Abstract

We present a method for computing waveguide eigenmodes based on complex coupled mode theory (CCMT). This approach generalizes Fourier transform methods by allowing an arbitrary but convenient basis set to be used for the expansion. In the presented approach, one is free to choose an arbitrary basis representation; for example, we show the use of electromagnetic modes of a cylindrical metal waveguide. CCMT-computed modes are compared with modes computed using analytic expressions and results obtained using a finite difference solver. In cases where the basis set is small, the method can efficiently re-compute modes after structural refinements are made, and can efficiently compute dispersion. The parallel nature of the algorithm makes it well suited to a graphics processing unit implementation, as demonstrated here.

© 2017 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2011 (1)

G. Colas des Francs, J.-P. Hugonin, and J. Čtyroký, “Mode solvers for very thin long–range plasmonic waveguides,” Opt. Quant. Electron. 42(8), 557–570 (2011).
[Crossref]

2010 (1)

J. Xiao and X. Sun, “Full-vectorial mode solver for anisotropic optical waveguides using multidomain spectral collocation method,” Opt. Commun. 283(14), 2835–2840 (2010).
[Crossref]

2009 (2)

2008 (2)

Y.-C. Lu, L. Yang, W.-P. Huang, and S.-S. Jian, “Improved Full-Vector Finite-Difference Complex Mode Solver for Optical Waveguides of Circular Symmetry,” J. Lightwave Technol. 26(13), 1868–1876 (2008)
[Crossref]

A. B. Fallahkhair, K. S. Li, and T. E. Murphy, “Vector Finite Difference Modesolver for Anisotropic Dielectric Waveguides,” J. Lightw. Technol. 26(11), 1423–1431 (2008).
[Crossref]

2007 (1)

2006 (3)

2005 (3)

A. Mohammadi, H. Nadgaran, and M. Agio, “Contour-path effective permittivities for the two-dimensional finite-difference time-domain method,” Opt. Express 13(25), 10367–10381 (2005).
[Crossref] [PubMed]

S. A. Maier and H. A. Atwater, “Plasmonics: Localization and guiding of electromagnetic energy in metal/dielectric structures,” J. Appl. Phys. 98(011101), 1–9 (2005).
[Crossref]

J. P. Hugonin, P. Lalanne, I. D. Villar, and I. R. Matias, “Fourier modal methods for modeling optical dielectric waveguides,” Opt. Quant. Electron. 37(1–3), 107–119 (2005).
[Crossref]

2004 (2)

C.-P. Yu and H.-C. Chang, “Applications of the finite difference mode solution method to photonic crystal structures,” Opt. Quant. Electron. 36(1–3), 145–163 (2004).
[Crossref]

C.-P. Yu and H.-C. Chang, “Yee-mesh-based finite difference eigenmode solver with PML absorbing boundary conditions for optical waveguides and photonic crystal fibers,” Opt. Express 12(25), 6165–6177 (2004).
[Crossref] [PubMed]

2003 (1)

P. Russell, “Photonic Crystal Fibers,” Science 299(5605), 358–362 (2003).
[Crossref] [PubMed]

2002 (5)

2001 (2)

W. Yu and R. Mittra, “A conformal finite difference time domain technique for modeling curved dielectric surfaces,” IEEE Microw. Wirel. Compon. Lett. 11(1), 25–27 (2001).
[Crossref]

S. Selleri, L. Vincetti, A. Cucinotta, and M. Zoboli, “Complex FEM modal solver of optical waveguides with PML boundary conditions,” Opt. Quant. Electron. 33(4–5), 359–371 (2001).
[Crossref]

2000 (2)

1999 (1)

1998 (1)

1997 (1)

C. Vassallo, “1993—1995 Optical mode solvers,” Opt. Quant. Electron. 29(2), 95–114 (1997).
[Crossref]

1996 (1)

Z. H. Wang, “Application of the Coupled Mode Theory to Eigenvalue Problems of Graded-Index Optical Fibers,” Microw. Opt. Technol. Lett. 12(2), 90–93 (1996).
[Crossref]

1995 (3)

S. Amari and J. Bornemann, “Efficient numerical computation of singular integrals with applications to electromagnetics,” IEEE Trans. Antennas Propag. 43(11), 1343–1348 (1995).
[Crossref]

M. G. Moharam, E. B. Grann, D. A. Pommet, and T. K. Gaylord, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A 12(5), 1068–1076 (1995).
[Crossref]

A. Weisshaar, J. Li, R. L. Gallawa, and I. C. Goyal, “Vector and quasi-vector solutions for optical waveguide modes using efficient Galerkin’s method with Hermite-Gauss basis functions,”J. Lightwave Tech. 13(8), 1795–1800 (1995).
[Crossref]

1994 (1)

K. S. Chiang, “Review of numerical and approximate methods for the modal analysis of general optical dielectric waveguides,” Opt. Quant. Electron. 26(3), S113–S134 (1994).
[Crossref]

1993 (2)

G. Sztefka and H. P. Nolting, “Bidirectional eigenmode propagation for large refractive index steps,” IEEE Photon. Technol. Lett. 5(5), 554–557 (1993).
[Crossref]

A. S. Sudbo, “Film mode matching: a versatile numerical method for vector mode field calculations in dielectric waveguides,” Pure Appl. Opt. 2(3), 211–233 (1993).
[Crossref]

1990 (1)

D. A. Goldberg, L. J. Laslett, and R. A. Rimmer, “Modes of Elliptical Waveguides: A Correction,” IEEE Trans. Microw. Theory Techn. 38(11), 1603–1608 (1990).
[Crossref]

1989 (2)

Y.-H. Wang and C. Vassallo, “Circular Fourier analysis of arbitrarily shaped optical fibers,” Opt. Lett. 14(24), 1377–1379 (1989).
[Crossref] [PubMed]

J. A. M. Svedin, “A Numerically Efficient Finite-Element Formulation for the General Waveguide Problem Without Spurious Modes,” IEEE Trans. Microw. Theory Techn. 37(11), 1708–1715 (1989).
[Crossref]

1986 (2)

K. Bierwirth, N. Shulz, and F. Arndt, “Finite-Difference Analysis of Rectangular Dielectric Waveguide Structures,” IEEE Trans. Microw. Theory Techn. 34(11) 1104–1114 (1986).
[Crossref]

J.-R. Qian and W.-P. Huang, “Coupled-mode theory for LP modes,” J. Lightwave Technol. 4(6), 619–625 (1986).
[Crossref]

1984 (1)

E. Schweig and W.B. Bridges, “Computer Analysis of Dielectric Waveguides: A Finite-Difference Method,” IEEE Trans. Microw. Theory Techn. 32(5), 531–541 (1984).
[Crossref]

1973 (1)

A. Yariv, “Coupled-mode theory for guided-wave optics,” IEEE J. Quant. Electron. 9(9), 919–933 (1973).
[Crossref]

1969 (1)

J. E. Goell, “A Circular-Harmonic Computer Analysis Of Rectangular Dielectric Waveguide,” Bell Syst. Tech. J. 48(7), 2133–2160 (1969).
[Crossref]

1962 (1)

C. Yeh, “Elliptical Dielectric Waveguides,” J. Appl. Phys. 33(11), 3235–3243 (1962).
[Crossref]

1896 (1)

A. Sommerfeld, “Mathematische theorie der diffraction,” Math. Ann. 47(2), 317–374 (1896).
[Crossref]

Agio, M.

Amari, S.

S. Amari and J. Bornemann, “Efficient numerical computation of singular integrals with applications to electromagnetics,” IEEE Trans. Antennas Propag. 43(11), 1343–1348 (1995).
[Crossref]

Andres, M. V.

Andres, P.

Andrés, M. V.

Andrés, P.

Arndt, F.

K. Bierwirth, N. Shulz, and F. Arndt, “Finite-Difference Analysis of Rectangular Dielectric Waveguide Structures,” IEEE Trans. Microw. Theory Techn. 34(11) 1104–1114 (1986).
[Crossref]

Atwater, H. A.

S. A. Maier and H. A. Atwater, “Plasmonics: Localization and guiding of electromagnetic energy in metal/dielectric structures,” J. Appl. Phys. 98(011101), 1–9 (2005).
[Crossref]

Azizur Rahman, B. M.

Bermel, P.

Bierwirth, K.

K. Bierwirth, N. Shulz, and F. Arndt, “Finite-Difference Analysis of Rectangular Dielectric Waveguide Structures,” IEEE Trans. Microw. Theory Techn. 34(11) 1104–1114 (1986).
[Crossref]

Bornemann, J.

S. Amari and J. Bornemann, “Efficient numerical computation of singular integrals with applications to electromagnetics,” IEEE Trans. Antennas Propag. 43(11), 1343–1348 (1995).
[Crossref]

Bridges, W.B.

E. Schweig and W.B. Bridges, “Computer Analysis of Dielectric Waveguides: A Finite-Difference Method,” IEEE Trans. Microw. Theory Techn. 32(5), 531–541 (1984).
[Crossref]

Brown, T. G.

Z. Zhu and T. G. Brown, “Multipole analysis of hole-assisted optical fibers,” Opt. Commun. 206(4), 333–339 (2002).
[Crossref]

Z. Zhu and T. G. Brown, “Full-vectorial finite-difference analysis of microstructured optical fibers,” Opt. Express 10(17), 853–864 (2002).
[Crossref] [PubMed]

Burr, G. W.

Chang, H.-C.

Chiang, K. S.

K. S. Chiang, “Review of numerical and approximate methods for the modal analysis of general optical dielectric waveguides,” Opt. Quant. Electron. 26(3), S113–S134 (1994).
[Crossref]

Chiang, Y.-C.

Chiou, Y.-P.

Colas des Francs, G.

G. Colas des Francs, J.-P. Hugonin, and J. Čtyroký, “Mode solvers for very thin long–range plasmonic waveguides,” Opt. Quant. Electron. 42(8), 557–570 (2011).
[Crossref]

Ctyroký, J.

G. Colas des Francs, J.-P. Hugonin, and J. Čtyroký, “Mode solvers for very thin long–range plasmonic waveguides,” Opt. Quant. Electron. 42(8), 557–570 (2011).
[Crossref]

Cucinotta, A.

A. Cucinotta, S. Selleri, L. Vincetti, and M. Zoboli, “Holey fiber analysis through the finite-element method,” IEEE Photon. Technol. Lett. 14(11), 1530–1532 (2002).
[Crossref]

S. Selleri, L. Vincetti, A. Cucinotta, and M. Zoboli, “Complex FEM modal solver of optical waveguides with PML boundary conditions,” Opt. Quant. Electron. 33(4–5), 359–371 (2001).
[Crossref]

El-Mikati, H. A.

Fallahkhair, A. B.

A. B. Fallahkhair, K. S. Li, and T. E. Murphy, “Vector Finite Difference Modesolver for Anisotropic Dielectric Waveguides,” J. Lightw. Technol. 26(11), 1423–1431 (2008).
[Crossref]

Farjadpour, A.

Ferrando, A.

Gallawa, R. L.

A. Weisshaar, J. Li, R. L. Gallawa, and I. C. Goyal, “Vector and quasi-vector solutions for optical waveguide modes using efficient Galerkin’s method with Hermite-Gauss basis functions,”J. Lightwave Tech. 13(8), 1795–1800 (1995).
[Crossref]

Gaylord, T. K.

Goell, J. E.

J. E. Goell, “A Circular-Harmonic Computer Analysis Of Rectangular Dielectric Waveguide,” Bell Syst. Tech. J. 48(7), 2133–2160 (1969).
[Crossref]

Goldberg, D. A.

D. A. Goldberg, L. J. Laslett, and R. A. Rimmer, “Modes of Elliptical Waveguides: A Correction,” IEEE Trans. Microw. Theory Techn. 38(11), 1603–1608 (1990).
[Crossref]

Gordon, R.

Goyal, I. C.

A. Weisshaar, J. Li, R. L. Gallawa, and I. C. Goyal, “Vector and quasi-vector solutions for optical waveguide modes using efficient Galerkin’s method with Hermite-Gauss basis functions,”J. Lightwave Tech. 13(8), 1795–1800 (1995).
[Crossref]

Grann, E. B.

Grattan, K. T. V.

Helmy, A. S.

Huang, W.-P.

Hugonin, J. P.

J. P. Hugonin, P. Lalanne, I. D. Villar, and I. R. Matias, “Fourier modal methods for modeling optical dielectric waveguides,” Opt. Quant. Electron. 37(1–3), 107–119 (2005).
[Crossref]

Hugonin, J.-P.

G. Colas des Francs, J.-P. Hugonin, and J. Čtyroký, “Mode solvers for very thin long–range plasmonic waveguides,” Opt. Quant. Electron. 42(8), 557–570 (2011).
[Crossref]

Ibanescu, M.

Jian, S.-S.

Joannopoulos, J. D.

Johnson, S. G.

Keiser, G.

G. Keiser, Optical Fiber Communications, 3rd ed. (McGraw-Hill, 2000).

Khavasi, A.

Lalanne, P.

J. P. Hugonin, P. Lalanne, I. D. Villar, and I. R. Matias, “Fourier modal methods for modeling optical dielectric waveguides,” Opt. Quant. Electron. 37(1–3), 107–119 (2005).
[Crossref]

P. Lalanne and E. Silberstein, “Fourier-modal methods applied to waveguide computational problems,” Opt. Lett. 25(15), 1092–1094 (2000).
[Crossref]

Laslett, L. J.

D. A. Goldberg, L. J. Laslett, and R. A. Rimmer, “Modes of Elliptical Waveguides: A Correction,” IEEE Trans. Microw. Theory Techn. 38(11), 1603–1608 (1990).
[Crossref]

Li, J.

A. Weisshaar, J. Li, R. L. Gallawa, and I. C. Goyal, “Vector and quasi-vector solutions for optical waveguide modes using efficient Galerkin’s method with Hermite-Gauss basis functions,”J. Lightwave Tech. 13(8), 1795–1800 (1995).
[Crossref]

Li, K. S.

A. B. Fallahkhair, K. S. Li, and T. E. Murphy, “Vector Finite Difference Modesolver for Anisotropic Dielectric Waveguides,” J. Lightw. Technol. 26(11), 1423–1431 (2008).
[Crossref]

Lu, Y.-C.

Maier, S. A.

S. A. Maier and H. A. Atwater, “Plasmonics: Localization and guiding of electromagnetic energy in metal/dielectric structures,” J. Appl. Phys. 98(011101), 1–9 (2005).
[Crossref]

Matias, I. R.

J. P. Hugonin, P. Lalanne, I. D. Villar, and I. R. Matias, “Fourier modal methods for modeling optical dielectric waveguides,” Opt. Quant. Electron. 37(1–3), 107–119 (2005).
[Crossref]

Mehrany, K.

Miret, J. J.

Mittra, R.

W. Yu and R. Mittra, “A conformal finite difference time domain technique for modeling curved dielectric surfaces,” IEEE Microw. Wirel. Compon. Lett. 11(1), 25–27 (2001).
[Crossref]

Mohammadi, A.

Moharam, M. G.

Mu, J.

Murphy, T. E.

A. B. Fallahkhair, K. S. Li, and T. E. Murphy, “Vector Finite Difference Modesolver for Anisotropic Dielectric Waveguides,” J. Lightw. Technol. 26(11), 1423–1431 (2008).
[Crossref]

Nadgaran, H.

Nolting, H. P.

G. Sztefka and H. P. Nolting, “Bidirectional eigenmode propagation for large refractive index steps,” IEEE Photon. Technol. Lett. 5(5), 554–557 (1993).
[Crossref]

Obayya, S. S. A.

Pommet, D. A.

Pozar, D. M.

D. M. Pozar, Microwave Engineering, 4th ed. (John Wiley & Sons, 2012).

Qian, J.-R.

J.-R. Qian and W.-P. Huang, “Coupled-mode theory for LP modes,” J. Lightwave Technol. 4(6), 619–625 (1986).
[Crossref]

Rahman, B. M. A.

Rashidian, B.

Rimmer, R. A.

D. A. Goldberg, L. J. Laslett, and R. A. Rimmer, “Modes of Elliptical Waveguides: A Correction,” IEEE Trans. Microw. Theory Techn. 38(11), 1603–1608 (1990).
[Crossref]

Rodriguez, A.

Roundy, D.

Russell, P.

P. Russell, “Photonic-Crystal Fibers,” J. Lightwave Technol. 24(12), 4729–4749 (2006).
[Crossref]

P. Russell, “Photonic Crystal Fibers,” Science 299(5605), 358–362 (2003).
[Crossref] [PubMed]

Schweig, E.

E. Schweig and W.B. Bridges, “Computer Analysis of Dielectric Waveguides: A Finite-Difference Method,” IEEE Trans. Microw. Theory Techn. 32(5), 531–541 (1984).
[Crossref]

Selleri, S.

A. Cucinotta, S. Selleri, L. Vincetti, and M. Zoboli, “Holey fiber analysis through the finite-element method,” IEEE Photon. Technol. Lett. 14(11), 1530–1532 (2002).
[Crossref]

S. Selleri, L. Vincetti, A. Cucinotta, and M. Zoboli, “Complex FEM modal solver of optical waveguides with PML boundary conditions,” Opt. Quant. Electron. 33(4–5), 359–371 (2001).
[Crossref]

Shulz, N.

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Figures (7)

Fig. 1
Fig. 1

Refractive maps n(ρ) (in (a)–(b)) and n(x, y) (in (c)) of three dielectric waveguides used to illustrate the coupled mode theory (CCMT) eigenmode solver. Each figure gives the refractive index in the transverse plane. The specified nf is the “fiber” index in the black region; the white background has an index of 1.0.

Fig. 2
Fig. 2

Electric field intensity of the eigenmodes of a step-index fiber (given in Fig. 1(a)) calculated using CCMT. The wavelength is λ = 1.5 μm, and the basis set size is N = 600. The corresponding effective indexes neff obtained using FDE are 1.5945 and 1.5750; using analytic expressions, they are 1.5947 and 1.5760.

Fig. 3
Fig. 3

Convergence and timings for the step-index fiber (given in Fig. 1(a)). In (a), the dependence of the effective index of the fundamental mode, neff, on the number of basis modes, N, is given. In (b) the times spent computing the eigenmode solutions in (a) are shown, as a function of N. The sample size for each data point is ten. The reason for the kink at N = 300 is discussed in the text.

Fig. 4
Fig. 4

Electric field intensity of the eigenmodes of a dielectric shell waveguide (given in Fig. 1(b)) calculated using CCMT. The wavelength is λ = 2.5 μm, and the basis set size is N = 1300. The corresponding effective indexes neff obtained using FDE (with a conformal mesh) are 1.4473+0.1970i, 1.4447+0.1972i and 1.4447+0.1972i.

Fig. 5
Fig. 5

Electric field intensity of the eigenmodes of a microstructured dielectric waveguide (given in Fig. 1(c)) calculated using CCMT. The wavelength is λ = 800 nm, and the basis set size is N = 1200. The corresponding effective indexes neff obtained using FDE are 1.4940, 1.4749 and 1.4621.

Fig. 6
Fig. 6

Electric field intensity of the Sommerfeld mode in a gold nanowire 6 nm in radius, calculated using CCMT. The wavelength is λ = 650 nm, and the basis set size is N = 550 (this is a modified basis set; see the text for details). The effective index neff calculated using CCMT, using the parameters given here and in the text, is 5.93+0.35i. For comparison, the effective index neff found using FDE (using a conformal mesh) is 5.92+0.33i and the analytic result is 5.81+0.34i.

Fig. 7
Fig. 7

A 1D slab waveguide system illustrating the use of a perfectly matched layer (PML) in the CCMT mode solver. In (a) an example structure is given; the details are given in the text. In (b)–(d) we see the field profile for the fundamental guided mode, the lowest order quasi-leaky mode, and the lowest order PML mode as solved using CCMT. The interfaces between the core, inner cladding, substrate and PML regions are marked with vertical lines.

Tables (4)

Tables Icon

Table 1 Computed effective indexes for the step-index fiber modes shown in Fig. 2.

Tables Icon

Table 2 Computed effective indexes for the lossy dielectric shell fiber modes shown in Fig. 4.

Tables Icon

Table 3 Computed effective indexes for the microstructured optical fiber modes shown in Fig. 5.

Tables Icon

Table 4 Computed effective indexes for the plasmonic nanowire shown in Fig. 6.

Equations (23)

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E t ( x , y , z ) = ( a k ( z ) + b k ( z ) ) e k t ( x , y )
H t ( x , y , z ) = ( a k ( z ) b k ( z ) ) h k t ( x , y ) .
E z ( x , y , z ) = ( a k ( z ) b k ( z ) ) ˜ e k z ( x , y )
H z ( x , y , z ) = ( a k ( z ) + b k ( z ) ) h k z ( x , y ) .
N j ( a j ( z ) z + i β j a j ( z ) ) = i κ j k a k ( z ) i χ j k b k ( z )
N j ( b j ( z ) z i β j b j ( z ) ) = i κ j k a k ( z ) + i χ j k b k ( z ) .
κ j k = ω 2 d A Δ ( e j t e k t ˜ e j z e k z )
χ j k = ω 2 d A Δ ( e j t e k t + ˜ e j z e k z )
N j = d A z ^ ( e j t × h j t ) ,
a k ( z ) = a k e i λ z and b k ( z ) = b k e i λ z .
( β + κ N χ N χ N β κ N ) ( a b ) = λ ( a b ) .
I t = ( x , y ) | k ( a k ( z ) + b k ( z ) ) e k t ( x , y ) | 2 + | k ( a k ( z ) b k ( z ) ) ˜ e k z ( x , y ) | 2 .
e ρ = i ω μ n k c 2 ρ ( A cos n ϕ B sin n ϕ ) J n ( k c ρ ) , e ϕ = i ω μ k c ( A sin n ϕ B cos n ϕ ) J n ( k c ρ ) , h ρ = i β k c ( A sin n ϕ + B cos n ϕ ) J n ( k c ρ ) , h ϕ = i β n k c 2 ρ ( A cos n ϕ B sin n ϕ ) J n ( k c ρ ) , e z = 0 , and , h z = ( A sin n ϕ B cos n ϕ ) J n ( k c ρ ) , with k c = p n m a .
e ρ = i β k c ( A sin n ϕ + B cos n ϕ ) J n ( k c ρ ) , e ϕ = i β n k c 2 ρ ( A cos n ϕ B sin n ϕ ) J n ( k c ρ ) , h ρ = i ω n k c 2 ρ ( A cos n ϕ B sin n ϕ ) J n ( k c ρ ) , h ϕ = i ω k c ( A sin n ϕ + B cos n ϕ ) J n ( k c ρ ) , e z = ( A sin n ϕ + B cos n ϕ ) J n ( k c ρ ) , and , H z = 0 , with k c = p n m a .
β = k 2 k c 2
J n ( ξ ) = 1 2 ( J n 1 ( ξ ) J n + 1 ( ξ ) ) = J n + 1 ( ξ ) + n ξ J n ( ξ ) .
d A Δ ( ρ ) ( e j t e k t ˜ e j z e k z ) d A Δ ( ρ ) I .
( e x e y ) = ( cos ϕ sin ϕ sin ϕ cos ϕ ) ( e ρ e ϕ ) .
d A Δ ( ρ ) I i = d ρ ρ Δ ( ρ ) R i ( ρ ) d ϕ Φ i ( ϕ ) .
e ρ = i ( A β u J ν ( u ρ ) + B μ ω u 2 ν ρ J ν ( u ρ ) ) sin ν ϕ , e ϕ = i ( A β u 2 ν ρ J ν ( u ρ ) + B μ ω u J ν ( u ρ ) ) cos ν ϕ .
e ρ = i ( C β w K ν ( w ρ ) D μ ω w 2 ν ρ K ν ( w ρ ) ) sin ν ϕ , e ϕ = i ( C β w 2 ν ρ K ν ( u r ) D μ ω w K ν ( w ρ ) ) cos ν ϕ .
( J ν ( u b ) u J ν ( u b ) + K ν ( u b ) w K ν ( u b ) ) ( k 1 2 J ν ( u b ) u J ν ( u b ) + k 2 2 K ν ( w b ) w K ν ( w b ) ) = ( β ν b ) 2 ( 1 u 2 + 1 w 2 ) 2 .
K 0 ( p d b ) I 1 ( p m b ) K 1 ( p d b ) I 0 ( p m b ) = d p m m p d .