; bÞ : p qa
(43) (44)
Therefore, Eq. (37) is rewritten as K ¼ p
ðqf ða; bÞ=qaÞ : f 2 ða; bÞ
This work was supported by the Foundation of Sichuan Provincial Natural Science and the Foundation of Science and Technology Development of Southwest Jiaotong University of China.
(45)
Making use of the integral formula (29), after tedious algebra, the finial result can be arranged as 2m2 þ 1 K ¼3 2 : ð1 þ xÞ=ð1 xÞ þ m
Acknowledgments
(46)
Eq. (46) implies that the K parameter of partially coherent MBGBs propagating through a paraxial optical ABCD system is dependent on the spectral degree of coherence x and order m, and is invariant upon propagation. It means that their shape is invariant upon propagation, which agrees with the above analysis in Section 2. Fig. 4 gives the K parameter of partially coherent MBGBs as a function of spectral degree of coherence x for different values of m ¼ 0; 2, 3 and 4. From Fig. 4 we see that, K increases with increasing x and decreasing m. The limiting case of K ¼ 2 (m ¼ 0 and x ¼ 0) corresponds to the fully coherent Gaussian beam in the polar coordinate system [7]. For the case of x ! 1 K approaches the asymptotic value of K ¼ 3:
References [1] S.A. Ponomarenko, A class of partially coherent beams carrying optical vortices, J. Opt. Soc. Am. A 18 (2001) 150–156. [2] K.M. Iftekharuddin, M.A. Karim, Heterodyne detection by using diffraction-free beam: tilt and offset effects, Appl. Opt. 31 (1992) 4853–4856. [3] J. Turunen, A.T. Friberg, Matrix representation of Gaussian–Schell-model beams in optical system, Opt. Laser Technol. 18 (1986) 259–266. [4] A. Erdelyhi (Ed.), Tables of Integral Transforms, vol. 2, McGraw-Hill, New York, 1954. [5] Y. Li, E. Wolf, Focal shift formula for focused apertured, Gaussian beams, J. Mod. Opt. 42 (1982) 151–156. [6] A.E. Siegman, New developments in resonators, Proc. SPIE 1224 (1990) 6–9. [7] B. Lu¨, X. Wang, Kurtosis parameter of Bessel-modulated Gaussian beams propagating through ABCD optical systems, Opt. Commun. 204 (2002) 91–97.