phys. stat. sol. (2003)
Depth resolved nonlinear optical nanoscopy W. Luis Mochán 1 , Catalina López-Bastidas2 , Jesús A. Maytorena1, Bernardo S. Mendoza3 , and Vera L. Brudny4 1 2 3 4
Centro de Ciencias Físicas, Universidad Nacional Autónoma de México, Apartado Postal 48-3, 62251 Cuernavaca, Morelos, México. Centro de Ciencias de la Materia Condensada, Universidad Nacional Autónoma de México, Apartado Postal 2681, Ensenada, Baja California, 22800, Mexico. Centro de Investigaciones en Optica, León, Guanajuato, México. Departamento de Física, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Ciudad Universitaria Pab. I, 1428 Buenos Aires, Argentina.
Received zzz, revised zzz, accepted zzz Published online zzz PACS 07.79.Fc, 42.65.Ky, 78.67.-n, 78.68.+m
An electromagnetic field forced to vary along a plane with a spatial scale d much smaller than its free space wavelength λ decays exponentially along its normal with a decay length d. This decay, similar to that of the wavefunction of tunneling electrons, has allowed the development of scanning near-field optical microscopes (SNOMs), reminiscent of scanning tunneling and atomic force microscopes, which have been able to resolve structures in the nanometer scale. However, existing SNOMs are unable to determine the depth below the surface from which the optical signals arise due to the monotonic decay of the optical evanescent probe fields. In this paper we study the optical second harmonic generation (SHG) produced by mixing of the evanescent fields produced by a SNOM tip. We show that employing an appropriately spatially-patterned tip, a non-monotonic non-linear probing field may be produced which has a maximum at a given distance beyond the tip, yielding a novel microscopy which may attain depth resolution with nanometric lengthscales. We estimate the size of the optical signal and we compare it with that arising in the usual SHG-based surface spectroscopy of centrosymmetric materials.
1 Introduction
The light collected by the imaging system of a conventional optical microscope has wavevector projections Q parallel to the image plane which are necessarily within the light cone nω c Q nω c, where ω is the frequency of the light and n the index of refraction of the ambient. With that finite range of wavevectors, the spatial resolution of the microscope is limited to spatial structures not much smaller than the wavelength λ [1]. Nevertheless, light constitutes a very attractive probe of matter, as it can be non-invasive, non-destructive, it is very finely tunable and many linear and nonlinear, elastic and inelastic spectroscopies have been built on it. Therefore, a large effort has been made in the last two decades to circumvent the Abbe resolution limit. In order to increment the resolving power, larger wavevectors have to be involved in the imaging processes. Writing the wave equation ∇ 2 φ n2 ω2 c2 φ for any component 2 2 2 2 2 φ of the electromagnetic field as ∂z φ n ω c Q φ, where we assumed that the field oscillates along the xy plane, we see that Q nω c implies an exponential decay along the normal z direction. Thus, for a fixed wavelength, better resolution may be attained by employing the evanescent near-field [2]. The exponential decay of the near-field is reminiscent of the exponential decay of the wavefunction of electrons
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2 W. L. Mochán, C. López-Bastidas, J. A. Maytorena, B. S. Mendoza, and V. L. Brudny: Nonlinear optical nanoscopy
that tunnel through classically inaccesible regions. This suggested the development [3] of scanning nearfield optical microscopes (SNOMs) using a technology similar to that employed in scanning tunneling microscopes (STMs). The first SNOMs employed apertures in metallic thin films [4, 5] and later optical fibre waveguides with a tapered tip covered with a metal layer with a small aperture through which the evanescent field could leak out towards a nearby surface. In one of its operation modes, the intensity I of the light reflected through the same waveguide is monitored as the tip is scanned along the xy directions while the height z above the nominal surface is kept fixed. Similarly, the reflected intensity could be kept fixed by raising or lowering the tip of the microscope through a feedback loop as it is scanned along the surface. The intensity I x y or the height z x y may be employed to draw three dimensional images of the surface which, if properly interpreted, can yield information on its composition and geometry. The horizontal resolving distance is of the order of the width of the aperture and the vertical decay length is about an order of magnitude smaller; both may be much smaller than the wavelength, to which they are quite insensitive.
There are disadvantages to the so called aperture SNOM described above. For instance, light can penetrate a finite distance into a metal. Thus, the field at the tip occupies an area somewhat larger than the actual size of the aperture, reducing the resolution. On the other hand, the signal is typically very small. Increasing the power to increase the signal is frequently not feasible, as the electromagnetic energy absorbed by the metallic screen heats the tip and may disrupt the sample under study [6]. To realize the highest resolutions the tip should be very close to the sample and thus topographical artifacts become entangled with variations in optical properties [2, 7, 8, 9, 10]. There have been many alternative configurations designed to overcome the limitations of the aperture SNOM. In apertureless SNOM one can eliminate the waveguide altogether by focusing the field propagating from a far away source onto the tip of a very sharp metallic needle. The strong near-field induced by the needle in the vicinity of its tip may be scattered by a nearby surface and the intensity and/or phase of the corresponding far field may be monitored as the tip is scanned over the surface. Another alternative is to use uncoated optical fibers with sharp dielectric tips used for both SNOM and atomic force microscopy (AFM) in the shear mode. The AFM allows the identification of topographic features and their separation from localized chemical and physical features [11]. Advantages of tip and aperture type nanoscopes have been combined by growing a tip within an aperture [12]. The signal may be further enhanced over the background by illuminating the tip with the evanescent fields that accompany totally reflected light [13, 14]. The resolution attained with scattering SNOMs has gone below 10nm [15, 16]. Field modulations with an even faster scale are present in the surface local field at crystalline surfaces. Thus, atomic resolution employing the crystalline near-field is feasible. This has been illustrated in an analysis of the optical spectra of adsorbate covered surfaces (though without a scanning instrument). As the field scattered by an adsorbate physisorbed over a crystalline surface is proportional to the square of the local field that polarizes it, incorporation of the near-field in the theoretical interpretation has enabled the extraction of geometrical information with a sub Å resolution [17]. SNOMs have been succesfully employed to observe many kinds of samples. Interferometric measurements [18] have yielded not only the amplitude but also the phase of the plasmon resonances of metallic nanoparticles optically excited with carbon nanotubes [19]. Beyond the relatively simple solid conducting and insulating surfaces, SNOMs have also been employed to investigate biological tissues in liquid ambients [20]. SNOMs may take advantage of the great variety of optical spectroscopies available. Thus, besides employing the intensity, phase and polarization [21, 22] of the linearly and elastically scattered near-fields, it can also use inelastical scattering to make images in the nanometer scale. For example, single-walled carbon nanotubes have been imaged with the Raman scattered fields [23]. Fluorescence and photobleaching have been used to study single adsorbed molecules [24]. Additional information may be obtained by combining other spectroscopies with a SNOM. For example, elemental and molecular analysis
phys. stat. sol. (2003)
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has been achieved by employing time-of-flight mass spectrometry on laser-desorbed molecules, where the laser light is delivered by a SNOM [25]. Non-linear optical processes have also been used both to image and to modify surfaces using nearfields. Metallic rough surfaces illuminated by a broad beam have been observed with a 100nm resolution by scanning a tappered optical waveguide to collect the generated second harmonic (SH) near-field[26]. The enhancement of the SH at a defect may be an order of magnitude larger than the enhancement of the fundamental field due to the lighting rod effect [27]. The near SH field generated by nanopads of nonlinear materials on a substrate have been shown to be more localized than the [28] fundamental fields. This allows a high contrast which allows the study of mesoscopic domains in ferroelectric films [29]. The theory of SHG in nonlinear SNOM is complicated by the fact that both the fundamental and SH fields scattered by the object under study, the SNOM probe, and the substrate, have to be obtained self-consistently [30, 31]. Besides producing SH by the system under study, the SH light may be generated at the tip taking advantage of the strong field enhancement [32], and afterwards used as a very confined light source whose near-field may be scattered by an underlying surface over which it may be scanned. The field enhancement at a metallic tip is larger when the fundamental field has a component along the tip axis, as when the tip is at the focus of a high order Hermite-Gaussian laser mode [33]. Other useful non-linear effects include twophoton litography. Using the near-field produced by the SNOM tip [34], patterns with a scale 100nm have been produced. One problem in surface science is the observation of buried structures. The usual surface probes (low energy electrons, ions, atoms, etc.) have a small penetration into matter and so they are sensitive mostly to the outermost layers. Thus, to study properties along the normal to the plane it is common to make a crater so that buried structures are brought to the surface where they can be conventionally examined. For example, secondary ion mass spectrometry (SIMS) is commonly used to study composition depth profiles as a surface is evaporated away [35]. The main disadvantage of this kind of techniques is that they destroy the sample. Optical fields may probe the interior of matter non-destructively, but their penetration depth is very large on an atomic scale. Nevertheless, the reduction in symmetry at interfaces has allowed the optical observation of buried interfaces [36] employing non-linear spectroscopy. Non planar interfaces such as that between Si nanospheres and the SiO2 matrix within which they were produced have also been observed with SHG [37, 38]. However, to obtain depth resolution it is not enough to reach the buried interface and identify its contribution to the signal but the depth itself has to be measured. Use of two photon optical processes such as SHG or two-photon optical-beam-induced current [39, 40] has permitted 3D imaging of circuit components in an integrated circuit with 1µm lateral and 100nm axial resolution. Differential confocal microscopy [41] has reached a depth resolution of 2nm. Much better lateral resolutions have been obtained in SNOMs. Thus, the purpose of the present paper is to propose a scheme that would allow the observation of structures buried below a surface employing the near-field produced by a SNOM tip permitting the extraction of depth information with a resolution in the nanometer scale. The paper is organized as follows. In Sec. 2 we present a scheme that employs sum frequency generation to obtain depth resolved information. In Sec. 3 we propose a simpler system based on SHG. A more realistic realization and calculation is developed in Sec. 4 and we devote Sec. 5 to conclusions.
2 Scheme One drawback of all the techniques associated so far with SNOM is that the sensitivity is a monotonically decaying function of distance. Thus, it is not possible to distinguish the near-field produced by a strong scatterer located relatively deep below a surface from that produced by a weak scatterer at the surface or at a very shallow position. A similar problem was the topographical masking of optical data at high resolutions [2, 7, 8, 9, 10]. Topographic artifacts have been minimized by monitoring the height above the surface with an AFM integrated with the SNOM [11]. However, this scheme is not useful for buried structures.
4 W. L. Mochán, C. López-Bastidas, J. A. Maytorena, B. S. Mendoza, and V. L. Brudny: Nonlinear optical nanoscopy
Fig. 1 Sum frequency generation. Photons with frequency ω3 ω1 ω2 are produced only in the region (dark gray) within the sample (light gray) where the beams with frequency ω1 and ω2 overlap. In principle, this region may be scanned over the 3D sample.
A solution to the problem of depth resolved nanoscopy may be found if we first look at a similar problem regarding the far field. Consider a three wave non-linear optical mixing effect such as sum frequency generation (SFG). In order to produce a photon of frequency ω 3 ω1 ω2 by mixing two photons with fundamental frequencies ω1 and ω2 , both photons must arrive simultaneously at the same place. Thus, SFG can only take place within a volume where the two beams with frequencies ω 1 and ω2 overlap (Fig. 1). In principle, this volume can be scanned over the interior of a sample in order to obtain a 3D image. A more practical idea based in a similar same principle has been to look at the SHG, two photon fluorescence, two photon induced current [39, 40, 42] or any two photon process which takes place mainly close to the focus of a tightly focused beam, where the fundamental intensity is largest. Consider then a SNOM tip made up of, or coated with, a periodic array of alternating layers of materials such that one of them is perfectly transparent to light of frequency ω 1 and opaque to light of frequency ω2 and the other one has the complementary behavior, as illustrated in Fig. 2. If the tip is illuminated simultaneously by two beams with frequencies ωa (a 1 2), we expect the corresponding field Ea to attain some finite value at positions immediately below the layers that are transparent to that frequency, and to be almost null immediately below the layers that are opaque to that frequency. Oversimplifying to some extent, the fields might look somewhat like the periodic step functions displayed in the left panel of Fig. 2. Although both E1 and E2 are relatively large close to the tip, one of them is almost null at those positions where the other is maximum, so that the product E 1 E2 is essentially zero. The tip acts for each field as a grating, inducing a diffracted field with diffraction wave-vectors G which are integer multiples of 2π d with d d1 d2 , where da is the width of a layer of type a. However, if we choose d in the nanometer scale,
all the diffracted fields are evanescent and decay along the normal direction with a decay length 2π G . Thus, field components with larger G decay faster and the field variations along the surface smoothen out as we move away from the tip. Thus, the overlap between E 1 and E2 , i.e., the product E1 E2 averaged along the surface, may be an increasing function of distance. Finally, far enough from the tip, most Fourier components would decay so that E1 E2 would again be zero. The main result of this qualitative discussion is that E1 E2 is not monotonic and we expect it to exhibit a maximum at a finite distance d. If the tip in Fig. 2 is brought close to the surface of a material, the evanescent fields E1 and E2 may mix to produce a propagating field E3 with frequency ω3 ω1 ω2 . However, unlike the usual SNOM signals, this instrument would be most sensitive at a distance from the tip and not at the tip itself. Thus, scanning the tip vertically and detecting the maximum in the SFG intensity we may infer the depth where the nonlinear signal is produced (Fig. 3).
phys. stat. sol. (2003)
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Fig. 2 SNOM tip (light gray) coated with a periodic array of alternating layers of widths d1 and d2 of materials which are either perfectly transparent at frequency ω1 and opaque at ω2 or the other way around. The frequency components of the field are either absorbed (short arrows) or transmitted (long arrows) through each layer, giving rise to the transmitted fields Ea at ωa (a 1 2) plotted immediately below the tip (left panel) and farther away from the tip (right panel).
Fig. 3 A typical SNOM with a sensitivity that decays monotonically from the tip (left) and a SNOM similar to that in Fig. 2 (right) with a sensitivity that has a maximum a distance below the tip. For the latter, the signal produced by a layer (shaded) buried a distance p below the surface is maximized when the tip is at a height h p above the surface.
3 SHG In order to avoid the need of illuminating the tip of the SNOM with two different beams to produce SFG, we consider now the possibility of generating SHG. Instead of separating at the tip fields E1 and E2 of different frequencies, we consider separating fields of different polarizations but with the same frequency. Thus, we assume that the layers in Fig. 2 are now perfect polarizers that either absorb light polarized along the x direction while letting through light polarized along the y direction or viceversa . As the field is scattered along the x direction we identify the xz plane as the plane of propagation, a 1 s layers as
6 W. L. Mochán, C. López-Bastidas, J. A. Maytorena, B. S. Mendoza, and V. L. Brudny: Nonlinear optical nanoscopy
s-polarizers and a 2 p layers with p polarizers. Therefore, we identify E 1 with Es and E2 with E p and write the fields below the tip as Esy and E px
∑ EsG eiGx e
Gz
(1)
G
∑ E pG eiGx e
Gz
(2)
G
which satisfy the wave equation for z 0 in the limit of a small period d, i.e., the non-retarded Laplace equation, and we assumed periodicity along x, ignoring the edge effects at the boundary of the tip. We assume that the tip is at z 0 and the system under study is somewhere at z 0, z points downwards in Fig. 2, and for simplicity, we calculate the fundamental fields as if in vacuum. The z component of E p may be obtained from Eq. (2) by applying Gauss law ∇ E p 0. The periodic step functions shown in Fig. 2 immediately below the tip yield the amplitudes
E0a iGxa Gda (3) e sin Gd 2 where xa denotes the center of a layer of type a. Here E0s and E0p are constants which depend on the absortance and transmittance of each layer and on the polarization of the incident field, which we assumed to be polarized along some intermediate direction between x and y. For our purposes we may take E 0s E0p E0 , which corresponds to 45 incident polarization and alternate layers with the same transmittance. We assume now that the nanoscope is brought to the neighborhood of an isotropic centrosymmetric material with nonlinear susceptibilities γ and δ , where the s and p fields (1) and (2) are mixed and produce a nonlinear polarization P at the second harmonic frequency [43] EaG
2
P
γ∇E 2 δ E ∇E
(4)
By filtering out the p component of the induced polarization we are left with only crossed terms of the form
δ E ∇E
Thus, in the previous section, we expect no signal from the immediate vicinity of the tip, where E and E areasnon-overlaping, and no signal originated far from the tip, where there are no evanescent fields left. Thus, the situation seems qualitatively similar to that illustrated in Fig. 3, and we expect a distance where the nonlinear polarization acquires a maximum. Unfortunately, a calculation of P employing Eqs. (1), Ps
p
s
(5)
s
p
s
(2), and (5), yield a null result. The reason for the null polarization is that our tip is symmetric under the inversion x x. Thus, the s SH polarization induced by mixing an s fundamental field scattered with any wavevector Gs G with a p polarized fundamental field with the wavevector Gp G has the same size and the opposite phase to the polarization induced by an s field with Gs G mixed with a p field with Gp G. We notice that only fields with Gp Gs contribute to Ps and thus to a far field. Combinations with Gp Gs produce polarizations which oscilate along the x direction with a large wavevector G p Gs and therefore produce only evanescent SH fields. Although they may be scattered into far fields by small structures, if not null, Ps should dominate. In order to produce a finite Ps we have to break the inversion symmetry of the surface. This may be done by asymmetrically introducing perfectly absorbing spacers between the polarizers, as indicated in Fig. 4. We have calculated the intensity of the SH light generated by a nonlinear layer buried beneath a surface as a function of the distance to the tip of the nanoscope. As exemplified by Fig. 5, there is a peak at a distance 0 08d, very close to the decay length d 4π of the intensity of an evanescent wave with wavelength d equal to the period of the polarizer array.
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Fig. 4 Tip coated with a periodic array of polarizing layers of widths ds and d p (shaded) separated by perfectly absorbing spacers (black). The period d, the distance between the centers of two layers r and the transmitted polarizations of each kind of layers are indicated. 1
Normalized Intensity
0.8 0.6 0.4 0.2 0 0
0.05
0.1 Distance/period
0.15
0.2
Fig. 5 Normalized intensity of the SHG generated by a nonlinear layer buried beneath a surface and excited by the near-fields produced by the tip of Fig. 4 with ds d p 0 2d and r 0 65d as a function of its distance to the tip normalized to the superlattice period d.
4 Realization Having shown that it is possible to produce a maximum in the sensitivity of an SHG nonlinear SNOM at a given distance from the tip, in this section we investigate a more realistic realization of the nanoscope tip. We consider a monolayer consisting of anisotropic molecules layed down in a periodic pattern such as that shown in Fig. 6. When illuminated by fundamental light polarized along an oblique direction with both x and y components, both kinds of molecules become polarized and produce fields with rapid spatial variations and with a polarization that alternates between the x and y directions. Therefore, we can expect that the SH field induced in a sample below the tip might have a maximum analogous to that in the previous section. Before calculating the field produced by a polarized array of molecules as that in Fig. 6 we realize that even though the period d may be very small, a fully nonretarded calculation would be inadequate. The reason is that, according to Eq. (4), the spatially averaged nonlinear polarization produced by a field E derived from a potential φ is
P
γ ∇ ∇φ
2
δ ∇φ ∇∇φ
(6)
8 W. L. Mochán, C. López-Bastidas, J. A. Maytorena, B. S. Mendoza, and V. L. Brudny: Nonlinear optical nanoscopy
Fig. 6 Bottom view of a SNOM tip on whose surface a periodic array consisting of alternating rows of anisotropic polarizable molecules with mutually perpendicular orientations. The principal direction with the largest linear polarizability is indicated for each molecule by double arrows. The period d along the alternation direction, the period s along the perpendicular direction and the distance r between succesive rows are indicated, as well as the coordinate axes.
Integrating by parts over one unit cell of the structure to calculate the averages we obtain P 0 as the integral over a period of a gradient is identically zero. Thus, we have to incorporate magnetic effects in our calculation. For simplicity, we will assume that s is much smaller that d, so that we approximate the system as if it were continuous along the y direction and take account of the spatial variations along x only, as in the previous section. Then, we can view the system as a collection of long flat capacitors with electrodes separated along the x direction alternating with wires oriented along the y direction. To lowest order in the retardation, the former produce an electric field
Ep
2π
pp G 0 iG eiGx e sd ∑ G
Gz
(7)
while the latter produce a magnetic field
Bs
2πiq
ps 1 0 i sgn G eiG x sd ∑ G
r
e
Gz
(8)
∇ E
where p p and ps are the amplitudes of the dipoles induced in the molecules which are aligned with the x and y axes respectively, q ω c and ω is the fundamental frequency. We write Eq. (5) as
δ E (9) we use ∇ E iqB and we substitute Eqs. (7) and (8) to calculate the intensity of SHG induced by the Ps
p
s
s
s
tip of Fig. 6 on a nonlinear material a distance z below. The nonlinear polarization (9) depends on the nonlinear response δ of the material and the driving term, which we write as
∇ E ζ qs s d P d E
p
2
s
2 2 1
(10)
where ζ is a dimensionless number which characterizes the potentiality of the near-field to generate an SH signal and P1 is the volumetric linear polarization P1 p s3 induced in the dipolar overlayer at the tip of the SNOM (we chose ps p p p). In Fig. 7 we show ζ as a function of the asymmetry parameter r d and
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|ζ| 2000 1500 1000 500 0
0.1 0.2 0.3
r/d
0.4
0.1
0.05
0.5 0
0.2
0.15
z/d
Fig. 7 SHG source strength ζ for different asymmetries r d as a function of the normalized distance z d to the tip.
of the normalized distance z d between the tip and the nonlinear material. As expected from the previous section, there is no SHG for an inversion invariant tip r 5d but ζ may acquire quite large values of the order of 103 for small r. The results are symmetrical around r 5d. An analysis of Fig. 7 shows that as r varies from 0 45d to 0 1d, both the position of the peak and its width move from d 10 to about d 40. The size of the SH signal may be estimated as follows: A thin region of width with an SH polarization Ps produces an SH far field Es q Ps. For our arrangement, Ps a3B e E ∇ E ζ a3B e qs 2 s d 2 P12 d, where we approximated the nonlinear quadrupolar response δ by its typical value a3B e, i.e., the inverse of a typical internal field divided by an atomic distance. We found that 0 1d and we may reasonably take s 10aB and d 10s. We finally write P1 χE. On the other hand, the typical polarization for the usual dipolar surface SHG is Ps a2B e E 2 and the polarized selvedge has a size aB . Putting all of these quantities together we obtain E SNOM E surf 10 1 qaB 2 χ2 ζ. Fig. 7 shows that ζ may be as large as 103 and at resonance χ might easily reach 10 or 100, so that the signal from the nonlinear nanoscope is expected to be comparable to that of the usual surface SHG. The calculations above have been performed for a perfectly periodic system, which is necessarily extended along the xy plane and therefore provides no lateral resolution. Preliminar calculations show that qualitatively similar results hold for finite tips, even when they have only one period, although the sensitivity would fall off as a power instead of exponentially. In this case the SH intensity would be smaller than the estimate above, but as in other SNOMs, it could be increased by tuning the fundamental field to a strong resonance of the tip.
5 Conclusions We have shown that three wave mixing may be employed to construct an optical near-field surface probe capable of observing buried structures with a depth resolution of the order of nanometers. By coating
10W. L. Mochán, C. López-Bastidas, J. A. Maytorena, B. S. Mendoza, and V. L. Brudny: Nonlinear optical nanoscopy
the tip with a small-period patterned material, evanescent fundamental fields may be produced which are spatially separated at the tip. Separated s and p fields may be produced by alternating linear chains of oriented anisotropic molecules. The arrangement has to lack inversion symmetry for this scheme to work. The source for the SHG has a peak at a distance close to a tenth of the modulation period and the generated signal is expected to be not smaller than that for ordinary surface SHG. Thus, we believe that the instrument we propose here is viable and worth of further research. Acknowledgements We acknowledge the support from DGAPA-UNAM under grant IN117402 (WLM, CLB and JM), from Conacyt under grants 36033-E (BM) and C01-41113 (CLB) and from Fundación Antorchas (VLB). VLB is also with CONICET, Argentina.
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