Abstract

A theory is presented for the interpretation of scanning near-field optical microscope measurements on pulses propagating in waveguiding structures. It is shown how the dispersion characteristics of the propagating guided modes may be derived from such experiments. Then it is demonstrated how to calibrate the scanning tip position and to derive experimental values for reflection and transmission of modes in identical single-mode waveguides connected to a photonic device such as a micro cavity.

© 2004 Optical Society of America

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  1. M. L. M. Balistreri, J. P. Korterik, G. J. Veldhuis, L. Kuipers, N. F. Van Hulst, “Quantitative photon tunneling and shear-force microscopy of planar WG splitters and mixers,” J. Appl. Phys. 89, 3307–3314 (2001).
    [CrossRef]
  2. R. Bachelot, P. Gleyzes, A. C. Boccara, “Near-field optical microscope based on local perturbation of a diffraction spot,” Opt. Lett. 20, 1924–1926 (1995).
    [CrossRef] [PubMed]
  3. M. L. M. Balistreri, J. P. Korterik, L. Kuipers, N. F. Van Hulst, “Tracking femtosecond laser pulses in space and time,” Science 294, 1080–1082 (2001).
    [CrossRef] [PubMed]
  4. S. I. Bozhevolnyi, B. Vohnsen, “Near-field imaging of optical phase and its singularities,” Opt. Commun. 212, 217–223 (2002).
    [CrossRef]
  5. D. J. W. Klunder, M. L. M. Balistreri, F. C. Blom, H. J. W. M. Hoekstra, A. Driessen, L. Kuipers, N. F. Van Hulst, “High-resolution photon-scanning tunneling microscope measurements of the whispering gallery modes in a cylindrical microresonator,” IEEE Photon. Technol. Lett. 12, 1531–1533 (2000).
    [CrossRef]
  6. E. Fluck, A. M. Otter, J. P. Korterik, M. L. M. Balistreri, L. Kuipers, N. F. Van Hulst, “Local phase measurements of light in a one-dimensional photonic crystal,” J. Microsc. 202, 104–109 (2001).
    [CrossRef] [PubMed]
  7. M. L. M. Balistreri, “Coherent imaging of guided optical fields,” ISBN 90-365-14924, Ph.D. thesis (University of Twente, Enschede, The Netherlands, 2000).
  8. D. J. W. Klunder, M. L. M. Balistreri, F. C. Blom, H. J. W. M. Hoekstra, A. Driessen, L. Kuipers, N. F. van Hulst, “Detailed analysis of the intracavity phenomena inside a cylindrical microresonator,” J. Lightwave Technol. 20, 519–529 (2002).
    [CrossRef]
  9. A. Hasegawa, Optical Solitons in Fibers, 2nd ed. (Springer-Verlag, Berlin, 1990).
  10. H. Kogelnik, “Theory of dielectric waveguides,” in Topics in Applied Optics, T. Tamir, ed. (Springer-Verlag, Berlin, 1979), Chap. 2.

2002

2001

M. L. M. Balistreri, J. P. Korterik, L. Kuipers, N. F. Van Hulst, “Tracking femtosecond laser pulses in space and time,” Science 294, 1080–1082 (2001).
[CrossRef] [PubMed]

E. Fluck, A. M. Otter, J. P. Korterik, M. L. M. Balistreri, L. Kuipers, N. F. Van Hulst, “Local phase measurements of light in a one-dimensional photonic crystal,” J. Microsc. 202, 104–109 (2001).
[CrossRef] [PubMed]

M. L. M. Balistreri, J. P. Korterik, G. J. Veldhuis, L. Kuipers, N. F. Van Hulst, “Quantitative photon tunneling and shear-force microscopy of planar WG splitters and mixers,” J. Appl. Phys. 89, 3307–3314 (2001).
[CrossRef]

2000

D. J. W. Klunder, M. L. M. Balistreri, F. C. Blom, H. J. W. M. Hoekstra, A. Driessen, L. Kuipers, N. F. Van Hulst, “High-resolution photon-scanning tunneling microscope measurements of the whispering gallery modes in a cylindrical microresonator,” IEEE Photon. Technol. Lett. 12, 1531–1533 (2000).
[CrossRef]

1995

Bachelot, R.

Balistreri, M. L. M.

D. J. W. Klunder, M. L. M. Balistreri, F. C. Blom, H. J. W. M. Hoekstra, A. Driessen, L. Kuipers, N. F. van Hulst, “Detailed analysis of the intracavity phenomena inside a cylindrical microresonator,” J. Lightwave Technol. 20, 519–529 (2002).
[CrossRef]

E. Fluck, A. M. Otter, J. P. Korterik, M. L. M. Balistreri, L. Kuipers, N. F. Van Hulst, “Local phase measurements of light in a one-dimensional photonic crystal,” J. Microsc. 202, 104–109 (2001).
[CrossRef] [PubMed]

M. L. M. Balistreri, J. P. Korterik, G. J. Veldhuis, L. Kuipers, N. F. Van Hulst, “Quantitative photon tunneling and shear-force microscopy of planar WG splitters and mixers,” J. Appl. Phys. 89, 3307–3314 (2001).
[CrossRef]

M. L. M. Balistreri, J. P. Korterik, L. Kuipers, N. F. Van Hulst, “Tracking femtosecond laser pulses in space and time,” Science 294, 1080–1082 (2001).
[CrossRef] [PubMed]

D. J. W. Klunder, M. L. M. Balistreri, F. C. Blom, H. J. W. M. Hoekstra, A. Driessen, L. Kuipers, N. F. Van Hulst, “High-resolution photon-scanning tunneling microscope measurements of the whispering gallery modes in a cylindrical microresonator,” IEEE Photon. Technol. Lett. 12, 1531–1533 (2000).
[CrossRef]

M. L. M. Balistreri, “Coherent imaging of guided optical fields,” ISBN 90-365-14924, Ph.D. thesis (University of Twente, Enschede, The Netherlands, 2000).

Blom, F. C.

D. J. W. Klunder, M. L. M. Balistreri, F. C. Blom, H. J. W. M. Hoekstra, A. Driessen, L. Kuipers, N. F. van Hulst, “Detailed analysis of the intracavity phenomena inside a cylindrical microresonator,” J. Lightwave Technol. 20, 519–529 (2002).
[CrossRef]

D. J. W. Klunder, M. L. M. Balistreri, F. C. Blom, H. J. W. M. Hoekstra, A. Driessen, L. Kuipers, N. F. Van Hulst, “High-resolution photon-scanning tunneling microscope measurements of the whispering gallery modes in a cylindrical microresonator,” IEEE Photon. Technol. Lett. 12, 1531–1533 (2000).
[CrossRef]

Boccara, A. C.

Bozhevolnyi, S. I.

S. I. Bozhevolnyi, B. Vohnsen, “Near-field imaging of optical phase and its singularities,” Opt. Commun. 212, 217–223 (2002).
[CrossRef]

Driessen, A.

D. J. W. Klunder, M. L. M. Balistreri, F. C. Blom, H. J. W. M. Hoekstra, A. Driessen, L. Kuipers, N. F. van Hulst, “Detailed analysis of the intracavity phenomena inside a cylindrical microresonator,” J. Lightwave Technol. 20, 519–529 (2002).
[CrossRef]

D. J. W. Klunder, M. L. M. Balistreri, F. C. Blom, H. J. W. M. Hoekstra, A. Driessen, L. Kuipers, N. F. Van Hulst, “High-resolution photon-scanning tunneling microscope measurements of the whispering gallery modes in a cylindrical microresonator,” IEEE Photon. Technol. Lett. 12, 1531–1533 (2000).
[CrossRef]

Fluck, E.

E. Fluck, A. M. Otter, J. P. Korterik, M. L. M. Balistreri, L. Kuipers, N. F. Van Hulst, “Local phase measurements of light in a one-dimensional photonic crystal,” J. Microsc. 202, 104–109 (2001).
[CrossRef] [PubMed]

Gleyzes, P.

Hasegawa, A.

A. Hasegawa, Optical Solitons in Fibers, 2nd ed. (Springer-Verlag, Berlin, 1990).

Hoekstra, H. J. W. M.

D. J. W. Klunder, M. L. M. Balistreri, F. C. Blom, H. J. W. M. Hoekstra, A. Driessen, L. Kuipers, N. F. van Hulst, “Detailed analysis of the intracavity phenomena inside a cylindrical microresonator,” J. Lightwave Technol. 20, 519–529 (2002).
[CrossRef]

D. J. W. Klunder, M. L. M. Balistreri, F. C. Blom, H. J. W. M. Hoekstra, A. Driessen, L. Kuipers, N. F. Van Hulst, “High-resolution photon-scanning tunneling microscope measurements of the whispering gallery modes in a cylindrical microresonator,” IEEE Photon. Technol. Lett. 12, 1531–1533 (2000).
[CrossRef]

Klunder, D. J. W.

D. J. W. Klunder, M. L. M. Balistreri, F. C. Blom, H. J. W. M. Hoekstra, A. Driessen, L. Kuipers, N. F. van Hulst, “Detailed analysis of the intracavity phenomena inside a cylindrical microresonator,” J. Lightwave Technol. 20, 519–529 (2002).
[CrossRef]

D. J. W. Klunder, M. L. M. Balistreri, F. C. Blom, H. J. W. M. Hoekstra, A. Driessen, L. Kuipers, N. F. Van Hulst, “High-resolution photon-scanning tunneling microscope measurements of the whispering gallery modes in a cylindrical microresonator,” IEEE Photon. Technol. Lett. 12, 1531–1533 (2000).
[CrossRef]

Kogelnik, H.

H. Kogelnik, “Theory of dielectric waveguides,” in Topics in Applied Optics, T. Tamir, ed. (Springer-Verlag, Berlin, 1979), Chap. 2.

Korterik, J. P.

E. Fluck, A. M. Otter, J. P. Korterik, M. L. M. Balistreri, L. Kuipers, N. F. Van Hulst, “Local phase measurements of light in a one-dimensional photonic crystal,” J. Microsc. 202, 104–109 (2001).
[CrossRef] [PubMed]

M. L. M. Balistreri, J. P. Korterik, L. Kuipers, N. F. Van Hulst, “Tracking femtosecond laser pulses in space and time,” Science 294, 1080–1082 (2001).
[CrossRef] [PubMed]

M. L. M. Balistreri, J. P. Korterik, G. J. Veldhuis, L. Kuipers, N. F. Van Hulst, “Quantitative photon tunneling and shear-force microscopy of planar WG splitters and mixers,” J. Appl. Phys. 89, 3307–3314 (2001).
[CrossRef]

Kuipers, L.

D. J. W. Klunder, M. L. M. Balistreri, F. C. Blom, H. J. W. M. Hoekstra, A. Driessen, L. Kuipers, N. F. van Hulst, “Detailed analysis of the intracavity phenomena inside a cylindrical microresonator,” J. Lightwave Technol. 20, 519–529 (2002).
[CrossRef]

E. Fluck, A. M. Otter, J. P. Korterik, M. L. M. Balistreri, L. Kuipers, N. F. Van Hulst, “Local phase measurements of light in a one-dimensional photonic crystal,” J. Microsc. 202, 104–109 (2001).
[CrossRef] [PubMed]

M. L. M. Balistreri, J. P. Korterik, G. J. Veldhuis, L. Kuipers, N. F. Van Hulst, “Quantitative photon tunneling and shear-force microscopy of planar WG splitters and mixers,” J. Appl. Phys. 89, 3307–3314 (2001).
[CrossRef]

M. L. M. Balistreri, J. P. Korterik, L. Kuipers, N. F. Van Hulst, “Tracking femtosecond laser pulses in space and time,” Science 294, 1080–1082 (2001).
[CrossRef] [PubMed]

D. J. W. Klunder, M. L. M. Balistreri, F. C. Blom, H. J. W. M. Hoekstra, A. Driessen, L. Kuipers, N. F. Van Hulst, “High-resolution photon-scanning tunneling microscope measurements of the whispering gallery modes in a cylindrical microresonator,” IEEE Photon. Technol. Lett. 12, 1531–1533 (2000).
[CrossRef]

Otter, A. M.

E. Fluck, A. M. Otter, J. P. Korterik, M. L. M. Balistreri, L. Kuipers, N. F. Van Hulst, “Local phase measurements of light in a one-dimensional photonic crystal,” J. Microsc. 202, 104–109 (2001).
[CrossRef] [PubMed]

van Hulst, N. F.

D. J. W. Klunder, M. L. M. Balistreri, F. C. Blom, H. J. W. M. Hoekstra, A. Driessen, L. Kuipers, N. F. van Hulst, “Detailed analysis of the intracavity phenomena inside a cylindrical microresonator,” J. Lightwave Technol. 20, 519–529 (2002).
[CrossRef]

E. Fluck, A. M. Otter, J. P. Korterik, M. L. M. Balistreri, L. Kuipers, N. F. Van Hulst, “Local phase measurements of light in a one-dimensional photonic crystal,” J. Microsc. 202, 104–109 (2001).
[CrossRef] [PubMed]

M. L. M. Balistreri, J. P. Korterik, L. Kuipers, N. F. Van Hulst, “Tracking femtosecond laser pulses in space and time,” Science 294, 1080–1082 (2001).
[CrossRef] [PubMed]

M. L. M. Balistreri, J. P. Korterik, G. J. Veldhuis, L. Kuipers, N. F. Van Hulst, “Quantitative photon tunneling and shear-force microscopy of planar WG splitters and mixers,” J. Appl. Phys. 89, 3307–3314 (2001).
[CrossRef]

D. J. W. Klunder, M. L. M. Balistreri, F. C. Blom, H. J. W. M. Hoekstra, A. Driessen, L. Kuipers, N. F. Van Hulst, “High-resolution photon-scanning tunneling microscope measurements of the whispering gallery modes in a cylindrical microresonator,” IEEE Photon. Technol. Lett. 12, 1531–1533 (2000).
[CrossRef]

Veldhuis, G. J.

M. L. M. Balistreri, J. P. Korterik, G. J. Veldhuis, L. Kuipers, N. F. Van Hulst, “Quantitative photon tunneling and shear-force microscopy of planar WG splitters and mixers,” J. Appl. Phys. 89, 3307–3314 (2001).
[CrossRef]

Vohnsen, B.

S. I. Bozhevolnyi, B. Vohnsen, “Near-field imaging of optical phase and its singularities,” Opt. Commun. 212, 217–223 (2002).
[CrossRef]

IEEE Photon. Technol. Lett.

D. J. W. Klunder, M. L. M. Balistreri, F. C. Blom, H. J. W. M. Hoekstra, A. Driessen, L. Kuipers, N. F. Van Hulst, “High-resolution photon-scanning tunneling microscope measurements of the whispering gallery modes in a cylindrical microresonator,” IEEE Photon. Technol. Lett. 12, 1531–1533 (2000).
[CrossRef]

J. Appl. Phys.

M. L. M. Balistreri, J. P. Korterik, G. J. Veldhuis, L. Kuipers, N. F. Van Hulst, “Quantitative photon tunneling and shear-force microscopy of planar WG splitters and mixers,” J. Appl. Phys. 89, 3307–3314 (2001).
[CrossRef]

J. Lightwave Technol.

J. Microsc.

E. Fluck, A. M. Otter, J. P. Korterik, M. L. M. Balistreri, L. Kuipers, N. F. Van Hulst, “Local phase measurements of light in a one-dimensional photonic crystal,” J. Microsc. 202, 104–109 (2001).
[CrossRef] [PubMed]

Opt. Commun.

S. I. Bozhevolnyi, B. Vohnsen, “Near-field imaging of optical phase and its singularities,” Opt. Commun. 212, 217–223 (2002).
[CrossRef]

Opt. Lett.

Science

M. L. M. Balistreri, J. P. Korterik, L. Kuipers, N. F. Van Hulst, “Tracking femtosecond laser pulses in space and time,” Science 294, 1080–1082 (2001).
[CrossRef] [PubMed]

Other

M. L. M. Balistreri, “Coherent imaging of guided optical fields,” ISBN 90-365-14924, Ph.D. thesis (University of Twente, Enschede, The Netherlands, 2000).

A. Hasegawa, Optical Solitons in Fibers, 2nd ed. (Springer-Verlag, Berlin, 1990).

H. Kogelnik, “Theory of dielectric waveguides,” in Topics in Applied Optics, T. Tamir, ed. (Springer-Verlag, Berlin, 1979), Chap. 2.

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Figures (7)

Fig. 1
Fig. 1

Schematic picture of the PSNOM (see text). BS, beam splitter; PMT, photomultiplier tube.

Fig. 2
Fig. 2

Schematic picture for the PSNOM measurements of frequency-dependent modal transmission and reflection coefficients of a device. For accurate determination of these coefficients, output and input WGs have to be identical.

Fig. 3
Fig. 3

Simulated gray-scale plots of VLI(zs, y) for a WG containing two modes. The absolute value (top picture) over the full relevant zs region, detailed plots of the real part (middle picture), and the absolute value of the phase in the interval [-π, π] (bottom picture) are shown.

Fig. 4
Fig. 4

Simulated curves of |VLI(zs)|, corresponding to a longitudinal scan along the WG, for two positions of the delay line, corresponding to the indicated zr values.

Fig. 5
Fig. 5

Simulated plot of |v(kz)|, corresponding to either one of the curves in Fig. 4 (bottom), and a comparison (top picture) of simulated dispersion curves (pluses) with curves used as input for the simulations (solid lines).

Fig. 6
Fig. 6

Simulated curve of |VLI(zs)| measurements on a fundamental mode after reflection by a Bragg grating.

Fig. 7
Fig. 7

Simulated reflection curve |r0(ω)| (pluses) compared with that used as an input for the simulations (solid curve).

Tables (1)

Tables Icon

Table 1 Parameters Used in the Numerical Experimentsa

Equations (47)

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E(x, y, z, t)=e(x, y)exp[i(ω1t-βz)],
Es(t, zs)=fp(t, zs)exp(iω0t),
Er(t, zr)=gp(t, zr)exp[i(ω0+Δω)t],
VPMT|Es+Er|2.
VPMT exp(iΔωt).
fp(t)gp*(t).
VLI(t1)=C1T-t1×exp[(t-t1)/T]pfp(t, zs)gp*(t, zr)dt,
VLI(zs, zr)=C1f(t, zs)g*(t, zr)dt,
f(t, zs)=Cs(ω)H(ω)exp[i(ωt-bs)]dω/2π,
g(t, zr)=Cr(ω)H(ω)exp[i(ωt-br)]dω/2π.
H(ω)h(t)exp(-iωt)dt/2π.
bs=βzs,br=k0zr.
Csexp(-iBs),BsSB,zs=0βs(ω, zs)dzs
Crexp(-iBr),BrRB,zr=0βr(ω, zr)dzr.
β=β0+βω+βω2/2,
bs=bs,0+bsω+bsω2/2
br=br,0+brω,br,0=ω0zr/c,br=zr/c.
Bs/r=B0,s/r+Bs/rω+Bs/rω2/2.
VLI(zs, zr)=C2dtdωdωH(ω)H*(ω)×exp{i[(ω-ω)t-bs(ω)+br(ω)+Br(ω)-Bs(ω)]}=2πC2dω|H(ω)|2 exp[i(b+B)],
v(kz)VLI exp(ikzzs)dzsdzsdω|H(ω)|2 exp{i[(kz-β)zs+br+B]}.
v(kz)dω|H(ω)|2 exp[i(br+B)]δ(kz-β),
δ{g(ω)}=zerosofgδ(ω-ωp)/|ωg(ωp)|,
g(ωp)=0,ωg(ωp)0,
v(kz)=C3{|H|2 exp[i(br+B)]/|β/ω|}β(ω)=kz,
v2(kz)/v1(kz)=exp(iϕ21),ϕ21k0(ω)Δzr,
Δzrzr,2-zr,1,
ω(kz)=[(ϕ21+2mπ)c/Δzr-ω0],
H(ω)exp(-ω2w2/4),
VLI(zs, zr)=C4dω exp[-ω2w2/2+i(b+B)]=C4 exp(id0)π/[2(w2-id)]×exp{-d2/[2(w2-id)]},
v(kz, zr)=C5{exp[-ω2w2/2+i(br+B)]/|β/ω|}β(ω)=kz=C5 exp[-ω2(kz)w2/2+i(br+B)β(ω)=kz]/|β+βω(kz)|.
ω(kz)=(kz-β0)/β-(kz-β0)2β/(2β3).
E(x, y, z, t)=pep(x, y)exp[i(ω1t-βpz+ϕp)],
VLI(zs, zr)=pdωCp|H(ω)|2 exp[i(bp+Bp)],
v(kz, zr)=pvp,
vp{C1,p|H|2 exp[i(br+Bp)]/|βp/ω|}βp(ω)=kz.
v0(kz, zr)={C0c|H(ω)|2 exp[i(br+Bp)]}ω=ckz-ω0.
|v0(kz, zr)||H(ω)|ω=ckz2,
VLI(zs, zr,1)=C2dω|H(ω)|2 exp[i(b1+B)],
VLI,p(zs, zr,2)=C2dωτp(ω)|H(ω)|2 exp[i(b2+B)].
τp(ω)exp[ik0(ω)(zr,2-zr,1)]=v(kz, zr,2)/v(kz, zr,1)|β(ω)=kz.
B0/1(Lair-2cm×ng,0/1-10cm×ng,f)/c0,
E0=cos(ky,0y)/cos(ky,0a2),-a2<y<a2,
E0=exp(γ0y),|y|>a2,
E1=sin(ky,1y)/sin(ky,1a2),-a2<y<a2,
E1=±exp(γ1y),|y|>a2.
r0=-iκ/[S coth(SLgr)+iδ],Sκ2-δ2.
b+B0=(zr,2-ng,0zs)/c+B0=0.

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