Abstract

The linear equations method is proposed to calculate the complete modal content of the partially coherent laser beam using only the intensity information. This method could give not only the incoherent expansion coefficients of the modal decomposition but also the cross-correlation expansion coefficients using the intensity profiles in several planes of finite distance along the propagation direction. A simulation is also presented to verify the validity of this theory. In our algorithm, the minimum and maximum mode orders should be known a priori, so we provide an estimation method for the two parameters.

© 2013 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. A. E. Siegman, Lasers (University Science Books, 1986).
  2. E. Wolf and G. S. Agarwal, “Coherence theory of laser resonator modes,” J. Opt. Soc. Am. A 1, 541–546 (1984).
    [CrossRef]
  3. C. Schulze, D. Naidoo, D. Flamm, O. A. Schmidt, A. Forbes, and M. Duparre, “Wavefront reconstruction by modal decomposition,” Opt. Express 20, 19714–19725 (2012).
    [CrossRef]
  4. O. A. Schmidt, C. Schulze, D. Flamm, R. Bruning, T. Kaiser, S. Schroter, and M. Duparre, “Real-time determination of laser beam quality by modal decomposition,” Opt. Express 19, 6741–6748 (2011).
    [CrossRef]
  5. R. Borghi, G. Piquero, and M. Santarsiero, “Use of biorthogonal functions for the modal decomposition of multimode beams,” Opt. Commun. 194, 235–242 (2001).
    [CrossRef]
  6. K. M. Du, G. Herziger, P. Loosen, and F. Ruhl, “Computation of the statistical properties of laser light,” Opt. Quantum Electron. 24, S1095–S1108 (1992).
    [CrossRef]
  7. F. Gori, M. Santarsiero, and G. Guattari, “Coherence and the spatial distribution of intensity,” J. Opt. Soc. Am. A 10, 673–679 (1993).
    [CrossRef]
  8. D. Dragoman, “Unambiguous coherence retrieval from intensity measurements,” J. Opt. Soc. Am. A 20, 290–295 (2003).
    [CrossRef]
  9. M. Santarsiero, F. Gori, R. Borghi, and G. Guattari, “Evaluation of the modal structure of light beams composed of incoherent mixtures of Hermite-Gaussian modes,” Appl. Opt. 38, 5272–5281 (1999).
    [CrossRef]
  10. X. Xue, H. Wei, and A. G. Kirk, “Intensity-based modal decomposition of optical beams in terms of Hermite-Gaussian functions,” J. Opt. Soc. Am. A 17, 1086–1091 (2000).
    [CrossRef]
  11. H. Laabs, B. Eppich, and H. Weber, “Modal decomposition of partially coherent beams using the ambiguity function,” J. Opt. Soc. Am. A 19, 497–504 (2002).
    [CrossRef]
  12. R. Borghi, G. Guattari, L. de la Torre, F. Gori, and M. Santarsiero, “Evaluation of the spatial coherence of a light beam through transverse intensity measurements,” J. Opt. Soc. Am. A 20, 1763–1770 (2003).
    [CrossRef]
  13. T. E. Gureyev, A. Roberts, and K. A. Nugent, “Partially coherent fields, the transport-of-intensity equation, and phase uniqueness,” J. Opt. Soc. Am. A 12, 1942–1946 (1995).
    [CrossRef]
  14. T. Kaiser, D. Flamm, S. Schroter, and M. Duparr, “Complete modal decomposition for optical fibers using CGH-based correlation filters,” Opt. Express 17, 9347–9356 (2009).
    [CrossRef]
  15. J. D. Schmidt, Numerical Simulation of Optical Wave Propagation with Examples in MATLAB (SPIE, 2010).
  16. H. Yanai, K. Takeuchi, and Y. Takane, Projection Matrices, Generalized Inverse Matrices, and Singular Value Decomposition (Springer, 2011).
  17. G. B. Arfken, H. J. Weber, and F. E. Harris, Mathematical Methods for Physicist (Academic, 2013).

2012 (1)

2011 (1)

2009 (1)

2003 (2)

2002 (1)

2001 (1)

R. Borghi, G. Piquero, and M. Santarsiero, “Use of biorthogonal functions for the modal decomposition of multimode beams,” Opt. Commun. 194, 235–242 (2001).
[CrossRef]

2000 (1)

1999 (1)

1995 (1)

1993 (1)

1992 (1)

K. M. Du, G. Herziger, P. Loosen, and F. Ruhl, “Computation of the statistical properties of laser light,” Opt. Quantum Electron. 24, S1095–S1108 (1992).
[CrossRef]

1984 (1)

Agarwal, G. S.

Arfken, G. B.

G. B. Arfken, H. J. Weber, and F. E. Harris, Mathematical Methods for Physicist (Academic, 2013).

Borghi, R.

Bruning, R.

de la Torre, L.

Dragoman, D.

Du, K. M.

K. M. Du, G. Herziger, P. Loosen, and F. Ruhl, “Computation of the statistical properties of laser light,” Opt. Quantum Electron. 24, S1095–S1108 (1992).
[CrossRef]

Duparr, M.

Duparre, M.

Eppich, B.

Flamm, D.

Forbes, A.

Gori, F.

Guattari, G.

Gureyev, T. E.

Harris, F. E.

G. B. Arfken, H. J. Weber, and F. E. Harris, Mathematical Methods for Physicist (Academic, 2013).

Herziger, G.

K. M. Du, G. Herziger, P. Loosen, and F. Ruhl, “Computation of the statistical properties of laser light,” Opt. Quantum Electron. 24, S1095–S1108 (1992).
[CrossRef]

Kaiser, T.

Kirk, A. G.

Laabs, H.

Loosen, P.

K. M. Du, G. Herziger, P. Loosen, and F. Ruhl, “Computation of the statistical properties of laser light,” Opt. Quantum Electron. 24, S1095–S1108 (1992).
[CrossRef]

Naidoo, D.

Nugent, K. A.

Piquero, G.

R. Borghi, G. Piquero, and M. Santarsiero, “Use of biorthogonal functions for the modal decomposition of multimode beams,” Opt. Commun. 194, 235–242 (2001).
[CrossRef]

Roberts, A.

Ruhl, F.

K. M. Du, G. Herziger, P. Loosen, and F. Ruhl, “Computation of the statistical properties of laser light,” Opt. Quantum Electron. 24, S1095–S1108 (1992).
[CrossRef]

Santarsiero, M.

Schmidt, J. D.

J. D. Schmidt, Numerical Simulation of Optical Wave Propagation with Examples in MATLAB (SPIE, 2010).

Schmidt, O. A.

Schroter, S.

Schulze, C.

Siegman, A. E.

A. E. Siegman, Lasers (University Science Books, 1986).

Takane, Y.

H. Yanai, K. Takeuchi, and Y. Takane, Projection Matrices, Generalized Inverse Matrices, and Singular Value Decomposition (Springer, 2011).

Takeuchi, K.

H. Yanai, K. Takeuchi, and Y. Takane, Projection Matrices, Generalized Inverse Matrices, and Singular Value Decomposition (Springer, 2011).

Weber, H.

Weber, H. J.

G. B. Arfken, H. J. Weber, and F. E. Harris, Mathematical Methods for Physicist (Academic, 2013).

Wei, H.

Wolf, E.

Xue, X.

Yanai, H.

H. Yanai, K. Takeuchi, and Y. Takane, Projection Matrices, Generalized Inverse Matrices, and Singular Value Decomposition (Springer, 2011).

Appl. Opt. (1)

J. Opt. Soc. Am. A (7)

Opt. Commun. (1)

R. Borghi, G. Piquero, and M. Santarsiero, “Use of biorthogonal functions for the modal decomposition of multimode beams,” Opt. Commun. 194, 235–242 (2001).
[CrossRef]

Opt. Express (3)

Opt. Quantum Electron. (1)

K. M. Du, G. Herziger, P. Loosen, and F. Ruhl, “Computation of the statistical properties of laser light,” Opt. Quantum Electron. 24, S1095–S1108 (1992).
[CrossRef]

Other (4)

A. E. Siegman, Lasers (University Science Books, 1986).

J. D. Schmidt, Numerical Simulation of Optical Wave Propagation with Examples in MATLAB (SPIE, 2010).

H. Yanai, K. Takeuchi, and Y. Takane, Projection Matrices, Generalized Inverse Matrices, and Singular Value Decomposition (Springer, 2011).

G. B. Arfken, H. J. Weber, and F. E. Harris, Mathematical Methods for Physicist (Academic, 2013).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (4)

Fig. 1.
Fig. 1.

Intensity profiles of the beam along the propagation direction.

Fig. 2.
Fig. 2.

Theoretical and numerical results of the amplitudes of the expansion coefficients cn*cn+m.

Fig. 3.
Fig. 3.

Theoretical and numerical results of the arguments of the expansion coefficients cn*cn+m.

Fig. 4.
Fig. 4.

Expansion results of the function A˜0(ξ) with different sampling numbers. (a) The sampling number K is 3. (b) The sampling number K is 5.

Tables (1)

Tables Icon

Table 1. Mode Coefficients cn of the nth HG Mode

Equations (36)

Equations on this page are rendered with MathJax. Learn more.

U(x,φ)=cosφexp(ikz)exp(ik2Rx2)×n=0cnGn(xcosφ)exp(inφ),
Gn(x)=(2πv02)1/412nn!Hn(x2v0)exp(x2v02),
I(x,φ)=U(x,φ)U*(x,φ)=n=0m=02cosφ1+δm,0Re{cn*cn+mexp(imφ)}×Gn(xcosφ)Gn+m(xcosφ),
I^(ξ,φ)=n=0m=021+δm,0Re{cn*cn+mexp(imφ)}×Gn(ξ)Gn+m(ξ).
cn*cn+m=Mn,mexp(iθn,m).
I^(ξ,φ)=m=0Nn0n=n0Nm21+δm,0Mn,mcos(mφθn,m)×Gn(ξ)Gn+m(ξ).
Am(ξ)=2n=n0Nm11+δm,0Mn,mcosθn,mGn(ξ)Gn+m(ξ),
Bm(ξ)=2n=n0NmMn,msinθn,mGn(ξ)Gn+m(ξ).
I^(ξ,φ)=m=0Nn0[Am(ξ)cos(mφ)+Bm(ξ)sin(mφ)].
P1=(cos(0φ1)cos[(Nn0)φ1]cos(0φT)cos[(Nn0)φT]),P2=(sin(0φ1)sin[(Nn0)φ1]sin(0φT)sin[(Nn0)φT]);
I=(I(ξ1,φ1)I(ξS,φ1)I(ξ1,φT)I(ξS,φT));
A=(A0,A1,ANn0)T,B=(B0,B1,BNn0)T,
Am=(Am(ξ1)Am(ξS)),Bm=(Bm(ξ1)Bm(ξS)).
I=P1A+P2B=(P1P2)(AB).
PΩ=I.
Ω=(AB)=P+I,
Gm=(Gn0(ξ1)Gn0+m(ξ1)GNm(ξ1)GN(ξ1)Gn0(ξS)Gn0+m(ξS)GNm(ξS)GN(ξS)),
Creal,m=(Mn0,mcosθn0,mMNm,mcosθNm,m)T,Cimag,m=(Mn0,msinθn0,mMNm,msinθNm,m)T.
GmCreal,m=1+δm,02AmT,
GmCimag,m=BmT.
Creal,m=1+δm,02Gm+AmT,
Cimag,m=Gm+BmT,
Cm=(cn0*cn0+mcn0+Nm*cn0+N)T=Creal,m+iCimag,m.
|cn|=cn*cn.
arg(cn)=12[arg(cn1*cn+1)arg(cn1*cn)arg(cn*cn+1)],
arg(cn0)=arg(cn0*cn0+1)arg(cn0+1),arg(cN)=arg(cN1*cN)arg(cN1).
I(x)=|ci|2Gi2(x)+|cj|2Gj2(x)+Re{ci*cj}Gi(x)Gj(x).
I^(ξ)φ=A0(ξ)+1Km=1Nn0k=1K[Am(ξ)cos(mφk)+Bm(ξ)sin(mφk)]=A0(ξ)+1Km=1Nn0k=1KAm(ξ)cos(mφk).
A˜0(ξ)=I^(ξ)φ+I^(ξ)φ2=A0(ξ)+1Kh=1[Nn02]A2h(ξ)k=1Kcos(mφk),
A2h(ξ)=Span{G0(ξ)G2h(ξ),G1(ξ)G2h+1(ξ),,GN2h(ξ)GN(ξ)}.
F{Gn(ξ)Gn+m(ξ)}(u)=(i)mψnm(π2v02u2),
ψnm(x)=Rnxm2Lnm(x)exp(x2),
xLn2h(x)=(2h1)m=0nLm2(h1)(x)(n+1)Ln+12(h1)(x).
Gn(ξ)Gn+2h(ξ)=Rn[m=0n(2h1)RmGm(ξ)Gm+2(h1)(ξ)n+1Rn+1Gn+1(ξ)G(n+1)+2(h1)(ξ)].
A2h(ξ)=Span{G0(ξ)2,G1(ξ)2,,GNh(ξ)2}.
A˜0(ξ)=Span{G0(ξ)2,G1(ξ)2,,GN(ξ)2}.

Metrics