Chapter 4 Spatial resolution and position accuracy Position accuracy refers to the precision with which an object can be localized in space. Spatial resolution, on the other hand, is a measure for the ability to distinguish two separated point-like objects from a single object. The diffraction limit implies that optical resolution is ultimately limited by the wavelength of light. Before the advent of near-field optics it was believed that the diffraction limit imposes a hard boundary and that physical laws strictly prohibit resolution significantly better than λ/2. It was found that this limit is not as strict as assumed and that various tricks allow us to access the evanescent modes of the spatial spectrum. In this chapter we will analyze the diffraction limit and discuss the principles of different imaging modes with resolutions near or beyond the diffraction limit. 4.1 The point spread function The point spread function is a measure of the resolving power of an optical system. The narrower the point spread function the better the resolution will be. As the name implies, the point spread function defines the spread of a point source. If we have a radiating point source then the image of that source will appear to have a finite size. This broadening is a direct consequence of spatial filtering. A point in space is characterized by a delta function which has an infinite spectrum of spatial frequen- cies kx, ky. On the way of propagation from the source to the image, high frequency 2 2 2 components are filtered out. Usually, the entire spectrum (kx +ky) > k associated with the evanescent waves is lost. Furthermore, not all plane wave components can be collected which leads to a further reduction in bandwidth. The reduced spectrum is not able to accurately reconstruct the original point source and the image of the point will have a finite size. The standard derivation of the point spread function is 47 48 CHAPTER 4. SPATIAL RESOLUTION AND POSITION ACCURACY based on scalar theory and the paraxial approximation. This theory is insufficient for many high resolution optical systems. With the so far established ‘angular spectrum’ framework we are in a position to rigorously investigate image formation in an optical system. Consider the situation in Fig. 4.1 which has been analyzed by Sheppard and Wil- son [1] and more recently by Enderlein [2]. An ideal electromagnetic point source is located in the focus of a high NA aplanatic objective lens with focal length f. This lens collimates the rays emanating from the point source and a second lens with focal length f ′ focuses the fields on the image plane at z =0. The situation is similar to the problem in Fig. 3.18. The only difference is that the source is a point source instead of the reflected field at an interface. The smallest radiating electromagnetic unit is a dipole. In the optical regime most subwavelength-sized particles scatter as electric dipoles. On the other hand, small apertures radiate as magnetic dipoles. In the microwave regime, paramagnetic materials exhibit magnetic transitions and in the infrared, small metal particles show magnetic dipole absorption caused by eddy currents of free carriers produced by the magnetic field. Nevertheless, we can restrict our analysis to an electric dipole since the field of a magnetic dipole is identical to the field of an electric dipole if we inter- change the electric and magnetic fields, i.e. E H and H E. → →− In its most general form, the electric field at a point r of an arbitrarily oriented electric dipole located at ro with dipole moment µ is defined by the dyadic Green’s ↔ function G (r, ro) as 2 ω ↔ E r G r r µ ( ) = 2 ( , o) . (4.1) εo c · n ns z µ ns' Θ Θ' z object image n sin f ' n' Θ = sin ' f f Θ f ' Figure 4.1: Configuration used for the calculation of the point spread function. The source is an arbitrarily oriented electric dipole with moment µ. The dipole radiation is collected with a high NA aplanatic objective lens and focused by a second lens on the image plane at z =0. 4.1. THEPOINTSPREADFUNCTION 49 We assume that the distance between dipole and objective lens is much larger than the wavelength of the emitted light. In this case, we do not need to consider the evanescent components of the dipole field. Furthermore, we choose the dipole to be located at the origin ro =0 and surrounded by a homogeneous medium with index n. ↔ In this case, we can use the free-space farfield form of G which, expressed in spherical coordinates (r, θ, φ) reads as (see Appendix ??) (1 cos2φ sin2 θ) sin φ cos φ sin2 θ cos φ sin θ cos θ ↔ exp(ikr) − 2 − 2 2 − G∞ (r, 0) = sin φ cos φ sin θ (1 sin φ sin θ) sin φ sin θ cos θ . 4πr − − − cos φ sin θ cos θ sin φ sin θ cos θ sin2 θ − − (4.2) This is simply a 3 3 matrix which has to be multiplied with the dipole moment vector × ∗ µ = (µx, µy, µz) to obtain the electric field. To describe refraction at the reference sphere f we have to project the electric field vector along the vectors nθ and nφ as already done in Section 3.5. After being refracted, the field propagates as a collimated beam to the second lens f ′ where it is refracted once again. For a dipole aligned with the x-axis (µ=µx nx) the field just after the second lens becomes ′ ′ 2 (1+cos θ cos θ ) (1 cosθ cos θ )cos2φ ′ ω µx exp(ikf) − − n cos θ E(x)(θ, φ)= (1 cos θ cos θ′)sin2φ , ∞ − ε c2 8πf − − n′ cos θ o 2cos θ sin θ′ cos φ r (4.3) where f sin θ′ = sin θ, cos θ′ = g(θ) = 1 (f/f ′)2 sin2θ . (4.4) f ′ − q The term (cos θ′/ cos θ)1/2 is a consequence of energy conservation as discussed in Section 3.5. In the limit f f ′ the contribution of cos θ′ can be ignored, but cos θ ≪ † cannot since we deal with a high NA objective lens. The fields for a dipole µy and a dipole µz can be derived in a similar way. For an arbitrarily oriented dipole µ=(µx, µy, µz) the field is simply obtained by the superposition (x) (y) (z) E∞(θ, φ) = E∞ + E∞ + E∞ . (4.5) To obtain the fields E near the focus of the second lens we insert the field E∞ into Eq. (3.47). We assume that f f ′ which allows us to use the approximations in ≪ Eq. (3.106). The integration with respect to φ can be carried out analytically and the result can be written as 2 ω ↔ E G µ (ρ,ϕ,z) = 2 P SF (ρ,ϕ,z) , (4.6) εo c · ∗ 2 The farfield at r of a dipole located at ro = 0 can also be expressed as E = ω µo [r r − × × µ] exp(ikr)/4πr3 . †The factor (cos θ)−1/2 has been omitted in Ref. [1] 50 CHAPTER 4. SPATIAL RESOLUTION AND POSITION ACCURACY where the dyadic point spread function is given by ˜ ˜ ˜ ˜ ′ (I00 +I02 cos2ϕ) I02 sin2ϕ 2iI01 cos ϕ ↔ k f ′ ′ − n G = ei (kf−k f ) I˜ sin2ϕ (I˜ I˜ cos2ϕ) 2iI˜ sin ϕ , P SF 8πi f ′ 02 00 − 02 − 01 n′ 0 0 0 r (4.7) and the integrals I˜00 - I˜02 are defined as θmax I˜ (ρ,z) = (cos θ)1/2 sin θ (1+cos θ) J (k′ ρ sin θf/f ′) (4.8) 00 0 × Z0 exp ik′z [1 1/2(f/f ′)2 sin2θ] dθ , − θmax I˜ (ρ,z) = (cos θ)1/2 sin2 θ J (k′ ρ sin θf/f ′) (4.9) 01 1 × Z0 exp ik′z [1 1/2(f/f ′)2 sin2θ] dθ , − θmax I˜ (ρ,z) = (cos θ)1/2 sin θ (1 cos θ) J (k′ ρ sin θf/f ′) (4.10) 02 − 2 × Z0 exp ik′z [1 1/2(f/f ′)2 sin2θ] dθ . − ↔ The first column of GP SF denotes the field of a dipole µx, the second column the field of a dipole µy and the third column the field of a dipole µz. The integrals I˜00 - I˜02 are similar to the integrals I00 - I02 encountered in conjunction with the focusing of a Gaussian beam [c.f. Eqs. (3.58 - 3.60)]. The main differences are the arguments of the Bessel functions and the exponential functions. Furthermore, the longitudinal field E is zero in the present case because we required f f ′. z ≪ Eqs. (4.6 - 4.10) describe the mapping of an arbitrarily oriented electric dipole from its source to its image. The result depends on the numerical aperture NA of the primary objective lens NA = n sin θmax (4.11) and the (transverse) magnification M of the optical system defined as n f ′ M = . (4.12) n′ f In the following, we will use the quantity E 2 to denote the point spread function, | | since it is the quantity relevant to optical detectors. We first consider the situation of a dipole with its axis perpendicular to the optical axis. Without loss of generality, we can define the x-axis to be parallel with the dipole axis, i.e. µ=µx nx. For a low NA objective lens, θmax is sufficiently small which allows us to make the approximations cos θ 1 and sin θ θ.