1. Introduction

White light supercontinuum, which can be generated by coupling short laser pulses into a highly nonlinear photonic crystal fiber [1, 2], is unique in that it not only covers a broad wavelength range (e.g., ~1000nm from visible to near infrared) but also has high spatial coherence. As a consequence, it can be focused to illuminate a single nano- or micro- scaled object and probe its properties via optical spectroscopy. For instance, Rayleigh scattering spectrum obtained with focused supercontinuum was used to probe the electronic structures of individual carbon nanotubes [3]. Focused supercontinuum was also used to perform spectroscopic measurement on single gold nanoparticles [4]. Recently, we have demonstrated a single particle spectroscopy technique [5] in which tightly focused supercontinuum is used not only to trap a particle but also at the same time to obtain its scattering spectrum. Unlike conventional spectroscopy, single particle spectroscopy can avoid ensemble averaging and have the capability to probe the properties of individual particles (e.g., size, shape, refractive index, resonant absorption, chemical composition), which can lead to many important applications particularly in nanoparticle characterization and sensing. To further explore the potential of single particle spectroscopy with focused supercontinuum illumination, theoretical modeling tools need to be developed.

Optical scattering in focused laser beam have been previously studied [6–12]. For instance, dipole radiation field approximation [7] was used to analyze scattering by a spherical particle in a converging beam. Diffraction theory[8] and generalized Lorenz-Mie theory [9–11] were applied to investigate scattering of Gaussian beams. However, the dipole field and paraxial Gaussian beam are not suitable for modeling tightly focused field produced by high numerical aperture objective lens. Fourier angular spectrum representation, on the other hand, can be used to describe strongly focused beam and, for example, has been applied to investigate scattering in optical tweezers [12], reflection at media interface [13], and confocal imaging[14]. In this paper, we apply the angular spectrum analysis and Mie’s scattering theory to investigate optical scattering spectroscopy of a single spherical scatterer illuminated with tightly focused supercontinuum. The theory is compared with carefully designed experiment. The effect of chromatic aberration is also considered.

2. Theory

The schematic diagram of our model system is shown in Fig. 1. We first consider a particular wavelength component (λ₀) of the incoming white light supercontinuum, which is polarized along the x direction and tightly focused by a high-numerical-aperture objective lens L1 (NA=nsinθ _m). The electric field near the focal point can be expressed by using angular spectrum decomposition [12–14].

where k _i =2πn(sinθcosϕ x̂ + sin θ sin ϕ ŷ + cos θ ẑ)/λ ₀ , n is the refractive index of the medium at λ₀, and θ, ϕ are the polar and azimuthal angles. E͂(k_i) is the spatial spectrum of the electrical field, which can be expressed in terms of the far field E _-f(θ,ϕ) immediately after the high-NA objective lens L1 [13, 15].

 

Fig. 1. Schematic diagram of optical scattering by using tightly focused supercontinuum. A linearly-polarized incoming supercontinuum is tightly focused by objective lens L1. The scattered light produced by a spherical scatterer is collected by objective lens L2 and analyzed by a spectrometer.

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where f is the focal length of the objective lens. To arrive at Eq. (2), far field approximation and the method of stationary phase [15] were used. If the incoming beam is a plane wave with a uniform electric field distribution E _inc = E ₀ x̂, E _-f (θ,ϕ) can be obtained by considering the refraction of the incoming rays [14].

If the refractive index of the medium is different from that of the coverslip and objective lens immersion oil, Eq. (2) should be modified to

where ψ = {n cos θ - n_c cos[sin ^-1 (n sin θ/ n_c)]}2πd / λ ₀, d is the distance from the nominal focal point to the coverslip/medium interface, T is the transmission coefficient at the interface of the media given by Fresnel’s law, and n _c is the refractive index of the coverslip at λ₀. Eq. (3) and (4) completely determine the field near the focal point. Now let us consider the optical scattering caused by a microsphere which is located at a position r ₀ (measured from the nominal focal point). Since the field near the focal point can be thought of as the superposition of many plane wavelets ( E͂(k_i)e^jk_i·r ), the scattered field can be obtained by linearly superimposing the scattered wavelets produced by each plane wavelet, which can be calculated by using Mie’s theory if the scatterer is spherical. Here we also assume that the sphere is either immersed in a uniform medium or far enough from any media interface. We shift the origin of our lab coordinate system from the nominal focal point to the center of the microsphere. This can be easily achieved by replacing r with r+r ₀ in Eq. (1). In addition, for each plane wavelet we define a rotated coordinate system (denoted with a prime) such that the plane wavelet is polarized along the x′ direction and travels along the z′ direction in the new coordinate system (the standard configuration used in Mie’s theory). The transformation from the rotated system to the lab (fixed) coordinate system is given by the Matrix M [14].

From Mie’s theory [14], the scattered wavelet in the rotated coordinate system is given by

where r'=r'(sinα'cosβ'x̂'+sinα'sinβ'ŷ'+cosα'ẑ'), α' and β' are the polar and azimuthal angles in the rotated coordinate system, and S ₁, S ₂ are the scattering functions for the in-plane and out-of-plane components [16] respectively which depend on the wavelength of the incident beam λ₀, radius of the microsphere, refractive index of the surrounding medium, refractive index of the microsphere, and the scattering angle α'. In the lab coordinate system, the scattered wavelet is then given by

where r = r(sinαcos β x̂ + sinαsin β ŷ + cosα ẑ). The total scattered wave is the superposition of all scattered wavelets.

The total field is then given by the superposition of the total scattered field and the incident field. Let us consider the far field at a distance r from the microsphere. The far field of the incident beam is given by

where k_o = 2πn(sin α cos β x̂ + sin α sin β ŷ + cos α ẑ)/ λ ₀ . Finally, an objective lens L2 (NA=nsinα _m) is used to collect the light in the forward direction. In the far field, the detected power is approximately given by

Since supercontinuum can cover an extremely broad wavelength range from UV, visible, to near infrared, chromatic aberration at the focal point should also be considered. Notice that

where ∆f is the chromatic aberration of the measurement system which can be calibrated. Eq. (11) together with Eq. (3)–(10) can be used to analyze the forward optical scattering spectrum of a spherical scatterer illuminated with tightly focused supercontinuum.

3. Experiment

To verify the theoretical analysis, we have also investigated scattering of tightly focused supercontinuum experimentally. Since the scattering spectrum depends on the position of the scatterer (c.f Eq. 6), in order to quantitatively compare theory and experiment we immobilized microspheres (Duke scientific) on the bottom of the sample cell (made from standard glass coverslip) so that we can precisely control and determine the position of microspheres. Immobilization was achieved by air-drying microsphere solution in a sample cell for a few days. The sample cell was then filled with immersion oil whose refractive index matches that of the sample cell. As a result, approximately speaking the microsphere is surrounded by a uniform medium. The experimental setup is shown in Fig. 2. White light supercontinuum, which was generated by coupling sub-nanosecond laser pulses (JDS Uniphase NP-10620-100) into a nonlinear photonic crystal fiber (BlazePhotonics SC-5.0-1040), was collimated and then tightly focused onto a single microsphere by an apochromatic objective lens L1 (100x /1.4NA). The full aperture of the objective lens was uniformly illuminated. Another objective lens L2 (10x /0.25N.A.) was used to collect the forward scattered light. In addition, an iris was used to further limit the effective numerical aperture of the collection system (effective N.A.~ 0.08). Finally, the spectrum of the forward-scattered light was measured by an optical spectrum analyzer. A CCD camera was used to monitor the position of the microsphere.

 

Fig. 2. Schematic diagram of experiment setup

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We normalized the forward scattering spectrum to the reference spectrum which was measured when no microsphere was present. This ratio is defined as scattering efficiency (denoted by q in Fig. 3 and Fig. 5.). By mechanically translating the sample cell, normalized scattering spectra (i.e., scattering efficiency) at different positions can be obtained. The results are plotted in Fig. 3. Figure 3(a) and (b) show the scattering efficiencies of two microspheres (diameter 1.5μm and 2μm respectively) at λ₀=600nm as a function of their axial position. The blue dots correspond to experimental data while the red lines represent calculation results. Intuitively, since the refractive index of the microsphere is larger than that of the surrounding medium it works like a positive lens approximately. A strong peak can be observed in the scattering efficiency curve if the sphere is moved away from the lens and collimates the incident beam. On the other hand, a minimum occurs if the sphere is moved closer to the lens and focuses/defocuses the incident beam more strongly. Finally, when the sphere is very far away from the focal region the scattering effect is weakened and the scattering efficiency approaches to unity as expected. Compared with that of the 1.5-μm-diameter microsphere, the maximum scattering efficiency of the 2-μm-diameter microsphere is slightly higher and the positions of its maximum and minimum are further away from the focal point.

 

Fig. 3 Dependence of scattering efficiency on axial position (a) and (b) are the scattering efficiencies of the 1.5- and 2-μm diameter microspheres respectively at λ₀=600nm as a function of axial position. Red lines are theoretical calculation results. Blue dots are experiment results.

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Due to chromatic aberration, different wavelengths focus at different axial positions. As a result, the z-dependence scattering efficiency curve for different wavelengths will center at different positions (their actual focal points). We correlated the experimentally measured scattering efficiency z-dependence curves with the corresponding theoretical curves (assuming no chromatic aberration) for each wavelength. The positions of correlation peaks indicate chromatic aberration for different wavelengths. We used both the 1.5- and 2.0-μm-diameter microspheres to calibrate the chromatic aberration of the whole system. The result ∆f = f(λ)-f(λ₀=600nm) is shown in Fig.4.

 

Fig. 4. Measured chromatic aberration of the experimental system. Circles and pluses represent the results using 1.5 and 2.0-μm-diameter microspheres respectively. The blue line is the fitted curve.

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Fig. 5. Scattering spectra of a 1.5-μm-diameter microsphere at two different positions. Dotted red lines are theoretical calculation results while solid blue lines are experiment data.

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Figure 5 shows the measured and calculated scattering efficiencies of the 1.5-μm-diameter microsphere as a function of wavelength at two different axial positions. Chromatic aberration was included in the calculation. The theory (dotted red line) in general agrees with the experimental results (solid blue line). However, there exists quite considerable discrepancy between the theory and experiment when the focused spot is moved near the interface of microsphere/cover slip. Several factors may have contributed to it: (1) although the refractive index of the immersion oil (1.515) is very close to that of the glass coverslip (1.523), they are not identical. As a matter of fact, the difference is about 10% of the refractive index difference between the microsphere (1.59) and the immersion oil. Boundary effects (e.g., multiple reflections and scatterings between the coverslip and the microsphere) may not be ignored. (2) dispersion of the microsphere, glass, and immersion oil was not considered in the theoretical calculation (3) possible lateral displacement of the microsphere from the center of the beam (4) noise in the supercontinuum. Since the polymer sphere is quite inert in the measurement wavelength range, the measured scattering spectrum does not show any pronounced features. We should point out that the method outlined here can also be used to model optical scattering of other types of particles including metallic particles. Since the scattering spectrum depends on particle size, shape, refractive index, and resonant absorption (if exists), optical spectroscopy with focused supercontinuum is a promising technique to non-invasively probe the properties of individual particles.

4. Summary and future work

In conclusion, we have studied optical scattering of a single spherical scatterer illuminated with tightly focused supercontinuum. The initial results are promising and show quite good agreement between the theory and the experiment. In the future, we plan to apply the theory to model optical scattering spectroscopy of a sphere three-dimensionally trapped by tightly focused supercontinuum (supercontinuum tweezers). In this case, the measured optical scattering spectra will be position averaged due to the Brownian motion of the trapped object. Although our analysis is primarily focused on the forward direction, scattering spectra at any arbitrary angle particularly in the backward direction and with different polarizations can also be obtained with some modification of the equations. Such analysis and experiments will also be part of our future work.