Abstract

Wave guidance is an important aspect of light trapping in thin film photovoltaics making it important to properly model the effects of loss on the field profiles. This paper derives the full-field solution for electromagnetic wave propagation in a symmetric dielectric slab with finite absorption. The functional form of the eigenvalue equation is identical to the lossless case except the propagation constants take on complex values. Additional loss-guidance and anti-guidance modes appear in the lossy model which do not normally exist in the analogous lossless case. An approximate solution for the longitudinal attenuation coefficient αz is derived from geometric optics and shows excellent agreement with the exact value. Lossy mode propagation is then explored in the context of photovoltaics by modeling a thin film solar cell made of amorphous silicon.

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References

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  1. C. A. Balanis, Advanced Engineering Electromagnetics (Wiley, New York, NY, 1989).
  2. J. A. Kong, Electromagnetic Wave Theory (EMW Publishing, Cambridge, MA, 2000).
  3. G. H. Owyang, Foundations of Optical Waveguides (Elsevier, New York, NY, 1981).
  4. C. R. Pollock and M. Lipson, Integrated Photonics (Kluwer Academic Publishers, Boston, MA, 2003).
  5. A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, New York, NY, 1983).
  6. S. Miyanaga and H. Fujiwara, “Effects of absorption on the propagation constants of guided modes in an asymmetric slab optical waveguide,” Opt. Commun. 64, 31–35 (1987).
    [CrossRef]
  7. K. Fujii, “Dispersion relations of TE-mode in a 3-layer slab waveguide with imaginary-part of refractive index,” Opt. Commun. 171, 245–251 (1999).
    [CrossRef]
  8. T. D. Visser, H. Blok, and D. Lenstra, “Modal analysis of a planar waveguide with gain and losses,” IEEE J. Quantum Electron. 31, 1803–1810 (1995).
    [CrossRef]
  9. A. E. Siegman, “Propagating modes in gain-guided optical fibers,” J. Opt. Soc. Am. A 20, 1617–1628 (2003).
    [CrossRef]
  10. A. W. Snyder and J. D. Love, “Goos-Hanchen shift,” Appl. Opt. 15, 236–238 (1973).
    [CrossRef]
  11. Lumerical Solutions Inc., http://www.lumerical.com/ .
  12. T. D. Visser, H. Blok, and D. Lenstra, “Confinement factors and gain in optical amplifiers,” IEEE J. Quantum Electron. 33, 1763–1766 (1997).
    [CrossRef]
  13. G. K. M. Thutupalli and S. G. Tomlin, “The optical properties of amorphous and crystalline silicon,” J. Phys. C Solid State 10, 467–477 (1977).
    [CrossRef]
  14. Z. Yu, A. Raman, and S. Fan, “Fundamental limit of nanophotonic light trapping in solar cells,” Proc. Natl. Acad. Sci. U.S.A. 107, 17491–17496 (2010).
    [CrossRef] [PubMed]
  15. J. A. Snyman, Practical Mathematical Optimization: An Introduction to Basic Optimization Theory and Classical and New Gradient-Based Algorithms (Applied Optimization) (Springer, 2005).
    [PubMed]

2010 (1)

Z. Yu, A. Raman, and S. Fan, “Fundamental limit of nanophotonic light trapping in solar cells,” Proc. Natl. Acad. Sci. U.S.A. 107, 17491–17496 (2010).
[CrossRef] [PubMed]

2003 (1)

1999 (1)

K. Fujii, “Dispersion relations of TE-mode in a 3-layer slab waveguide with imaginary-part of refractive index,” Opt. Commun. 171, 245–251 (1999).
[CrossRef]

1997 (1)

T. D. Visser, H. Blok, and D. Lenstra, “Confinement factors and gain in optical amplifiers,” IEEE J. Quantum Electron. 33, 1763–1766 (1997).
[CrossRef]

1995 (1)

T. D. Visser, H. Blok, and D. Lenstra, “Modal analysis of a planar waveguide with gain and losses,” IEEE J. Quantum Electron. 31, 1803–1810 (1995).
[CrossRef]

1987 (1)

S. Miyanaga and H. Fujiwara, “Effects of absorption on the propagation constants of guided modes in an asymmetric slab optical waveguide,” Opt. Commun. 64, 31–35 (1987).
[CrossRef]

1977 (1)

G. K. M. Thutupalli and S. G. Tomlin, “The optical properties of amorphous and crystalline silicon,” J. Phys. C Solid State 10, 467–477 (1977).
[CrossRef]

1973 (1)

Balanis, C. A.

C. A. Balanis, Advanced Engineering Electromagnetics (Wiley, New York, NY, 1989).

Blok, H.

T. D. Visser, H. Blok, and D. Lenstra, “Confinement factors and gain in optical amplifiers,” IEEE J. Quantum Electron. 33, 1763–1766 (1997).
[CrossRef]

T. D. Visser, H. Blok, and D. Lenstra, “Modal analysis of a planar waveguide with gain and losses,” IEEE J. Quantum Electron. 31, 1803–1810 (1995).
[CrossRef]

Fan, S.

Z. Yu, A. Raman, and S. Fan, “Fundamental limit of nanophotonic light trapping in solar cells,” Proc. Natl. Acad. Sci. U.S.A. 107, 17491–17496 (2010).
[CrossRef] [PubMed]

Fujii, K.

K. Fujii, “Dispersion relations of TE-mode in a 3-layer slab waveguide with imaginary-part of refractive index,” Opt. Commun. 171, 245–251 (1999).
[CrossRef]

Fujiwara, H.

S. Miyanaga and H. Fujiwara, “Effects of absorption on the propagation constants of guided modes in an asymmetric slab optical waveguide,” Opt. Commun. 64, 31–35 (1987).
[CrossRef]

Kong, J. A.

J. A. Kong, Electromagnetic Wave Theory (EMW Publishing, Cambridge, MA, 2000).

Lenstra, D.

T. D. Visser, H. Blok, and D. Lenstra, “Confinement factors and gain in optical amplifiers,” IEEE J. Quantum Electron. 33, 1763–1766 (1997).
[CrossRef]

T. D. Visser, H. Blok, and D. Lenstra, “Modal analysis of a planar waveguide with gain and losses,” IEEE J. Quantum Electron. 31, 1803–1810 (1995).
[CrossRef]

Lipson, M.

C. R. Pollock and M. Lipson, Integrated Photonics (Kluwer Academic Publishers, Boston, MA, 2003).

Love, J. D.

A. W. Snyder and J. D. Love, “Goos-Hanchen shift,” Appl. Opt. 15, 236–238 (1973).
[CrossRef]

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, New York, NY, 1983).

Miyanaga, S.

S. Miyanaga and H. Fujiwara, “Effects of absorption on the propagation constants of guided modes in an asymmetric slab optical waveguide,” Opt. Commun. 64, 31–35 (1987).
[CrossRef]

Owyang, G. H.

G. H. Owyang, Foundations of Optical Waveguides (Elsevier, New York, NY, 1981).

Pollock, C. R.

C. R. Pollock and M. Lipson, Integrated Photonics (Kluwer Academic Publishers, Boston, MA, 2003).

Raman, A.

Z. Yu, A. Raman, and S. Fan, “Fundamental limit of nanophotonic light trapping in solar cells,” Proc. Natl. Acad. Sci. U.S.A. 107, 17491–17496 (2010).
[CrossRef] [PubMed]

Siegman, A. E.

Snyder, A. W.

A. W. Snyder and J. D. Love, “Goos-Hanchen shift,” Appl. Opt. 15, 236–238 (1973).
[CrossRef]

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, New York, NY, 1983).

Snyman, J. A.

J. A. Snyman, Practical Mathematical Optimization: An Introduction to Basic Optimization Theory and Classical and New Gradient-Based Algorithms (Applied Optimization) (Springer, 2005).
[PubMed]

Thutupalli, G. K. M.

G. K. M. Thutupalli and S. G. Tomlin, “The optical properties of amorphous and crystalline silicon,” J. Phys. C Solid State 10, 467–477 (1977).
[CrossRef]

Tomlin, S. G.

G. K. M. Thutupalli and S. G. Tomlin, “The optical properties of amorphous and crystalline silicon,” J. Phys. C Solid State 10, 467–477 (1977).
[CrossRef]

Visser, T. D.

T. D. Visser, H. Blok, and D. Lenstra, “Confinement factors and gain in optical amplifiers,” IEEE J. Quantum Electron. 33, 1763–1766 (1997).
[CrossRef]

T. D. Visser, H. Blok, and D. Lenstra, “Modal analysis of a planar waveguide with gain and losses,” IEEE J. Quantum Electron. 31, 1803–1810 (1995).
[CrossRef]

Yu, Z.

Z. Yu, A. Raman, and S. Fan, “Fundamental limit of nanophotonic light trapping in solar cells,” Proc. Natl. Acad. Sci. U.S.A. 107, 17491–17496 (2010).
[CrossRef] [PubMed]

Appl. Opt. (1)

IEEE J. Quantum Electron. (2)

T. D. Visser, H. Blok, and D. Lenstra, “Confinement factors and gain in optical amplifiers,” IEEE J. Quantum Electron. 33, 1763–1766 (1997).
[CrossRef]

T. D. Visser, H. Blok, and D. Lenstra, “Modal analysis of a planar waveguide with gain and losses,” IEEE J. Quantum Electron. 31, 1803–1810 (1995).
[CrossRef]

J. Opt. Soc. Am. A (1)

J. Phys. C Solid State (1)

G. K. M. Thutupalli and S. G. Tomlin, “The optical properties of amorphous and crystalline silicon,” J. Phys. C Solid State 10, 467–477 (1977).
[CrossRef]

Opt. Commun. (2)

S. Miyanaga and H. Fujiwara, “Effects of absorption on the propagation constants of guided modes in an asymmetric slab optical waveguide,” Opt. Commun. 64, 31–35 (1987).
[CrossRef]

K. Fujii, “Dispersion relations of TE-mode in a 3-layer slab waveguide with imaginary-part of refractive index,” Opt. Commun. 171, 245–251 (1999).
[CrossRef]

Proc. Natl. Acad. Sci. U.S.A. (1)

Z. Yu, A. Raman, and S. Fan, “Fundamental limit of nanophotonic light trapping in solar cells,” Proc. Natl. Acad. Sci. U.S.A. 107, 17491–17496 (2010).
[CrossRef] [PubMed]

Other (7)

J. A. Snyman, Practical Mathematical Optimization: An Introduction to Basic Optimization Theory and Classical and New Gradient-Based Algorithms (Applied Optimization) (Springer, 2005).
[PubMed]

Lumerical Solutions Inc., http://www.lumerical.com/ .

C. A. Balanis, Advanced Engineering Electromagnetics (Wiley, New York, NY, 1989).

J. A. Kong, Electromagnetic Wave Theory (EMW Publishing, Cambridge, MA, 2000).

G. H. Owyang, Foundations of Optical Waveguides (Elsevier, New York, NY, 1981).

C. R. Pollock and M. Lipson, Integrated Photonics (Kluwer Academic Publishers, Boston, MA, 2003).

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, New York, NY, 1983).

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Figures (10)

Fig. 1
Fig. 1

A dielectric slab with thickness 2h is surrounded by two cladding layers. The slab is a lossy material with complex index of refraction nf = n + . The cladding layers are both lossless and symmetric with real-valued index nc.

Fig. 2
Fig. 2

(a) Logarithmic power of the misfit function, 10 log10 ϕ, for the even modes. Mode solutions manifest as zeros in ϕ (dark blue regions) with the M = 0 and M = 2 modes indicated. Model parameters are nf = 2 + j0.5, nc = 1.5 and h/λ0 = 0.5. The “X” mark indicates the initial trial solution (kx,0λ0 = 7.27). The “O” mark indicates the lossy solution for the M = 2 mode (kxλ0 = 7.89 + j0.77). (b) Normalized electric field profile along both x and z (V/m). Horizontal bars indicate the waveguide boundaries.

Fig. 3
Fig. 3

(a) Electric field profiles (normalized to unit amplitude) for the first four modes of the lossy waveguide example. (b) M = 2 profile under increasing values of κ. Vertical bars indicate the waveguide boundaries.

Fig. 4
Fig. 4

Ray diagram of the Goos-Hanchen effect for a lossy film. The ray R1 is more intense than the ray R2, leading to a net flow of power into the film at any given point along z. Consequently, the time-averaged Poynting’s vector S in the cladding region has an x-component that points toward the film.

Fig. 5
Fig. 5

(a) Positive and (b) negative solutions to the misfit function with respect to the complex radical in Equation (4). The negative misfit reveals a new set of solutions to the eigenvalue equation, though such solutions are not physically admissible.

Fig. 6
Fig. 6

(a) Log-power of the even misfit using nf = 2.0 + j0.1, nc = 2.25, and h/λ0 = 0.5, demonstrating an anti-guidance mode at kxλ0 = 2.8 + j0.77. (b) Normalized field profile of anti-guidance mode. The “skewing” effect on the evanescent fields is much more dramatic in these modes.

Fig. 7
Fig. 7

(a) Longitudinal attenuation coefficient (αz) versus mode number using nf = 2.5 + j0.01, nc = 1.5, h = 1.5 μm and λ0 = 1.0 μm. Exact computations (black) are compared against numerical simulation (red) and the low-loss approximation (blue). (b) Ray propagation in the film according to geometric optics. High-order modes (black) experience greater path-length than low-order modes (red) for a given displacement along z. Some fraction of the overall path-length is also spent within the cladding region due to the evanescent field propagation. Since this region is lossless, it does not contribute to the ray attenuation.

Fig. 8
Fig. 8

Configuration for a lossy dielectric waveguide backed by a PEC ground plane. Solutions are equivalent to the odd modes of a symmetric dielectric slab with twice the width.

Fig. 9
Fig. 9

(a) Electric field profile of the M = 4 mode for thin (h = 500 nm) film of amorphous silicon at λ0 = 600 nm. Indices are given by nf = 4.6 + j0.3 and nc = 1. (b) Longitudinal attenuation coefficient versus mode number. The low-loss approximation is applied to the first seven modes, but does not exist for the extra five in the lossy model.

Fig. 10
Fig. 10

(a) Electric field profile of the M = 7 mode from the amorphous silicon model. (b) Full 2D profile, showing dramatic longitudinal absorption of the loss-guided mode.

Equations (35)

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2 E ( x , z ) + k 2 E ( x , z ) = 0 .
E ( x , z ) = y ^ E 0 ( e + j β x x e α x x ± e j β x x e + α x x ) e + j β z e α z z .
E ( x , z ) = y ^ E 0 ( e + j k x x ± e j k x x ) e + j k z .
k x tan ( k x h ) = k f 2 k c 2 k x 2 ( TE , Even ) ,
k x cot ( k x h ) = k f 2 k c 2 k x 2 ( TE , Odd ) .
k x tan ( k x h ) = n f 2 n c 2 k f 2 k c 2 k x 2 ( TM , Even ) ,
k x cot ( k x h ) = n f 2 n c 2 k f 2 k c 2 k x 2 ( TM , Odd ) .
f e ( k x ) = k x tan ( k x h ) [ k f 2 k c 2 k x 2 ] 1 / 2 ,
f o ( k x ) = k x cot ( k x h ) [ k f 2 k c 2 k x 2 ] 1 / 2 .
ϕ ( k x ) = f ( k x ) f * ( k x ) = f ( k x ) 2 .
k z λ 0 = 9.89 + j 3.38 , γ λ 0 = 5.89 + j 5.67.
α x = k 0 2 n κ β x .
k x , BP = k f 2 k c 2 .
α z α 0 Γ sec θ ,
Γ = h + h | m ( x ) | 2 dx + | m ( x ) | 2 dx ,
E ( x , z ) = y ^ E 0 ( e + j k x x ± e j k x x ) e + j k z .
E e ( x , z ) = y ^ E 0 cos ( k x x ) e + j k z ( even )
E o ( x , z ) = y ^ E 0 sin ( k x x ) e + j k z ( odd ) .
k x 2 + k z 2 = k f 2 .
E ( x , z ) = y ^ C E 0 e + j k z z { e + j γ ( x h ) ( x > h ) ± e j γ ( x + h ) ( x < h ) .
γ 2 = k z 2 = k c 2 ,
E e ( x , z ) = y ^ E 0 e + j k z z { C e + j γ ( x h ) ( x > h ) cos ( k x x ) ( | x | h ) C e j γ ( x + h ) ( x < h ) ,
H e ( x , z ) = E 0 ω μ 0 e + j k z z { C ( γ z ^ k z x ^ ) e + j γ ( x h ) ( x > h ) j k x z ^ sin ( k x x ) k z x ^ cos ( k x x ) ( | x | h ) C ( γ z ^ k z x ^ ) e j γ ( x + h ) ( x < h ) ,
cos ( k x h ) = C .
j k x sin ( k x h ) = C γ .
j k x tan ( k x x ) = γ .
γ = j k f 2 k c 2 k x 2 .
f e ( k x ) = f k x = k x h sec 2 ( k x h ) + tan ( k x h ) + k x [ k f 2 k c 2 k x 2 ] 1 / 2 ,
f n = f ( k x , n )
F n = f ( k x , n )
n = F n * f n
g n = F n n
u n = n 2 g n 2
k x , n + 1 = k x , n u n n ,
ϕ ( k x , n + 1 ) < ϕ ( k x , n ) .

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