Abstract

Light-trapping techniques can be used to improve the efficiency of thin silicon solar cells. We report on numerical investigation of a light trapping design consisting of a 2D back-side diffraction grating in combination with an aluminum mirror and a spacing layer of low permittivity to minimize parasitic absorption in the aluminum. The light-trapping design was compared to a planar reference design with antireflection coating and back-side aluminum mirror. Both normally and obliquely incident light was investigated. For normal incidence, the light trapping structure increases the short circuit current density with 17% from 30.4 mA/cm2 to 35.5 mA/cm2 for a 20 µm thick silicon solar cell. Our design also increases the current density in thinner cells, and yields higher current density than two recently published designs for cell thickness of 2 and 5 µm, respectively. The increase in current may be attributed to two factors; increased path length due to in-coupling of light, and decreased parasitic absorption in the aluminum due to the spacing layer.

© 2010 OSA

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2009 (3)

H. Sai, H. Fujiwara, and M. Kondo, “Back surface reflectors with periodic textures fabricated by self-ordering process for light trapping in thin-film microcrystalline silicon solar cells,” Sol. Energy Mater. Sol. Cells 93(6-7), 1087–1090 (2009).
[CrossRef]

R. Dewan and D. Knipp, “Light-trapping in thin-film silicon solar cells with integrated diffraction grating,” J. Appl. Phys. 106(7), 074901 (2009).
[CrossRef]

F.-J. Haug, T. Söderström, M. Python, V. Terrazzoni-Daudrix, X. Niquille, and C. Ballif, “Development of micromorph tandem solar cells on flexible low-cost plastic substrates,” Sol. Energy Mater. Sol. Cells 93(6-7), 884–887 (2009).
[CrossRef]

2008 (3)

J. G. Mutitu, S. Shi, C. Chen, T. Creazzo, A. Barnett, C. Honsberg, and D. W. Prather, “Thin film solar cell design based on photonic crystal and diffractive grating structures,” Opt. Express 16(19), 15238–15248 (2008).
[CrossRef] [PubMed]

A. Čampa, J. Krč, F. Smole, and M. Topič, “Potential of diffraction gratings for implementation as a metal back reflector in thin-film silicon solar cells,” Thin Solid Films 516(20), 6963–6967 (2008).
[CrossRef]

H. Sai, H. Fujiwara, M. Kondo, and Y. Kanamori, “Enhancement of light trapping in thin-film hydrogenated microcrystalline Si solar cells using back reflectors with self-ordered dimple pattern,” Appl. Phys. Lett. 93(14), 143501 (2008).
[CrossRef]

2007 (4)

H. Sai, Y. Kanamori, K. Arafune, Y. Ohshita, and M. Yamaguchi, “Light trapping effect of submicron surface textures in crystalline Si solar cells,” Prog. Photovoltaics 15(5), 415–423 (2007).
[CrossRef]

N.-N. Feng, J. Michel, L. Zeng, J. Liu, C.-Y. Hong, L. C. Kimerling, and X. Duan, “Design of Highly Efficient Light-Trapping Structures of Thin-Film Crystalline Silicon Solar Cells,” IEEE Trans. Electron. Dev. 54(8), 1926–1933 (2007).
[CrossRef]

P. Bermel, C. Luo, L. Zeng, L. C. Kimerling, and J. D. Joannopoulos, “Improving thin-film crystalline silicon solar cell efficiencies with photonic crystals,” Opt. Express 15(25), 16986–17000 (2007).
[CrossRef] [PubMed]

C. Haase and H. Stiebig, “Thin-film silicon solar cells with efficient periodic light trapping texture,” Appl. Phys. Lett. 91(6), 061116 (2007).
[CrossRef]

2006 (2)

C. Haase and H. Stiebig, “Optical Properties of Thin-film Silicon Solar Cells with Grating Couplers,” Prog. Photovoltaics 14(7), 629–641 (2006).
[CrossRef]

L. Zeng, Y. Yi, C. Hong, J. Liu, N. Feng, X. Duan, L. C. Kimerling, and B. A. Alamariu, “Efficiency enhancement in Si solar cells by textured photonic crystal back reflector,” Appl. Phys. Lett. 89(11), 111111 (2006).
[CrossRef]

2002 (1)

E. Schneiderlöchner, R. Preu, R. Lüdemann, and S. W. Glunz, “Laser-Fired Rear Contacts for Crystalline Silicon Solar Cells,” Prog. Photovoltacis 10, 29–34 (2002).
[CrossRef]

1998 (1)

C. M. Herzinger, B. Johs, W. McGahan, J. Woollam, and W. Paulson, “Ellipsometric determination of optical constants for silicon and thermally grown silicon dioxide via a multi-sample, multi-wavelength, multi-angle investigation,” J. Appl. Phys. 83(6), 3323–3336 (1998).
[CrossRef]

1996 (1)

1995 (1)

1990 (1)

M. T. Gale, B. Curtis, H. Kiess, and R. H. Morf, “Design and fabrication of submicron structures for light trapping in silicon solar cells,” Proc. SPIE 1272, 60–66 (1990).
[CrossRef]

1983 (1)

P. Sheng, A. N. Bloch, and R. S. Stepleman, “Wavelength-selective absorption enhancement in thin-film solar cells,” Appl. Phys. Lett. 43(6), 579–581 (1983).
[CrossRef]

1982 (1)

1980 (1)

C. H. Henry, “Limiting efficiencies of ideal single and multiple energy gap terrestrial solar cells,” J. Appl. Phys. 51(8), 4494–4500 (1980).
[CrossRef]

Alamariu, B. A.

L. Zeng, Y. Yi, C. Hong, J. Liu, N. Feng, X. Duan, L. C. Kimerling, and B. A. Alamariu, “Efficiency enhancement in Si solar cells by textured photonic crystal back reflector,” Appl. Phys. Lett. 89(11), 111111 (2006).
[CrossRef]

Arafune, K.

H. Sai, Y. Kanamori, K. Arafune, Y. Ohshita, and M. Yamaguchi, “Light trapping effect of submicron surface textures in crystalline Si solar cells,” Prog. Photovoltaics 15(5), 415–423 (2007).
[CrossRef]

Ballif, C.

F.-J. Haug, T. Söderström, M. Python, V. Terrazzoni-Daudrix, X. Niquille, and C. Ballif, “Development of micromorph tandem solar cells on flexible low-cost plastic substrates,” Sol. Energy Mater. Sol. Cells 93(6-7), 884–887 (2009).
[CrossRef]

Barnett, A.

Bermel, P.

Bloch, A. N.

P. Sheng, A. N. Bloch, and R. S. Stepleman, “Wavelength-selective absorption enhancement in thin-film solar cells,” Appl. Phys. Lett. 43(6), 579–581 (1983).
[CrossRef]

Campa, A.

A. Čampa, J. Krč, F. Smole, and M. Topič, “Potential of diffraction gratings for implementation as a metal back reflector in thin-film silicon solar cells,” Thin Solid Films 516(20), 6963–6967 (2008).
[CrossRef]

Chen, C.

Creazzo, T.

Curtis, B.

M. T. Gale, B. Curtis, H. Kiess, and R. H. Morf, “Design and fabrication of submicron structures for light trapping in silicon solar cells,” Proc. SPIE 1272, 60–66 (1990).
[CrossRef]

Dewan, R.

R. Dewan and D. Knipp, “Light-trapping in thin-film silicon solar cells with integrated diffraction grating,” J. Appl. Phys. 106(7), 074901 (2009).
[CrossRef]

Duan, X.

N.-N. Feng, J. Michel, L. Zeng, J. Liu, C.-Y. Hong, L. C. Kimerling, and X. Duan, “Design of Highly Efficient Light-Trapping Structures of Thin-Film Crystalline Silicon Solar Cells,” IEEE Trans. Electron. Dev. 54(8), 1926–1933 (2007).
[CrossRef]

L. Zeng, Y. Yi, C. Hong, J. Liu, N. Feng, X. Duan, L. C. Kimerling, and B. A. Alamariu, “Efficiency enhancement in Si solar cells by textured photonic crystal back reflector,” Appl. Phys. Lett. 89(11), 111111 (2006).
[CrossRef]

Feng, N.

L. Zeng, Y. Yi, C. Hong, J. Liu, N. Feng, X. Duan, L. C. Kimerling, and B. A. Alamariu, “Efficiency enhancement in Si solar cells by textured photonic crystal back reflector,” Appl. Phys. Lett. 89(11), 111111 (2006).
[CrossRef]

Feng, N.-N.

N.-N. Feng, J. Michel, L. Zeng, J. Liu, C.-Y. Hong, L. C. Kimerling, and X. Duan, “Design of Highly Efficient Light-Trapping Structures of Thin-Film Crystalline Silicon Solar Cells,” IEEE Trans. Electron. Dev. 54(8), 1926–1933 (2007).
[CrossRef]

Fujiwara, H.

H. Sai, H. Fujiwara, and M. Kondo, “Back surface reflectors with periodic textures fabricated by self-ordering process for light trapping in thin-film microcrystalline silicon solar cells,” Sol. Energy Mater. Sol. Cells 93(6-7), 1087–1090 (2009).
[CrossRef]

H. Sai, H. Fujiwara, M. Kondo, and Y. Kanamori, “Enhancement of light trapping in thin-film hydrogenated microcrystalline Si solar cells using back reflectors with self-ordered dimple pattern,” Appl. Phys. Lett. 93(14), 143501 (2008).
[CrossRef]

Gale, M. T.

M. T. Gale, B. Curtis, H. Kiess, and R. H. Morf, “Design and fabrication of submicron structures for light trapping in silicon solar cells,” Proc. SPIE 1272, 60–66 (1990).
[CrossRef]

Glunz, S. W.

E. Schneiderlöchner, R. Preu, R. Lüdemann, and S. W. Glunz, “Laser-Fired Rear Contacts for Crystalline Silicon Solar Cells,” Prog. Photovoltacis 10, 29–34 (2002).
[CrossRef]

Haase, C.

C. Haase and H. Stiebig, “Thin-film silicon solar cells with efficient periodic light trapping texture,” Appl. Phys. Lett. 91(6), 061116 (2007).
[CrossRef]

C. Haase and H. Stiebig, “Optical Properties of Thin-film Silicon Solar Cells with Grating Couplers,” Prog. Photovoltaics 14(7), 629–641 (2006).
[CrossRef]

Haug, F.-J.

F.-J. Haug, T. Söderström, M. Python, V. Terrazzoni-Daudrix, X. Niquille, and C. Ballif, “Development of micromorph tandem solar cells on flexible low-cost plastic substrates,” Sol. Energy Mater. Sol. Cells 93(6-7), 884–887 (2009).
[CrossRef]

Heine, C.

Henry, C. H.

C. H. Henry, “Limiting efficiencies of ideal single and multiple energy gap terrestrial solar cells,” J. Appl. Phys. 51(8), 4494–4500 (1980).
[CrossRef]

Herzinger, C. M.

C. M. Herzinger, B. Johs, W. McGahan, J. Woollam, and W. Paulson, “Ellipsometric determination of optical constants for silicon and thermally grown silicon dioxide via a multi-sample, multi-wavelength, multi-angle investigation,” J. Appl. Phys. 83(6), 3323–3336 (1998).
[CrossRef]

Hong, C.

L. Zeng, Y. Yi, C. Hong, J. Liu, N. Feng, X. Duan, L. C. Kimerling, and B. A. Alamariu, “Efficiency enhancement in Si solar cells by textured photonic crystal back reflector,” Appl. Phys. Lett. 89(11), 111111 (2006).
[CrossRef]

Hong, C.-Y.

N.-N. Feng, J. Michel, L. Zeng, J. Liu, C.-Y. Hong, L. C. Kimerling, and X. Duan, “Design of Highly Efficient Light-Trapping Structures of Thin-Film Crystalline Silicon Solar Cells,” IEEE Trans. Electron. Dev. 54(8), 1926–1933 (2007).
[CrossRef]

Honsberg, C.

Joannopoulos, J. D.

Johs, B.

C. M. Herzinger, B. Johs, W. McGahan, J. Woollam, and W. Paulson, “Ellipsometric determination of optical constants for silicon and thermally grown silicon dioxide via a multi-sample, multi-wavelength, multi-angle investigation,” J. Appl. Phys. 83(6), 3323–3336 (1998).
[CrossRef]

Kanamori, Y.

H. Sai, H. Fujiwara, M. Kondo, and Y. Kanamori, “Enhancement of light trapping in thin-film hydrogenated microcrystalline Si solar cells using back reflectors with self-ordered dimple pattern,” Appl. Phys. Lett. 93(14), 143501 (2008).
[CrossRef]

H. Sai, Y. Kanamori, K. Arafune, Y. Ohshita, and M. Yamaguchi, “Light trapping effect of submicron surface textures in crystalline Si solar cells,” Prog. Photovoltaics 15(5), 415–423 (2007).
[CrossRef]

Kiess, H.

M. T. Gale, B. Curtis, H. Kiess, and R. H. Morf, “Design and fabrication of submicron structures for light trapping in silicon solar cells,” Proc. SPIE 1272, 60–66 (1990).
[CrossRef]

Kimerling, L. C.

P. Bermel, C. Luo, L. Zeng, L. C. Kimerling, and J. D. Joannopoulos, “Improving thin-film crystalline silicon solar cell efficiencies with photonic crystals,” Opt. Express 15(25), 16986–17000 (2007).
[CrossRef] [PubMed]

N.-N. Feng, J. Michel, L. Zeng, J. Liu, C.-Y. Hong, L. C. Kimerling, and X. Duan, “Design of Highly Efficient Light-Trapping Structures of Thin-Film Crystalline Silicon Solar Cells,” IEEE Trans. Electron. Dev. 54(8), 1926–1933 (2007).
[CrossRef]

L. Zeng, Y. Yi, C. Hong, J. Liu, N. Feng, X. Duan, L. C. Kimerling, and B. A. Alamariu, “Efficiency enhancement in Si solar cells by textured photonic crystal back reflector,” Appl. Phys. Lett. 89(11), 111111 (2006).
[CrossRef]

Knipp, D.

R. Dewan and D. Knipp, “Light-trapping in thin-film silicon solar cells with integrated diffraction grating,” J. Appl. Phys. 106(7), 074901 (2009).
[CrossRef]

Kondo, M.

H. Sai, H. Fujiwara, and M. Kondo, “Back surface reflectors with periodic textures fabricated by self-ordering process for light trapping in thin-film microcrystalline silicon solar cells,” Sol. Energy Mater. Sol. Cells 93(6-7), 1087–1090 (2009).
[CrossRef]

H. Sai, H. Fujiwara, M. Kondo, and Y. Kanamori, “Enhancement of light trapping in thin-film hydrogenated microcrystalline Si solar cells using back reflectors with self-ordered dimple pattern,” Appl. Phys. Lett. 93(14), 143501 (2008).
[CrossRef]

Krc, J.

A. Čampa, J. Krč, F. Smole, and M. Topič, “Potential of diffraction gratings for implementation as a metal back reflector in thin-film silicon solar cells,” Thin Solid Films 516(20), 6963–6967 (2008).
[CrossRef]

Li, L.

Liu, J.

N.-N. Feng, J. Michel, L. Zeng, J. Liu, C.-Y. Hong, L. C. Kimerling, and X. Duan, “Design of Highly Efficient Light-Trapping Structures of Thin-Film Crystalline Silicon Solar Cells,” IEEE Trans. Electron. Dev. 54(8), 1926–1933 (2007).
[CrossRef]

L. Zeng, Y. Yi, C. Hong, J. Liu, N. Feng, X. Duan, L. C. Kimerling, and B. A. Alamariu, “Efficiency enhancement in Si solar cells by textured photonic crystal back reflector,” Appl. Phys. Lett. 89(11), 111111 (2006).
[CrossRef]

Lüdemann, R.

E. Schneiderlöchner, R. Preu, R. Lüdemann, and S. W. Glunz, “Laser-Fired Rear Contacts for Crystalline Silicon Solar Cells,” Prog. Photovoltacis 10, 29–34 (2002).
[CrossRef]

Luo, C.

McGahan, W.

C. M. Herzinger, B. Johs, W. McGahan, J. Woollam, and W. Paulson, “Ellipsometric determination of optical constants for silicon and thermally grown silicon dioxide via a multi-sample, multi-wavelength, multi-angle investigation,” J. Appl. Phys. 83(6), 3323–3336 (1998).
[CrossRef]

Michel, J.

N.-N. Feng, J. Michel, L. Zeng, J. Liu, C.-Y. Hong, L. C. Kimerling, and X. Duan, “Design of Highly Efficient Light-Trapping Structures of Thin-Film Crystalline Silicon Solar Cells,” IEEE Trans. Electron. Dev. 54(8), 1926–1933 (2007).
[CrossRef]

Morf, R. H.

C. Heine and R. H. Morf, “Submicrometeer gratings for solar energy applications,” Appl. Opt. 34(14), 2476–2482 (1995).
[CrossRef] [PubMed]

M. T. Gale, B. Curtis, H. Kiess, and R. H. Morf, “Design and fabrication of submicron structures for light trapping in silicon solar cells,” Proc. SPIE 1272, 60–66 (1990).
[CrossRef]

Mutitu, J. G.

Niquille, X.

F.-J. Haug, T. Söderström, M. Python, V. Terrazzoni-Daudrix, X. Niquille, and C. Ballif, “Development of micromorph tandem solar cells on flexible low-cost plastic substrates,” Sol. Energy Mater. Sol. Cells 93(6-7), 884–887 (2009).
[CrossRef]

Ohshita, Y.

H. Sai, Y. Kanamori, K. Arafune, Y. Ohshita, and M. Yamaguchi, “Light trapping effect of submicron surface textures in crystalline Si solar cells,” Prog. Photovoltaics 15(5), 415–423 (2007).
[CrossRef]

Paulson, W.

C. M. Herzinger, B. Johs, W. McGahan, J. Woollam, and W. Paulson, “Ellipsometric determination of optical constants for silicon and thermally grown silicon dioxide via a multi-sample, multi-wavelength, multi-angle investigation,” J. Appl. Phys. 83(6), 3323–3336 (1998).
[CrossRef]

Prather, D. W.

Preu, R.

E. Schneiderlöchner, R. Preu, R. Lüdemann, and S. W. Glunz, “Laser-Fired Rear Contacts for Crystalline Silicon Solar Cells,” Prog. Photovoltacis 10, 29–34 (2002).
[CrossRef]

Python, M.

F.-J. Haug, T. Söderström, M. Python, V. Terrazzoni-Daudrix, X. Niquille, and C. Ballif, “Development of micromorph tandem solar cells on flexible low-cost plastic substrates,” Sol. Energy Mater. Sol. Cells 93(6-7), 884–887 (2009).
[CrossRef]

Sai, H.

H. Sai, H. Fujiwara, and M. Kondo, “Back surface reflectors with periodic textures fabricated by self-ordering process for light trapping in thin-film microcrystalline silicon solar cells,” Sol. Energy Mater. Sol. Cells 93(6-7), 1087–1090 (2009).
[CrossRef]

H. Sai, H. Fujiwara, M. Kondo, and Y. Kanamori, “Enhancement of light trapping in thin-film hydrogenated microcrystalline Si solar cells using back reflectors with self-ordered dimple pattern,” Appl. Phys. Lett. 93(14), 143501 (2008).
[CrossRef]

H. Sai, Y. Kanamori, K. Arafune, Y. Ohshita, and M. Yamaguchi, “Light trapping effect of submicron surface textures in crystalline Si solar cells,” Prog. Photovoltaics 15(5), 415–423 (2007).
[CrossRef]

Schneiderlöchner, E.

E. Schneiderlöchner, R. Preu, R. Lüdemann, and S. W. Glunz, “Laser-Fired Rear Contacts for Crystalline Silicon Solar Cells,” Prog. Photovoltacis 10, 29–34 (2002).
[CrossRef]

Sheng, P.

P. Sheng, A. N. Bloch, and R. S. Stepleman, “Wavelength-selective absorption enhancement in thin-film solar cells,” Appl. Phys. Lett. 43(6), 579–581 (1983).
[CrossRef]

Shi, S.

Smole, F.

A. Čampa, J. Krč, F. Smole, and M. Topič, “Potential of diffraction gratings for implementation as a metal back reflector in thin-film silicon solar cells,” Thin Solid Films 516(20), 6963–6967 (2008).
[CrossRef]

Söderström, T.

F.-J. Haug, T. Söderström, M. Python, V. Terrazzoni-Daudrix, X. Niquille, and C. Ballif, “Development of micromorph tandem solar cells on flexible low-cost plastic substrates,” Sol. Energy Mater. Sol. Cells 93(6-7), 884–887 (2009).
[CrossRef]

Stepleman, R. S.

P. Sheng, A. N. Bloch, and R. S. Stepleman, “Wavelength-selective absorption enhancement in thin-film solar cells,” Appl. Phys. Lett. 43(6), 579–581 (1983).
[CrossRef]

Stiebig, H.

C. Haase and H. Stiebig, “Thin-film silicon solar cells with efficient periodic light trapping texture,” Appl. Phys. Lett. 91(6), 061116 (2007).
[CrossRef]

C. Haase and H. Stiebig, “Optical Properties of Thin-film Silicon Solar Cells with Grating Couplers,” Prog. Photovoltaics 14(7), 629–641 (2006).
[CrossRef]

Terrazzoni-Daudrix, V.

F.-J. Haug, T. Söderström, M. Python, V. Terrazzoni-Daudrix, X. Niquille, and C. Ballif, “Development of micromorph tandem solar cells on flexible low-cost plastic substrates,” Sol. Energy Mater. Sol. Cells 93(6-7), 884–887 (2009).
[CrossRef]

Topic, M.

A. Čampa, J. Krč, F. Smole, and M. Topič, “Potential of diffraction gratings for implementation as a metal back reflector in thin-film silicon solar cells,” Thin Solid Films 516(20), 6963–6967 (2008).
[CrossRef]

Woollam, J.

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Figures (7)

Fig. 1
Fig. 1

Optical solar cell model structure used in the computations (not to scale). A single layer of Si3N4 is used as AR-coating. The reference structure is similar except that the SiO2 and the grating layer are removed, and the Si is in direct contact with a planar Al-mirror.

Fig. 2
Fig. 2

Power fraction DHO in higher-order diffracted beams (a), defined in Eq. (4) as a function of tg and tox calculated with GD-calc using a period of 2 µm. (b) shows the phase difference between grating peaks and valleys calculated independently for the peaks and valleys by the transfer matrix method. A phase difference close to + π or -π indicates destructive interference at normal incidence, resulting in a corresponding maximum DHO.

Fig. 3
Fig. 3

(a) DHO defined in Eq. (4) as a function of tg and tox calculated with GD-calc using a period of 0.3 µm. Note that the magnitude of the higher order reflections is much lower than in figure Fig. 2(a). Figure 3(b) shows the phase plot of a reflected wave from a one dimensional stack of homogenous slabs where the grating layer have been substituted by a homogenous slab with an effective refractive index neff = 3. The calculation of phase is performed with the transfer matrix method.

Fig. 4
Fig. 4

The figure shows Jsc on the left axis, and parasitic absorption on the right axis as a function of thickness of the SiO2 spacing layer tox between the grating and the Al mirror.

Fig. 5
Fig. 5

The figure shows Jsc for different combinations of grating period Λ and fill factor for a grating thickness tg = 0.23 µm and a spacing layer thickness tox = 0.2 µm.

Fig. 6
Fig. 6

The figure shows absorption as a function of wavelength for the light-trapping design (solid line) and the reference (dotted line), compared to the limitation imposed by the front-side reflectance from a single layer AR-coating on top of an infinite Si slab (dashed line). (a) shows the simulated spectral response, and (b) shows a moving average of the same data. A Λ of 0.7 µm, fill factor of 0.6, tg of 0.23 µm and tox of 0.2 µm are used for the light trapping structure.

Fig. 7
Fig. 7

The figure shows the effect of incidence angle on Jsc for the cell with light-trapping structure and for the reference cell. Both p-polarization and s-polarization and their mean are shown.

Tables (1)

Tables Icon

Table 1 Comparison of Jsc for different model structures assuming all excited charge carriers reach the contacts. The spectral response of the 20 µm reference and the light-trapping structure with Al-reflector is shown in Fig. 6. The same parameters are used for all light trapping three cells, varying only the back side reflector. The AM 1.5 absorbance is the percentage of the photons in the wavelength interval 300-1100 nm which are absorbed in the Si.

Equations (4)

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

J s c = q 0 A ( λ ) Φ ( λ ) d λ
n o sin ( θ m ) = n i sin ( θ i ) + m λ Λ
λ ¯ = 0 λ ( A b a s e ( λ ) A o p t ( λ ) ) Φ ( λ ) d λ 0 ( A b a s e ( λ ) A o p t ( λ ) ) Φ ( λ ) d λ
D H O = m x = m x = m y = m y = D m x m y D 00

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