17 Microsphere-Assisted Interference Microscopy
445
by using an even illumination of the object using an arrangement of lenses and
diaphragms [8].
In addition in [6], H. von Helmholtz also discussed a second spatial resolution
criterion which was determined previously by Lord Rayleigh in 1874 [9]. Unlike E.
Abbe who assumed the object to consist of diffraction gratings, Lord Rayleigh and
H. von Helmholtz considered the object as a sum of emitting point light sources.
Lord Rayleigh defined the lateral resolution in the focal plane of an imaging system as the distance between the intensity maximum and the first intensity minimum
of the diffraction pattern of a bright point source. In 1896, Lord Rayleigh significantly described his criterion of lateral resolution [10] using the previous work of
G. B. Airy on the diffraction of light by a circular aperture [11].
δ x,y ≈ 1.22
f λ
D
≈ 0.61
λ
N A
(17.3)
From (17.3), N A can be retrieved through the focal length of the imaging system
f and its pupil diameter D. Moreover, Lord Rayleigh highlighted the influence of
imperfections on the imaging quality. In reality, the microscopes of the time were
not able to reach the theoretical resolution due to the associated optical aberrations.
Concurrently with these optical studies occurred the manufacturing of high-quality
optical components (e.g. with the development of new types of glass by the German
company Glastechnische Laboratorium Schott & Genossen). In 1879, O. Schott and
E. Abbe achieved in making the first abnormal dispersion glass in order to correct the
spherical aberrations for several wavelengths [12] and, in 1886, the first apochromatic
microscope objective [13]. Nowadays, the majority of optical microscopes are able
to form aberration-free images in the so-called diffraction-limited imaging systems.
Other resolution criteria exist such as the Sparrow criterion based on the contrast
response of a photodetector [14] and the Houston criterion based on the full width at
half maximum (FWHM) of the point spread function (PSF) of the imaging system
[15]. One of them, in particular, became prominent in optical microscopy (and confirmed by the work of E. Abbe): the Fourier criterion through the measurement of
the system’s frequency response [16]. Introduced in 1946 by P. M. Duffieux, Fourier
optics makes the analogy with analogue electronic systems and is based on diffraction as well as the propagation of light [17]. Indeed, this powerful method allows the
characterization of an entire imaging system through the measurement of its transfer
function, enabling both performance (e.g. the resolution) and imaging quality (e.g.
the optical aberrations) retrieval. In 1960, V. Ronchi highlighted the importance of
considering both the sensitivity of the sensor and the illumination conditions, in order
to determine the resolving power of an imaging system [18] (Fig. 17.1).
Despite a decrease in the wavelength of light λ or an increase in the refractive
index of the surrounding medium in order to achieve a better lateral resolution, this
key parameter in the classical optical microscope is still limited by the diffraction
of light, i.e. a maximum of λ/2 in air. Typically, a perfect diffraction-limited optical
microscope is able to resolve details with 300 nm of size over a lateral field of view
(FOV) of 100 µm. The problem is that the higher frequency components of the details
445
by using an even illumination of the object using an arrangement of lenses and
diaphragms [8].
In addition in [6], H. von Helmholtz also discussed a second spatial resolution
criterion which was determined previously by Lord Rayleigh in 1874 [9]. Unlike E.
Abbe who assumed the object to consist of diffraction gratings, Lord Rayleigh and
H. von Helmholtz considered the object as a sum of emitting point light sources.
Lord Rayleigh defined the lateral resolution in the focal plane of an imaging system as the distance between the intensity maximum and the first intensity minimum
of the diffraction pattern of a bright point source. In 1896, Lord Rayleigh significantly described his criterion of lateral resolution [10] using the previous work of
G. B. Airy on the diffraction of light by a circular aperture [11].
δ x,y ≈ 1.22
f λ
D
≈ 0.61
λ
N A
(17.3)
From (17.3), N A can be retrieved through the focal length of the imaging system
f and its pupil diameter D. Moreover, Lord Rayleigh highlighted the influence of
imperfections on the imaging quality. In reality, the microscopes of the time were
not able to reach the theoretical resolution due to the associated optical aberrations.
Concurrently with these optical studies occurred the manufacturing of high-quality
optical components (e.g. with the development of new types of glass by the German
company Glastechnische Laboratorium Schott & Genossen). In 1879, O. Schott and
E. Abbe achieved in making the first abnormal dispersion glass in order to correct the
spherical aberrations for several wavelengths [12] and, in 1886, the first apochromatic
microscope objective [13]. Nowadays, the majority of optical microscopes are able
to form aberration-free images in the so-called diffraction-limited imaging systems.
Other resolution criteria exist such as the Sparrow criterion based on the contrast
response of a photodetector [14] and the Houston criterion based on the full width at
half maximum (FWHM) of the point spread function (PSF) of the imaging system
[15]. One of them, in particular, became prominent in optical microscopy (and confirmed by the work of E. Abbe): the Fourier criterion through the measurement of
the system’s frequency response [16]. Introduced in 1946 by P. M. Duffieux, Fourier
optics makes the analogy with analogue electronic systems and is based on diffraction as well as the propagation of light [17]. Indeed, this powerful method allows the
characterization of an entire imaging system through the measurement of its transfer
function, enabling both performance (e.g. the resolution) and imaging quality (e.g.
the optical aberrations) retrieval. In 1960, V. Ronchi highlighted the importance of
considering both the sensitivity of the sensor and the illumination conditions, in order
to determine the resolving power of an imaging system [18] (Fig. 17.1).
Despite a decrease in the wavelength of light λ or an increase in the refractive
index of the surrounding medium in order to achieve a better lateral resolution, this
key parameter in the classical optical microscope is still limited by the diffraction
of light, i.e. a maximum of λ/2 in air. Typically, a perfect diffraction-limited optical
microscope is able to resolve details with 300 nm of size over a lateral field of view
(FOV) of 100 µm. The problem is that the higher frequency components of the details