2
C. Hu and G. Popescu
Combing these pioneer works, quantitative phase imaging (QPI) appears and has
become a powerful label-free imaging modality for studying biospecimens [1, 8–11].
In general, QPI produces a map of optical pathlength shift associated with the specimen, and thus, reports information about both the local thickness as well as the
refractive index, which are directly related to a sample’s biophysical properties. In
addition, because it employs interferometry, QPI exhibits nanoscale sensitivity to
morphological and dynamic changes of transparent features. Since the early demonstrations in the 1990s [12–15], multiple QPI imaging platforms have been proposed,
which enable 2D, 3D, and 4D investigation capabilities [16–19]. These instruments
provide the remarkable performance of QPI in a range of applications in biomedicine
and material science.
In this chapter, we overview several important aspects of QPI, from basic physics
to system instrumentation. The material is organized as follows. Section 1.2 presents
the physical interpretation of the QPI data in transmission versus backscattering,
using the first-order Born approximation. Section 1.3 focuses on the typical QPI
configurations, phase retrieval methods and novel QPI systems. Section 1.4 presents
QPI applications in a few aspects. Section 1.5 discusses concepts related to the
phase measurability and resolution. We summarize and discuss future directions in
Sect. 1.6.
1.2 Physical Interpretation of Phase Imaging
in Transmission Versus Reflection Measurements
Let us consider the physics meaning of the measurable phase associated with an
optical field interacting with a weakly scattering medium. Given an inhomogeneous
medium, the scattered field, U 1 , satisfies the inhomogeneous Helmholtz equation
[20–24]
∇
2 U 1 (r) + n
2
0 β
2
0 U 1 (r) = −β
2
0 χ(r)U (r)
(1.1)
where r is the spatial coordinate, n 0 the refractive index in the background, β 0 =
2π/λ with λ the wavelength, and χ(r) = n
2
(r)−n
2
0 the scattering potential. The total
field, U, is the sum of the incident field (U 0 ) and the linear scattering component (U 1 ),
i.e., U = U 0 +U 1 . Let us consider the object is illuminated by a monochromatic plane
wave traveling in z-axis, U 0 = Ae
iβ 0 z with A the amplitude. Under the first-order
Born approximation, which assumes the incident wave is dominant in the medium,
(1.1) becomes
∇
2 U 1 (r) + β
2 U 1 (r) = −β
2
0 χ(r)Ae
iβ 0 z
(1.2)
In (1.2), β = n 0 β 0 . Performing a Fourier transform with respect to r, the scattering
field can be easily obtained in the spatial frequency (k) space [25, 26] as
C. Hu and G. Popescu
Combing these pioneer works, quantitative phase imaging (QPI) appears and has
become a powerful label-free imaging modality for studying biospecimens [1, 8–11].
In general, QPI produces a map of optical pathlength shift associated with the specimen, and thus, reports information about both the local thickness as well as the
refractive index, which are directly related to a sample’s biophysical properties. In
addition, because it employs interferometry, QPI exhibits nanoscale sensitivity to
morphological and dynamic changes of transparent features. Since the early demonstrations in the 1990s [12–15], multiple QPI imaging platforms have been proposed,
which enable 2D, 3D, and 4D investigation capabilities [16–19]. These instruments
provide the remarkable performance of QPI in a range of applications in biomedicine
and material science.
In this chapter, we overview several important aspects of QPI, from basic physics
to system instrumentation. The material is organized as follows. Section 1.2 presents
the physical interpretation of the QPI data in transmission versus backscattering,
using the first-order Born approximation. Section 1.3 focuses on the typical QPI
configurations, phase retrieval methods and novel QPI systems. Section 1.4 presents
QPI applications in a few aspects. Section 1.5 discusses concepts related to the
phase measurability and resolution. We summarize and discuss future directions in
Sect. 1.6.
1.2 Physical Interpretation of Phase Imaging
in Transmission Versus Reflection Measurements
Let us consider the physics meaning of the measurable phase associated with an
optical field interacting with a weakly scattering medium. Given an inhomogeneous
medium, the scattered field, U 1 , satisfies the inhomogeneous Helmholtz equation
[20–24]
∇
2 U 1 (r) + n
2
0 β
2
0 U 1 (r) = −β
2
0 χ(r)U (r)
(1.1)
where r is the spatial coordinate, n 0 the refractive index in the background, β 0 =
2π/λ with λ the wavelength, and χ(r) = n
2
(r)−n
2
0 the scattering potential. The total
field, U, is the sum of the incident field (U 0 ) and the linear scattering component (U 1 ),
i.e., U = U 0 +U 1 . Let us consider the object is illuminated by a monochromatic plane
wave traveling in z-axis, U 0 = Ae
iβ 0 z with A the amplitude. Under the first-order
Born approximation, which assumes the incident wave is dominant in the medium,
(1.1) becomes
∇
2 U 1 (r) + β
2 U 1 (r) = −β
2
0 χ(r)Ae
iβ 0 z
(1.2)
In (1.2), β = n 0 β 0 . Performing a Fourier transform with respect to r, the scattering
field can be easily obtained in the spatial frequency (k) space [25, 26] as