OSA's Digital Library

Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 13, Iss. 4 — Feb. 21, 2005
  • pp: 1107–1112
« Show journal navigation

Propagation law for the generating function of Hermite-Gaussian-type modes in first-order optical systems

Martin J. Bastiaans and Tatiana Alieva  »View Author Affiliations


Optics Express, Vol. 13, Issue 4, pp. 1107-1112 (2005)
http://dx.doi.org/10.1364/OPEX.13.001107


View Full Text Article

Acrobat PDF (85 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

Abstract

Based on the common Hermite-Gaussian modes, a general class of orthonormal sets of Hermite-Gaussian-type modes is introduced. Such modes can most easily be defined by means of their generating function. It is shown that these modes remain in their class of orthonormal Hermite-Gaussian-type modes, when they propagate through first-order optical systems. A propagation law for the generating function is formulated.

© 2005 Optical Society of America

1. Introduction

In this paper we introduce a general class of orthonormal sets of Hermite-Gaussian type modes, by generalizing the quadratic form that arises in the generating function of the common Hermite-Gaussian modes. We study how these modes propagate through first-order optical systems and express the generating function of the set of output modes in terms of the generating function of the set of input modes.

The requirement of orthonormality yields some additional conditions for the quadratic form of the generating function. As a result of that, we will be able to express the elements of this quadratic form in terms of four matrices that can be combined into a symplectic matrix.

The main result of this paper is that this symplectic matrix propagates through a first-order optical system by a mere multiplication with the system’s ray transformation matrix. From this simple propagation law we can easily derive how different members from the class of the Hermite-Gaussian-type modes (such as the common Hermite-Gaussian and Laguerre-Gaussian modes) can be converted into each other.

2. Hermite-Gaussian-type modes

The complex field amplitude of the common Hermite-Gaussian modes takes the form

𝓗n,m(r;wx,wy)=𝓗n(x;wx)𝓗m(y;wy)
(1)

with

𝓗n(x;w)=214(2nn!w)12Hn(2πxw)exp(πx2w2),
(2)

where Hn(°) are the Hermite polynomials [1

1. M. Abramowitz and I. A. Stegun, eds., Pocketbook of Mathematical Functions (Deutsch, Frankfurt am Main, Germany, 1984).

, Section 22] and where the column vector r=(x,y)t is a short-hand notation for the spatial variables x and y, with the superscript t denoting transposition. Note that 𝓗n(x;w) has been defined such that we have the orthonormality relationship

𝓗n(x;w)𝓗l(x;w)dx=δnl
(3)

with δnl the Kronecker delta. (All integrals in this paper extend from -∞ to +∞.) From the generating function of the Hermite polynomials [1

1. M. Abramowitz and I. A. Stegun, eds., Pocketbook of Mathematical Functions (Deutsch, Frankfurt am Main, Germany, 1984).

, Eq. (22.9.17)],

exp(s2+2sz)=n=0Hn(z)skn!,
(4)

we can easily find the generating function of the Hermite-Gaussian modes 𝓗n,m(r;wx,wy):

212(wxwy)12exp[(sx2+sy2)+22π(sxxwx+syywy)π(x2wx2+y2wy2)]
=n=0m=0𝓗n,m(r;wx,wy)(2n+mn!m!)12sxnsym.
(5)

The general class of orthonormal sets of Hermite-Gaussian-type modes 𝓗n,m(r;K,L) that we propose, is most easily defined by the generating function

212(detK)12exp(stMs+22πstKrπrtLr)
=n=0m=0𝓗n,m(r;K,L)(2n+mn!m!)12sxnsym,
(6)

cf. Eq. (5), where we have introduced the column vector s=(sx, sy)t and three (possibly complex) 2×2-matrices K, L=L t, and M=M t. For the common Hermite-Gaussian modes 𝓗n,m(r;wx,wy) we have, see Eq. (5),

K=(wx00wy)1=W1,L=W2,M=I.
(7)

We will observe later that the matrix M is completely determined by K, see Eq. (19), and therefore does not have to be included as a parameter in 𝓗n,m(r;K,L).

3. Propagation through first-order optical systems

We let Hermite-Gaussian-type modes 𝓗n,m(r;K,L) propagate through a lossless, first-order optical system – also called an ABCD-system – and determine the generating function of the set of modes to which the beam that appears at the output of this system belongs. Any lossless, first-order optical system can be described by its ray transformation matrix [2

2. R. K. Luneburg, Mathematical Theory of Optics (University of California Press, Berkeley and Los Angeles, CA, USA, 1966).

], which relates the position ri and direction qi of an incoming ray to the position r o and direction qo of the outgoing ray:

(roqo)=(ABCD)(riqi).
(8)

The ray transformation matrix is real and symplectic, yielding the relations

ABt=BAt,CDt=DCt,ADtBCt=I,
AtC=CtA,BtD=DtB,AtDCtB=I.
(9)

Using the matrices A, B, and D, and assuming that B is a non-singular matrix, we can represent the first-order optical system by the Collins integral [3

3. S. A. Collins Jr., “Lens-system diffraction integral written in terms of matrix optics,” J. Opt. Soc. Am. 60, 1168–1177 (1970). [CrossRef]

]

fo(ro)=exp(iϕ)detiBfi(ri)exp[iπ(ritB1Ari2ritB1ro+rotDB1ro)]dri,
(10)

where the output amplitude fo(r) is expressed in terms of the input amplitude f i(r). The phase factor exp() in Eq. (10) is rather irrelevant and can often be chosen arbitrarily. We remark that the signal transformation fi(r)→fo(r) that corresponds to a lossless, first-order optical system, is unitary, i.e.

fi,1(r)fi,2*(r)dr=fo,1(r)fo,2*(r)dr,
(11)

where * denotes complex conjugation.

With a Hermite-Gaussian-type mode 𝓗n,m(r;K i,L i) at the input of an ABCD-system, we denote the output beam by 𝓗n,m(r;K o,L o). To find the generating function of the set of modes to which this output beam belongs, we write

n=0m=0𝓗n,m(r;Ko,Lo)(2n+mn!m!)12sxnsym=n=0m=0(2n+mn!m!)12sxnsymexp(iϕ)detiB ×𝓗n,m(ri;Ki,Li)exp[iπ(ritB1Ari2ritB1ro+rotDB1ro)]dri,

where Collins integral (10) has been used. After substituting from the generating function (6) and proceeding along the same lines as in [4

4. M. J. Bastiaans and T. Alieva, “Generating function for Hermite-Gaussian modes propagating through first-order optical systems,” J. Phys. A: Math. Gen 38, L73–L78 (2005). [CrossRef]

], we get for the generating function

exp(iϕ)212[detKidetiBdet(LiiB1A)]12exp(stMis+iπrotDB1ro) ×exp[π(B1ro+i2πKits)t(LiiB1A)1(B1ro+i2πKits)] exp(iϕ)212(detKo)12exp(stMos+22πstKoroπrotLoro)

with

Ko=Ki(A+BiLi)1,
(12)
iLo=(C+DiLi)(A+BiLi)1,
(13)
Mo=Mi2iKi(A+BiLi)1BKit.
(14)

We conclude that the generating function (6) keeps its form when the associated Hermite-Gaussian-type modes propagate through a first-order optical system; we only have to replace the input matrices K i, L i, and M i by the output matrices K o, L o, and M o, respectively, in accordance with the input-output relationships (12–14). Note that Eq. (13) is in fact the wellknown ABCD-law, and that Eqs. (12) and (13) can be combined into

(IiLo)Ko1=(ABCD)(IiLi)Ki1.
(15)

4. Conditions resulting from orthonormality

From the requirement that the Hermite-Gaussian-type modes are orthonormal,

𝓗n,m(r;K,L)𝓗l,k*(r;K,L)dr=δnlδmk,
(16)

we get additional conditions for the three matrices K, L, and M. To derive these, we consider the expression

n=0m=0l=0k=0(2n+mn!m!)12sxnsym(2l+kl!k!)12txltyk𝓗n,m(r;K,L)𝓗l,k*(r;K,L)dr =2detKexp(stMsttM*t)exp[22π(stK+ttK*)rπrt(L+L*)r]dr,

where t=(tx,ty)t on the analogy of s=(sx, sy)t, and where we have substituted from the generating function (6). We note that the integral in this expression equals

[det(L+L*)]12exp{st[K(L+L*2)1Kt]s+tt[K(L+L*2)1Kt]*t} ×exp{2st[K(L+L*2)1K*t]t}

and we get

n=0m=0l=0k=0(2n+mn!m!)12sxnsym(2l+kl!k!)12txltyk𝓗n,m(r;K,L)𝓗l,k*(r;K,L)dr =2detK[det(L+L*)]12exp{2st[K(L+L*2)1K*t]t} ×exp{st[MK(L+L*2)1Kt]stt[MK(L+L*2)1Kt]*t}.

To get to the orthonormality condition (16), we have to require

MK[(L+L*)2]1Kt=0,
(17)
K[(L+L*)2]1K*t=I,
(18)

leading to the conditions

M1=M*=K*K1=(K*K1)t,
(19)
(L+L*)2=KtK*=(KtK*)t,
(20)

where we have also expressed the symmetry of the matrices L and M once again. Note that M=KK*-1 is completely determined by K, see Eq. (19), which is the reason why we did not includeM as a parameter in 𝓗n,m(r;K,L).

If we express K -1 in its real and imaginary parts, K -1=a+i b, we immediately get from the realness of (K*t K)-1, see Eq. (20), that the matrix ab t is symmetric. If we then express L as L=(d-ic)K=(d-i c)(a+i b)-1, the symmetry of L leads to

atd+btc=dta+ctbandatcbtd=ctadtb,

while Eq. (20) leads to the requirements

atdbtc+dtactb=2Iandatc+btd=cta+dtb,

From these four conditions we conclude that the 4×4-matrix

(abcd)

is symplectic and thus satisfies relations of the form (9).

The results in this paper resemble those derived by Wünsche [5

5. A. Wünsche, “General Hermite and Laguerre two-dimensional polynomials,” J. Phys. A: Math Gen. 33, 1603–1629 (2000). [CrossRef]

]. The main difference is that we use as the Gaussian part exp(-π r t Lr), with a matrix L that can be chosen freely if we would only require Eq. (17) and not necessarily Eq. (18), whereasWünsche uses a fixed expression of the form exp(-π r t r). Wünsche’s results arise indeed from ours for the special choice L=I, in which case Eq. (17) leads to M=KK t, yielding the generating function

exp[stKKts+2stK(2πr)(2πr)t(2πr)2],

which is compatible to [5

5. A. Wünsche, “General Hermite and Laguerre two-dimensional polynomials,” J. Phys. A: Math Gen. 33, 1603–1629 (2000). [CrossRef]

, Eq. (8.4)]. Eq. (18) would yield the additional condition KK*t=I.

Special cases of Hermite-Gaussian-type modes can easily be recognized. We mention the (separable) Hermite-Gaussian modes (with curvatures in the x and y directions determined by γx and γy, respectively), for which the matrices a, b, c, and d are given by

W1(a+ib)=(cosγxexp(iγx)00cosγyexp(iγy))W(dic)=(exp(iγ1)00exp(iγ2));
(21)

the common Hermite-Gaussian modes with which we started this paper [see Eqs. (15)], arise for the special choice γxy12=0. Note that for Hermite-Gaussian modes the matrix

L=((1+itanγx)wx200(1+itanγy)wy2)

is a diagonal matrix, and that the ABCD-law (13) is useful when such modes propagate through separable systems (for which A, B, C, and D are diagonal matrices). For (rotationally symmetric) Laguerre-Gaussian modes (with its curvature determined by γ) we have

w1(a+ib)=cosγexp(iγ)w(dic)=12(exp(iγ1)iexp(iγ2)iexp(iγ1)exp(iγ2));
(22)

the special case γ=γ12=0 has been reported, for instance, in [6

6. M.W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993). [CrossRef]

]. Note that for Laguerre- Gaussian modes the matrix L=(1+itanγ)w -2 I is a scalar matrix, and that the ABCD-law (13) is useful when such modes propagate through isotropic systems (for which A, B, C, and D are scalar matrices). We remark that the discriminating parameters in the above expressions are the widths (wx, wy, w) and the curvatures (γx, γy, γ) of the modes; the parameters γ1 and γ2 lead to a mere multiplication of the complex field amplitude by a phase factor that depends on the mode-number (n,m) but not on the space variables r.

5. An alternate propagation law

Now that we know that the input matrices Ki, Li, Mi and the output matrices K o, L o, M o can be expressed in the special forms

Ki,o=(ai,o+ibi,o)1,
Li,o=(di,oici,o)(ai,o+ibi,o)1,
Mi,o=(ai,o+ibi,o)1(ai,oibi,o),
(23)

where the real matrices a i, b i, c i, d i, and the real matrices a o, b o, c o, d o constitute two real symplectic matrices, we can – after some straightforward but rather lengthy calculations, in which we make extensive use of the symplecticity properties, cf. Eqs. (9) – bring the input-output relationships (12–14) into a different form and formulate the elegant propagation law

(aobocodo)=(ABCD)(aibicidi).
(24)

This propagation law resembles Eqs. (12) and (29) in Ref. [7

7. J. A. Arnaud, “Mode coupling in first-order optics,” J. Opt. Soc. Am. 61, 751–758 (1971). [CrossRef]

], split up into their real and imaginary parts, where i(a+ib),d-ic correspond to the ‘matricial rays’ Q,P [7

7. J. A. Arnaud, “Mode coupling in first-order optics,” J. Opt. Soc. Am. 61, 751–758 (1971). [CrossRef]

, Eq. (11)]: Qπ=i K -1=i(a+i b) and PπLK -1=λ(d-ic), with λ the wavelength of the light. The treatment in [7

7. J. A. Arnaud, “Mode coupling in first-order optics,” J. Opt. Soc. Am. 61, 751–758 (1971). [CrossRef]

] is based on a so-called ‘mode-generating system,’ which is excited by an off-axis point source at its input plane; in our case of lossless, first-order optics, this system is determined by the matricial rays Q,P that are associated with the mode that is to be generated. The modes in [7

7. J. A. Arnaud, “Mode coupling in first-order optics,” J. Opt. Soc. Am. 61, 751–758 (1971). [CrossRef]

] then arise by expanding the resulting output field in a power series of the coordinates of the point source at the input. The kernel in the Collins integral that describes the mode-generating system thus plays the role of a generating function. The present treatment is directly based on the general form (6) of the generating function, and we get the additional result that the parameters a,b and c,d that characterize this generating function correspond to the real and imaginary parts ofQ and P, and constitute a real, symplectic matrix. With r=x and λs=πy√2, there is indeed a one-to-one correspondence between our generating function (6) and the one used in [7

7. J. A. Arnaud, “Mode coupling in first-order optics,” J. Opt. Soc. Am. 61, 751–758 (1971). [CrossRef]

], see in particular [7

7. J. A. Arnaud, “Mode coupling in first-order optics,” J. Opt. Soc. Am. 61, 751–758 (1971). [CrossRef]

, Eq. (22)].

We remark that all sets of Hermite-Gaussian-type modes can be converted into each other by means of properly chosen first-order optical systems, and we conclude that knowledge of the generating function and in particular its propagation law (24) may be valuable in the design of mode converters. Further work is in progress, see for instance [8

8. T. Alieva and M. J. Bastiaans, “Mode mapping in paraxial lossless optics,” submitted to Opt. Lett. (2005).

], and future papers may also take lossy mode converters [7

7. J. A. Arnaud, “Mode coupling in first-order optics,” J. Opt. Soc. Am. 61, 751–758 (1971). [CrossRef]

] into account.

6. Conclusion

A general class of orthonormal sets of Hermite-Gaussian-type modes has been introduced by formulating a generalized version of the generating function that yields the common Hermite-Gaussian modes. These sets of Hermite-Gaussian-type modes remain in their class when they propagate through first-order optical systems, and a propagation law for their generating function has been formulated. The propagation law is in a form that suits itself for the design of mode converters.

Acknowledgments

T. Alieva thanks the Spanish Ministry of Education and Science for financial support (‘Ramon y Cajal’ grant and projects TIC 2002-01846 and TIC 2002-11581-E).

References and links

1.

M. Abramowitz and I. A. Stegun, eds., Pocketbook of Mathematical Functions (Deutsch, Frankfurt am Main, Germany, 1984).

2.

R. K. Luneburg, Mathematical Theory of Optics (University of California Press, Berkeley and Los Angeles, CA, USA, 1966).

3.

S. A. Collins Jr., “Lens-system diffraction integral written in terms of matrix optics,” J. Opt. Soc. Am. 60, 1168–1177 (1970). [CrossRef]

4.

M. J. Bastiaans and T. Alieva, “Generating function for Hermite-Gaussian modes propagating through first-order optical systems,” J. Phys. A: Math. Gen 38, L73–L78 (2005). [CrossRef]

5.

A. Wünsche, “General Hermite and Laguerre two-dimensional polynomials,” J. Phys. A: Math Gen. 33, 1603–1629 (2000). [CrossRef]

6.

M.W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993). [CrossRef]

7.

J. A. Arnaud, “Mode coupling in first-order optics,” J. Opt. Soc. Am. 61, 751–758 (1971). [CrossRef]

8.

T. Alieva and M. J. Bastiaans, “Mode mapping in paraxial lossless optics,” submitted to Opt. Lett. (2005).

OCIS Codes
(070.2580) Fourier optics and signal processing : Paraxial wave optics
(070.4690) Fourier optics and signal processing : Morphological transformations
(080.2730) Geometric optics : Matrix methods in paraxial optics
(120.4820) Instrumentation, measurement, and metrology : Optical systems
(140.3300) Lasers and laser optics : Laser beam shaping
(350.5500) Other areas of optics : Propagation

ToC Category:
Research Papers

History
Original Manuscript: November 30, 2004
Revised Manuscript: November 26, 2004
Published: February 21, 2005

Citation
Martin Bastiaans and Tatiana Alieva, "Propagation law for the generating function of Hermite-Gaussian-type modes in first-order optical systems," Opt. Express 13, 1107-1112 (2005)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-4-1107


Sort:  Journal  |  Reset  

References

  1. M. Abramowitz and I. A. Stegun, eds., Pocketbook of Mathematical Functions (Deutsch, Frankfurt am Main, Germany, 1984).
  2. R. K. Luneburg, Mathematical Theory of Optics (University of California Press, Berkeley and Los Angeles, CA, USA, 1966).
  3. S. A. Collins Jr., “Lens-system diffraction integral written in terms of matrix optics,” J. Opt. Soc. Am. 60, 1168–1177 (1970). [CrossRef]
  4. M. J. Bastiaans and T. Alieva, “Generating function for Hermite-Gaussian modes propagating through first-order optical systems,” J. Phys. A: Math. Gen 38, L73–L78 (2005). [CrossRef]
  5. A. Wünsche, “General Hermite and Laguerre two-dimensional polynomials,” J. Phys. A: Math Gen. 33, 1603–1629 (2000). [CrossRef]
  6. M.W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P.Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993). [CrossRef]
  7. J. A. Arnaud, “Mode coupling in first-order optics,” J. Opt. Soc. Am. 61, 751–758 (1971). [CrossRef]
  8. T. Alieva and M. J. Bastiaans, “Mode mapping in paraxial lossless optics,” submitted to Opt. Lett. (2005).

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.


« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited