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Microscopy: Science, Technology, Applications and Education
A. Méndez-Vilas and J. Díaz (Eds.)
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Partially Coherent Image Formation Theory for X-ray Microscopy
Chang Chang1,2 and Takashi Nakamura2
1
2
Department of Radiation Oncology, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
School of Biomedical Engineering, Science & Health Systems, Drexel University, Philadelphia, Pennsylvania 19104, USA
Partially coherent image formation theory for a full-field imaging microscope is introduced. Propagation of mutual coherence
is presented following Huygens-Fresnel principle. The concept of mutual intensity, together with the quasimonochromatic
condition under which it is valid, is also introduced. Detailed analysis of a microscope illuminated by a partially coherent
source is presented. Mutual intensity distributions at each stage of the image formation process, i.e. at the condenser, object,
objective lens and image planes, are examined thoroughly. The overall imaging process is summarized using amplitude
spread function and illumination mutual intensity. The resultant image intensity distribution is obtained analytically. Furthermore, Fourier domain techniques such as modulation transfer function and pupil function are also introduced, together
with the associated space invariance conditions. These techniques greatly simplify the analysis and provide valuable insight
into the imaging system. Formulations for both critical and Köhler illuminations are provided. The partially coherent image
formation theory presented here follows a simple x-ray microscope setup while the derivation is true in general for all wavelengths.
keywords: image formation theory; partially coherent imaging; illumination design; x-ray microscopy; statistical optics
1
Introduction
Full-field x-ray microscopes are widely used in many fields of science [1–3]. Advances in nanofabrication tehniques
enabled development of short wavelenth focusing elements and significantly improved the spatial resolution [4–9]. In
the soft x-ray spectral region, samples as small as a few tens of nanometers can be resolved using micro zone-plates
(MZP) as the objective lens [3]. In addition to conventional x-ray microscopy in which x-ray absorption difference
provides the image contrast, phase contrast mechanisms such as differential phase contrast (DIC) [10–14] and Zernike
phase contrast [10, 15, 16] are also available. These phase contrast imaging mechanisms are especially attractive at the
x-ray wavelengths where phase contrast of most materials is typically 10 times stronger than its absorption contrast. With
recent progresses in plasma-based x-ray sources [17–19] and increasing accessibility to synchrotron user facilities, x-ray
microscopes are quickly becoming a routine measurement equipment in the laboratory. Understanding of the underlying
image formation theory and the associated characterization techniques of x-ray microscopy is therefore of increasing
interest.
This chapter details the mathematical descriptions required for the analysis of a full-field microscope. Section 2
reviews the partially coherent image formation theory which lays the foundation for the transfer function method to
be introduced in Section 3. Mutual intensities at each stage of the image formation process, i.e. at the condenser,
object, objective lens and image planes, are derived based on the propagation of mutual coherence. Section 3 introduces
the transfer function method and examines the space invariance condition required for such analyses. The concept of
amplitude spread function is also introduced and the resultant convolution integral is presented. Section 4 summarizes the
chapter by examining a typical full-field microscope with an incoherent source used for illumination. The overall image
intensity distribution is obtained analytically. Fourier domain techniques such as modulation transfer function (MTF) and
pupil function are also introduced.
2
2.1
Partially Coherent Imaging Systems
Propagation of Mutual Coherence under Quasimonochromatic and Paraxial Approximations
Here we introduce the fundamental concept that governs the propagation of mutual coherence and examine the mutual
intensity distributions at each stage of the imaging forming process of a microscope. Using Huygens-Fresnel principle,
the propagation of mutual coherence can be described using the mutual coherence function, i.e.
ZZ ZZ
∂2
r2 − r1 χ(θ1 ) χ(θ2 )
Γ P1 , P2 ; τ +
dS1 dS2 ,
(1)
Γ(Q1 , Q2 ; τ ) = −
∂τ 2
c
2πcr1 2πcr2
Σ1
r2 −r1
c
Σ1
and Γ(Q1 , Q2 ; τ ) are the mutual coherence functions on the starting Σ1 and ending Σ2
where Γ P1 , P2 ; τ +
surfaces, respectively, c is the velocity of light, and τ is the time delay. Under quasimonochromatic condition, effect of
time delay τ becomes deterministic and the mutual coherence between any two given points in space is a function of
spatial coordinates only. In this case, the mutual coherence function can be expressed as,
Γ(P1 , P2 ; τ ) = Γ(P1 , P2 ; 0) exp −jωτ ,
(2)
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Q1
P1
P2
θ1
θ2
Q2
Σ1
Σ2
Fig. 1 Variables used for describing propagation of mutual coherence.
where ω = 2πc/λ and λ is the wavelength of the quasimonochromatic field. Define mutual intensity J(P1 , P2 ) as the
mutual coherence function with zero time delay τ , i.e.
J(P1 , P2 ) , Γ(P1 , P2 ; 0),
Eq. (1) can therefore be simplified under quasimonochromatic condition to,
ZZ ZZ
2π
χ(θ1 ) χ(θ2 )
J(P1 , P2 ) exp −j (r2 − r1 )
J(Q1 , Q2 ) =
dS1 dS2 ,
λ
λr1 λr2
Σ1
(3)
(4)
Σ1
where Pi=1,2 and Qi=1,2 are the coordinates and J(P1 , P2 ) and J(Q1 , Q2 ) are the mutual intensity functions at the
starting Σ1 and ending Σ2 surfaces, respectively. As illustrated in Fig. 1, χ(θ1 ) and χ(θ2 ) are the obliquity factors, ri is
the distance between Pi and Qi , and θi is the angle between Pi Qi and the normal of Σ1 at Pi .
For optical imaging systems where paraxial condition is satisfied, we have,
χ(θ1 ) ∼
= χ(θ2 ) ∼
=1
1 ∼ 1 ∼ 1
=
=
λr1
λr2
λz
2
2
2
2
(x
−
ξ
2
2 ) + (y2 − η2 ) − (x1 − ξ1 ) − (y1 − η1 )
r2 − r1 ∼
,
=
2z
(5)
where z is the axial distance between the starting and ending surfaces. This further simplifies Eq. (4) into a mathematically
manageable form,
i
1
πh 2
2
2
2
J(x1 , y1 ; x2 , y2 ) =
exp −j
(x2 + y2 ) − (x1 + y1 )
(λz)2
λz
Z Z+∞
ZZ
i
πh 2
2
2
2
×
J(ξ1 , η1 ; ξ2 , η2 ) exp −j
(ξ + η2 ) − (ξ1 + η1 )
(6)
λz 2
−∞
i
2π h
× exp j
x2 ξ2 + y2 η2 − x1 ξ1 − y1 η1 dξ1 dη1 dξ2 dη2 .
λz
Note that Pi and Qi have been explicitly written as Pi = (ξi , ηi ) and Qi = (xi , yi ). This equation serves as the foundation
in the study of the effect of partial coherence on imaging systems. The three exponential phase terms in Eq. (6) need
to be treated carefully in order to achieve space invariance. Examining the two exponential phase terms within the
integral
n of Eq. (6), one finds that the cross
o term between the “starting plane (ξ, η)” and the “ending plane (x, y)”, i.e.
exp j 2π
x
ξ
+
y
η
−
x
ξ
−
y
η
, can be used for subsequent Fourier analysis if the squared phase terms in (ξ, η)
2 2
2 2
1 1
1 1
λz
can be eliminated. Note that similar constraints can also be found in the derivation of generalized Van Cittert-Zernike
theorem. Specifically, the propagation distance z needs to be greater than 4ξ∆ξ/λ and 4η∆η/λ where ξ = (ξ1 + ξ2 ) /2,
η = (η1 + η2 ) /2, ∆ξ = ξ2 − ξ1 , and ∆η = η2 − η1 [20], in order for the effect of this “starting plane phase term” to
be ignored. The square phase terms involving the ending plane (x, y) requires less attention since it does not affect the
image intensity distribution. Equation (6) will be used throughout this chapter and unless otherwise stated the imaging
system under study is quasimonochromatic and satisfies paraxial approximation.
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Ima
Ob
jec
Co
nd
en
(x˜, ser Le
y˜)
ns
Ob
je
(ξ, ct
η)
S
ou
(α, rce
β)
z1
Jc J'c
z2
Jo J'o
ti
(x, ve Len
s
y)
zo
zi
g
(u, e
v)
Ji
Fig. 2 A partially coherent cascade imaging
system examined in this chapter.
Js
2.2
Illumination
Figure 2 depicts a typical partially coherent imaging system where the object is trans-illuminated via a condenser, and
imaged by an objective lens onto the image plane. Coordinate systems for the source, condenser, object, objective lens,
and image planes, together with their relative distances along the optical axis, are defined and used throughout this chapter.
Specifically, the relative axial distances between the source, condenser, object, objective lens, and the image planes are
designated as z1 , z2 , zo , and zi ; transverse coordinate systems for these elements are defined as (α, β), (x̃, ỹ), (ξ, η),
(x, y), and (u, v), respectively. The propagation of mutual intensity is examined in cascade from the source to the image
plane.
Source to Condenser Given a partially coherent cascade imaging system as shown in Fig. 2, mutual intensity incident
onto the condenser from the source is given by Eq. (6) as,
i
1
π h 2
2
2
2
exp −j
x̃ + ỹ2 − x̃1 − ỹ1
Jc (x̃1 , ỹ1 ; x̃2 , ỹ2 ) =
(λz1 )2
λz1 2
Z Z+∞
ZZ
i
π h 2
α2 + β22 − α12 − β12
×
Js (α1 , β1 ; α2 , β2 ) exp −j
(7)
λz1
−∞
i
2π h
× exp j
x̃2 α2 + ỹ2 β2 − x̃1 α1 − ỹ1 β1 dα1 dβ1 dα2 dβ2 ,
λz1
where Jc (x̃1 , ỹ1 ; x̃2 , ỹ2 ) is the mutual intensity in front of the condenser and Js (α1 , β1 ; α2 , β2 ) is that of the source.
Effect of the Condenser Assuming that the condenser is large and aberration-free, mutual intensity leaving the back of
the condenser is given by,
i
π h 2
2
2
2
0
x̃ + ỹ2 − x̃1 − ỹ1 ,
(8)
J c (x̃1 , ỹ1 ; x̃2 , ỹ2 ) = Jc (x̃1 , ỹ1 ; x̃2 , ỹ2 ) exp j
λfc 2
where fc is the condenser focal length. Note that the condenser lens aperture is large enough such that the extent of Jc is
not constricted as it passes the condenser.
Condenser to Object Plane The propagation of mutual coherence from the condenser to the object plane is again given
by Eq. (6), i.e.
i
1
π h 2
2
2
2
Jo (ξ1 , η1 ; ξ2 , η2 ) =
exp −j
ξ + η 2 − ξ1 − η 1
(λz2 )2
λz2 2
Z Z+∞
ZZ
i
π h 2
×
J0c (x̃1 , ỹ1 ; x̃2 , ỹ2 ) exp −j
x̃2 + ỹ22 − x̃21 − ỹ12
(9)
λz2
−∞
i
2π h
× exp j
ξ2 x̃2 + η2 ỹ2 − ξ1 x̃1 − η1 ỹ1 dx̃1 dỹ1 dx̃2 dỹ2 ,
λz2
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and the mutual intensity incident onto the object can be obtained by combining Eqs. (7), (8) and (9) as,
i
π h 2
1
2
2
2
exp −j
ξ + η 2 − ξ1 − η 1
Jo (ξ1 , η1 ; ξ2 , η2 ) =
(λz1 )2 (λz2 )2
λz2 2
Z Z+∞
ZZ
i
π h 2
dα1 dβ1 dα2 dβ2 Js (α1 , β1 ; α2 , β2 ) exp −j
×
α2 + β22 − α12 − β12
λz1
−∞
Z Z+∞
ZZ
×
−∞
(10)
i
πh 1
1
1 ih 2
2
2
2
x̃2 + ỹ2 − x̃1 − ỹ1
dx̃1 dỹ1 dx̃2 dỹ2 exp −j
+
−
λ z1
z2
fc
2π h
α2
ξ2 η2 ξ1 η1 i
β2
α1
β1
× exp j
x̃2
.
+
+ ỹ2
+
− x̃1
+
− ỹ1
+
λ
z1
z2
z1
z2
z1
z2
z1
z2
RRRR
Note that the order of integration has been changed and the resultant integral
dx̃1 dỹ1 dx̃2 dỹ2 is separable. In the
case of critical illumination, z1 and z2 satisfy the Gaussian lens formula z11 + z12 − f1c = 0, and Jo reduces to a scaled
version of Js with additional quadratic phase terms, as expected in the case of a large aberration-free condenser aperture.
Specifically, putting the Gaussian lens formula into Eq. (10) reduces it to the critical illumination, i.e.
i
π h 2
1
2
2
2
exp
−j
Jo (ξ1 ,η1 ; ξ2 , η2 ) =
ξ
+
η
−
ξ
−
η
2
1
1
(λz1 )2 (λz2 )2
λz2 2
Z Z+∞
ZZ
i
π h 2
2
2
2
×
dα1 dβ1 dα2 dβ2 Js (α1 , β1 ; α2 , β2 ) exp −j
α + β2 − α1 − β1
λz1 2
−∞
Z Z+∞
ZZ
×
=
−∞
2
z1
z2
(11)
α2
β2
α1
β1
2π h
ξ2 η2 ξ1 η1 i
x̃2
+ ỹ2
− x̃1
− ỹ1
dx̃1 dỹ1 dx̃2 dỹ2 exp j
+
+
+
+
λ
z1
z2
z1
z2
z1
z2
z1
z2
−z
π z1 2
−z1
−z1
−z1 1
2
2
2
exp −j
1+
ξ2 + η 2 − ξ1 − η 1
ξ1 ,
η1 ;
ξ2 ,
η2 .
Js
λz2
z2
z2
z2
z2
z2
On the other hand when Köhler illumination is used, z1 and z2 are both set to the focal length of the condenser [20], i.e.
z1 = z2 = fc and Eq. (10) reduces to the well-known formula for the propagation of mutual intensity from the front focal
plane to the rear focal plane of a thin lens, i.e.
1
Jo (ξ1 , η1 ; ξ2 , η2 ) =
(λfc )2
2.3
Z Z+∞
ZZ
−∞
2π dα1 dβ1 dα2 dβ2 Js (α1 , β1 ; α2 , β2 ) exp j
α2 ξ2 +β2 η2 −α1 ξ1 −β1 η1 . (12)
λfc
Image Formation
The coherence relationship between a conjugated object and image pair of a thin lens can be described as [20],
Z Z+∞
ZZ
Ji (u1 , v1 ; u2 , v2 ) =
Jo (ξ1 , η1 ; ξ2 , η2 )to (ξ1 , η1 )t∗o (ξ2 , η2 )K(u1 , v1 ; ξ1 , η1 )K∗(u2 , v2 ; ξ2 , η2 )dξ1 dη1 dξ2 dη2 , (13)
−∞
where Ji is the mutual intensity at the image plane, to is the amplitude transmittance of a trans-illuminated sample, Jo is
given by Eq. (10), and the amplitude spread function K is defined as,
h
h
i
i
π
π
2
2
2
2
exp j λz
u
+
v
exp
j
ξ
+
η
λz
i
o
K(u, v; ξ, η) =
λ2 zi zo
(
(14)
)
Z+∞
Z
2π zi zi ×
P(x, y) exp −j
u + ξ x + v + η y dxdy,
λzi
zo
zo
−∞
where P(x, y) is the pupil function of the objective lens. Note that since the object and the image planes are conjugate of
each other with respect to the objective lens, zo and zi satisfy the Gaussian lens formula, z1o + z1i − f1m = 0, where fm is the
focal length of the objective lens. The superposition integral in Eq. (13) describes a space-variant four-dimensional (4D)
linear system, with its impulse response given by the space-variant amplitude spread function K(u, v; ξ, η) in Eq. (14).
One can regard the input function of this 4D linear system as Jo · to · t∗o combined and the output as the image mutual
intensity Ji .
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3
Space Invariant Imaging Systems
This section introduces the space invariant amplitude spread function that enables transfer function descriptions of the
imaging system. Transfer function is a powerful tool widely used in optical imaging system analysis. However, the
imaging system under study needs to be linear and space invariant in order for the transfer function description to be
valid. To achieve space invariance, one can place an extra phase-correcting lens of proper focal length in front of the
object to cancel the quadratic phase factors. Alternatively, one can minimize the effect of the quadratic phase terms by
limiting the degree of coherence of the illumination such that the phase factors in the amplitude spread function can be
approximated by unity over the range of separation of interest [20–22]. The partially coherent imaging system in these
cases can then be regarded as space invariant and transfer function descriptions then apply.
3.1
Space Invariant Amplitude Spread Functions
Assuming that the quadratic phase terms in the amplitude spread function is eliminated by the methods described above,
image mutual intensity Ji in these cases can be expressed as,
n
n π
o
o
π
u21 + v12 exp −j
u22 + v22
Ji (u1 , v1 ; u2 , v2 ) = exp j
λzi
λzi
Z Z+∞
ZZ
(15)
×
dξ1 dη1 dξ2 dη2 Jo (ξ1 , η1 ; ξ2 , η2 )to (ξ1 , η1 )t∗o (ξ2 , η2 )K(u1 , v1 ; ξ1 , η1 )K∗ (u2 , v2 ; ξ2 , η2 ),
−∞
where K(u, v; ξ, η) is the effective amplitude spread function given by,
(
)
Z+∞
Z
1
zi zi 2π K(u, v; ξ, η) = 2
u + ξ x + v + η y dxdy.
P(x, y) exp −j
λ zi zo
λzi
zo
zo
(16)
−∞
n
n
o
o
π
π
2
2
2
2
u
+
v
and
exp
−j
u
+
v
,
The square phase terms in front of the integral in Eq. (15), i.e. exp j λz
1
1
2
2
λzi
i
can be ignored without loss of intensity information since these phase terms disappear in intensity distribution, i.e. when
setting (u1 , v1 ) = (u2 , v2 ) = (u, v). To make Eq. (16) formally space invariant, we define the magnification as M = zzoi
and a new coordinate system (ξ 0 , η 0 ) as ξ 0 = − zzoi ξ = −M ξ and η 0 = −M η. The amplitude spread function K is now,
1
K(u, v; ξ, η) = K(u − ξ , v − η ) = 2
λ zi zo
0
0
Z+∞
Z
−∞
i
2π h
P(x, y) exp −j
u − ξ 0 x + v − η 0 y dxdy.
λzi
(17)
Note that the amplitude spread function in Eq. (17) is space invariant since it depends solely on the coordinate difference
(u − ξ 0 , v − η 0 ). The integral for image mutual intensity Ji in Eq. (13) therefore becomes a convolution, i.e.
Z Z+∞
ZZ
Ji (u1 , v1 ; u2 , v2 ) =
0
0
0
0
dξ dη dξ dη
J0 o (ξ10 , η10 ; ξ20 , η20 )K(u1 − ξ10 , v1 − η10 )K∗(u2 − ξ20 , v2 − η20 ) 1 1 4 2 2 ,
M
(18)
−∞
= Jo (ξ10 , η10 ; ξ20 , η20 )to (ξ10 , η10 )t∗o (ξ20 , η20 ) is expressed in the (ξ 0 , η 0 ) coordinate as well. Note that
where J
0 0
to (ξ , η ) is the geometric image of the sample as viewed at the image plane.
Four-dimensional Fourier transform of the convolution integral in Eq. (18) is now,
Ji (ν1 , ν2 , ν3 , ν4 ) = J0o (ν1 , ν2 , ν3 , ν4 )K(ν1 , ν2 )K∗ (−ν3 , −ν4 ),
(19)
0
0
0
0
0
o (ξ1 , η1 ; ξ2 , η2 )
where Ji and J0o are the 4D Fourier spectra of Ji and J0o , respectively. Transfer function of this 4D linear space invariant
system is given by the Fourier transform of the space invariant amplitude spread function in Eq. (17) which is a scaled
pupil function of the objective lens, i.e.
zi
K(p, q) = P(λzi p, λzi q)
zo
(20)
= M P(λzo M p, λzo M q).
4
Transfer Function Analysis: an example using incoherent source
Here we examine the use of transfer functions on the analysis of a typical partially coherent imaging system. An incoherent source is used for illumination and the overall image intensity is obtained under the space-invariant condition.
Modulation transfer functions (MTFs) of two partially coherent imaging systems are numerically evaluated to demonstrate the application of the image formation theory developed in this chapter. Effect of coherence on image contrast is
also examined.
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4.1
Incoherent Illumination Mutual Intensity
Incoherent sources are most commonly employed for illumination in partially coherent imaging systems. Mutual intensity
of an incoherent source can be expressed as [20, 23],
Js (α1 , β1 ; α2 , β2 ) = κIs (α1 , β1 )δ α1 − α2 , β1 − β2 ,
(21)
where κ = λ2 /π. Assuming Köhler illumination, mutual intensity incident onto the object can be obtained from Eq. (12)
as,
Z+∞
Z
n 2π o
κ
I
(α,
β)
exp
j
dαdβ,
(22)
α∆ξ
+
β∆η
Jo (∆ξ, ∆η) =
s
(λfc )2
λfc
−∞
where (∆ξ, ∆η) = (ξ2 − ξ1 , η2 − η1 ). Define ∆ξ 0 = −M ∆ξ and ∆η 0 = −M ∆η, illumination mutual intensity Jo in
Eq. (22) can now be expressed in terms of (ξ 0 , η 0 )
κ
Jo (∆ξ , ∆η ) =
(λfc )2
0
0
Z+∞
Z
Is (α, β) exp j
−∞
2π
0
0
(α∆ξ + β∆η ) dαdβ
λfc (−M )
(23)
and its Fourier transform is given by,
Jo (p, q) = κM 2 Is (fc M λp, fc M λq) .
(24)
The object’s amplitude transmittance functions to (ξ1 , η1 ) and t∗o (ξ2 , η2 ) are converted into the (ξ 0 , η 0 ) coordinate and
Eq. (18) now becomes,
Z Z+∞
ZZ
dξ 0 dη 0 dξ 0 dη 0
Ji (u1 , v1 ; u2 , v2 ) =
J0o (ξ10 , η10 ; ξ20 , η20 )K(u1 − ξ10 , v1 − η10 )K∗ (u2 − ξ20 , v2 − η20 ) 1 1 4 2 2 ,
M
(25)
−∞
where the mutual intensity leaving the sample J0o is given by,
J0o (ξ10 , η10 ; ξ20 , η20 ) = Jo (∆ξ 0 , ∆η 0 )to (ξ10 , η10 )t∗o (ξ20 , η20 ),
(26)
for an incoherent source.
The 4D Fourier transform of the mutual intensity leaving the sample is, from Eq. (26),
J0o (ν1 , ν2 , ν3 , ν4 )
Z+∞
Z
=
Jo (p, q)To (p + ν1 , q + ν2 )T∗o (p − ν3 , q − ν4 )dpdq,
(27)
−∞
where Jo is given by Eq. (24) and To is the two-dimensional Fourier transformation of the sample amplitude transmittance
as seen in the image plane, i.e. the Fourier transform of to (ξ 0 , η 0 ).
4.2
Image Intensity Distribution
Image intensity I(u, v) can now be obtained by taking the inverse Fourier transform of Eq. (19) with J0o given by Eq. (27)
and setting (u1 , v1 ) = (u2 , v2 ) = (u, v),
Z+∞
Z
Z+∞
Z
dpdqJo (p, q) ×
I(u, v) =
−∞
Z+∞
Z
×
dz1 dz2 To (w1 , w2 )K(w1 − p, w2 − q)
−∞
(28)
n
o
dνU dνV exp −j2π(uνU + vνV ) T∗o (w1 − νU , w2 − νV )K∗ (w1 − p − νU , w2 − q − νV ),
−∞
where w1 = p + ν1 , w2 = q + ν2 , νU = ν1 + ν3 , and νV = ν2 + ν4 . Note that Jo is given by the intensity distribution of
the incoherent source in Eq. (24) and K in the pupil function of the objective lens given in Eq. (20).
1902
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1
1
σ=0
σ=0.5
σ=1
σ=1.5
σ=2
0.9
0.8
0.7
0.8
0.7
0.6
MTF
MTF
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
σ=0
σ=0.5
σ=1
σ=1.5
σ=2
0.9
0
0.5
1
νo/νc
1.5
0
2
0
0.5
1
νo/νc
1.5
2
(b) DIC objective with bias retardation equal to π/2.
(a) MZP objective (Conventional microscope).
Fig. 3 Modulation transfer functions of a sinusoidal absorption grating under different illumination coherence defined
by various σ values. An illumination is coherent when σ = 0 and the degree of coherence decreases as σ increases.
4.3
Modulation Transfer Function (MTF)
Here we examine the MTF of a partially coherent imaging system using methods developed in this chapter. Modulation
transfer function of an imaging system is defined as [20, 22],
M T F (νo ) ,
Vimage (νo )
,
Vobject (νo )
(29)
where Vimage and Vobject are the intensity visibilities of the image and the object, respectively. Visibility of an intensity
distribution is defined as,
Imax − Imin
V ,
,
(30)
Imax + Imin
where Imax and Imin are the maximum and minimum image intensity. Note that Vimage and Vobject are both functions
of sample spatial frequency νo . Modulation transfer function allows one to quantitatively characterize the performance of
a full-field microscope and the spatial resolution of the imaging system can therefore be quantitatively defined [3].
Modulation transfer functions of imaging systems using a regular thin lens objective and a DIC objective [24] are
shown in Fig. 3 with a sinusoidal absorption grating as the object being imaged. Effect of coherence on image formation
is quantified by varying the values of σ, defined as,
σ,
Rs /fc
,
Rm /zo
(31)
where Rs is the source radius, Rm is the radius of the objective lens aperture, fc is the focal length of the condenser, and
zo is the distance between the object and the objective lens. The value of σ ranges from zero to infinity, with complete
coherence designated as σ = 0 and incoherence as σ = ∞. Alternatively, one can define the cut-off frequency νc as,
νc ,
Rm
,
λzo
(32)
νs ,
Rs
,
λfc
(33)
and the extent of source angular distribution νs as,
and σ is then given by σ = νs /νc . Note that the MTFs shown in the figure are both functions of νo and plotted against
the normalized spatial frequency νo /νc . When a regular thin lens is employed as the objective, MTF drops sharply from
1 to 0 at νo /νc = 1 in the coherent case, as shown in Fig. 3(a). As σ increases, MTF extends beyond νo /νc = 1
while the visibility within the νo /νc < 1 region decreases. The performance of a DIC objective is also evaluated using
the same sinusoidal absorption grating as the sample and the result is shown in Fig. 3(b). Image visibility is lower in
DIC microscopes especially when the sample spatial frequency approaches νc /2. As expected, since DIC microscopy is
designed to image phase objects, its performance on absorption samples are not as good as the thin lens objective.
©FORMATEX 2010
1903
Microscopy: Science, Technology, Applications and Education
A.
Méndez-Vilas and J. Díaz (Eds.)
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5
Conclusion
This chapter introduced the image formation theory of a full-field imaging microscope under partially coherent illumination. Transfer functions are used to characterize the system and simplify the analysis. Propagation of partial coherence is
derived in Sec. 2 following Huygens-Fresnel principle. Section 3 discussed the space invariance condition which allows
the use of transfer function method for partially coherent imaging systems. Based on the theories presented in Secs. 2 and
3, overall image intensity distribution was derived analytically in Sec. 4 with an incoherent source used for illumination.
Modulation transfer functions for a thin lens objective and a DIC objective were also present as examples. The comprehensive mathematical description presented here lays the foundation for further studies on the partially coherent image
formation properties of various types of absorption and phase contrast full-field microscopes.
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