418
A. V. Maslov and V. N. Astratov
represented as
P =
2π
0
dϕ p(ϕ), p(ϕ) =
c
2
2πωε b
| f (ϕ)|
2
,
(16.5)
where p(ϕ) is the angular power density and ϕ is the angle with respect to the x axis.
Figure 16.4(b) shows p(ϕ). In the absence of the particle, the power distribution
is p(ϕ) ∼ cos
2
ϕ. In the nonresonant case, the power emission in the upper halfspace becomes narrow and multiple lobes appear. In the resonant case, the emission
distribution also shows rather narrow but small peaks, which can be attributed to the
reemission of the standing WGMs excited in the particle.
16.4 Microsphere as a Solid Immersion Lens: Resolution
Analysis
16.4.1 Imaging of a Point Source
A basic characteristic of any imaging system is the response to a point source, called
the point spread function (PSF). For incoherent imaging, it is sufficient to know
the image intensity created by a point source. For coherent imaging, one needs the
complex field distribution which includes the magnitude and phase of the created
field on the detector. The knowledge of PSF allows one to find the incoherent images
created by extended objects by performing the convolution of the PSF and the objects.
Here we discuss the PSF of a microsphere lens within our 2D model. To achieve
this we can take the far-field distribution f (ϕ) created by current source (16.3). The
propagating field in the background material can be represented as
H z (x, y) =
|g|
ihx+igy ˜
H z (g),
(16.6)
where ˜
H z (g) is the Fourier component and g
2
+ h
2
= k
2
b . Evaluating the integral
in (16.6) asymptotically at ρ → ∞ and comparing with (16.4) give us the Fourier
component
˜
H z (g) =
f (ϕ)
√
2π k b cos ϕ
, g = k b sin ϕ.
(16.7)
An objective lens collects the far field and forms an image on the detector. For an
ideal objective lens with NA = 1 and without magnification the image is equivalent
to the intensity distribution of the backpropagated far field in the plane of focus
(defined by x). The knowledge of ˜
H z allows us to formally backpropagate the field
to any point in space and find the image intensity
A. V. Maslov and V. N. Astratov
represented as
P =
2π
0
dϕ p(ϕ), p(ϕ) =
c
2
2πωε b
| f (ϕ)|
2
,
(16.5)
where p(ϕ) is the angular power density and ϕ is the angle with respect to the x axis.
Figure 16.4(b) shows p(ϕ). In the absence of the particle, the power distribution
is p(ϕ) ∼ cos
2
ϕ. In the nonresonant case, the power emission in the upper halfspace becomes narrow and multiple lobes appear. In the resonant case, the emission
distribution also shows rather narrow but small peaks, which can be attributed to the
reemission of the standing WGMs excited in the particle.
16.4 Microsphere as a Solid Immersion Lens: Resolution
Analysis
16.4.1 Imaging of a Point Source
A basic characteristic of any imaging system is the response to a point source, called
the point spread function (PSF). For incoherent imaging, it is sufficient to know
the image intensity created by a point source. For coherent imaging, one needs the
complex field distribution which includes the magnitude and phase of the created
field on the detector. The knowledge of PSF allows one to find the incoherent images
created by extended objects by performing the convolution of the PSF and the objects.
Here we discuss the PSF of a microsphere lens within our 2D model. To achieve
this we can take the far-field distribution f (ϕ) created by current source (16.3). The
propagating field in the background material can be represented as
H z (x, y) =
|g|
ihx+igy ˜
H z (g),
(16.6)
where ˜
H z (g) is the Fourier component and g
2
+ h
2
= k
2
b . Evaluating the integral
in (16.6) asymptotically at ρ → ∞ and comparing with (16.4) give us the Fourier
component
˜
H z (g) =
f (ϕ)
√
2π k b cos ϕ
, g = k b sin ϕ.
(16.7)
An objective lens collects the far field and forms an image on the detector. For an
ideal objective lens with NA = 1 and without magnification the image is equivalent
to the intensity distribution of the backpropagated far field in the plane of focus
(defined by x). The knowledge of ˜
H z allows us to formally backpropagate the field
to any point in space and find the image intensity