Abstract

The waist parameter is a particularly important factor for functional expansion in terms of localized orthogonal basis functions. We present a systematic approach to evaluate an asymptotic trend for the optimum waist parameter in truncated orthogonal localized bases satisfying several general conditions. This asymptotic behavior is fully introduced and verified for Hermite–Gauss and Laguerre– Gauss bases. As a special case of importance, a good estimate for the optimum waist in projection of discontinuous profiles on localized basis functions is proposed. The importance and application of the proposed estimation is demonstrated via several optical applications.

© 2010 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. R. Borghi, F. Gori, and M. Santarsiero, “Optimization of Laguerre-Gauss truncated series,” Opt. Commun. 125, 197-203 (1996).
    [CrossRef]
  2. Y. Liu and B. Lu, “Truncated Hermite-Gauss series expansion and its application,” Optik (Jena) 117, 437-442 (2006).
    [CrossRef]
  3. A. C. den Brinker and H. J. W. Belt, “Optimal free parameters in orthonormal approximations,” IEEE Trans. Signal Process. 46, 2081-2087 (1998).
    [CrossRef]
  4. P. Lazaridis, G. Debarge, and P. Gallion, “Discrete orthogonal Gauss-Hermite transform for optical pulse propagation analysis,” J. Opt. Soc. Am. B 20, 1508-1513 (2003).
    [CrossRef]
  5. F. Chiadini, G. Panariello, and A. Scaglione, “Variational analysis of matched-clad optical fibers,” J. Lightwave Technol. 21, 96-105 (2003).
    [CrossRef]
  6. T. Rasmussen, J. H. Povlsen, A. Bjarklev, O. Lumholt, B. Pedersen, and K. Rottwitt, “Detailed comparison of two approximate methods for the solution of the scalar wave equation for a rectangular optical waveguide,” J. Lightwave Technol. 11, 429-433 (1993).
    [CrossRef]
  7. I. A. Erteza and J. W. Goodman, “A scalar variational analysis of rectangular dielectric waveguides using Hermite-Gaussian modal approximations,” J. Lightwave Technol. 13, 493-506(1995).
    [CrossRef]
  8. R. L. Gallawa, I. C. Goyal, and A. K. Ghatak, “Fiber spot size: a simple method of calculation,” J. Lightwave Technol. 11, 192-197 (1993);
    [CrossRef]
  9. Z. H. Wang and J. P. Meunier, “Comments on 'Fiber spot size: a simple method of calculation,'” J. Lightwave Technol. 13, 1593 (1995).
    [CrossRef]
  10. R. L. Gallawa, I. C. Goyal, and A. K. Ghatak, “Optical waveguide modes: an approximate solution using Galerkin's method with Hermite-Gauss basis functions,” IEEE J. Quantum Electron. 27, 518-522 (1991).
    [CrossRef]
  11. J. P. Meunier, Z. H. Wang, and S. I. Hosain, “Evaluation of splice loss between two non identical single mode graded fibers,” IEEE Photonics Technol. Lett. 6, 998-1000 (1994).
    [CrossRef]
  12. D. G. Dudley, Mathematical Foundations for Electromagnetic Theory (Oxford U. Press, 1994).
    [CrossRef]
  13. G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists (Academic, 2005).
  14. S. Elaydi, An Introduction to Difference Equations (Springer, 2005).
  15. M. Martinelli and P. Martelli, “Laguerre mathematics in optical communications,” Opt. Photonics News 19, 30-35(2008).
    [CrossRef]
  16. T. M. Monro, D. J. Richardson, N. G. R. Broderick, and P. J. Bennett, “Holey optical fibers: an efficient modal model,” J. Lightwave Technol. 17, 1093-1102 (1999).
    [CrossRef]
  17. S. Guo and S. Albin, “Comparative analysis of Bragg fibers,” Opt. Express 12, 198-207 (2004).
    [CrossRef] [PubMed]
  18. V. García-Muñoz and M. A. Muriel, “Hermite-Gauss series expansions applied to arrayed waveguide gratings,” IEEE Photonics Technol. Lett. 17, 2331-2333 (2005).
    [CrossRef]
  19. R. McDuff, “Matrix method for beam propagation using Gaussian Hermite polynomials,” Appl. Opt. 29, 802-808(1990).
    [CrossRef] [PubMed]
  20. A. Sharma and J. P. Meunier, “Cutoff frequencies in planar optical waveguides with arbitrary index profiles: an efficient numerical method,” Opt. Quantum Electron. 34, 377-392(2002).
    [CrossRef]
  21. A. Sharma and J-P. Meunier, “On the scalar modal analysis of optical waveguides using approximate methods,” Opt. Commun. 281, 592-599 (2008).
    [CrossRef]
  22. P. Lazaridis, G. Debarge, and P. Gallion, “Exact solutions for linear propagation of chirped pulses using a chirped Gauss-Hermite orthogonal basis,” Opt. Lett. 22, 685-687(1997).
    [CrossRef] [PubMed]

2008

M. Martinelli and P. Martelli, “Laguerre mathematics in optical communications,” Opt. Photonics News 19, 30-35(2008).
[CrossRef]

A. Sharma and J-P. Meunier, “On the scalar modal analysis of optical waveguides using approximate methods,” Opt. Commun. 281, 592-599 (2008).
[CrossRef]

2006

Y. Liu and B. Lu, “Truncated Hermite-Gauss series expansion and its application,” Optik (Jena) 117, 437-442 (2006).
[CrossRef]

2005

V. García-Muñoz and M. A. Muriel, “Hermite-Gauss series expansions applied to arrayed waveguide gratings,” IEEE Photonics Technol. Lett. 17, 2331-2333 (2005).
[CrossRef]

2004

2003

2002

A. Sharma and J. P. Meunier, “Cutoff frequencies in planar optical waveguides with arbitrary index profiles: an efficient numerical method,” Opt. Quantum Electron. 34, 377-392(2002).
[CrossRef]

1999

1998

A. C. den Brinker and H. J. W. Belt, “Optimal free parameters in orthonormal approximations,” IEEE Trans. Signal Process. 46, 2081-2087 (1998).
[CrossRef]

1997

1996

R. Borghi, F. Gori, and M. Santarsiero, “Optimization of Laguerre-Gauss truncated series,” Opt. Commun. 125, 197-203 (1996).
[CrossRef]

1995

I. A. Erteza and J. W. Goodman, “A scalar variational analysis of rectangular dielectric waveguides using Hermite-Gaussian modal approximations,” J. Lightwave Technol. 13, 493-506(1995).
[CrossRef]

Z. H. Wang and J. P. Meunier, “Comments on 'Fiber spot size: a simple method of calculation,'” J. Lightwave Technol. 13, 1593 (1995).
[CrossRef]

1994

J. P. Meunier, Z. H. Wang, and S. I. Hosain, “Evaluation of splice loss between two non identical single mode graded fibers,” IEEE Photonics Technol. Lett. 6, 998-1000 (1994).
[CrossRef]

1993

R. L. Gallawa, I. C. Goyal, and A. K. Ghatak, “Fiber spot size: a simple method of calculation,” J. Lightwave Technol. 11, 192-197 (1993);
[CrossRef]

T. Rasmussen, J. H. Povlsen, A. Bjarklev, O. Lumholt, B. Pedersen, and K. Rottwitt, “Detailed comparison of two approximate methods for the solution of the scalar wave equation for a rectangular optical waveguide,” J. Lightwave Technol. 11, 429-433 (1993).
[CrossRef]

1991

R. L. Gallawa, I. C. Goyal, and A. K. Ghatak, “Optical waveguide modes: an approximate solution using Galerkin's method with Hermite-Gauss basis functions,” IEEE J. Quantum Electron. 27, 518-522 (1991).
[CrossRef]

1990

Albin, S.

Arfken, G. B.

G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists (Academic, 2005).

Belt, H. J. W.

A. C. den Brinker and H. J. W. Belt, “Optimal free parameters in orthonormal approximations,” IEEE Trans. Signal Process. 46, 2081-2087 (1998).
[CrossRef]

Bennett, P. J.

Bjarklev, A.

T. Rasmussen, J. H. Povlsen, A. Bjarklev, O. Lumholt, B. Pedersen, and K. Rottwitt, “Detailed comparison of two approximate methods for the solution of the scalar wave equation for a rectangular optical waveguide,” J. Lightwave Technol. 11, 429-433 (1993).
[CrossRef]

Borghi, R.

R. Borghi, F. Gori, and M. Santarsiero, “Optimization of Laguerre-Gauss truncated series,” Opt. Commun. 125, 197-203 (1996).
[CrossRef]

Broderick, N. G. R.

Chiadini, F.

Debarge, G.

den Brinker, A. C.

A. C. den Brinker and H. J. W. Belt, “Optimal free parameters in orthonormal approximations,” IEEE Trans. Signal Process. 46, 2081-2087 (1998).
[CrossRef]

Dudley, D. G.

D. G. Dudley, Mathematical Foundations for Electromagnetic Theory (Oxford U. Press, 1994).
[CrossRef]

Elaydi, S.

S. Elaydi, An Introduction to Difference Equations (Springer, 2005).

Erteza, I. A.

I. A. Erteza and J. W. Goodman, “A scalar variational analysis of rectangular dielectric waveguides using Hermite-Gaussian modal approximations,” J. Lightwave Technol. 13, 493-506(1995).
[CrossRef]

Gallawa, R. L.

R. L. Gallawa, I. C. Goyal, and A. K. Ghatak, “Fiber spot size: a simple method of calculation,” J. Lightwave Technol. 11, 192-197 (1993);
[CrossRef]

R. L. Gallawa, I. C. Goyal, and A. K. Ghatak, “Optical waveguide modes: an approximate solution using Galerkin's method with Hermite-Gauss basis functions,” IEEE J. Quantum Electron. 27, 518-522 (1991).
[CrossRef]

Gallion, P.

García-Muñoz, V.

V. García-Muñoz and M. A. Muriel, “Hermite-Gauss series expansions applied to arrayed waveguide gratings,” IEEE Photonics Technol. Lett. 17, 2331-2333 (2005).
[CrossRef]

Ghatak, A. K.

R. L. Gallawa, I. C. Goyal, and A. K. Ghatak, “Fiber spot size: a simple method of calculation,” J. Lightwave Technol. 11, 192-197 (1993);
[CrossRef]

R. L. Gallawa, I. C. Goyal, and A. K. Ghatak, “Optical waveguide modes: an approximate solution using Galerkin's method with Hermite-Gauss basis functions,” IEEE J. Quantum Electron. 27, 518-522 (1991).
[CrossRef]

Goodman, J. W.

I. A. Erteza and J. W. Goodman, “A scalar variational analysis of rectangular dielectric waveguides using Hermite-Gaussian modal approximations,” J. Lightwave Technol. 13, 493-506(1995).
[CrossRef]

Gori, F.

R. Borghi, F. Gori, and M. Santarsiero, “Optimization of Laguerre-Gauss truncated series,” Opt. Commun. 125, 197-203 (1996).
[CrossRef]

Goyal, I. C.

R. L. Gallawa, I. C. Goyal, and A. K. Ghatak, “Fiber spot size: a simple method of calculation,” J. Lightwave Technol. 11, 192-197 (1993);
[CrossRef]

R. L. Gallawa, I. C. Goyal, and A. K. Ghatak, “Optical waveguide modes: an approximate solution using Galerkin's method with Hermite-Gauss basis functions,” IEEE J. Quantum Electron. 27, 518-522 (1991).
[CrossRef]

Guo, S.

Hosain, S. I.

J. P. Meunier, Z. H. Wang, and S. I. Hosain, “Evaluation of splice loss between two non identical single mode graded fibers,” IEEE Photonics Technol. Lett. 6, 998-1000 (1994).
[CrossRef]

Lazaridis, P.

Liu, Y.

Y. Liu and B. Lu, “Truncated Hermite-Gauss series expansion and its application,” Optik (Jena) 117, 437-442 (2006).
[CrossRef]

Lu, B.

Y. Liu and B. Lu, “Truncated Hermite-Gauss series expansion and its application,” Optik (Jena) 117, 437-442 (2006).
[CrossRef]

Lumholt, O.

T. Rasmussen, J. H. Povlsen, A. Bjarklev, O. Lumholt, B. Pedersen, and K. Rottwitt, “Detailed comparison of two approximate methods for the solution of the scalar wave equation for a rectangular optical waveguide,” J. Lightwave Technol. 11, 429-433 (1993).
[CrossRef]

Martelli, P.

M. Martinelli and P. Martelli, “Laguerre mathematics in optical communications,” Opt. Photonics News 19, 30-35(2008).
[CrossRef]

Martinelli, M.

M. Martinelli and P. Martelli, “Laguerre mathematics in optical communications,” Opt. Photonics News 19, 30-35(2008).
[CrossRef]

McDuff, R.

Meunier, J. P.

A. Sharma and J. P. Meunier, “Cutoff frequencies in planar optical waveguides with arbitrary index profiles: an efficient numerical method,” Opt. Quantum Electron. 34, 377-392(2002).
[CrossRef]

Z. H. Wang and J. P. Meunier, “Comments on 'Fiber spot size: a simple method of calculation,'” J. Lightwave Technol. 13, 1593 (1995).
[CrossRef]

J. P. Meunier, Z. H. Wang, and S. I. Hosain, “Evaluation of splice loss between two non identical single mode graded fibers,” IEEE Photonics Technol. Lett. 6, 998-1000 (1994).
[CrossRef]

Meunier, J-P.

A. Sharma and J-P. Meunier, “On the scalar modal analysis of optical waveguides using approximate methods,” Opt. Commun. 281, 592-599 (2008).
[CrossRef]

Monro, T. M.

Muriel, M. A.

V. García-Muñoz and M. A. Muriel, “Hermite-Gauss series expansions applied to arrayed waveguide gratings,” IEEE Photonics Technol. Lett. 17, 2331-2333 (2005).
[CrossRef]

Panariello, G.

Pedersen, B.

T. Rasmussen, J. H. Povlsen, A. Bjarklev, O. Lumholt, B. Pedersen, and K. Rottwitt, “Detailed comparison of two approximate methods for the solution of the scalar wave equation for a rectangular optical waveguide,” J. Lightwave Technol. 11, 429-433 (1993).
[CrossRef]

Povlsen, J. H.

T. Rasmussen, J. H. Povlsen, A. Bjarklev, O. Lumholt, B. Pedersen, and K. Rottwitt, “Detailed comparison of two approximate methods for the solution of the scalar wave equation for a rectangular optical waveguide,” J. Lightwave Technol. 11, 429-433 (1993).
[CrossRef]

Rasmussen, T.

T. Rasmussen, J. H. Povlsen, A. Bjarklev, O. Lumholt, B. Pedersen, and K. Rottwitt, “Detailed comparison of two approximate methods for the solution of the scalar wave equation for a rectangular optical waveguide,” J. Lightwave Technol. 11, 429-433 (1993).
[CrossRef]

Richardson, D. J.

Rottwitt, K.

T. Rasmussen, J. H. Povlsen, A. Bjarklev, O. Lumholt, B. Pedersen, and K. Rottwitt, “Detailed comparison of two approximate methods for the solution of the scalar wave equation for a rectangular optical waveguide,” J. Lightwave Technol. 11, 429-433 (1993).
[CrossRef]

Santarsiero, M.

R. Borghi, F. Gori, and M. Santarsiero, “Optimization of Laguerre-Gauss truncated series,” Opt. Commun. 125, 197-203 (1996).
[CrossRef]

Scaglione, A.

Sharma, A.

A. Sharma and J-P. Meunier, “On the scalar modal analysis of optical waveguides using approximate methods,” Opt. Commun. 281, 592-599 (2008).
[CrossRef]

A. Sharma and J. P. Meunier, “Cutoff frequencies in planar optical waveguides with arbitrary index profiles: an efficient numerical method,” Opt. Quantum Electron. 34, 377-392(2002).
[CrossRef]

Wang, Z. H.

Z. H. Wang and J. P. Meunier, “Comments on 'Fiber spot size: a simple method of calculation,'” J. Lightwave Technol. 13, 1593 (1995).
[CrossRef]

J. P. Meunier, Z. H. Wang, and S. I. Hosain, “Evaluation of splice loss between two non identical single mode graded fibers,” IEEE Photonics Technol. Lett. 6, 998-1000 (1994).
[CrossRef]

Weber, H. J.

G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists (Academic, 2005).

Appl. Opt.

IEEE J. Quantum Electron.

R. L. Gallawa, I. C. Goyal, and A. K. Ghatak, “Optical waveguide modes: an approximate solution using Galerkin's method with Hermite-Gauss basis functions,” IEEE J. Quantum Electron. 27, 518-522 (1991).
[CrossRef]

IEEE Photonics Technol. Lett.

J. P. Meunier, Z. H. Wang, and S. I. Hosain, “Evaluation of splice loss between two non identical single mode graded fibers,” IEEE Photonics Technol. Lett. 6, 998-1000 (1994).
[CrossRef]

V. García-Muñoz and M. A. Muriel, “Hermite-Gauss series expansions applied to arrayed waveguide gratings,” IEEE Photonics Technol. Lett. 17, 2331-2333 (2005).
[CrossRef]

IEEE Trans. Signal Process.

A. C. den Brinker and H. J. W. Belt, “Optimal free parameters in orthonormal approximations,” IEEE Trans. Signal Process. 46, 2081-2087 (1998).
[CrossRef]

J. Lightwave Technol.

F. Chiadini, G. Panariello, and A. Scaglione, “Variational analysis of matched-clad optical fibers,” J. Lightwave Technol. 21, 96-105 (2003).
[CrossRef]

T. Rasmussen, J. H. Povlsen, A. Bjarklev, O. Lumholt, B. Pedersen, and K. Rottwitt, “Detailed comparison of two approximate methods for the solution of the scalar wave equation for a rectangular optical waveguide,” J. Lightwave Technol. 11, 429-433 (1993).
[CrossRef]

I. A. Erteza and J. W. Goodman, “A scalar variational analysis of rectangular dielectric waveguides using Hermite-Gaussian modal approximations,” J. Lightwave Technol. 13, 493-506(1995).
[CrossRef]

R. L. Gallawa, I. C. Goyal, and A. K. Ghatak, “Fiber spot size: a simple method of calculation,” J. Lightwave Technol. 11, 192-197 (1993);
[CrossRef]

Z. H. Wang and J. P. Meunier, “Comments on 'Fiber spot size: a simple method of calculation,'” J. Lightwave Technol. 13, 1593 (1995).
[CrossRef]

T. M. Monro, D. J. Richardson, N. G. R. Broderick, and P. J. Bennett, “Holey optical fibers: an efficient modal model,” J. Lightwave Technol. 17, 1093-1102 (1999).
[CrossRef]

J. Opt. Soc. Am. B

Opt. Commun.

R. Borghi, F. Gori, and M. Santarsiero, “Optimization of Laguerre-Gauss truncated series,” Opt. Commun. 125, 197-203 (1996).
[CrossRef]

A. Sharma and J-P. Meunier, “On the scalar modal analysis of optical waveguides using approximate methods,” Opt. Commun. 281, 592-599 (2008).
[CrossRef]

Opt. Express

Opt. Lett.

Opt. Photonics News

M. Martinelli and P. Martelli, “Laguerre mathematics in optical communications,” Opt. Photonics News 19, 30-35(2008).
[CrossRef]

Opt. Quantum Electron.

A. Sharma and J. P. Meunier, “Cutoff frequencies in planar optical waveguides with arbitrary index profiles: an efficient numerical method,” Opt. Quantum Electron. 34, 377-392(2002).
[CrossRef]

Optik (Jena)

Y. Liu and B. Lu, “Truncated Hermite-Gauss series expansion and its application,” Optik (Jena) 117, 437-442 (2006).
[CrossRef]

Other

D. G. Dudley, Mathematical Foundations for Electromagnetic Theory (Oxford U. Press, 1994).
[CrossRef]

G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists (Academic, 2005).

S. Elaydi, An Introduction to Difference Equations (Springer, 2005).

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 (6)

Fig. 1
Fig. 1

Expansion of a sphere defect using the 3D LG basis. (a) Solid curve, error by w = 1 / M 3 ; dashed curve, error by optimum waist. (b) Enlarged view of (a). (c) Solid curve, w = 1 / M 3 ; dots, optimum waist. (d) Enlarged view of (c).

Fig. 2
Fig. 2

Error versus waist: (a)  order = 4 , (b)  order = 100 .

Fig. 3
Fig. 3

Error versus order: solid curve, optimum waist; dashed curve, w = 0.5 , dashed-dotted curve, w = 0.25 .

Fig. 4
Fig. 4

Expansion of Bragg fiber structure in the 2D LG basis. (a) Bragg structure. (b) Dashed curve, error by optimum waist; solid curve, error by w = 3 / M . (c) Enlarged view of (b) for small orders.

Fig. 5
Fig. 5

Expansion of raised-cosine slab in 1D-HG basis. Error is plotted versus order. Solid curve, optimum waist; dashed curve, w = 2 / M ; dashed-dotted curve, w = 0.5 ; circles, minimized upper bound [3].

Fig. 6
Fig. 6

Expansion of raised-cosine slab in the 1D HG basis: dots, optimum waist; solid curve, 1.93 / M .

Tables (1)

Tables Icon

Table 1 Optimum Waist in Fig. 3 of [20] (Middle Row) and Estimated Optimum Waist (Lower Row)

Equations (30)

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

e 2 = | f ( r ) m = 0 M 1 α m ψ m ( r , w ) | 2 d r .
G t M A = B M , G M = [ g i , k ] ; g i , k ( w ) = ψ i , ψ k = ψ i ( r , w ) ψ k * ( r , w ) d r , B M = [ b i ] ; b i = f , ψ i , A = [ α i ] ; i = 0 , , M 1.
e 2 = f , f B M t G M 1 B M * .
e 2 w = B M t ( D M T G M 1 + G M 1 D M ) B M * B r t D r T G M 1 B M * B M t G M 1 D r B r * ,
G M 1 D M + D M T G M 1 = 0.
e 2 w = 2 Re { B M t G M 1 D r B r * } = 2 Re { i = 0 M 1 k = M f , ψ i f , ψ k * ( g M 1 ) i d i , k } .
F M ( w ) a M ( w w M ) ,
w F M ( w ) = w f , ψ M = i d M , i F i ( w ) .
1 D a M = i d ˜ M , i a i ,
0 = i d ˜ M , i a i w i ,
h m ( x , w ) = 1 w π 1 2 m m ! H m ( x w ) exp ( x 2 2 w 2 ) ( x ) ,
a m = 1 2 2 m ( 2 m 1 ) a m 1 + 1 2 ( 2 m + 1 ) ( 2 m + 2 ) a m + 1 ,
w M + 2 w M 2 = M ( M 1 ) ( M + 1 ) ( M + 2 ) a M 2 a M + 2 = M ( M 1 ) ( M + 1 ) ( M + 2 ) a m 1 a m + 1 .
a m 1 a m + 1 1 + O ( 1 m 2 ) 1 + O ( 1 M 2 ) ,
w M + 2 w M 2 ( 1 1 2 M ) M 2 M + 2 .
w M μ M .
e 2 w = 2 Re { 1 ε i W / 2 0 ε k W / 2 f , ψ M ε i f , ψ M + ε k * ( g M 1 ) M ε i d M ε i M + ε k } ,
φ m ( x ) = 2 n ! ( n + k ) ! ( 2 x ) k / 2 L m k ( 2 x ) exp ( x ) ( x + ) , φ m ( D ) ( r , w ) = c D 1 w D / 2 φ m ( ( | r | w ) D , 1 ) .
a M = D 2 M ( M + k ) a M 1 + D 2 ( M + 1 ) ( M + k + 1 ) a M + 1 ,
0 = D 2 M ( M + k ) a M 1 w M 1 + D 2 ( M + 1 ) ( M + k + 1 ) a M + 1 w M + 1 .
w M μ M 1 / D ,
f ( x ) = { 1 | x | < 1 cos 2 ( π 2 ( | x | 1 ) ) 1 < | x | < 2 .
F ( w ) M n = 0 M 1 n a n 2 ( w ) M f 2 ,
w Ψ = D w Ψ , Ψ = [ ψ i ] ,
D w = [ D M D r D s D p ] .
B = [ B M B r ] ,
G = Ψ , Ψ t ,
G w = D w G + GD w T .
G 1 D w + D w T G 1 = 0.
G M 1 D M + D M T G M 1 = 0.

Metrics