Abstract

We numerically investigate plasmon-resonance absorption of incident light energy by a bisinusoidal metal grating, i.e., one whose surface profile is sinusoidally corrugated in two orthogonal directions with a common period. Employing Yasuura’s modal expansion method, we solve the problem of plane-wave diffraction by the grating and evaluate the absorption, which is observed as dips in diffraction efficiency curves. We examine the field distribution and energy flow in detail at the angles of incidence at which the absorption occurs. We show that the absorption is caused by coupling of the TM component of an evanescent order with surface plasmons. A phase-matching condition is used in the prediction of the incident angle at which the absorption occurs. This, together with the field profile in the presence of the resonance absorption, explains the mechanism of the absorption. We then illustrate interesting features of the absorption: enhancement of polarization conversion between the incident light and the reflected light and simultaneous excitation of two plasmon waves in directions that are symmetric with respect to the plane of incidence.

© 2002 Optical Society of America

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References

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  1. H. Raeter, “Surface plasmon and roughness,” in Surface Polaritons, V. M. Argranovich, D. L. Mills, eds. (North-Holland, New York, 1982), Chap. 9, pp. 331–403.
  2. M. Nevièr, “The homogenous problem,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), Chap. 5, pp. 123–157.
  3. S. J. Elston, G. P. Bryan-Brown, J. R. Sambles, “Polarisation conversion from diffraction gratings,” Phys. Rev. B 44, 6393–6399 (1991).
    [CrossRef]
  4. A. Nemetz, U. Fernandez, W. Knoll, “Surface plasmon field-enhanced Raman spectroscopy with double gratings,” J. Appl. Phys. 75, 1582–1585 (1994).
    [CrossRef]
  5. W. L. Barnes, T. W. Preist, S. C. Kitson, J. R. Sambles, N. P. K. Cotter, D. J. Nash, “Photonic gaps in the dispersion of surface plasmons on gratings,” Phys. Rev. B 51, 11164–11167 (1995).
    [CrossRef]
  6. N. E. Glass, A. A. Maradudin, V. Celli, “Surface plasmons on a large-amplitude doubly periodic corrugated surface,” Phys. Rev. B 26, 5357–5365 (1982).
    [CrossRef]
  7. N. E. Glass, A. A. Maradudin, V. Celli, “Theory of surface-polariton resonances and field enhancements in light scattering from bigratings,” J. Opt. Soc. Am. 73, 1240–1248 (1983).
    [CrossRef]
  8. T. Inagaki, J. P. Goudonnet, J. W. Little, E. T. Arakawa, “Photoacoustic study of plasmon-resonance absorption in a bigrating,” J. Opt. Soc. Am. B 2, 432–439 (1985).
    [CrossRef]
  9. J. B. Harris, T. W. Preist, J. R. Sambles, R. N. Thorpe, R. A. Watts, “Optical response of bigratings,” J. Opt. Soc. Am. A 13, 2041–2049 (1996).
    [CrossRef]
  10. T. Matsuda, Y. Okuno, “Numerical evaluation of plane-wave diffraction by a doubly periodic grating,” Radio Sci. 31, 1791–1798 (1996).
    [CrossRef]
  11. H. Ikuno, K. Yasuura, “Improved point-matching method with application to scattering from a periodic surface,” IEEE Trans. Antennas Propag. AP-21, 657–662 (1973).
    [CrossRef]
  12. C. C. Chen, “Transmission of microwave through perforated flat plates of finite thickness,” IEEE Trans. Microwave Theory Tech. MTT-21, 1–6 (1973).
    [CrossRef]
  13. C. L. Lawson, R. J. Hanson, Solving Least Squares Problems (Prentice-Hall, Englewood Cliffs, N. J., 1974).
  14. Theoretical foundations of the Yasuura method can be found in Ref. 15-17.
  15. K. Yasuura, T. Itakura, “Approximation method for wave functions (I),” , Vol. 38, No. 1 (Kyushu University, Fukuoka, Japan, 1965), pp. 72–77.
  16. K. Yasuura, T. Itakura, “Complete set of wave functions,” , Vol. 38, No. 4 (Kyushu University, Fukuoka, Japan, 1966), pp. 378–385.
  17. K. Yasuura, T. Itakura, “Approximate algorithm by complete set of wave functions,” , Vol. 39, No. 1 (Kyushu University, Fukuoka, Japan, 1966), pp. 51–56.
  18. G. Hass, L. Hadley, “Optical properties of metals,” 2nd ed., in American Institute of Physics HandbookD. E. Gray, ed. (McGraw-Hill, New York, 1963), pp. 6–107.
  19. R. H. Ritchie, E. T. Arakawa, J. J. Cowan, R. N. Hamm, “Surface-plasmon resonance effect in grating diffraction,” Phys. Rev. Lett. 21, 1530–1533 (1968).
    [CrossRef]

1996 (2)

T. Matsuda, Y. Okuno, “Numerical evaluation of plane-wave diffraction by a doubly periodic grating,” Radio Sci. 31, 1791–1798 (1996).
[CrossRef]

J. B. Harris, T. W. Preist, J. R. Sambles, R. N. Thorpe, R. A. Watts, “Optical response of bigratings,” J. Opt. Soc. Am. A 13, 2041–2049 (1996).
[CrossRef]

1995 (1)

W. L. Barnes, T. W. Preist, S. C. Kitson, J. R. Sambles, N. P. K. Cotter, D. J. Nash, “Photonic gaps in the dispersion of surface plasmons on gratings,” Phys. Rev. B 51, 11164–11167 (1995).
[CrossRef]

1994 (1)

A. Nemetz, U. Fernandez, W. Knoll, “Surface plasmon field-enhanced Raman spectroscopy with double gratings,” J. Appl. Phys. 75, 1582–1585 (1994).
[CrossRef]

1991 (1)

S. J. Elston, G. P. Bryan-Brown, J. R. Sambles, “Polarisation conversion from diffraction gratings,” Phys. Rev. B 44, 6393–6399 (1991).
[CrossRef]

1985 (1)

T. Inagaki, J. P. Goudonnet, J. W. Little, E. T. Arakawa, “Photoacoustic study of plasmon-resonance absorption in a bigrating,” J. Opt. Soc. Am. B 2, 432–439 (1985).
[CrossRef]

1983 (1)

1982 (1)

N. E. Glass, A. A. Maradudin, V. Celli, “Surface plasmons on a large-amplitude doubly periodic corrugated surface,” Phys. Rev. B 26, 5357–5365 (1982).
[CrossRef]

1973 (2)

H. Ikuno, K. Yasuura, “Improved point-matching method with application to scattering from a periodic surface,” IEEE Trans. Antennas Propag. AP-21, 657–662 (1973).
[CrossRef]

C. C. Chen, “Transmission of microwave through perforated flat plates of finite thickness,” IEEE Trans. Microwave Theory Tech. MTT-21, 1–6 (1973).
[CrossRef]

1968 (1)

R. H. Ritchie, E. T. Arakawa, J. J. Cowan, R. N. Hamm, “Surface-plasmon resonance effect in grating diffraction,” Phys. Rev. Lett. 21, 1530–1533 (1968).
[CrossRef]

Arakawa, E. T.

T. Inagaki, J. P. Goudonnet, J. W. Little, E. T. Arakawa, “Photoacoustic study of plasmon-resonance absorption in a bigrating,” J. Opt. Soc. Am. B 2, 432–439 (1985).
[CrossRef]

R. H. Ritchie, E. T. Arakawa, J. J. Cowan, R. N. Hamm, “Surface-plasmon resonance effect in grating diffraction,” Phys. Rev. Lett. 21, 1530–1533 (1968).
[CrossRef]

Barnes, W. L.

W. L. Barnes, T. W. Preist, S. C. Kitson, J. R. Sambles, N. P. K. Cotter, D. J. Nash, “Photonic gaps in the dispersion of surface plasmons on gratings,” Phys. Rev. B 51, 11164–11167 (1995).
[CrossRef]

Bryan-Brown, G. P.

S. J. Elston, G. P. Bryan-Brown, J. R. Sambles, “Polarisation conversion from diffraction gratings,” Phys. Rev. B 44, 6393–6399 (1991).
[CrossRef]

Celli, V.

N. E. Glass, A. A. Maradudin, V. Celli, “Theory of surface-polariton resonances and field enhancements in light scattering from bigratings,” J. Opt. Soc. Am. 73, 1240–1248 (1983).
[CrossRef]

N. E. Glass, A. A. Maradudin, V. Celli, “Surface plasmons on a large-amplitude doubly periodic corrugated surface,” Phys. Rev. B 26, 5357–5365 (1982).
[CrossRef]

Chen, C. C.

C. C. Chen, “Transmission of microwave through perforated flat plates of finite thickness,” IEEE Trans. Microwave Theory Tech. MTT-21, 1–6 (1973).
[CrossRef]

Cotter, N. P. K.

W. L. Barnes, T. W. Preist, S. C. Kitson, J. R. Sambles, N. P. K. Cotter, D. J. Nash, “Photonic gaps in the dispersion of surface plasmons on gratings,” Phys. Rev. B 51, 11164–11167 (1995).
[CrossRef]

Cowan, J. J.

R. H. Ritchie, E. T. Arakawa, J. J. Cowan, R. N. Hamm, “Surface-plasmon resonance effect in grating diffraction,” Phys. Rev. Lett. 21, 1530–1533 (1968).
[CrossRef]

Elston, S. J.

S. J. Elston, G. P. Bryan-Brown, J. R. Sambles, “Polarisation conversion from diffraction gratings,” Phys. Rev. B 44, 6393–6399 (1991).
[CrossRef]

Fernandez, U.

A. Nemetz, U. Fernandez, W. Knoll, “Surface plasmon field-enhanced Raman spectroscopy with double gratings,” J. Appl. Phys. 75, 1582–1585 (1994).
[CrossRef]

Glass, N. E.

N. E. Glass, A. A. Maradudin, V. Celli, “Theory of surface-polariton resonances and field enhancements in light scattering from bigratings,” J. Opt. Soc. Am. 73, 1240–1248 (1983).
[CrossRef]

N. E. Glass, A. A. Maradudin, V. Celli, “Surface plasmons on a large-amplitude doubly periodic corrugated surface,” Phys. Rev. B 26, 5357–5365 (1982).
[CrossRef]

Goudonnet, J. P.

T. Inagaki, J. P. Goudonnet, J. W. Little, E. T. Arakawa, “Photoacoustic study of plasmon-resonance absorption in a bigrating,” J. Opt. Soc. Am. B 2, 432–439 (1985).
[CrossRef]

Hadley, L.

G. Hass, L. Hadley, “Optical properties of metals,” 2nd ed., in American Institute of Physics HandbookD. E. Gray, ed. (McGraw-Hill, New York, 1963), pp. 6–107.

Hamm, R. N.

R. H. Ritchie, E. T. Arakawa, J. J. Cowan, R. N. Hamm, “Surface-plasmon resonance effect in grating diffraction,” Phys. Rev. Lett. 21, 1530–1533 (1968).
[CrossRef]

Hanson, R. J.

C. L. Lawson, R. J. Hanson, Solving Least Squares Problems (Prentice-Hall, Englewood Cliffs, N. J., 1974).

Harris, J. B.

Hass, G.

G. Hass, L. Hadley, “Optical properties of metals,” 2nd ed., in American Institute of Physics HandbookD. E. Gray, ed. (McGraw-Hill, New York, 1963), pp. 6–107.

Ikuno, H.

H. Ikuno, K. Yasuura, “Improved point-matching method with application to scattering from a periodic surface,” IEEE Trans. Antennas Propag. AP-21, 657–662 (1973).
[CrossRef]

Inagaki, T.

T. Inagaki, J. P. Goudonnet, J. W. Little, E. T. Arakawa, “Photoacoustic study of plasmon-resonance absorption in a bigrating,” J. Opt. Soc. Am. B 2, 432–439 (1985).
[CrossRef]

Itakura, T.

K. Yasuura, T. Itakura, “Approximate algorithm by complete set of wave functions,” , Vol. 39, No. 1 (Kyushu University, Fukuoka, Japan, 1966), pp. 51–56.

K. Yasuura, T. Itakura, “Complete set of wave functions,” , Vol. 38, No. 4 (Kyushu University, Fukuoka, Japan, 1966), pp. 378–385.

K. Yasuura, T. Itakura, “Approximation method for wave functions (I),” , Vol. 38, No. 1 (Kyushu University, Fukuoka, Japan, 1965), pp. 72–77.

Kitson, S. C.

W. L. Barnes, T. W. Preist, S. C. Kitson, J. R. Sambles, N. P. K. Cotter, D. J. Nash, “Photonic gaps in the dispersion of surface plasmons on gratings,” Phys. Rev. B 51, 11164–11167 (1995).
[CrossRef]

Knoll, W.

A. Nemetz, U. Fernandez, W. Knoll, “Surface plasmon field-enhanced Raman spectroscopy with double gratings,” J. Appl. Phys. 75, 1582–1585 (1994).
[CrossRef]

Lawson, C. L.

C. L. Lawson, R. J. Hanson, Solving Least Squares Problems (Prentice-Hall, Englewood Cliffs, N. J., 1974).

Little, J. W.

T. Inagaki, J. P. Goudonnet, J. W. Little, E. T. Arakawa, “Photoacoustic study of plasmon-resonance absorption in a bigrating,” J. Opt. Soc. Am. B 2, 432–439 (1985).
[CrossRef]

Maradudin, A. A.

N. E. Glass, A. A. Maradudin, V. Celli, “Theory of surface-polariton resonances and field enhancements in light scattering from bigratings,” J. Opt. Soc. Am. 73, 1240–1248 (1983).
[CrossRef]

N. E. Glass, A. A. Maradudin, V. Celli, “Surface plasmons on a large-amplitude doubly periodic corrugated surface,” Phys. Rev. B 26, 5357–5365 (1982).
[CrossRef]

Matsuda, T.

T. Matsuda, Y. Okuno, “Numerical evaluation of plane-wave diffraction by a doubly periodic grating,” Radio Sci. 31, 1791–1798 (1996).
[CrossRef]

Nash, D. J.

W. L. Barnes, T. W. Preist, S. C. Kitson, J. R. Sambles, N. P. K. Cotter, D. J. Nash, “Photonic gaps in the dispersion of surface plasmons on gratings,” Phys. Rev. B 51, 11164–11167 (1995).
[CrossRef]

Nemetz, A.

A. Nemetz, U. Fernandez, W. Knoll, “Surface plasmon field-enhanced Raman spectroscopy with double gratings,” J. Appl. Phys. 75, 1582–1585 (1994).
[CrossRef]

Nevièr, M.

M. Nevièr, “The homogenous problem,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), Chap. 5, pp. 123–157.

Okuno, Y.

T. Matsuda, Y. Okuno, “Numerical evaluation of plane-wave diffraction by a doubly periodic grating,” Radio Sci. 31, 1791–1798 (1996).
[CrossRef]

Preist, T. W.

J. B. Harris, T. W. Preist, J. R. Sambles, R. N. Thorpe, R. A. Watts, “Optical response of bigratings,” J. Opt. Soc. Am. A 13, 2041–2049 (1996).
[CrossRef]

W. L. Barnes, T. W. Preist, S. C. Kitson, J. R. Sambles, N. P. K. Cotter, D. J. Nash, “Photonic gaps in the dispersion of surface plasmons on gratings,” Phys. Rev. B 51, 11164–11167 (1995).
[CrossRef]

Raeter, H.

H. Raeter, “Surface plasmon and roughness,” in Surface Polaritons, V. M. Argranovich, D. L. Mills, eds. (North-Holland, New York, 1982), Chap. 9, pp. 331–403.

Ritchie, R. H.

R. H. Ritchie, E. T. Arakawa, J. J. Cowan, R. N. Hamm, “Surface-plasmon resonance effect in grating diffraction,” Phys. Rev. Lett. 21, 1530–1533 (1968).
[CrossRef]

Sambles, J. R.

J. B. Harris, T. W. Preist, J. R. Sambles, R. N. Thorpe, R. A. Watts, “Optical response of bigratings,” J. Opt. Soc. Am. A 13, 2041–2049 (1996).
[CrossRef]

W. L. Barnes, T. W. Preist, S. C. Kitson, J. R. Sambles, N. P. K. Cotter, D. J. Nash, “Photonic gaps in the dispersion of surface plasmons on gratings,” Phys. Rev. B 51, 11164–11167 (1995).
[CrossRef]

S. J. Elston, G. P. Bryan-Brown, J. R. Sambles, “Polarisation conversion from diffraction gratings,” Phys. Rev. B 44, 6393–6399 (1991).
[CrossRef]

Thorpe, R. N.

Watts, R. A.

Yasuura, K.

H. Ikuno, K. Yasuura, “Improved point-matching method with application to scattering from a periodic surface,” IEEE Trans. Antennas Propag. AP-21, 657–662 (1973).
[CrossRef]

K. Yasuura, T. Itakura, “Complete set of wave functions,” , Vol. 38, No. 4 (Kyushu University, Fukuoka, Japan, 1966), pp. 378–385.

K. Yasuura, T. Itakura, “Approximation method for wave functions (I),” , Vol. 38, No. 1 (Kyushu University, Fukuoka, Japan, 1965), pp. 72–77.

K. Yasuura, T. Itakura, “Approximate algorithm by complete set of wave functions,” , Vol. 39, No. 1 (Kyushu University, Fukuoka, Japan, 1966), pp. 51–56.

IEEE Trans. Antennas Propag. (1)

H. Ikuno, K. Yasuura, “Improved point-matching method with application to scattering from a periodic surface,” IEEE Trans. Antennas Propag. AP-21, 657–662 (1973).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

C. C. Chen, “Transmission of microwave through perforated flat plates of finite thickness,” IEEE Trans. Microwave Theory Tech. MTT-21, 1–6 (1973).
[CrossRef]

J. Appl. Phys. (1)

A. Nemetz, U. Fernandez, W. Knoll, “Surface plasmon field-enhanced Raman spectroscopy with double gratings,” J. Appl. Phys. 75, 1582–1585 (1994).
[CrossRef]

J. Opt. Soc. Am. (1)

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

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

T. Inagaki, J. P. Goudonnet, J. W. Little, E. T. Arakawa, “Photoacoustic study of plasmon-resonance absorption in a bigrating,” J. Opt. Soc. Am. B 2, 432–439 (1985).
[CrossRef]

Phys. Rev. B (3)

W. L. Barnes, T. W. Preist, S. C. Kitson, J. R. Sambles, N. P. K. Cotter, D. J. Nash, “Photonic gaps in the dispersion of surface plasmons on gratings,” Phys. Rev. B 51, 11164–11167 (1995).
[CrossRef]

N. E. Glass, A. A. Maradudin, V. Celli, “Surface plasmons on a large-amplitude doubly periodic corrugated surface,” Phys. Rev. B 26, 5357–5365 (1982).
[CrossRef]

S. J. Elston, G. P. Bryan-Brown, J. R. Sambles, “Polarisation conversion from diffraction gratings,” Phys. Rev. B 44, 6393–6399 (1991).
[CrossRef]

Phys. Rev. Lett. (1)

R. H. Ritchie, E. T. Arakawa, J. J. Cowan, R. N. Hamm, “Surface-plasmon resonance effect in grating diffraction,” Phys. Rev. Lett. 21, 1530–1533 (1968).
[CrossRef]

Radio Sci. (1)

T. Matsuda, Y. Okuno, “Numerical evaluation of plane-wave diffraction by a doubly periodic grating,” Radio Sci. 31, 1791–1798 (1996).
[CrossRef]

Other (8)

H. Raeter, “Surface plasmon and roughness,” in Surface Polaritons, V. M. Argranovich, D. L. Mills, eds. (North-Holland, New York, 1982), Chap. 9, pp. 331–403.

M. Nevièr, “The homogenous problem,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), Chap. 5, pp. 123–157.

C. L. Lawson, R. J. Hanson, Solving Least Squares Problems (Prentice-Hall, Englewood Cliffs, N. J., 1974).

Theoretical foundations of the Yasuura method can be found in Ref. 15-17.

K. Yasuura, T. Itakura, “Approximation method for wave functions (I),” , Vol. 38, No. 1 (Kyushu University, Fukuoka, Japan, 1965), pp. 72–77.

K. Yasuura, T. Itakura, “Complete set of wave functions,” , Vol. 38, No. 4 (Kyushu University, Fukuoka, Japan, 1966), pp. 378–385.

K. Yasuura, T. Itakura, “Approximate algorithm by complete set of wave functions,” , Vol. 39, No. 1 (Kyushu University, Fukuoka, Japan, 1966), pp. 51–56.

G. Hass, L. Hadley, “Optical properties of metals,” 2nd ed., in American Institute of Physics HandbookD. E. Gray, ed. (McGraw-Hill, New York, 1963), pp. 6–107.

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Figures (11)

Fig. 1
Fig. 1

Bisinusoidal grating.

Fig. 2
Fig. 2

Specification of the incident wave and coordinate system.

Fig. 3
Fig. 3

Comparison of resonance absorption curves calculated by the present method with other existing theoretical results.7 Solid curves show our results, and dotted curves are taken from Fig. 2 of Ref. 7.

Fig. 4
Fig. 4

Total diffraction efficiencies ρtotal as functions of θ.

Fig. 5
Fig. 5

A10-1TM as functions of θ.

Fig. 6
Fig. 6

Field distributions for the total and (0, -1)st-order evanescent mode field when plasmon-resonance absorption occurs.

Fig. 7
Fig. 7

Energy flows of the total field when plasmon-resonance absorption occurs ϕ=30°, θ=23.3°.

Fig. 8
Fig. 8

Diffraction efficiencies of (a) ρ00 and (b) ρ00TE and ρ00TM as functions of θ; parameters are the same as in Fig. 4.

Fig. 9
Fig. 9

Resonance angles θr as functions of azimuthal angle ϕ.

Fig. 10
Fig. 10

Energy flows of the total field when simultaneous resonance absorption occurs ϕ=45°, θ=12.2°.

Fig. 11
Fig. 11

Total diffraction efficiencies ρtotal as functions of θ.

Tables (1)

Tables Icon

Table 1 Propagation Constants and Estimated Resonance Anglesa

Equations (23)

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

z=η(x, y)=(h/4)[sin(2πx/d)+sin(2πy/d)].
EiHi(P)=eihiexp(iki·P-iωt),
hi=(1/ωμ0)k×ei.
ki=(α, β,-γ)T,
ei=cos δ eTE+sin δeTM,
φqmnTE(P)=eqmnTE exp(ikqmn·P),
eqmnTE=kqmn×iZ/|kqmn×iZ|,
φqmnTM(P)=eqmnTM exp(ikqmn·P),
eqmnTM=eqmnTE×kqmn/|eqmnTE×kqmn|,m, n=0, ±1, ±2,(PVq; q=1, 2),
k1mn=(αm, βn, γ1mn),k2mn=(αm, βn,-γ2mn),
αm=α+2mπ/d,βn=β+2nπ/d,
γqmn=[(nqk)2-(αm2+βn2)]1/2
Re(γqmn)0,Im(γqmn)0.
EqNdHqNd(P)=m=-NNn=-NNAqmnTE(N)φqmnTEψqmnTE(P)+m=-NNn=-NNAqmnTM(N)φqmnTMψqmnTE(P)(PVq, q=1, 2),
ψqmnl=(1/ωμ0)kqmn×φqmnl(l=TE, TM).
IN=S0|v×[E1Nd+Ei-E2Nd](s)|2dS+W2 S0|v×[H1Nd+Hi-H2Nd](s)|2dS,
ρmn=ρmnTE+ρmnTM,
ρmnTE=γ1mnγ|A1mnTE(N)|2,ρmnTM=γ1mnγ|A1mnTM(N)|2.
{P(x, y, z):0xd, 0yd, z=(h/4)[sin(2πx/d)+sin(2πy/d)]+0.01d}.
αˆSP=sin θc cos ϕ+mλ/d,
βˆSP=sin θc sin ϕ+nλ/d.
Re{αˆSP}=sin θr cos ϕ+mλ/d,
Re{βˆSP}=sin θr sin ϕ+nλ/d.

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