Simple image-dipole method for photon scanning tunneling microscopy

ultramicroscopy ELSEVIER

Ultramicroscopy61 (1995) 81-84

Simple image-dipole method for photon scanning tunneling microscopy W. Jhe *, K. Jang Department of Physics, Seoul National University, Seoul 151-742, South Korea

Received 9 May 1995;accepted 10 August 1995

Abstract We present a simple intuitive model of a Photon Scanning Tunneling Microscope (PSTM) using the image-dipole method of classical electrodynamics. In particular, the polarization dependences of the nanometric PSTM images can be qualitatively understood and agree well with recent experimental results.

1. Introduction The Photon Scanning Tunneling Microscope (PSTM) is a nanometric high-resolution optical microscope. Its operation is based on the short-range electromagnetic interaction between a small dielectric probe and a small sample which supports a strongly localized optical near-field [1]. With recent progress in the improved resolution of the PSTM images, their polarization dependences became of considerable interest and study. In this paper, we consider a collection-mode (cmode) PSTM where a nanometric tip is used as a scatterer: when the tip is introduced into the evanescent field generated on the sample surface by the incident light under the total internal reflection (TIR) condition, the non-radiative evanescent light is converted into the scattered light which is guided to the detection optics. In addition to its high signal-to-noise ratio, it also has an advantage of arbitrary selection

* Corresponding author.

of the s- or p-polarization states of the illuminating light. It is our main object to present a simple intuitive model of the high-resolution images obtained by a c-mode PSTM. In particular, the dependence of the images on the polarization of the incident light can be qualitatively explained in terms of the simple image-dipole model.

2. Image-dipole model for PSTM Recently, Naya et al. [2] obtained high-resolution images of the straight-type flagellar filaments of salmonella (FFS) whose shape is a cylindrical-rod with a diameter of 25 nm. It was about the same size as the characteristic dimensions of the metal-apertured probe tip. In the case of s-polarization of the illuminating light, the full-width at half-maximum (FWHM) of the image of the FFS was 50 nm. On the other hand, they observed a two-peak structure due to the same FFS in the case of the p-polarization light. Moreover, the FFS lying perpendicular to the

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82

IV. Jhe, K. J a n g / U l t r a m i c r o s c o p y 61 (1995) 8 1 - 8 4

ff is the oscillating dipole moment and the electric field at ~' due to an oscillating dipole at /~ is given by, in free space [5]: image dipole

dielectric sphere

eikro

=

Ca)

2(go

× go--

ro 1

-),. P

+ [3 o(go

ik)

ro3

eikr° '

(3) dielectric sphere

image dipole (b)

Fig. 1. Dipole-image-dipole model of PSTM.

direction of the incident beam was better imaged with high contrast, which is in conflict with the predictions of the recent field propagator theory [3]. The interaction of the laser-excited small sample and the small probe can be qualitatively understood by simple electromagnetic consideration: the dipole-image-dipole interaction model. In other words, the laser-excited small sample can be considered as a radiating (physical) dipole, whereas the spherical dielectric probe-tip nearby can be substituted by an image dipole. It has been found [4] that the radiative properties of a dipole near a dielectric microsphere can be approximated by the dipoleimage-dipole interaction in the non-retarded- and retarded-regime (see Fig. 1) therefore, a proper image dipole can be placed inside the sphere (or probe tip) satisfying the electromagnetic boundary conditions. Consider a radiating dipole located outside the sphere at a distance R from the center of the sphere. The electric field at a position 7 (outside the sphere) consists of the free field

where go is the unit vector of r e = r - R, r o is its magnitude, and k is the wavevector ( w / c ) . As usual, the physical dipole can be considered as a system of two point charges q (at R + 3 / 2 ) and - q (at r - 3//2) separated by a small distance 6 in such a way that 6 is negligibly small compared to the radius of the sphere a but dipole moment p = q6 remains finite. When the dipole is polarized tangential to the surface (/~= p0), the image charges and their positions from the center are found as in [6], -a q1

+a q2

(1)

q,

(4)

F1

i"2

~R 2 -t- 6 2 / / 4

for the charges q and - q, respectively. In the dipole limit 6 / R << 1, these quantities can be approximated as, to first order in 6 / R , -

qa

q l "~ - -

R qa

--

R'

a 2

'

rl'~ R ' az r 2 "~ - -

R

(5) •

Therefore the image of a tangential dipole consists of an image dipole/Timag e located at Rimag e given by

and the reflected field /~R(V, w) = ,fiR(V,/~; w) -if,

v/R2 + 6 2/4 a2

q2 ~

E°(7, o)) =/F°(V,/?; o)) .ff

~R 2 + 6 2//4 q'

(2) P~image = ( ql rl -}- q2 r2) ? = -- ( a/R)3ff,

which can be considered as coming from an image dipole, b~ is the field susceptibility function. Hence

eimag e = ( r 1 + r 2 ) / 2 = a2/R.

(6)

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W. Jhe, K. Jang / Ultramicroscopy 61 (1995) 81-84

From Eqs. (3) and (6), the field at ?', radiated by the dielectric sphere, can be written as

(r

, -

]

eik(r-aZ/R).

a2//R)

(7)

3

In the image-dipole model, this is equivalent to the radiation field scattered by the spherical tip, guided by the optical fiber probe and detected by a photodetector. For the dipole polarized in the radial direction ( f f = p?), the image charges and their positions are obtained in the similar way

ql

R + 6 / 2 q'

ra

R + 6/2 '

a q2

r2

p,

R - 6/2 "

=

(9)

Note that, unlike the tangential-dipole case, the unequal magnitudes of ql and q2 to first order in 6 / R result in a net image point-charge given by a

qimage = q1 + q2 = ~ P ,

(10)

at the atom's position. However, it is well known [6] that another image point charge of equal magnitude

2wL (a)

s-polarization

e e ik(r-a2/R).

(11)

(8)

Therefore the image dipole and its position are given by

p-polarization

a2/R)3

(r-- 1

(r-a2/R)

a2

R - 6 / ~ q'

/~m age =

Esplaere(~) = 2 ( e ) 3 p [

a2

a

-

but opposite sign is added to the center of the sphere so that the electrostatic boundary conditions are met and charge conservation holds. Moreover, in the self-radiation theory [7] as considered in the image method, radiative corrections for the oscillating dipole come from the image-dipole terms, but not from the image "monopole" terms. In other words, the net image charge, despite its time-dependence, accompanies no current flow and consequently does not contribute to the radiation field. Therefore the radiation field reflected due to the sphere can be considered as coming only from the image dipole. From Eqs. (3) and (9) we obtain the radiation field

(b)

Fig. 2. The polarization dependences of the high-resolution PSTM images can be qualitatively understood by the simple dipoleimage-dipole interaction model.

Eqs. (7) and (11) are our main results in the imagedipole model. Note that, in general, a radiative problem for a sphere cannot be exactly solved by the image-dipole method. In other words, the radiation fields produced by the image dipoles [Eqs. (7) and (11)] do not satisfy the boundary conditions (i.e. vanishing tangential components of electric fields) at all distances unlike the plane-mirror case [7]. However, it turns out that the tangential components become negligible (i.e. image method becomes valid) in the short-range and long-range limits. Therefore, the image-dipole model is valid in the PSTM process since the shortrange limit corresponds to the electrostatic near-field limit. Now, one can give qualitative understanding of the general images for the PSTM processes by this image model and furthermore the high-resolution images of FFS [2]. As in Fig. 2, let us consider a nanometric sample on a glass substrate under the TIR condition. When the exciting laser is s-(p)polarized, the sample can be considered as a radiating dipole polarized parallel (normal) to the substrate plane. Now let us approach an apertured fiber probe, which can be approximated by a dielectric sphere, near the sample. Then the tip can be replaced by an image dipole whose direction of polarization depends on its position relative to the sample (physical

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W. Jhe, K. Jang / Ultramicroscopy 61 (1995) 81-84

dipole) due to the boundary conditions. Let us consider the s-polarization case first. When the tip is right over the sample, the image dipole is also polarized parallel to the substrate plane but with the opposite phase. In this case, the detected light intensity radiated by the tip and transmitted through the optical fiber becomes maximum. As the tip is scanned over the sample in the lateral direction, the orientation of the image dipole should rotate to meet the boundary conditions and as a result the measured power decreases. We thus predict a single peak resembling the topology of the sample object. Now let us consider the p-polarization case. When the tip is right over the sample dipole, the probe becomes also a p-polarized dipole for which the radiated power through the fiber is minimum (the detected power may vanish if the fiber has a negligible solid angle which is unrealistic, however). When the tip is scanned laterally, the orientation of the image dipole rotates as in the s-polarization case and therefore the detected power increases. It will however fall down soon due to the short-range nature of the distribution of the evanescent wave generated near the sample. We thus expect an image of double peak structure. These results agree well with the high-resolution image recently obtained [2]. The strong point of the simple image-dipole consideration is that it includes the effects of the specific configuration of the guiding optical fiber as well as the probe tip. This approach may be a good approximation for a small tip and a small sample, otherwise the interaction of the multipole terms may be nonnegligible.

3. Conclusion We have found that the simple image-dipole model of electromagnetics provides qualitative understanding of the high-resolution PSTM images and in particular, the polarization dependence of the images obtained by a c-mode PSTM.

Acknowledgements W.J. is grateful to S. Eah, Professor M. Ohtsu, and Professor H. Hori for helpful discussions. He is also grateful to JRDC (Research Development Corporation of Japan) for its support during his stay when part of the work was performed.

References [1] D.W. Pohl and D. Courjon, Eds., Near-field Optics (Kluwer, Dordrecht, 1993); M. Ohtsu, IEEE J. Lightwave Technol. (1995) to be published. [2] M. Naya, S. Mononobe, R. Uma Maheswari, T. Saiki and M. Ohtsu, Scanning Probe Microscopes 3, Proc. SPIE 2384 (1995); Opt. Commun. 124 (1996) 9. [3] OJ. Martin, C. Girard and A. Dereux, Phys. Rev. Lett. 74 (1995) 526. [4] W. Jhe and J.W. Kim, Phys. Rev. A 51 (1995) 1150; Phys. Lett. A 197 (1995) 192. [5] J.D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975) pp. 141 and 395. [6] M.H. Nayfeh and M.K. Brussel, Electricity and Magnetism (Wiley, New York, 1985) p. 106. [7] D. Meschede, W. Jhe and E.A. Hinds, Phys. Rev. A. 40 (1990) 1587.