Abstract

The majority of photons emitted in organic light-emitting diodes are either trapped in the substrate or emitted into lossy waveguided modes. We show how optimized, non-periodic dielectric stacks inserted between the substrate and transparent anode can be used to improve the photon outcoupling efficiency by tailoring of the local photon density of states. Unlike previously demonstrated outcoupling schemes, this method does not lead to pixel blurring and maintains a Lambertian angular emission profiles within a specified cone. For small molecular weight, green-emitting devices, a 2.5-fold uniformly distributed increase in brightness is achievable for a viewing angle of 60°.

© 2007 Optical Society of America

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  1. P. A. Hobson, S. Wedge, J. A. E. Wasey, I. Sage, and W. L. Barnes, “Surface plasmon mediated emission from organic light-emitting diodes,” Adv. Mater. 14, 1393–1396 (2002).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]

2004 (1)

L. H. Smith, J. A. Wasey, and W. L. Barnes, “Light outcoupling efficiency of top-emitting organic light-emitting diodes,” Appl. Phys. Lett. 84, 2986–2988 (2004).
[CrossRef]

2003 (1)

M. Gerken and D. A. B. Miller, “Wavelength demultiplexer using the spatial dispersion of multilayer thin-film structures,” Photonic Technol Lett.  15, 1097–1099 (2003).
[CrossRef]

2002 (2)

P. A. Hobson, S. Wedge, J. A. E. Wasey, I. Sage, and W. L. Barnes, “Surface plasmon mediated emission from organic light-emitting diodes,” Adv. Mater. 14, 1393–1396 (2002).
[CrossRef]

S. Moller and S. R. Forrest, “Improved light out-coupling in organic light emitting diodes employing ordered microlens arrays,” J. Appl. Phys. 91, 3324–3327 (2002).
[CrossRef]

2001 (3)

T. Tsutsui, M. Yashiro, H. Yokogawa, K. Kawano, and M. Yokoyama, “Doubling coupling-out efficiency in organic light-emitting devices using a thin silica aerogel layer,” Adv. Mater., 13, 1149–1152 (2001)
[CrossRef]

P. T. Worthing and W. L. Barnes, “Efficient coupling of surface plasmon polaritons to radiation using a bigrating,” Appl. Phys. Lett.  79, 3035–3037(2001).
[CrossRef]

M. Deopura, C. K. Ullal, B. Temelkuran, and Y Fink,” Dielectric omnidirectional visible reflector,” Opt. Lett. 26, 1197–1199 (2001).
[CrossRef]

2000 (2)

C. F. Madigan, M. H. Lu, and J. C. Sturm, “Improvement of output coupling efficiency of organic lightemitting diodes by backside substrate modification,” Appl. Phys. Lett. 76, 1650–1652 (2000).
[CrossRef]

J. Vuckovic, M. Loncar, and A. Scherer, “Surface plasmon enhanced light-emitting diodes,” IEEE J. Quantum Electron. 36, 1131–1144 (2000).
[CrossRef]

1999 (2)

Y. Xu, J. S. Vuckovic, R. K. Lee, O. J. Painter, A. Scherer, and A Yariv, “Finite-difference time-domain calculation of spontaneous emission lifetime in a microcavity,” J. Opt. Soc. Am. B, 16, 465–474 (1999).
[CrossRef]

M. A. Baldo, S. Lamansky, P. E. Burrows, M. E. Thompson, and S. R. Forrest, “Very high-efficiency green organic light-emitting devices based on electrophosphorescence,” Appl. Phys. Lett. 75, 4–6 (1999).
[CrossRef]

1998 (1)

N. Matuschek, Franz X. Kartner, and Ursula Keller, “Theory of double-chirped mirrors,” IEEE J. Sel. Top. Quantum Electron. 4, 197–208 (1998).
[CrossRef]

1997 (2)

S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High extraction efficiency of spontaneous emission from slabs of photonic crystals,” Phys. Rev. Lett. 78, 3294–3297 (1997)
[CrossRef]

K. V. Popov, J. A. Dobrowolski, A. V. Tikhonravov, and B. T. Sullivan, “Broadband high-reflection multilayer coatings at oblique angles of incidence,” Appl. Opt. 36, 2139–2151 (1997).
[CrossRef] [PubMed]

1996 (3)

1994 (2)

E. A. HindsP. R. Berman, ed., (Academic, New York, 1994).

T. Tsutsui, N. Takada, and Shogo Saito, “Sharply directed emission in organic electroluminescent diodes with an optical-microcavity structure,” Appl. Phys. Lett.  65, 1868–1870 (1994).
[CrossRef]

1993 (2)

T. Nakayama, Y. Itoh, and A. Kakuta, “Organic photo- and electroluminescent devices with double mirrors,” Appl. Phys. Lett. 63, 594–595 (1993).
[CrossRef]

A. V. Tikhonravov, “Some theoretical aspects of thin-film optics and their applications,” Appl. Opt. 32, 5417–5426 (1993).
[CrossRef] [PubMed]

1991 (1)

R. J. Glauber and M. L. Lewenstein, “Quantum optics of dielectric media,” Phys. Rev. A 43, 467–491 (1991).
[CrossRef] [PubMed]

1984 (1)

G. W. Ford and W. H. Weber, “Electromagnetic interactions of molecules with metal surfaces,” Phys. Rep. 113, 195–287 (1984).
[CrossRef]

1978 (1)

R. R. Chance, A. Prock, and R. Silbey, “Molecular fluorescence and energy transfer near interfaces,” Adv. Chem. Phys. 37, 1–65 (1978).
[CrossRef]

1969 (1)

P. A. Lee, G. Said, R. Davis, and T. H. Lim, “On the optical properties of some layer compounds,” J. Phys. Chem. Solids 30, 2719–2729 (1969).
[CrossRef]

1965 (1)

J. A. Nelder and R. Mead, “A simplex method for function minimization,” Comput. J. 7, 308–313 (1965).

Baldo, M. A.

M. A. Baldo, S. Lamansky, P. E. Burrows, M. E. Thompson, and S. R. Forrest, “Very high-efficiency green organic light-emitting devices based on electrophosphorescence,” Appl. Phys. Lett. 75, 4–6 (1999).
[CrossRef]

Barnes, W. L.

L. H. Smith, J. A. Wasey, and W. L. Barnes, “Light outcoupling efficiency of top-emitting organic light-emitting diodes,” Appl. Phys. Lett. 84, 2986–2988 (2004).
[CrossRef]

P. A. Hobson, S. Wedge, J. A. E. Wasey, I. Sage, and W. L. Barnes, “Surface plasmon mediated emission from organic light-emitting diodes,” Adv. Mater. 14, 1393–1396 (2002).
[CrossRef]

P. T. Worthing and W. L. Barnes, “Efficient coupling of surface plasmon polaritons to radiation using a bigrating,” Appl. Phys. Lett.  79, 3035–3037(2001).
[CrossRef]

Burrows, P. E.

M. A. Baldo, S. Lamansky, P. E. Burrows, M. E. Thompson, and S. R. Forrest, “Very high-efficiency green organic light-emitting devices based on electrophosphorescence,” Appl. Phys. Lett. 75, 4–6 (1999).
[CrossRef]

Chance, R. R.

R. R. Chance, A. Prock, and R. Silbey, “Molecular fluorescence and energy transfer near interfaces,” Adv. Chem. Phys. 37, 1–65 (1978).
[CrossRef]

Cox, M. J

M. J Cox, Visual ergonomics, (University of Bradford1999).

Davis, R.

P. A. Lee, G. Said, R. Davis, and T. H. Lim, “On the optical properties of some layer compounds,” J. Phys. Chem. Solids 30, 2719–2729 (1969).
[CrossRef]

DeBell, G. W.

Deopura, M.

Dobrowolski, J. A.

Dodabalapur, A.

A. Dodabalapur, L. J. Rothberg, R. H. Jordan, T. M. Miller, R. E. Slusher, and Julia M. Phillipse, “Physics and applications of organic microcavity light emitting diodes,” J. Appl. Phys. 80, 6954–6964 (1996).
[CrossRef]

Fan, S.

S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High extraction efficiency of spontaneous emission from slabs of photonic crystals,” Phys. Rev. Lett. 78, 3294–3297 (1997)
[CrossRef]

Fink, Y

Ford, G. W.

G. W. Ford and W. H. Weber, “Electromagnetic interactions of molecules with metal surfaces,” Phys. Rep. 113, 195–287 (1984).
[CrossRef]

Forrest, S. R.

S. Moller and S. R. Forrest, “Improved light out-coupling in organic light emitting diodes employing ordered microlens arrays,” J. Appl. Phys. 91, 3324–3327 (2002).
[CrossRef]

M. A. Baldo, S. Lamansky, P. E. Burrows, M. E. Thompson, and S. R. Forrest, “Very high-efficiency green organic light-emitting devices based on electrophosphorescence,” Appl. Phys. Lett. 75, 4–6 (1999).
[CrossRef]

Gerken, M.

M. Gerken and D. A. B. Miller, “Wavelength demultiplexer using the spatial dispersion of multilayer thin-film structures,” Photonic Technol Lett.  15, 1097–1099 (2003).
[CrossRef]

Glauber, R. J.

R. J. Glauber and M. L. Lewenstein, “Quantum optics of dielectric media,” Phys. Rev. A 43, 467–491 (1991).
[CrossRef] [PubMed]

Hinds, E. A.

E. A. HindsP. R. Berman, ed., (Academic, New York, 1994).

Hobson, P. A.

P. A. Hobson, S. Wedge, J. A. E. Wasey, I. Sage, and W. L. Barnes, “Surface plasmon mediated emission from organic light-emitting diodes,” Adv. Mater. 14, 1393–1396 (2002).
[CrossRef]

Itoh, Y.

T. Nakayama, Y. Itoh, and A. Kakuta, “Organic photo- and electroluminescent devices with double mirrors,” Appl. Phys. Lett. 63, 594–595 (1993).
[CrossRef]

Joannopoulos, J. D.

S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High extraction efficiency of spontaneous emission from slabs of photonic crystals,” Phys. Rev. Lett. 78, 3294–3297 (1997)
[CrossRef]

Jordan, R. H.

A. Dodabalapur, L. J. Rothberg, R. H. Jordan, T. M. Miller, R. E. Slusher, and Julia M. Phillipse, “Physics and applications of organic microcavity light emitting diodes,” J. Appl. Phys. 80, 6954–6964 (1996).
[CrossRef]

Kakuta, A.

T. Nakayama, Y. Itoh, and A. Kakuta, “Organic photo- and electroluminescent devices with double mirrors,” Appl. Phys. Lett. 63, 594–595 (1993).
[CrossRef]

Kartner, Franz X.

N. Matuschek, Franz X. Kartner, and Ursula Keller, “Theory of double-chirped mirrors,” IEEE J. Sel. Top. Quantum Electron. 4, 197–208 (1998).
[CrossRef]

Kawano, K.

T. Tsutsui, M. Yashiro, H. Yokogawa, K. Kawano, and M. Yokoyama, “Doubling coupling-out efficiency in organic light-emitting devices using a thin silica aerogel layer,” Adv. Mater., 13, 1149–1152 (2001)
[CrossRef]

Keller, Ursula

N. Matuschek, Franz X. Kartner, and Ursula Keller, “Theory of double-chirped mirrors,” IEEE J. Sel. Top. Quantum Electron. 4, 197–208 (1998).
[CrossRef]

Lamansky, S.

M. A. Baldo, S. Lamansky, P. E. Burrows, M. E. Thompson, and S. R. Forrest, “Very high-efficiency green organic light-emitting devices based on electrophosphorescence,” Appl. Phys. Lett. 75, 4–6 (1999).
[CrossRef]

Lee, P. A.

P. A. Lee, G. Said, R. Davis, and T. H. Lim, “On the optical properties of some layer compounds,” J. Phys. Chem. Solids 30, 2719–2729 (1969).
[CrossRef]

Lee, R. K.

Y. Xu, J. S. Vuckovic, R. K. Lee, O. J. Painter, A. Scherer, and A Yariv, “Finite-difference time-domain calculation of spontaneous emission lifetime in a microcavity,” J. Opt. Soc. Am. B, 16, 465–474 (1999).
[CrossRef]

Lewenstein, M. L.

R. J. Glauber and M. L. Lewenstein, “Quantum optics of dielectric media,” Phys. Rev. A 43, 467–491 (1991).
[CrossRef] [PubMed]

Lim, T. H.

P. A. Lee, G. Said, R. Davis, and T. H. Lim, “On the optical properties of some layer compounds,” J. Phys. Chem. Solids 30, 2719–2729 (1969).
[CrossRef]

Loncar, M.

J. Vuckovic, M. Loncar, and A. Scherer, “Surface plasmon enhanced light-emitting diodes,” IEEE J. Quantum Electron. 36, 1131–1144 (2000).
[CrossRef]

Lu, M. H.

C. F. Madigan, M. H. Lu, and J. C. Sturm, “Improvement of output coupling efficiency of organic lightemitting diodes by backside substrate modification,” Appl. Phys. Lett. 76, 1650–1652 (2000).
[CrossRef]

Madigan, C. F.

C. F. Madigan, M. H. Lu, and J. C. Sturm, “Improvement of output coupling efficiency of organic lightemitting diodes by backside substrate modification,” Appl. Phys. Lett. 76, 1650–1652 (2000).
[CrossRef]

Matuschek, N.

N. Matuschek, Franz X. Kartner, and Ursula Keller, “Theory of double-chirped mirrors,” IEEE J. Sel. Top. Quantum Electron. 4, 197–208 (1998).
[CrossRef]

Mead, R.

J. A. Nelder and R. Mead, “A simplex method for function minimization,” Comput. J. 7, 308–313 (1965).

Miller, D. A. B.

M. Gerken and D. A. B. Miller, “Wavelength demultiplexer using the spatial dispersion of multilayer thin-film structures,” Photonic Technol Lett.  15, 1097–1099 (2003).
[CrossRef]

Miller, T. M.

A. Dodabalapur, L. J. Rothberg, R. H. Jordan, T. M. Miller, R. E. Slusher, and Julia M. Phillipse, “Physics and applications of organic microcavity light emitting diodes,” J. Appl. Phys. 80, 6954–6964 (1996).
[CrossRef]

Moller, S.

S. Moller and S. R. Forrest, “Improved light out-coupling in organic light emitting diodes employing ordered microlens arrays,” J. Appl. Phys. 91, 3324–3327 (2002).
[CrossRef]

Nakayama, T.

T. Nakayama, Y. Itoh, and A. Kakuta, “Organic photo- and electroluminescent devices with double mirrors,” Appl. Phys. Lett. 63, 594–595 (1993).
[CrossRef]

Nelder, J. A.

J. A. Nelder and R. Mead, “A simplex method for function minimization,” Comput. J. 7, 308–313 (1965).

Painter, O. J.

Y. Xu, J. S. Vuckovic, R. K. Lee, O. J. Painter, A. Scherer, and A Yariv, “Finite-difference time-domain calculation of spontaneous emission lifetime in a microcavity,” J. Opt. Soc. Am. B, 16, 465–474 (1999).
[CrossRef]

Phillipse, Julia M.

A. Dodabalapur, L. J. Rothberg, R. H. Jordan, T. M. Miller, R. E. Slusher, and Julia M. Phillipse, “Physics and applications of organic microcavity light emitting diodes,” J. Appl. Phys. 80, 6954–6964 (1996).
[CrossRef]

Popov, K. V.

Prock, A.

R. R. Chance, A. Prock, and R. Silbey, “Molecular fluorescence and energy transfer near interfaces,” Adv. Chem. Phys. 37, 1–65 (1978).
[CrossRef]

Rothberg, L. J.

A. Dodabalapur, L. J. Rothberg, R. H. Jordan, T. M. Miller, R. E. Slusher, and Julia M. Phillipse, “Physics and applications of organic microcavity light emitting diodes,” J. Appl. Phys. 80, 6954–6964 (1996).
[CrossRef]

Sage, I.

P. A. Hobson, S. Wedge, J. A. E. Wasey, I. Sage, and W. L. Barnes, “Surface plasmon mediated emission from organic light-emitting diodes,” Adv. Mater. 14, 1393–1396 (2002).
[CrossRef]

Said, G.

P. A. Lee, G. Said, R. Davis, and T. H. Lim, “On the optical properties of some layer compounds,” J. Phys. Chem. Solids 30, 2719–2729 (1969).
[CrossRef]

Saito, Shogo

T. Tsutsui, N. Takada, and Shogo Saito, “Sharply directed emission in organic electroluminescent diodes with an optical-microcavity structure,” Appl. Phys. Lett.  65, 1868–1870 (1994).
[CrossRef]

Scherer, A.

J. Vuckovic, M. Loncar, and A. Scherer, “Surface plasmon enhanced light-emitting diodes,” IEEE J. Quantum Electron. 36, 1131–1144 (2000).
[CrossRef]

Y. Xu, J. S. Vuckovic, R. K. Lee, O. J. Painter, A. Scherer, and A Yariv, “Finite-difference time-domain calculation of spontaneous emission lifetime in a microcavity,” J. Opt. Soc. Am. B, 16, 465–474 (1999).
[CrossRef]

Silbey, R.

R. R. Chance, A. Prock, and R. Silbey, “Molecular fluorescence and energy transfer near interfaces,” Adv. Chem. Phys. 37, 1–65 (1978).
[CrossRef]

Slusher, R. E.

A. Dodabalapur, L. J. Rothberg, R. H. Jordan, T. M. Miller, R. E. Slusher, and Julia M. Phillipse, “Physics and applications of organic microcavity light emitting diodes,” J. Appl. Phys. 80, 6954–6964 (1996).
[CrossRef]

Smith, L. H.

L. H. Smith, J. A. Wasey, and W. L. Barnes, “Light outcoupling efficiency of top-emitting organic light-emitting diodes,” Appl. Phys. Lett. 84, 2986–2988 (2004).
[CrossRef]

Sturm, J. C.

C. F. Madigan, M. H. Lu, and J. C. Sturm, “Improvement of output coupling efficiency of organic lightemitting diodes by backside substrate modification,” Appl. Phys. Lett. 76, 1650–1652 (2000).
[CrossRef]

Sullivan, B. T.

Takada, N.

T. Tsutsui, N. Takada, and Shogo Saito, “Sharply directed emission in organic electroluminescent diodes with an optical-microcavity structure,” Appl. Phys. Lett.  65, 1868–1870 (1994).
[CrossRef]

Temelkuran, B.

Thompson, M. E.

M. A. Baldo, S. Lamansky, P. E. Burrows, M. E. Thompson, and S. R. Forrest, “Very high-efficiency green organic light-emitting devices based on electrophosphorescence,” Appl. Phys. Lett. 75, 4–6 (1999).
[CrossRef]

Tikhonravov, A. V.

Trubetskov, M. K.

Tsutsui, T.

T. Tsutsui, M. Yashiro, H. Yokogawa, K. Kawano, and M. Yokoyama, “Doubling coupling-out efficiency in organic light-emitting devices using a thin silica aerogel layer,” Adv. Mater., 13, 1149–1152 (2001)
[CrossRef]

T. Tsutsui, N. Takada, and Shogo Saito, “Sharply directed emission in organic electroluminescent diodes with an optical-microcavity structure,” Appl. Phys. Lett.  65, 1868–1870 (1994).
[CrossRef]

Ullal, C. K.

Villeneuve, P. R.

S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High extraction efficiency of spontaneous emission from slabs of photonic crystals,” Phys. Rev. Lett. 78, 3294–3297 (1997)
[CrossRef]

Vuckovic, J.

J. Vuckovic, M. Loncar, and A. Scherer, “Surface plasmon enhanced light-emitting diodes,” IEEE J. Quantum Electron. 36, 1131–1144 (2000).
[CrossRef]

Vuckovic, J. S.

Y. Xu, J. S. Vuckovic, R. K. Lee, O. J. Painter, A. Scherer, and A Yariv, “Finite-difference time-domain calculation of spontaneous emission lifetime in a microcavity,” J. Opt. Soc. Am. B, 16, 465–474 (1999).
[CrossRef]

Wasey, J. A.

L. H. Smith, J. A. Wasey, and W. L. Barnes, “Light outcoupling efficiency of top-emitting organic light-emitting diodes,” Appl. Phys. Lett. 84, 2986–2988 (2004).
[CrossRef]

Wasey, J. A. E.

P. A. Hobson, S. Wedge, J. A. E. Wasey, I. Sage, and W. L. Barnes, “Surface plasmon mediated emission from organic light-emitting diodes,” Adv. Mater. 14, 1393–1396 (2002).
[CrossRef]

Weber, W. H.

G. W. Ford and W. H. Weber, “Electromagnetic interactions of molecules with metal surfaces,” Phys. Rep. 113, 195–287 (1984).
[CrossRef]

Wedge, S.

P. A. Hobson, S. Wedge, J. A. E. Wasey, I. Sage, and W. L. Barnes, “Surface plasmon mediated emission from organic light-emitting diodes,” Adv. Mater. 14, 1393–1396 (2002).
[CrossRef]

Worthing, P. T.

P. T. Worthing and W. L. Barnes, “Efficient coupling of surface plasmon polaritons to radiation using a bigrating,” Appl. Phys. Lett.  79, 3035–3037(2001).
[CrossRef]

Xu, Y.

Y. Xu, J. S. Vuckovic, R. K. Lee, O. J. Painter, A. Scherer, and A Yariv, “Finite-difference time-domain calculation of spontaneous emission lifetime in a microcavity,” J. Opt. Soc. Am. B, 16, 465–474 (1999).
[CrossRef]

Yariv, A

Y. Xu, J. S. Vuckovic, R. K. Lee, O. J. Painter, A. Scherer, and A Yariv, “Finite-difference time-domain calculation of spontaneous emission lifetime in a microcavity,” J. Opt. Soc. Am. B, 16, 465–474 (1999).
[CrossRef]

Yashiro, M.

T. Tsutsui, M. Yashiro, H. Yokogawa, K. Kawano, and M. Yokoyama, “Doubling coupling-out efficiency in organic light-emitting devices using a thin silica aerogel layer,” Adv. Mater., 13, 1149–1152 (2001)
[CrossRef]

Yokogawa, H.

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Figures (4)

Fig. 1.
Fig. 1.

OLED device structure with a non-periodic dielectric stack between the substrate and ITO anode. The various bound and unbound optical modes are identified: unbound modes (UB, kxyc/ω<1), substrate trapped modes (ST, 1<kxyc/ω<1.5), waveguided modes (WG, 1.5<kxyc/ω<1.7) and surface plasmon polariton modes (SPP,kxyc/ω>1.7).

Fig. 2.
Fig. 2.

(a). Spectrally integrated power emitted per unit solid angle vs. azimuthal angle for OLEDs with non-periodic dielectric stacks designs optimized for viewing cones of 20° (A), 60° (B) and 160° (C) (solid lines). The ideal Lambertian target responses (gray lines) and that of the control OLED (dotted line) are also shown (gray lines). (b) Thicknesses of the dielectric layers for the three designs shown in. All three structures have alternate layers of high (n=3.0) and low (n=1.3) index materials. The first layer is in contact with the substrate and is a high index layer in all three designs.

Fig. 3.
Fig. 3.

(a). Fraction of emitted power into UB, ST, WG and SPP modes vs. wavelength for design C (120° viewing cone). (b) Same information as (a) for the control OLED. (c) Spectral power density of emitted radiation the various optical modes design C (120° viewing cone). The spectral power density distribution of the emitter in free space (FS, see text) is also shown. (d) Same information as (c) for the control OLED.

Fig. 4.
Fig. 4.

(a). Relation between targeted viewing cone and total ηOC (filled squares) and apparent ηOC (open squares). The lines are a guide to the eye. For comparison purposes,ηOC for the control OLED is also shown (dashed line). For designs that are not subject to the Lambertian profile-constraint, total outcoupling efficiencies of 45% for a small molecular weight (SMW) and 73% for a polymer OLED can be obtained (dotted lines). (b) CIE vs. angle for the control OLED, and designs B (60° viewing cone) and C (120° viewing cone).

Tables (1)

Tables Icon

Table 1. Fraction of power emitted into the UB, ST, WG and SPP modes as a function of the angular range of the targeted viewing cone. The apparent ηOC within the viewing cone is also listed.

Equations (4)

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P = r ex p , s ω k xy ω 2 Im ( p * . E l ( r ex ) ) f ( r ex ) g ( ω ) ,
E s = ( k 2 4 π ε 0 ε ) i 2 k xy k z η 1 s p xy
E p = ( k 2 4 π ε 0 ε ) ( i 2 k xy k z k 2 η 2 p p xy + i 2 k xy 3 k 2 k z η 1 p p z )
δ = [ 1 θ view 0 θ view ( P ( θ ) β OC f ( θ ) f ( θ ) ) 2 d θ ] 1 2

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