April 1, 2015 / Vol. 40, No. 7 / OPTICS LETTERS
1587
Twisted Gaussian Schell-model beams as series of partially coherent modified Bessel–Gauss beams F. Gori and M. Santarsiero* Dipartimento di Ingegneria, Università Roma Tre Via V. Volterra, 62, 00146 Rome, Italy *Corresponding author: [email protected] Received February 11, 2015; accepted February 27, 2015; posted March 6, 2015 (Doc. ID 234136); published March 30, 2015 We show that twisted Gaussian Schell-model (TGSM) beams can be represented through an incoherent superposition of partially coherent beams carrying optical vortices and whose cross-spectral densities are expressed in terms of modified Bessel functions. Moreover, starting from this result, we show that the modal expansion of the crossspectral density of a TGSM source can be directly obtained through simple mathematics. © 2015 Optical Society of America OCIS codes: (030.0030) Coherence and statistical optics; (030.1640) Coherence; (030.4070) Modes. http://dx.doi.org/10.1364/OL.40.001587
Twisted Gaussian Schell-model (TGSM) beams were introduced by Simon and Mukunda [1] as a generalization of the well-known Gaussian Schell-model (GSM) beams [2]. The cross-spectral density (CSD) of this class of beams differs from the one of a standard GSM for the presence of a position-dependent nonseparable quadratic phase factor, which is responsible for a twist of the beam around its axis in the course of propagation. Since their introduction, TGSM beams have received great attention [3–6], due to their peculiar properties and their connection with fields carrying optical vortices [7–13]. As a matter of fact, while GSM beams continue to represent an incomparable tool for the study of conventional partially coherent light, both in the scalar [14–16] and in the vectorial domain [17–22], for both stationary and nonstationary fields [23–25], TGSM beams can provide a powerful model in the study of partially coherent twisting beams [26–28]. The expansion of the CSD of TGSM beams in terms of its coherent modes [2] was carried out by Simon et al. [3] using rather demanding mathematical tools borrowed from quantum mechanics and group theory. However, although in many cases modal expansions provide a useful way for solving problems with partially coherent light [29–31], alternative representations can also be of help [32–34]. For instance, it has been shown that TGSM beams can also be represented as the superposition of a continuous set of mutually uncorrelated fundamental Gaussian beams, suitably displaced and tilted [5]. Some years ago, another class of partially coherent beams carrying optical vortices was introduced by Ponomarenko [35], as a significant example of fields that, differently from TGSM beams, possess a CSD with separable phase, so that their analysis turns out to be considerably simplified. Since the radial part of their CSD is expressed in terms of the product between a modified Bessel function and a Gaussian, we shall refer to such fields as Modified Bessel–Gauss (henceforth denoted by MGB, for brevity) beams. In this Letter, we will show that MBG beams can be used to give a new expansion to the CSD of TGSM beams. Furthermore, as an added bonus, we will derive the modal expansion of the CSD of TGSM beams in a very simple and straightforward way. 0146-9592/15/071587-04$15.00/0
Let us recall that a TGSM source has a CSD of the form [1] 2 ρ ρ02 ρ − ρ0 2 0 0 − − iuρ × ρ ⊥ ; (1) W ρ; ρ A exp − 4σ 2 2δ2 where the suffix ⊥ denotes the orthogonal component with respect to the source plane of the cross product. The polar coordinates ρ; φ specify the typical position vector ρ across the source plane, while the positive constants σ 2 and δ2 are the variances of the intensity and of the degree of coherence, respectively. Further, u is the twist parameter, whose modulus is bounded by the inequality juj ≤ 1∕δ2 . For brevity, we write Eq. (1) as W ρ; ρ0 A exp−aρ2 ρ02 − bρ − ρ0 2 − iuρ × ρ0 ⊥ ; (2) with a 1∕4σ 2 and b 1∕2δ2 . In terms of the parameters introduced here, the above constraint about the twist parameter turns out to be juj ≤ 2b:
(3)
We want to show that such CSD can be expressed as a series of MBG functions. Let us begin by expanding the exponent of Eq. (2), which becomes −a bρ2 ρ02 2bρ · ρ0 − iuρ × ρ0 ⊥ −a bρ2 ρ02 2ρρ0 bC − iuS∕2;
(4)
where C and S stand for the cosine and sine of φ − φ0 , respectively. Then, we consider the coefficient of 2ρρ0 and write it as u −iφ−φ0 u iφ−φ0 b e e b− 2 2 r 1 r 1 0 0 τ ; (5) te−iφ−φ eiφ−φ 2 t 2 τ
1 bC − iuS∕2 2
with © 2015 Optical Society of America
1588
OPTICS LETTERS / Vol. 40, No. 7 / April 1, 2015
s b u∕2 ; t b − u∕2
r u2 r b2 − ; 4 and τ te
−iφ−φ0
Nm
(6)
:
(7)
When inserting this result into Eqs. (4) and (2), for the CSD we obtain W ρ; ρ0 A exp−sρ2 ρ02 exprρρ0 τ 1∕τ;
p!jmj 2 p s2 − r 2 2 2 s ; s −r r π
is chosen in such a way that the integral of the intensity across the source plane turns out to be unit for any choice of the index m (see Eq. (6.611.4) of Ref. [37]). In terms of the normalized modified Bessel–Gauss CSDs, the expansion in Eq. (12) reads
(8)
where we put, for brevity, with (9)
If we now recall the connection between the modified Bessel functions and their generating function (see relation (9.6.33) of [36]), i.e., exp
hz 2
ϑ
∞ X 1i ϑm I m z; ϑ m−∞
ϑ ≠ 0;
(10)
we can write the second exponential in Eq. (8) as ∞ h X 1i τm I m 2rρρ0 : exp rρρ0 τ τ m−∞
(11)
On using this result together with Eq. (7), from Eq. (8) we finally obtain W ρ; ρ0 A
∞ X m−∞
2
02
0
tm I m 2rρρ0 e−sρ ρ e−imφ−φ :
(12)
According to Eq. (12), any assigned CSD of the form (2) can be uniquely represented as a series of MBG CSDs, each of which is of the form introduced in [35]. It can be easily shown that the converse is also true: from Eq. (12), for a given choice of the quantities s, t, and r, a CDS of the TGSM type is uniquely determined, having parameters as−
r 1 t ; 2 t
b
∞ X
W ρ; ρ0
m−∞
s a b:
r 1 t ; 2 t
1 ur t− ; t (13)
provided that r 1 s≥ t ; 2 t
(14)
as required from the condition a ≥ 0. To provide a clearer physical interpretation to Eq. (12), we introduce a normalized version of the MBG CDSs, defining them as M m ρ; ρ0 N m I m 2rρρ0 e−sρ
2 ρ02
0
e−imφ−φ ;
where the normalization factor, namely,
(15)
(16)
μm M m ρ; ρ0 ;
jmj πtm r p : μm p 2 s2 − r 2 s s2 − r 2
(17)
(18)
Taking into account the definitions in Eq. (6) together with the condition in Eq. (3), it can be shown that the weights μm tend to zero when m → ∞. As a consequence, the expansion in Eq. (17) can be read as the incoherent superposition of a set of partially coherent sources, each of them characterized by a MBG CSD and carrying the finite power μm . This is the main result of the present report. It provides a new superposition scheme for TGSM CSDs and may suggest further procedures for the synthesis of the corresponding sources. An added bonus of the previous result is that, as we shall show, the modal expansion of a TGSM CSD can be straightaway derived. Modes and eigenvalues of TGSM sources were evaluated in Ref. [3] but, as we shall see, the present approach leads to the same result in a surprisingly simple way. To this aim, following the approach of [35], we introduce two new parameters, namely w and ξ (with w > 0 and 0 < ξ < 1), so that Eq. (12) takes the form p 0 4 ξ ρρ W ρ; ρ A t I jmj 1 − ξ w2 m−∞ ∞ X
0
−
×e
1ξ 1−ξ
m
ρ2 ρ02 w2
0
e−imφ−φ ;
(19)
where use has also been made of the fact that I m I −m I jmj (see (9.6.6) of [36]). On comparing Eqs. (12) and (19), it turns out that w and ξ can be expressed as functions of r and s by inverting the relations p 2 ξ 1 ; r 1 − ξ w2
s
1ξ 1 : 1 − ξ w2
(20)
After algebraic manipulation of the above equations we obtain: 1 c ; 2 2 w ξ
s − c∕2 ; s c∕2
where, following [3], we introduced the notation
(21) (22)
April 1, 2015 / Vol. 40, No. 7 / OPTICS LETTERS
q p c 2 s2 − r 2 2 a2 2ab u2 ∕4:
(23)
which corresponds to the modal expansion of the CSD: the eigenfunctions are the LG modes and the eigenvalues,
The inverse relations, aimed at deducing the parameters a, b, and u from w, ξ, and t, are easily derived and read p 1 ξ − ξt 1∕t ; a w2 1 − ξ b
p ξt 1∕t ; w2 1 − ξ
p 2 ξt − 1∕t u ; w2 1 − ξ
(24)
(25)
(26)
and the condition a ≥ 0 becomes p 1 ξ ≤ t ≤ p : ξ
(27)
Now, the modified Bessel function of any order m can be expressed as series of products of Laguerre polynomials. In fact, using Eq. (8.976.1) of [37] we can write p 0 2ξ ρ2 ρ02 0 jmj 4 ξ ρρ 2ρρ 1−ξ jmj∕2 w2 e 1 − ξξ I jmj 1 − ξ w2 w2 2 02 ∞ X n! jmj 2ρ n jmj 2ρ ξ Ln × Ln jmj n! w2 w2 n0 (28) and on inserting the latter expression into Eq. (19), we obtain 2 ρ02
−ρ
W ρ; ρ0 A1 − ξe w2 ∞ X ∞ X × tm ξjmj∕2n
n! 2ρρ0 jmj jmj n! w2 m−∞ n0 2 02 2ρ 0 jmj 2ρ L e−imφ−φ : × Ljmj (29) n n w2 w2
Since the expression of the normalized Laguerre– Gauss (LG) modes Φm n ρ; φ is Φm n ρ; φ
s p 2n! ρ 2 jmj πn jmj! w 2 2ρ 2 2 × Ljmj e−ρ ∕w eimφ ; n w2
1 w
(30)
we see that Eq. (29) becomes W ρ; ρ0
∞ X ∞ X m−∞ n0
m 0 0 λnm Φm n ρ; φΦn ρ ; φ
(31)
1589
λmn
jmj Aπ 2 w 1 − ξtm ξ 2 n ; 2
(32)
with m 0; 1; …. and n 0; 1; …, are arranged in a doubly indexed sequence. The condition in Eq. (27) ensures that the eigenvalues tend to zero geometrically for either m → ∞ and m → −∞, so that the Mercer series in Eq. (29) converges. Using the symbols a; b; c, and u, the eigenvalues read [see Eqs. (20), (6), (22), (23) and (9)] λmn
jmj Aπ b u∕2 m∕2 a b − c∕2 2 n ; a b c∕2 b − u∕2 a b c∕2 (33)
which coincide with the ones derived in Ref. [3], if the different ordering used there is taken into account. MBG beams had a considerable impact on optics of partially coherent vortex beams [26–28]. Their connection to TGSM beams, which was somehow foreseen by Ponomarenko [35], has been demonstrated in this Letter. Our procedure lent itself to derive the modal structure of twisted GSM beams in an unexpectedly simple way. By virtue of its elementary nature, such a derivation is likely to suggest further results and applications. References 1. R. Simon and N. Mukunda, J. Opt. Soc. Am. A 10, 95 (1993). 2. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995). 3. R. Simon, K. Sundar, and N. Mukunda, J. Opt. Soc. Am. A 10, 2008 (1993). 4. A. T. Friberg, E. Tervonen, and J. Turunen, J. Opt. Soc. Am. A 11, 1818 (1994). 5. D. Ambrosini, V. Bagini, F. Gori, and M. Santarsiero, J. Mod. Opt. 41, 1391 (1994). 6. M. J. Bastiaans, J. Opt. Soc. Am. A 17, 2475 (2000). 7. F. Gori, V. Bagini, M. Santarsiero, F. Frezza, G. Schettini, and G. Schirripa Spagnolo, Opt. Rev. 1, 143 (1994). 8. Y. Y. Schechner, R. Piestun, and J. Shamir, Phys. Rev. E 54, R50 (1996). 9. L. Allen, M. J. Padgett, and M. Babikier, in Progress in Optics, E. Wolf, ed. (Elsevier, 2000), Vol. 34, p. 291. 10. Y. Gu, J. Opt. Soc. Am. A 30, 708 (2013). 11. J. Lin, R. Chen, H. Yu, P. Jin, Y. Ma, and M. Cada, J. Opt. Soc. Am. A 31, 1395 (2014). 12. X. Wang, Z. Liu, and D. Zhao, J. Opt. Soc. Am. A 31, 2268 (2014). 13. R. Sharma, J. Ivan, and C. Narayanamurthy, J. Opt. Soc. Am. A 31, 2185 (2014). 14. S. Zhu, Y. Chen, and Y. Cai, J. Opt. Soc. Am. A 30, 171 (2013). 15. M. Lahiri and E. Wolf, J. Opt. Soc. Am. A 30, 1107 (2013). 16. Y. Cai, Y. Chen, and F. Wang, J. Opt. Soc. Am. A 31, 2083 (2014). 17. T. Voipio, T. Setälä, and A. Friberg, J. Opt. Soc. Am. A 30, 71 (2013). 18. M. Lahiri and E. Wolf, J. Opt. Soc. Am. A 30, 2547 (2013). 19. S. B. Raghunathan, H. F. Schouten, and T. D. Visser, J. Opt. Soc. Am. A 30, 582 (2013). 20. G. Gbur, J. Opt. Soc. Am. A 31, 2038 (2014).
1590
OPTICS LETTERS / Vol. 40, No. 7 / April 1, 2015
21. Y. Shen, L. Liu, C. Zhao, Y. Yuan, and Y. Cai, J. Opt. Soc. Am. A 31, 238 (2014). 22. L. Leppänen, A. Friberg, and T. Setälä, J. Opt. Soc. Am. A 31, 1627 (2014). 23. L. Mokhtarpour and S. Ponomarenko, J. Opt. Soc. Am. A 30, 627 (2013). 24. S. Zhu, L. Liu, Y. Chen, and Y. Cai, J. Opt. Soc. Am. A 30, 2306 (2013). 25. R. Dutta, M. Korhonen, A. Friberg, G. Genty, and J. Turunen, J. Opt. Soc. Am. A 31, 637 (2014). 26. Y. Cai, Q. Lin, and O. Korotkova, Opt. Express 17, 2453 (2009). 27. T. Alieva, J. A. Rodrigo, A. Cámara, and E. Abramochkin, J. Opt. Soc. Am. A 30, 2237 (2013). 28. J. Shu, Z. Chen, and J. Pu, J. Opt. Soc. Am. A 30, 916 (2013).
29. T. Voipio, T. Setälä, and A. Friberg, J. Opt. Soc. Am. A 30, 2433 (2013). 30. Y. Li, J. Guo, L. Liu, T. Wang, and J. Shao, J. Opt. Soc. Am. A 30, 2502 (2013). 31. J. Auñón, F. Valdivia-Valero, and M. Nieto-Vesperinas, J. Opt. Soc. Am. A 31, 206 (2014). 32. F. Gori and M. Santarsiero, Opt. Lett. 32, 3531 (2007). 33. R. Martínez-Herrero, P. M. Mejías, and F. Gori, Opt. Lett. 34, 1399 (2009). 34. M. Singh, J. Tervo, and J. Turunen, J. Opt. Soc. Am. A 30, 2611 (2013). 35. S. A. Ponomarenko, J. Opt. Soc. Am. A 18, 150 (2001). 36. M. Abramovitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1972). 37. I. S. Gradstein and I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic, 1965).
1587
Twisted Gaussian Schell-model beams as series of partially coherent modified Bessel–Gauss beams F. Gori and M. Santarsiero* Dipartimento di Ingegneria, Università Roma Tre Via V. Volterra, 62, 00146 Rome, Italy *Corresponding author: [email protected] Received February 11, 2015; accepted February 27, 2015; posted March 6, 2015 (Doc. ID 234136); published March 30, 2015 We show that twisted Gaussian Schell-model (TGSM) beams can be represented through an incoherent superposition of partially coherent beams carrying optical vortices and whose cross-spectral densities are expressed in terms of modified Bessel functions. Moreover, starting from this result, we show that the modal expansion of the crossspectral density of a TGSM source can be directly obtained through simple mathematics. © 2015 Optical Society of America OCIS codes: (030.0030) Coherence and statistical optics; (030.1640) Coherence; (030.4070) Modes. http://dx.doi.org/10.1364/OL.40.001587
Twisted Gaussian Schell-model (TGSM) beams were introduced by Simon and Mukunda [1] as a generalization of the well-known Gaussian Schell-model (GSM) beams [2]. The cross-spectral density (CSD) of this class of beams differs from the one of a standard GSM for the presence of a position-dependent nonseparable quadratic phase factor, which is responsible for a twist of the beam around its axis in the course of propagation. Since their introduction, TGSM beams have received great attention [3–6], due to their peculiar properties and their connection with fields carrying optical vortices [7–13]. As a matter of fact, while GSM beams continue to represent an incomparable tool for the study of conventional partially coherent light, both in the scalar [14–16] and in the vectorial domain [17–22], for both stationary and nonstationary fields [23–25], TGSM beams can provide a powerful model in the study of partially coherent twisting beams [26–28]. The expansion of the CSD of TGSM beams in terms of its coherent modes [2] was carried out by Simon et al. [3] using rather demanding mathematical tools borrowed from quantum mechanics and group theory. However, although in many cases modal expansions provide a useful way for solving problems with partially coherent light [29–31], alternative representations can also be of help [32–34]. For instance, it has been shown that TGSM beams can also be represented as the superposition of a continuous set of mutually uncorrelated fundamental Gaussian beams, suitably displaced and tilted [5]. Some years ago, another class of partially coherent beams carrying optical vortices was introduced by Ponomarenko [35], as a significant example of fields that, differently from TGSM beams, possess a CSD with separable phase, so that their analysis turns out to be considerably simplified. Since the radial part of their CSD is expressed in terms of the product between a modified Bessel function and a Gaussian, we shall refer to such fields as Modified Bessel–Gauss (henceforth denoted by MGB, for brevity) beams. In this Letter, we will show that MBG beams can be used to give a new expansion to the CSD of TGSM beams. Furthermore, as an added bonus, we will derive the modal expansion of the CSD of TGSM beams in a very simple and straightforward way. 0146-9592/15/071587-04$15.00/0
Let us recall that a TGSM source has a CSD of the form [1] 2 ρ ρ02 ρ − ρ0 2 0 0 − − iuρ × ρ ⊥ ; (1) W ρ; ρ A exp − 4σ 2 2δ2 where the suffix ⊥ denotes the orthogonal component with respect to the source plane of the cross product. The polar coordinates ρ; φ specify the typical position vector ρ across the source plane, while the positive constants σ 2 and δ2 are the variances of the intensity and of the degree of coherence, respectively. Further, u is the twist parameter, whose modulus is bounded by the inequality juj ≤ 1∕δ2 . For brevity, we write Eq. (1) as W ρ; ρ0 A exp−aρ2 ρ02 − bρ − ρ0 2 − iuρ × ρ0 ⊥ ; (2) with a 1∕4σ 2 and b 1∕2δ2 . In terms of the parameters introduced here, the above constraint about the twist parameter turns out to be juj ≤ 2b:
(3)
We want to show that such CSD can be expressed as a series of MBG functions. Let us begin by expanding the exponent of Eq. (2), which becomes −a bρ2 ρ02 2bρ · ρ0 − iuρ × ρ0 ⊥ −a bρ2 ρ02 2ρρ0 bC − iuS∕2;
(4)
where C and S stand for the cosine and sine of φ − φ0 , respectively. Then, we consider the coefficient of 2ρρ0 and write it as u −iφ−φ0 u iφ−φ0 b e e b− 2 2 r 1 r 1 0 0 τ ; (5) te−iφ−φ eiφ−φ 2 t 2 τ
1 bC − iuS∕2 2
with © 2015 Optical Society of America
1588
OPTICS LETTERS / Vol. 40, No. 7 / April 1, 2015
s b u∕2 ; t b − u∕2
r u2 r b2 − ; 4 and τ te
−iφ−φ0
Nm
(6)
:
(7)
When inserting this result into Eqs. (4) and (2), for the CSD we obtain W ρ; ρ0 A exp−sρ2 ρ02 exprρρ0 τ 1∕τ;
p!jmj 2 p s2 − r 2 2 2 s ; s −r r π
is chosen in such a way that the integral of the intensity across the source plane turns out to be unit for any choice of the index m (see Eq. (6.611.4) of Ref. [37]). In terms of the normalized modified Bessel–Gauss CSDs, the expansion in Eq. (12) reads
(8)
where we put, for brevity, with (9)
If we now recall the connection between the modified Bessel functions and their generating function (see relation (9.6.33) of [36]), i.e., exp
hz 2
ϑ
∞ X 1i ϑm I m z; ϑ m−∞
ϑ ≠ 0;
(10)
we can write the second exponential in Eq. (8) as ∞ h X 1i τm I m 2rρρ0 : exp rρρ0 τ τ m−∞
(11)
On using this result together with Eq. (7), from Eq. (8) we finally obtain W ρ; ρ0 A
∞ X m−∞
2
02
0
tm I m 2rρρ0 e−sρ ρ e−imφ−φ :
(12)
According to Eq. (12), any assigned CSD of the form (2) can be uniquely represented as a series of MBG CSDs, each of which is of the form introduced in [35]. It can be easily shown that the converse is also true: from Eq. (12), for a given choice of the quantities s, t, and r, a CDS of the TGSM type is uniquely determined, having parameters as−
r 1 t ; 2 t
b
∞ X
W ρ; ρ0
m−∞
s a b:
r 1 t ; 2 t
1 ur t− ; t (13)
provided that r 1 s≥ t ; 2 t
(14)
as required from the condition a ≥ 0. To provide a clearer physical interpretation to Eq. (12), we introduce a normalized version of the MBG CDSs, defining them as M m ρ; ρ0 N m I m 2rρρ0 e−sρ
2 ρ02
0
e−imφ−φ ;
where the normalization factor, namely,
(15)
(16)
μm M m ρ; ρ0 ;
jmj πtm r p : μm p 2 s2 − r 2 s s2 − r 2
(17)
(18)
Taking into account the definitions in Eq. (6) together with the condition in Eq. (3), it can be shown that the weights μm tend to zero when m → ∞. As a consequence, the expansion in Eq. (17) can be read as the incoherent superposition of a set of partially coherent sources, each of them characterized by a MBG CSD and carrying the finite power μm . This is the main result of the present report. It provides a new superposition scheme for TGSM CSDs and may suggest further procedures for the synthesis of the corresponding sources. An added bonus of the previous result is that, as we shall show, the modal expansion of a TGSM CSD can be straightaway derived. Modes and eigenvalues of TGSM sources were evaluated in Ref. [3] but, as we shall see, the present approach leads to the same result in a surprisingly simple way. To this aim, following the approach of [35], we introduce two new parameters, namely w and ξ (with w > 0 and 0 < ξ < 1), so that Eq. (12) takes the form p 0 4 ξ ρρ W ρ; ρ A t I jmj 1 − ξ w2 m−∞ ∞ X
0
−
×e
1ξ 1−ξ
m
ρ2 ρ02 w2
0
e−imφ−φ ;
(19)
where use has also been made of the fact that I m I −m I jmj (see (9.6.6) of [36]). On comparing Eqs. (12) and (19), it turns out that w and ξ can be expressed as functions of r and s by inverting the relations p 2 ξ 1 ; r 1 − ξ w2
s
1ξ 1 : 1 − ξ w2
(20)
After algebraic manipulation of the above equations we obtain: 1 c ; 2 2 w ξ
s − c∕2 ; s c∕2
where, following [3], we introduced the notation
(21) (22)
April 1, 2015 / Vol. 40, No. 7 / OPTICS LETTERS
q p c 2 s2 − r 2 2 a2 2ab u2 ∕4:
(23)
which corresponds to the modal expansion of the CSD: the eigenfunctions are the LG modes and the eigenvalues,
The inverse relations, aimed at deducing the parameters a, b, and u from w, ξ, and t, are easily derived and read p 1 ξ − ξt 1∕t ; a w2 1 − ξ b
p ξt 1∕t ; w2 1 − ξ
p 2 ξt − 1∕t u ; w2 1 − ξ
(24)
(25)
(26)
and the condition a ≥ 0 becomes p 1 ξ ≤ t ≤ p : ξ
(27)
Now, the modified Bessel function of any order m can be expressed as series of products of Laguerre polynomials. In fact, using Eq. (8.976.1) of [37] we can write p 0 2ξ ρ2 ρ02 0 jmj 4 ξ ρρ 2ρρ 1−ξ jmj∕2 w2 e 1 − ξξ I jmj 1 − ξ w2 w2 2 02 ∞ X n! jmj 2ρ n jmj 2ρ ξ Ln × Ln jmj n! w2 w2 n0 (28) and on inserting the latter expression into Eq. (19), we obtain 2 ρ02
−ρ
W ρ; ρ0 A1 − ξe w2 ∞ X ∞ X × tm ξjmj∕2n
n! 2ρρ0 jmj jmj n! w2 m−∞ n0 2 02 2ρ 0 jmj 2ρ L e−imφ−φ : × Ljmj (29) n n w2 w2
Since the expression of the normalized Laguerre– Gauss (LG) modes Φm n ρ; φ is Φm n ρ; φ
s p 2n! ρ 2 jmj πn jmj! w 2 2ρ 2 2 × Ljmj e−ρ ∕w eimφ ; n w2
1 w
(30)
we see that Eq. (29) becomes W ρ; ρ0
∞ X ∞ X m−∞ n0
m 0 0 λnm Φm n ρ; φΦn ρ ; φ
(31)
1589
λmn
jmj Aπ 2 w 1 − ξtm ξ 2 n ; 2
(32)
with m 0; 1; …. and n 0; 1; …, are arranged in a doubly indexed sequence. The condition in Eq. (27) ensures that the eigenvalues tend to zero geometrically for either m → ∞ and m → −∞, so that the Mercer series in Eq. (29) converges. Using the symbols a; b; c, and u, the eigenvalues read [see Eqs. (20), (6), (22), (23) and (9)] λmn
jmj Aπ b u∕2 m∕2 a b − c∕2 2 n ; a b c∕2 b − u∕2 a b c∕2 (33)
which coincide with the ones derived in Ref. [3], if the different ordering used there is taken into account. MBG beams had a considerable impact on optics of partially coherent vortex beams [26–28]. Their connection to TGSM beams, which was somehow foreseen by Ponomarenko [35], has been demonstrated in this Letter. Our procedure lent itself to derive the modal structure of twisted GSM beams in an unexpectedly simple way. By virtue of its elementary nature, such a derivation is likely to suggest further results and applications. References 1. R. Simon and N. Mukunda, J. Opt. Soc. Am. A 10, 95 (1993). 2. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995). 3. R. Simon, K. Sundar, and N. Mukunda, J. Opt. Soc. Am. A 10, 2008 (1993). 4. A. T. Friberg, E. Tervonen, and J. Turunen, J. Opt. Soc. Am. A 11, 1818 (1994). 5. D. Ambrosini, V. Bagini, F. Gori, and M. Santarsiero, J. Mod. Opt. 41, 1391 (1994). 6. M. J. Bastiaans, J. Opt. Soc. Am. A 17, 2475 (2000). 7. F. Gori, V. Bagini, M. Santarsiero, F. Frezza, G. Schettini, and G. Schirripa Spagnolo, Opt. Rev. 1, 143 (1994). 8. Y. Y. Schechner, R. Piestun, and J. Shamir, Phys. Rev. E 54, R50 (1996). 9. L. Allen, M. J. Padgett, and M. Babikier, in Progress in Optics, E. Wolf, ed. (Elsevier, 2000), Vol. 34, p. 291. 10. Y. Gu, J. Opt. Soc. Am. A 30, 708 (2013). 11. J. Lin, R. Chen, H. Yu, P. Jin, Y. Ma, and M. Cada, J. Opt. Soc. Am. A 31, 1395 (2014). 12. X. Wang, Z. Liu, and D. Zhao, J. Opt. Soc. Am. A 31, 2268 (2014). 13. R. Sharma, J. Ivan, and C. Narayanamurthy, J. Opt. Soc. Am. A 31, 2185 (2014). 14. S. Zhu, Y. Chen, and Y. Cai, J. Opt. Soc. Am. A 30, 171 (2013). 15. M. Lahiri and E. Wolf, J. Opt. Soc. Am. A 30, 1107 (2013). 16. Y. Cai, Y. Chen, and F. Wang, J. Opt. Soc. Am. A 31, 2083 (2014). 17. T. Voipio, T. Setälä, and A. Friberg, J. Opt. Soc. Am. A 30, 71 (2013). 18. M. Lahiri and E. Wolf, J. Opt. Soc. Am. A 30, 2547 (2013). 19. S. B. Raghunathan, H. F. Schouten, and T. D. Visser, J. Opt. Soc. Am. A 30, 582 (2013). 20. G. Gbur, J. Opt. Soc. Am. A 31, 2038 (2014).
1590
OPTICS LETTERS / Vol. 40, No. 7 / April 1, 2015
21. Y. Shen, L. Liu, C. Zhao, Y. Yuan, and Y. Cai, J. Opt. Soc. Am. A 31, 238 (2014). 22. L. Leppänen, A. Friberg, and T. Setälä, J. Opt. Soc. Am. A 31, 1627 (2014). 23. L. Mokhtarpour and S. Ponomarenko, J. Opt. Soc. Am. A 30, 627 (2013). 24. S. Zhu, L. Liu, Y. Chen, and Y. Cai, J. Opt. Soc. Am. A 30, 2306 (2013). 25. R. Dutta, M. Korhonen, A. Friberg, G. Genty, and J. Turunen, J. Opt. Soc. Am. A 31, 637 (2014). 26. Y. Cai, Q. Lin, and O. Korotkova, Opt. Express 17, 2453 (2009). 27. T. Alieva, J. A. Rodrigo, A. Cámara, and E. Abramochkin, J. Opt. Soc. Am. A 30, 2237 (2013). 28. J. Shu, Z. Chen, and J. Pu, J. Opt. Soc. Am. A 30, 916 (2013).
29. T. Voipio, T. Setälä, and A. Friberg, J. Opt. Soc. Am. A 30, 2433 (2013). 30. Y. Li, J. Guo, L. Liu, T. Wang, and J. Shao, J. Opt. Soc. Am. A 30, 2502 (2013). 31. J. Auñón, F. Valdivia-Valero, and M. Nieto-Vesperinas, J. Opt. Soc. Am. A 31, 206 (2014). 32. F. Gori and M. Santarsiero, Opt. Lett. 32, 3531 (2007). 33. R. Martínez-Herrero, P. M. Mejías, and F. Gori, Opt. Lett. 34, 1399 (2009). 34. M. Singh, J. Tervo, and J. Turunen, J. Opt. Soc. Am. A 30, 2611 (2013). 35. S. A. Ponomarenko, J. Opt. Soc. Am. A 18, 150 (2001). 36. M. Abramovitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1972). 37. I. S. Gradstein and I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic, 1965).
