Abstract

Light transmission through a Fabry–Perot resonator (FPR) holding a dielectric cylinder rod is considered. For the cylinder parallel to mirrors of the FPR and the mirrors mimicked by the δ functions we present an exact analytical theory. It is shown that light transmits only for resonant incident angles, αm, similar to the empty FPR. However after transmission the light scatters into different resonant angles, αm, performing resonant angular conversion. We compare the theory with experiment in the FPR, exploring multilayer films as the mirrors and glass cylinder with diameter coincided with the distance between the FPR mirrors. The measured values of angular light conversion agree qualitatively with the theoretical results.

© 2014 Optical Society of America

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References

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  1. M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University, 1999).
  2. D. Maystre, S. Enoch, and G. Tayeb, “Scattering matrix method applied to photonic crystals,” in Electromagnetic Theory and Applications for Photonic Crystals, K. Yasumoto, ed. (Taylor & Francis, 2006).
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    [CrossRef]
  4. D. E. Lawrence and K. Sarabandi, “Electromagnetic scattering from a dielectric cylinder buried beneath a slightly rough surface,” IEEE Trans. Antennas Propag. 50, 1368–1376 (2002).
    [CrossRef]
  5. S.-C. Lee, “Scattering by a radially stratified infinite cylinder buried in an absorbing half-space,” J. Opt. Soc. Am. A 30, 565–572 (2013).
    [CrossRef]
  6. S. Flügge, Practical Quantum Mechanics I (Springer-Verlag, 1971).
  7. P. Markoš and C. M. Soukoulis, Wave Propagation From Electrons to Photonic Crystals and Left-Handed Materials (Princeton University, 2008).
  8. V. G. Arkhipkin, V. A. Gunyakov, S. A. Myslivets, V. Y. Zyryanov, and V. F. Shabanov, “Angular tuning of defect modes spectrum in the one-dimensional photonic crystal with liquid-crystal layer,” Eur. Phys. J. E 24, 297–302 (2007).
    [CrossRef]

2013 (1)

2007 (1)

V. G. Arkhipkin, V. A. Gunyakov, S. A. Myslivets, V. Y. Zyryanov, and V. F. Shabanov, “Angular tuning of defect modes spectrum in the one-dimensional photonic crystal with liquid-crystal layer,” Eur. Phys. J. E 24, 297–302 (2007).
[CrossRef]

2002 (1)

D. E. Lawrence and K. Sarabandi, “Electromagnetic scattering from a dielectric cylinder buried beneath a slightly rough surface,” IEEE Trans. Antennas Propag. 50, 1368–1376 (2002).
[CrossRef]

1997 (1)

Arkhipkin, V. G.

V. G. Arkhipkin, V. A. Gunyakov, S. A. Myslivets, V. Y. Zyryanov, and V. F. Shabanov, “Angular tuning of defect modes spectrum in the one-dimensional photonic crystal with liquid-crystal layer,” Eur. Phys. J. E 24, 297–302 (2007).
[CrossRef]

Borghi, R.

Born, M.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University, 1999).

Enoch, S.

D. Maystre, S. Enoch, and G. Tayeb, “Scattering matrix method applied to photonic crystals,” in Electromagnetic Theory and Applications for Photonic Crystals, K. Yasumoto, ed. (Taylor & Francis, 2006).

Flügge, S.

S. Flügge, Practical Quantum Mechanics I (Springer-Verlag, 1971).

Frezza, F.

Gunyakov, V. A.

V. G. Arkhipkin, V. A. Gunyakov, S. A. Myslivets, V. Y. Zyryanov, and V. F. Shabanov, “Angular tuning of defect modes spectrum in the one-dimensional photonic crystal with liquid-crystal layer,” Eur. Phys. J. E 24, 297–302 (2007).
[CrossRef]

Lawrence, D. E.

D. E. Lawrence and K. Sarabandi, “Electromagnetic scattering from a dielectric cylinder buried beneath a slightly rough surface,” IEEE Trans. Antennas Propag. 50, 1368–1376 (2002).
[CrossRef]

Lee, S.-C.

Markoš, P.

P. Markoš and C. M. Soukoulis, Wave Propagation From Electrons to Photonic Crystals and Left-Handed Materials (Princeton University, 2008).

Maystre, D.

D. Maystre, S. Enoch, and G. Tayeb, “Scattering matrix method applied to photonic crystals,” in Electromagnetic Theory and Applications for Photonic Crystals, K. Yasumoto, ed. (Taylor & Francis, 2006).

Myslivets, S. A.

V. G. Arkhipkin, V. A. Gunyakov, S. A. Myslivets, V. Y. Zyryanov, and V. F. Shabanov, “Angular tuning of defect modes spectrum in the one-dimensional photonic crystal with liquid-crystal layer,” Eur. Phys. J. E 24, 297–302 (2007).
[CrossRef]

Santarsiero, M.

Sarabandi, K.

D. E. Lawrence and K. Sarabandi, “Electromagnetic scattering from a dielectric cylinder buried beneath a slightly rough surface,” IEEE Trans. Antennas Propag. 50, 1368–1376 (2002).
[CrossRef]

Schettini, G.

Shabanov, V. F.

V. G. Arkhipkin, V. A. Gunyakov, S. A. Myslivets, V. Y. Zyryanov, and V. F. Shabanov, “Angular tuning of defect modes spectrum in the one-dimensional photonic crystal with liquid-crystal layer,” Eur. Phys. J. E 24, 297–302 (2007).
[CrossRef]

Soukoulis, C. M.

P. Markoš and C. M. Soukoulis, Wave Propagation From Electrons to Photonic Crystals and Left-Handed Materials (Princeton University, 2008).

Tayeb, G.

D. Maystre, S. Enoch, and G. Tayeb, “Scattering matrix method applied to photonic crystals,” in Electromagnetic Theory and Applications for Photonic Crystals, K. Yasumoto, ed. (Taylor & Francis, 2006).

Wolf, E.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University, 1999).

Zyryanov, V. Y.

V. G. Arkhipkin, V. A. Gunyakov, S. A. Myslivets, V. Y. Zyryanov, and V. F. Shabanov, “Angular tuning of defect modes spectrum in the one-dimensional photonic crystal with liquid-crystal layer,” Eur. Phys. J. E 24, 297–302 (2007).
[CrossRef]

Eur. Phys. J. E (1)

V. G. Arkhipkin, V. A. Gunyakov, S. A. Myslivets, V. Y. Zyryanov, and V. F. Shabanov, “Angular tuning of defect modes spectrum in the one-dimensional photonic crystal with liquid-crystal layer,” Eur. Phys. J. E 24, 297–302 (2007).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

D. E. Lawrence and K. Sarabandi, “Electromagnetic scattering from a dielectric cylinder buried beneath a slightly rough surface,” IEEE Trans. Antennas Propag. 50, 1368–1376 (2002).
[CrossRef]

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

Other (4)

S. Flügge, Practical Quantum Mechanics I (Springer-Verlag, 1971).

P. Markoš and C. M. Soukoulis, Wave Propagation From Electrons to Photonic Crystals and Left-Handed Materials (Princeton University, 2008).

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University, 1999).

D. Maystre, S. Enoch, and G. Tayeb, “Scattering matrix method applied to photonic crystals,” in Electromagnetic Theory and Applications for Photonic Crystals, K. Yasumoto, ed. (Taylor & Francis, 2006).

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Figures (4)

Fig. 1.
Fig. 1.

Single dielectric cylinder shown by red (left circle) and its fictitious image (yellow, right circle) spaced symmetrically relative to the single mirror.

Fig. 2.
Fig. 2.

Single dielectric cylinder shown by the red circle (above the d) is spaced between two parallel mirrors (a view from above). The cylinder is parallel to the mirrors. The first sequence of FSs, shown by yellow and labeled by the index jR, begins after the reflection of the actual cylinder at the right mirror to produce the jR=1 FS, then it reflects at the left mirror to produce the jR=2 FS, etc. The second set of FSs, shown by green and labeled by the index jL, begins after the reflection at the left mirror.

Fig. 3.
Fig. 3.

(a) Optical setup of a laser beam transmission through the FPR holding the dielectric cylinder and (b) transmission spectra of single mirror (dash line) and double mirrors (solid line) for green light, λ=532nm.

Fig. 4.
Fig. 4.

Transmittance versus scattering angles of the light beams, for d=7.09μm. The solid line shows the intensity, |ψVP|2, normalized by the input intensity in a log scale, calculated by use of Eq. (26), while diamonds show the experimental light transmittance in a log scale, measured for the resonant angles [Eq. (1)].

Equations (26)

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cosαm=mλ2d,
2ψ(x,y)+k2ϵ(x,y)ψ(x,y)=0.
ψ(0,y)=ψ(+0,y),ψ(0,y)xψ(+0,y)x=μψ(0,y).
ψ(cyl)(r,ϕ)=mbm(s)Hm(1)(kr)exp(imϕ),
bm(s)=imSm=qJm(kR)Jm(qR)Jm(kR)Jm(qR)Hm(kR)Jm(qR)qHm(kR)Jm(qR),
ψ(t)(r,ϕ)=nbn(t)Hn(1)(kr)exp(inϕ),
ψ(r)(r,ϕ)=nbn(r)Hn(kr)exp[in(πϕ)],
bn(t)=mFnmbm(s),bn(r)=bn(s)+bn(t)
Fm(1)={γ1+γ(q2q1)q1m+1,m1,γ1+γ(q2q1)q2m1,m1,
Fm(2)={γ1+γ(q2q1)q1m+1,m1,γ1+γ(q2q1)q2m1,m1,
q1=11+4γ22γ,q2=2γ1+1+4γ2,
[Aexp(ikxx)+Bexp(ikxx)]exp(ikyy),<d/2<x<d/2.
A=1+iμ2kx+μ22kx2coskxd+μ2kxsinkxd(1+iμkx)eikxd+μ2kx2(eikxdeikxd),B=iμ2kx(1iμ2kx)iμ2kxcoskxd+μ2kxsinkxd(1+iμkx)eikxd+μ2kx2(eikxdeikxd).
ψempty(r,ϕ)=mimJm(kr)[Aeim(ϕα)+(1)mBeim(ϕ+α)]=mam(0)Jm(kr)eimϕ,
ψinc(r,ϕ)=ψempty(r,ϕ)+ψfict(r,ϕ)=m(am(0)+am(fict))Jm(kr)eimϕ=mam(tot)Jm(kr)eimϕ,
ψscat(r,ϕ)=mbm(tot)Hm(1)(kr)eimϕ,
ψ(1)(r1R,ϕ1R)=nbn(1)Hn(1)(kr1R)exp(inϕ1R),
bn(1)=mDnmbm(tot),
ψ(2)(r2R,ϕ2R)=nbn(2)Hn(1)(kr2R)exp(inϕ2R),
bn(2)=mDnmbm(1)=m(D^2)nmbm(tot).
ψ=njR=1m(D^k)nmbm(tot)Hn(1)(krjR)exp[in(πjR+(1)jRϕjR)].
Hn(1)(krjR)einϕjR=meiπ(nm)(jR+1)Hmn(1)(jRkd)Jm(kr)eimϕ,
am=am(0)+nG^mnbn(tot),
Gm(R)=jRn(1)jRHmn(1)(jRkd)(D^k)n.
bm(tot)=Smam(0)+n(Gmn(R)+Gmn(L))bn(tot),
ψVP=jmn(F^D^j)mnbn(tot)Hm(1)(qRj)eimϕj,

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