Abstract

We theoretically studied the influence of an infinite set of waveguides on the evanescent field of Bloch waves at the surface of a one-dimensional photonic crystal (PC) excited by a TE Gaussian beam undergoing total internal reflection. The set of waveguides is regarded as a periodic inhomogeneous medium with period a. Numerical results are presented for the case in which a is greater than λ, which is the wavelength used to excite the surface mode. When the waveguide tip is very close to the surface of the PC, a fraction of the surface wave is reflected by the tip, producing an interference pattern that can be observed in the near field. In this case, the system simulates scanning tunneling optical microscopy in 2D geometry, and an image of the field distribution of the surface mode is obtained by quantifying the flux energy throughout the waveguide.

© 2014 Optical Society of America

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References

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  1. R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Electromagnetic Bloch waves at the surface of a photonic crystal,” Phys. Rev. B 44, 10961–10964 (1991).
    [CrossRef]
  2. W. M. Robertson, G. Arjavalingam, R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Observation of surface photons on periodic dielectric arrays,” Opt. Lett. 18, 528–530 (1993).
    [CrossRef]
  3. F. Ramos-Mendieta and P. Halevi, “Electromagnetic surface modes of a dielectric superlatice: the supercell method,” J. Opt. Soc. Am. B 14, 370–381 (1997).
    [CrossRef]
  4. W. M. Robertson and M. S. May, “Surface electromagnetic wave excitation on one-dimensional photonic band-gap arrays,” Appl. Phys. Lett. 74, 1800–1802 (1999).
    [CrossRef]
  5. F. Ramos-Mendieta and P. Halevi, “Surface electromagnetic waves in two-dimensional photonic crystals: effect of the position of the surface plane,” Phys. Rev. B 59, 15112–15120 (1999).
    [CrossRef]
  6. A. S. Ramírez-Duverger, J. Gaspar-Armenta, and R. García-Llamas, “Surface wave effect on light scattering from one dimensional photonic crystals,” Opt. Commun. 277, 302–309 (2007).
    [CrossRef]
  7. L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University, 2006).
  8. D. Courjon, Near-Field Microscopy and Near-Field Optics (Imperial College, 2003).
  9. D. W. Pohl and D. Courjon, eds. Near Field Optics (Kluwer, 1992).
  10. Y. Kulchin, O. Vitrik, and A. Kuchmizhak, “Near-field optical probe based on the fiber Fabry–Perot interferometer with protruding evanescent light source,” J. Opt. Soc. Am. B 30, 598–602 (2013).
    [CrossRef]
  11. C. C. Yeh, K. F. Casey, and Z. A. Kaprielian, “Transverse magnetic wave propagation in sinusoidally dielectric stratified media,” IEEE Trans. Microwave Theor. Tech. MTT-13, 297–302 (1965).
    [CrossRef]
  12. C. B. Burckhardt, “Diffraction of a plane wave at a sinusoidally stratified dielectric grating,” J. Opt. Soc. Am. 56, 1502–1509 (1966).
    [CrossRef]
  13. F. G. Kaspar, “Diffraction by thick, periodically stratified gratings with complex dielectric constant,” J. Opt. Soc. Am. 63, 37–45 (1973).
    [CrossRef]
  14. E. Glass and A. A. Maradudin, “Diffraction of light by a periodically modulated dielectric half-space,” Phys. Rev. B 29, 1840–1847 (1984).
    [CrossRef]

2013 (1)

2007 (1)

A. S. Ramírez-Duverger, J. Gaspar-Armenta, and R. García-Llamas, “Surface wave effect on light scattering from one dimensional photonic crystals,” Opt. Commun. 277, 302–309 (2007).
[CrossRef]

1999 (2)

W. M. Robertson and M. S. May, “Surface electromagnetic wave excitation on one-dimensional photonic band-gap arrays,” Appl. Phys. Lett. 74, 1800–1802 (1999).
[CrossRef]

F. Ramos-Mendieta and P. Halevi, “Surface electromagnetic waves in two-dimensional photonic crystals: effect of the position of the surface plane,” Phys. Rev. B 59, 15112–15120 (1999).
[CrossRef]

1997 (1)

1993 (1)

1991 (1)

R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Electromagnetic Bloch waves at the surface of a photonic crystal,” Phys. Rev. B 44, 10961–10964 (1991).
[CrossRef]

1984 (1)

E. Glass and A. A. Maradudin, “Diffraction of light by a periodically modulated dielectric half-space,” Phys. Rev. B 29, 1840–1847 (1984).
[CrossRef]

1973 (1)

1966 (1)

1965 (1)

C. C. Yeh, K. F. Casey, and Z. A. Kaprielian, “Transverse magnetic wave propagation in sinusoidally dielectric stratified media,” IEEE Trans. Microwave Theor. Tech. MTT-13, 297–302 (1965).
[CrossRef]

Arjavalingam, G.

Brommer, K. D.

W. M. Robertson, G. Arjavalingam, R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Observation of surface photons on periodic dielectric arrays,” Opt. Lett. 18, 528–530 (1993).
[CrossRef]

R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Electromagnetic Bloch waves at the surface of a photonic crystal,” Phys. Rev. B 44, 10961–10964 (1991).
[CrossRef]

Burckhardt, C. B.

Casey, K. F.

C. C. Yeh, K. F. Casey, and Z. A. Kaprielian, “Transverse magnetic wave propagation in sinusoidally dielectric stratified media,” IEEE Trans. Microwave Theor. Tech. MTT-13, 297–302 (1965).
[CrossRef]

Courjon, D.

D. Courjon, Near-Field Microscopy and Near-Field Optics (Imperial College, 2003).

García-Llamas, R.

A. S. Ramírez-Duverger, J. Gaspar-Armenta, and R. García-Llamas, “Surface wave effect on light scattering from one dimensional photonic crystals,” Opt. Commun. 277, 302–309 (2007).
[CrossRef]

Gaspar-Armenta, J.

A. S. Ramírez-Duverger, J. Gaspar-Armenta, and R. García-Llamas, “Surface wave effect on light scattering from one dimensional photonic crystals,” Opt. Commun. 277, 302–309 (2007).
[CrossRef]

Glass, E.

E. Glass and A. A. Maradudin, “Diffraction of light by a periodically modulated dielectric half-space,” Phys. Rev. B 29, 1840–1847 (1984).
[CrossRef]

Halevi, P.

F. Ramos-Mendieta and P. Halevi, “Surface electromagnetic waves in two-dimensional photonic crystals: effect of the position of the surface plane,” Phys. Rev. B 59, 15112–15120 (1999).
[CrossRef]

F. Ramos-Mendieta and P. Halevi, “Electromagnetic surface modes of a dielectric superlatice: the supercell method,” J. Opt. Soc. Am. B 14, 370–381 (1997).
[CrossRef]

Hecht, B.

L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University, 2006).

Joannopoulos, J. D.

W. M. Robertson, G. Arjavalingam, R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Observation of surface photons on periodic dielectric arrays,” Opt. Lett. 18, 528–530 (1993).
[CrossRef]

R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Electromagnetic Bloch waves at the surface of a photonic crystal,” Phys. Rev. B 44, 10961–10964 (1991).
[CrossRef]

Kaprielian, Z. A.

C. C. Yeh, K. F. Casey, and Z. A. Kaprielian, “Transverse magnetic wave propagation in sinusoidally dielectric stratified media,” IEEE Trans. Microwave Theor. Tech. MTT-13, 297–302 (1965).
[CrossRef]

Kaspar, F. G.

Kuchmizhak, A.

Kulchin, Y.

Maradudin, A. A.

E. Glass and A. A. Maradudin, “Diffraction of light by a periodically modulated dielectric half-space,” Phys. Rev. B 29, 1840–1847 (1984).
[CrossRef]

May, M. S.

W. M. Robertson and M. S. May, “Surface electromagnetic wave excitation on one-dimensional photonic band-gap arrays,” Appl. Phys. Lett. 74, 1800–1802 (1999).
[CrossRef]

Meade, R. D.

W. M. Robertson, G. Arjavalingam, R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Observation of surface photons on periodic dielectric arrays,” Opt. Lett. 18, 528–530 (1993).
[CrossRef]

R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Electromagnetic Bloch waves at the surface of a photonic crystal,” Phys. Rev. B 44, 10961–10964 (1991).
[CrossRef]

Novotny, L.

L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University, 2006).

Ramírez-Duverger, A. S.

A. S. Ramírez-Duverger, J. Gaspar-Armenta, and R. García-Llamas, “Surface wave effect on light scattering from one dimensional photonic crystals,” Opt. Commun. 277, 302–309 (2007).
[CrossRef]

Ramos-Mendieta, F.

F. Ramos-Mendieta and P. Halevi, “Surface electromagnetic waves in two-dimensional photonic crystals: effect of the position of the surface plane,” Phys. Rev. B 59, 15112–15120 (1999).
[CrossRef]

F. Ramos-Mendieta and P. Halevi, “Electromagnetic surface modes of a dielectric superlatice: the supercell method,” J. Opt. Soc. Am. B 14, 370–381 (1997).
[CrossRef]

Rappe, A. M.

W. M. Robertson, G. Arjavalingam, R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Observation of surface photons on periodic dielectric arrays,” Opt. Lett. 18, 528–530 (1993).
[CrossRef]

R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Electromagnetic Bloch waves at the surface of a photonic crystal,” Phys. Rev. B 44, 10961–10964 (1991).
[CrossRef]

Robertson, W. M.

W. M. Robertson and M. S. May, “Surface electromagnetic wave excitation on one-dimensional photonic band-gap arrays,” Appl. Phys. Lett. 74, 1800–1802 (1999).
[CrossRef]

W. M. Robertson, G. Arjavalingam, R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Observation of surface photons on periodic dielectric arrays,” Opt. Lett. 18, 528–530 (1993).
[CrossRef]

Vitrik, O.

Yeh, C. C.

C. C. Yeh, K. F. Casey, and Z. A. Kaprielian, “Transverse magnetic wave propagation in sinusoidally dielectric stratified media,” IEEE Trans. Microwave Theor. Tech. MTT-13, 297–302 (1965).
[CrossRef]

Appl. Phys. Lett. (1)

W. M. Robertson and M. S. May, “Surface electromagnetic wave excitation on one-dimensional photonic band-gap arrays,” Appl. Phys. Lett. 74, 1800–1802 (1999).
[CrossRef]

IEEE Trans. Microwave Theor. Tech. (1)

C. C. Yeh, K. F. Casey, and Z. A. Kaprielian, “Transverse magnetic wave propagation in sinusoidally dielectric stratified media,” IEEE Trans. Microwave Theor. Tech. MTT-13, 297–302 (1965).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. B (2)

Opt. Commun. (1)

A. S. Ramírez-Duverger, J. Gaspar-Armenta, and R. García-Llamas, “Surface wave effect on light scattering from one dimensional photonic crystals,” Opt. Commun. 277, 302–309 (2007).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. B (3)

R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Electromagnetic Bloch waves at the surface of a photonic crystal,” Phys. Rev. B 44, 10961–10964 (1991).
[CrossRef]

E. Glass and A. A. Maradudin, “Diffraction of light by a periodically modulated dielectric half-space,” Phys. Rev. B 29, 1840–1847 (1984).
[CrossRef]

F. Ramos-Mendieta and P. Halevi, “Surface electromagnetic waves in two-dimensional photonic crystals: effect of the position of the surface plane,” Phys. Rev. B 59, 15112–15120 (1999).
[CrossRef]

Other (3)

L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University, 2006).

D. Courjon, Near-Field Microscopy and Near-Field Optics (Imperial College, 2003).

D. W. Pohl and D. Courjon, eds. Near Field Optics (Kluwer, 1992).

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Figures (4)

Fig. 1.
Fig. 1.

(a) Set of equally spaced waveguides with dielectric constant εc and width d surrounded by nonabsorbing medium with dielectric constant εe<εc. The distance of separation between waveguides is a. (b) The PC is composed of 10 films, and it has a period p=d2+d3, with d2=d4=d6=d8=d10 and d3=d5=d7=d9. The thickness of the last film, d11, breaks the symmetry of the PC in order to obtain the surface mode. The waveguides are separated by a distance d12 from the truncated 1D PC for a vacuum gap.

Fig. 2.
Fig. 2.

(a) Reflection as a function of the energy (wavelength) for a fixed angle of incidence θi=60.13°. The curve with red solid circles (blue solid squares) corresponds to the reflection of the 1D PC near to (far from) a waveguide. The air gap is λ0/63, where λ is the wavelength of 633 nm. The curve was calculated with energy steps of 0.00025 eV. (b) Log total power (red solid squares) in the IM and the power in the waveguide (blue solid circles) as functions of the incident energy at a 60.13° fixed angle of incidence.

Fig. 3.
Fig. 3.

Modulus of the electric field projected into the yz plane. The horizontal black lines correspond to the interfaces between each film that form the 1D PC. The vertical lines above the top of the 1D PC help to visualize one of the waveguides in the set; the remainder are outside of the limits of this graph. The image was calculated using steps of Δy=80.6nm and Δz=20.1nm.

Fig. 4.
Fig. 4.

Near-field power (power flowing inside of the waveguide) plotted as a function of the waveguide end-tip position for both distances, perpendicular (z) and parallel (y) to the surface, at resonant conditions, i.e., E=1.92325eV and θi=60.13°.

Tables (1)

Tables Icon

Table 1. 1D PC is Composed of Alternate Materials That Have High and Low Refractive Indexesa

Equations (46)

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E13tx(y,z)=n=+An,k(13)exp[+i(qy+gyn)y]exp(+iξk(13)z),
1ε13(y)×[×E⃗13t(r⃗)]=ω2μ0ε0E⃗13t(r⃗).
1ε13(y)=n=[ε131]nexp(+igyny).
n=[q02δn,n[ε131]nn(qy+gyn)(qy+gyn)]An,k(13)=ξk(13)ξk(13)n=[ε131]nnAn,k(13),
n=[q02ηn,nl=ηn,l[ε131]ln(qy+gyn)(qy+gyn)]An,k(13)=ξk(13)ξk(13)An,k(13),
Elx(y,z)=n=eltx(βn)exp{+i[βny+γl(βn)(zzl1)]}+n=elrx(βn)exp{+i[βnyγl(βn)(zzl1)]},
βn=μ1ε1ωsinθi+(2π/a)n,
e1tx(βn)={exp[(βn2β02)w2/2]exp(+iβnyd)for|βn|β00for|βn|>β0.
e2tx(βn)+e2rx(βn)=e1tx(βn)+e1rx(βn),
γ2(βn)μ2e2tx(βn)γ2(βn)μ2e2rx(βn)=γ1(βn)μ1e1tx(βn)γ1(βn)μ1e1rx(βn).
e2tx(βn)=+C1(βn)e1tx(βn)D1(βn)e1rx(βn)F2(βn),
e2rx(βn)=D1(βn)e1tx(βn)C1(βn)e1rx(βn)F2(βn),
C1(βn)=γ1(βn)μ1+γ2(βn)μ2,
D1(βn)=γ1(βn)μ1γ2(βn)μ2,
F2(βn)=2γ2(βn)μ2.
e3tx(βn)+e3rx(βn)=e2tx(βn)exp[+iγ2(βn)z2]+e2rx(βn)exp[iγ2(βn)z2],
γ3(βn)μ3e3tx(βn)γ3(βn)μ3e3rx(βn)=γ2(βn)μ2e2tx(βn)exp[+iγ2(βn)z2]γ2(βn)μ2e2rx(βn)exp[iγ2(βn)z2].
e3tx(βn)=C2(βn)e2tx(βn)exp[+iγ2(βn)z2]D2(βn)e2rx(βn)exp[iγ2(βn)z2]F3(βn),
e3rx(βn)=D2(βn)e2tx(βn)exp[+iγ2(βn)z2]C2(βn)e2rx(βn)exp[iγ2(βn)z2]F3(βn),
e3tx(βn)=U2(βn)e1tx(βn)V2(βn)e1rx(βn)]F3r(βn),
e3rx(βn)=U2(βn)e1tx(βn)V2(βn)e1rx(βn)]F3r(βn),
V2(βn)=D1(βn)D2(βn)exp[+iγ2(βn)d2]+C1(βn)C2(βn)exp[iγ2(βn)d2],
U2(βn)=C1(βn)D2(βn)exp[+iγ2(βn)d2]+D1(βn)C2(βn)exp[iγ2(βn)d2],
U2(βn)=C1(βn)C2(βn)exp[+iγ2(βn)d2]+D1(βn)D2(βn)exp[iγ2(βn)d2],
V2(βn)=D1(βn)C2(βn)exp[+iγ2(βn)d2]+C1(βn)D2(βn)exp[iγ2(βn)d2],
F3r(βn)=F2(βn)F3(βn).
e12tx(βn)=U11(βn)e1tx(βn)V11(βn)e1rx(βn)]F12r(βn),
e12rx(βn)=U11(βn)e1tx(βn)V11(βn)e1rx(βn)]F12r(βn),
V11(βn)=V10(βn)D11(βn)exp[+iγ11(βn)d11]+V10(βn)C11(βn)exp[iγ11(βn)d11],
U11(βn)=U10(βn)D11(βn)exp[+iγ11(βn)d11]+U10(βn)C11(βn)exp[iγ11(βn)d11],
U11(βn)=U10(βn)C11(βn)exp[+iγ11(βn)d11]+U10(βn)D11(βn)exp[iγ11(βn)d11],
V11(βn)=V10(βn)C11(βn)exp[+iγ11(βn)d11]+V10(βn)D11(βn)exp[iγ11(βn)d11],
F12r(βn)=F11r(βn)F12(βn).
E13tx(r⃗)=k=1n=+An,k(13)ek(13tx)exp{i[βny+ξk(13)(zz12)]},
k=1An,k(13)ek(13tx)=e12tx(βn)exp[+iγ12(βn)z12]+e12rx(βn)exp[iγ12(βn)z12],
k=1ξk(13)μ13An,k(13)ek(13tx)=γ12(βn)μ12e12tx(βn)exp[+iγ12(βn)z12]γ12(βn)μ12e12rx(βn)exp[iγ12(βn)z12].
F12(βn)F11r(βn)k=1An,k(13)ek(13tx)=Ht(12)(βn)e1tx(βn)Hr(12)(βn)e1rx(βn),
F11r(βn)k=12ξk(13)μ13An,k(13)ek(13tx)=Gt(12)(βn)e1tx(βn)Gr(12)(βn)e1rx(βn),
Gt(12)(βn)=U11(βn)exp[+iγ12(βn)d12]+U11(βn)exp[iγ12(βn)d12],
Gr(12)(βn)=V11(βn)exp[+iγ12(βn)d12]+V11(βn)exp[iγ12(βn)d12],
Hr(12)(βn)=V11(βn)exp[+iγ12(βn)d12]V11(βn)exp[iγ12(βn)d12],
Ht(12)(βn)=U11(βn)exp[+iγ12(βn)d12]U11(βn)exp[iγ12(βn)d12].
k=1Tn,k(12)ek(13tx)=Un(12),
Tn,k(12)=[Gr(12)(βn)2γ12(βn)μ12Hr(12)(βn)2ξk(13)μ13]An,k(13)
Un(12)=[Gr(12)(βn)Ht(12)(βn)Hr(12)(βn)Gr(12)(βn)]F11r(βn)e1tx(βn).
P13wz=μ1μ13ξu(13)n,nAn,u(13)eu(13tx)(An,u(13)eu(13tx))*×fsinc[π(nn)f]/Pi,

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