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Imaging with straight-edge phase plates in the TEM
Ultramicroscopy 182 (2017) 124–130
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Imaging with straight-edge phase plates in the TEM C.J. Edgcombe TFM Group, Department of Physics, University of Cambridge, CB3 0HE Cambridge, UK
a r t i c l e
i n f o
Article history: Received 24 March 2017 Revised 15 June 2017 Accepted 19 June 2017 Available online 22 June 2017 Keywords: Phase plate Foucault plate Hilbert plate Fourier–Bessel transform 2D transform
a b s t r a c t The image of a simple phase object produced by a round lens with a Foucault or Hilbert phase plate can be determined with Abbe imaging theory and a 2D transform expressed in cylindrical coordinates. The contributions to the image amplitude from a uniform disc object and an azimuthally varying plate can then be distinguished and their phases relative to the incident wave can be compared. It appears that the usual choice of added phase for a Hilbert plate causes the image of a weak disc object to vanish as the plate edge approaches the axis, but a different choice of plate thickness can enable a weak phase object to provide a linear contribution to image intensity. © 2017 Elsevier B.V. All rights reserved.
1. Introduction Phase plates are currently of great interest for cryomicroscopy of biomedical materials since they offer the possibility of reducing the electron dose to objects that produce a weak phase shift. The two major types that have been widely investigated are the Zernike plate, which has rotational symmetry, and the Hilbert plate, consisting of a half-plane of dielectric material with a straight edge (Fig. 1), [1,2]. A recent survey [3] reviews the many forms in which these plates have been realized, and also gives some analysis of the operation of both types by applying Abbe’s wave theory to a rotationally symmetric lens and object. The method was also used in a separate paper [4] to analyse the effect of a Zernike phase plate in the TEM. Here we extend the previous analysis to include azimuthal variation and so to describe the behaviour of straight-edge (Foucault and Hilbert) plates. To do this, relations for the 2D transform have been written using cylindrical coordinates, after which the images produced by certain combinations of a simple phase object and a phase plate can be readily deduced. This process yields information about the components of the image (both the copy of the object and artefacts due to the plate), their amplitudes, and in addition their relative phases. The detailed analysis suggests how the performance of the Hilbert plate can be improved. This plate was so named by Lowenthal and Belvaux [5] because the Hilbert transform was used in processing the image. The phase added by the plate was chosen in [5] as π , without discussion of other choice of angle. The detailed
E-mail address: [email protected] http://dx.doi.org/10.1016/j.ultramic.2017.06.020 0304-3991/© 2017 Elsevier B.V. All rights reserved.
analysis given here shows that a different choice of thickness of the Hilbert plate may lead to substantially improved performance. 2. Method for post-specimen phase plates Analysis of imaging in the TEM often considers the phase contrast transfer function (PCTF), which defines the relative phase introduced by the lens system as a function of spatial frequency. It is well known (for example, Section 8.6.3(c) of [6]) that, if the argument of the PCTF can be changed by π /2, then the contrast at low frequencies can be increased. The established transfer functions show how the system responds to a single spatial frequency (of infinite transverse extent), but they do not show the image produced by a bounded object that contains many sources of scattering and many spatial frequencies. However, theory for the transfer of a spectrum of frequency components has been defined for optical systems by Abbe (Section 8.6.3(b) of [6]), and for TEM by Hawkes and Kasper (Section 65.2 of [7]). This theory, known elsewhere as Fourier optics [8], shows how the image can be obtained from the object exit wave and the frequency response of intermediate parts of the system. Here we apply this transfer theory to a system in which the object and the lens each have rotational symmetry and there is a Foucault or Hilbert phase plate at the standard post-lens location. Since the behaviour is periodic in azimuth, we use cylindrical coordinates: (r, φ ) at exit from the object, (q, θ ) at the phase plate and (ri , φ i ) at the image plane. According to Abbe’s theory, the object wave can be specified by its spectrum in the spatial frequency q. The lens redistribute the wave so that all parts that have the same magnitude of q before reaching the lens converge after the lens to one physical radius that is proportional to q, on a specific plane, the ‘diffraction’ plane, transverse to the cylindrical axis. If the illumination is par-
C.J. Edgcombe / Ultramicroscopy 182 (2017) 124–130
125
Fig. 2. Unit step, used to describe illumination and phase of disc object.
to object space. The inversion of the modified spectrum then gives the image functions directly in these scaled coordinates. In the 2D transforms as defined in Appendix A, each function f (r) is both a coefficient of a Fourier series and a source for a Hankel transform; similarly for each function t used below. In previous work [4] the name ‘Fourier–Bessel’ was used for the radial transform with no azimuthal variation; either this name, or possibly ‘Fourier-Hankel’, now seems more suitable for the general 2D transform in cylindrical coordinates. 2.1. Illumination Fig. 1. Location and form of a conventional Hilbert plate (from Fig. 1a,b, Principle of difference-contrast TEM, in reference [1], © 2002 Kluwer Academic Publishers, with permission of Springer.)
allel, this plane is the back focal plane. It is the relation between q and physical radius at the diffraction plane that allows a phase plate to modify particular ranges of q (Eqs. (65.15)–(65.17) of [7]). The modified spectrum then propagates further, and its inversion defines the focused image. The complex amplitude of the image can be converted to intensity and then shows the contributions of the component terms to the detectable intensity. The spectrum of the simple object considered here can be found analytically. The present work describes the transmission of a Foucault or Hilbert plate as a series of azimuthal harmonics. Here we use the 2D transform expressed in cylindrical coordinates; many functional relations can be deduced and a consistent set, based on harmonics of exp iθ , is given in Appendix A. A benefit of Abbe’s method is that it shows not only the spatial distribution but also the relative phases of components of the image amplitude. Analysis shows that the image intensity obtained with the Foucault plate contains a copy of the object’s spatial distribution, whose magnitude is proportional to the square of the phase change due to the object. Objects with phase change less than one radian (‘weak phase objects’) then contribute less to the image intensity than if it were linear in the object phase. The Foucault intensity also contains an approximately linear gradation perpendicular to the plate edge, with magnitude proportional to the phase of a weak uniform object. The Hilbert plate produces a similar gradation of intensity, but can generate a copy of the object shape that varies linearly with the object phase, as is desired. Analysis shows how the image depends on the thickness of the plate, from which some new deductions can be made. The theory given here omits known contributions to the PCTF of a TEM such as defocus, aberrations, energy spread, signal-to-noise ratio and other effects. The object is assumed to produce zero absorption and so to be a ‘pure phase object’. The plate is assumed to be located at the axial plane at which each value of q occurs at a single radius. The coordinates (ri , φ i ) of the image are assumed to be scaled by reduction and rotation as in Section 65.2 of [7]; or, as an alternative statement, the image coordinates are referred
An easily defined form of illumination is a transversely infinite plane wave, but a question arises about its transform. When Cartesian coordinates are used, the Fourier transform in each coordinate direction is represented by a Dirac delta function and justified by distribution theory [9]. When we use cylindrical coordinates, it is not immediately clear how to determine the Hankel transform of a radially uniform wave of infinite extent. We can avoid using distribution theory by noting that in practice the illumination will always be limited at some outer radius. We therefore assume that at the plane of the object exit wave, the illumination is uniform within a radius a much greater than the specimen radius, and is zero outside radius a. Here we omit the factor exp i(kz − ωt), and represent the illuminating wave at the specimen by
ψ = u1 (r/a ) where u1 is a unit step function (Fig. 2):
u1 ( ρ ) =
0<ρ<1 1<ρ
1, 0,
The Hankel transform P0 of this rotationally symmetric wave is (using Eq. (A.4) with = 0)
P0 (q, a ) =
0 2
∞
u1 (r/a ) J0 (qr )r dr
= a J1 (qa )/qa Then when a is not infinite, P0 is well-behaved as q → 0. 2.2. Object The object is assumed to advance the phase of the wave (relative to free-space propagation) by γ h(r, φ ), where γ is a constant angle, h is a specified function of approximately unit magnitude and is zero for r greater than some value b which is less than a, and r and φ are the cylindrical coordinates transverse to the axis of the system. Then, with the incident wave as in Section 2.1, the exit wave is described by
ψ = u1 (r/a ) exp iγ h(r, φ ) (With this definition, the phase change γ h is defined only modulo 2π .) For convenience in discussing weak-phase behaviour, ψ is written as the sum of the incident wave and a scattered wave f:
ψ = u1 (r/a ) + f (r, φ )
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C.J. Edgcombe / Ultramicroscopy 182 (2017) 124–130
intersect the material of the plate. Then as q0 becomes more positive, more of the beam is clear of the plate. For a given (q, θ ), transmission outside the plate occurs when
q cos θ > − q0 The magnitude q is non-negative, but q0 may be positive, zero or negative.
Fig. 3. Geometry for straight-edge plates, looking along the axis of the system.
from which
f = u1 (r/a ) eiγ h − 1 = eiγ h − 1 = iγ h(r, φ ) + O
γ2
(1)
The second equality follows because u1 is unity over the range r < a, and h is zero for r > b. Here we consider only a notional disc object. Although this is not typical of most biological objects, it can be used to analyse many types of plate. 2.2.1. Disc object This object is assumed to be a circular disc that produces uniform phase change γ (relative to free-space propagation) over its diameter, 2b, while outside this diameter the phase change is zero (Fig. 2). The scattered wave at this object can be written with another unit step function:
h = u (ρ ) where u is as defined in Section 2.1 but now a function of ρ , defined by
ρ = r/b
2.3.2. Relation of q to physical radius at plate By equating the scattering angle for a wave with phase constant k, acquiring transverse wave vector q, to the angle for a particle scattered to radius x at the focal distance f from a thin lens, we obtain the relation q/k ≈ x/f. This is only approximate since the objective usually has thick-lens properties. The simple relation suggests that the value of q0 corresponding to a physical spacing x0 of the plate edge from the axis is q0 ∼ 2π x0 /λf, so for 200 kV electrons, a focal length of 3 mm and a spacing x0 of 1 micrometre, the corresponding value of q0 is about 0.84 nm−1 . 2.3.3. Intercepting plates The Foucault plate is opaque to the beam and blocks part of the spectrum. The transmission function T then has the value 0 or 1 as a function of q and θ . These plates, like apertures and beam stops, are found to produce weak-phase image amplitudes that differ in phase by π /2 from the direct beam. As described in Section 3.1, the intensities of desired images are then proportional only to the square of the object phase γ and so are undesirably small for weak-phase objects. 2.3.4. Hilbert plate The Hilbert plate is also a half-plane plate but instead of being opaque, it consists of dielectric material that produces a phase change, additional to that of free space, denoted here by α . Any absorption by the plate is ignored here. The transmission T at (q, θ ) depends on the relative magnitudes of q and q0 , as shown in Fig. 3. In the following theory, it will be convenient to use the symbol p0 defined by
p0 ( q0 ) =
The wave at the plane of exit from the object is then, from (1),
f d (ρ ) = e
iγ h
−1= e
iγ
− 1 u (ρ )
(2)
The object is rotationally symmetric so its azimuthal periodicity is zero. The Hankel transform of (2) according to (A.4) for = 0 can be written as
Fd (q ) = eiγ − 1 b2 J1 (Q )/Q
(3)
where
Q = qb and J1 () is a Bessel function of the first kind.
1, exp iα ,
0 < q0 q0 < 0
(4)
Those elements of the electron wave that have q ≤ |q0 | are transmitted with the same phase change for all θ . For q0 > 0, the phase change is zero, while for q0 < 0, all transmitted electrons suffer phase change α . Thus
T (q ≤ |q0 |, θ ) = p0 For parts of the wave with q > |q0 |, the phase change depends on θ :
T (q > |q0 |, θ ) =
1, exp iα ,
−θ0 < θ < θ0 −π < θ < −θ0 ,
θ0 < θ < π
(5)
where θ 0 is defined as in Fig. 3 by
θ0 (q ) = arccos (−q0 /q ) 2.3. Plate The phase added by the Hilbert plate is represented (as for the Zernike plate) by the argument of a complex number. 2.3.1. Location of a straight-edge plate The plate lies in a plane perpendicular to the axis of the lens system. The intersection of the axis with this plane is taken as the origin of transverse coordinates (q, θ ) (Fig. 3). The direction of the perpendicular outward from the plate edge is chosen to define θ = 0. The perpendicular spacing of the plate edge from the axis is denoted by q0 and is defined as positive when the axis does not
These T are either constant or periodic in θ with period 2π , so they can be expressed as Fourier series of the form
T (q, θ ) =
∞
t (q ) exp iθ
=−∞
By the usual method for Fourier series, the t can be written as
t (q ≤ |q0 | ) =
0, p0 ,
l = 0 l=0
1 − eiα sin θ0 (q )/π , t (q > |q0 | ) = t0R + i t0I ,
(6a,b) l = 0 l=0
(6c,d)
C.J. Edgcombe / Ultramicroscopy 182 (2017) 124–130
× cos φi J (ρi Q ) J1 (Q )
where
t0R + i t0I = [θ0 /π + {1 − (θ0 /π )} cos α ] + i{1 − (θ0 /π )} sin α
3. Images with Hilbert plates
The terms in the image function will be shown to have complex amplitudes which define their relative phases. Real parts of these amplitudes have the phase of the incident beam. If the images of the incident and scattered waves can be represented as u and (iη), with phases differing by π /2, then their total image intensity is |u + iη|2 = u2 + η2 and the intensity is modulated as η2 . However, if part of the image of the scattered wave is in phase with the incident wave and has the real amplitude η, then the total intensity |u + η|2 = u2 + 2 ηu + η2 , and the intensity is modulated by η linearly. 3.2. A disc object with a Hilbert plate The spectrum of a disc object, Fd (q) as in (3), is independent of azimuthal angle θ . The transmission of the Hilbert plate was defined by relations (5) and (6). The spectrum at exit from this plate then has the form ∞
eiθ t (q )
=−∞
This spectrum corresponds to the form of (A.8), so by equating terms
P (q ) = i t (q ) Fd (q )
(7)
We can now use (A.10) to find the image amplitude. After substituting in (7) for Fd from (3), inserting P in (A.10), writing g(ri , φ i ) for the image amplitude f in (A.10) including the unscattered part u1 (ri /a) (with the phase shift (4) that occurs when q0 < 0), and noting that t and (i J ) are independent of the sign of ,
g ( r i , φi ) = p 0 u 1 + e iγ − 1
v0 ( ri , b ) + 2
∞
v (ri , b) cos φi (8)
v ( r i , b ) = i b
∞
0
t (q ) J1 (qb) J (qri )dq
(9)
To evaluate the integrals in (9), it is necessary to sum over the two ranges 0 < q < |q0 | and |q0 | < q < ∞, using the specific value of t for each range. The tedious calculation for the general case is relocated to Appendix B. The result is
g(ri , φi ) = p0 u1 + eiγ − 1 [ p0 I0 (ρi , B ) + w0R + i w0I sinα
+ 1 − e iα S
I0 (ρi , B ) =
|B| 0
(10)
J1 (Q ) J0 (ρi Q ) dQ; B = q0 b, Q = qb
w0R + i w0I sinα =
∞
|B|
θ 0 iα θ0 + 1− e J1 (Q ) J0 (ρi Q )dQ π π
and, with S arranged as an integral of a sum,
S ( ρ i , φi , B ) = s R + i s I =
2
π
∞ ∞ B
=1
i
1 sin θ0 (Q, B )
|g|2 = u1 2 − 2 γ u1 [(w0I − sR ) sin α + sI (1 − cos α )] + O(γ 2 ) (12) and for q0 < 0 (including the effect of the phase change at q = 0 produced by the plate)
|g|2 = u1 2 − 2 γ u1 [(−w0R + w0I cos α − sR ) sin α − sI (1 − cos α )] + O(γ 2 )
(13)
The image intensity thus contains the intensity of the incident beam (u1 2 , equal to u1 ) and a term linear in the object phase, γ . This linear contribution includes w0 I , which is an integral of the object spectrum independent of φ i , together with φ i -dependent artefacts sR and sI . The forms of w0 I for B = 0.5, 1, 2 and 5 are shown in Fig. 4 as functions of ρ i B = q0 ri . The central region of the disc object is imaged only if B = q0 b is less than about 1, that is if the object radius is less than a limit set by system parameters. Larger disc objects show up in the image by the discontinuities at their edges. Approximate calculations of sR and sI as functions of ρ i = ri /b for B = 1 are shown in Fig. 5. Like w0 I , sI produces a response mainly at the edge of a disc, but sI varies with φ i . Eq. (12) shows that the relative sizes of these two contributions to the intensity depend on the plate phase change α (see Section 3.2.3). 3.2.2. Special case q0 b = ε (0 < ε << 1) When the edge of the plate is close to the axis, θ 0 becomes independent of q and equal to π /2. Then B, I0 and sR become negligible, and the image amplitude and intensity simplify to
g ( r i , φi ) = u 1 + e iγ − 1 q0 = 0+
|g|2 = u1 2 − 2 γ u1
=1
where
(11)
3.2.1. Weak-phase disc object and Hilbert plate On approximating (10) to first order in the object phase γ and forming the intensity, the result for q0 > 0 is
3.1. Linear and quadratic components of intensity
T (q, θ )Fd (q ) = Fd (q )
dQ
The function I0 appeared in earlier work on Zernike plates with no azimuthal variation [4], where it was named Id . It has no relation to the modified Bessel function In ().
These coefficients t are independent of the sign of .
where
127
1 2
1 + eiα u(ρi )/2 + i 1 − eiα sI0 ,
u(ρi ) sin α + sI0 (1 − cos α ) + O(γ 2 ) (14)
where
sI0 (ρi , φi ) =
2
π
∞ ∞ 0
m=0
cos (2m + 1 )φi J2m+1 (ρi Q )J1 (Q ) dQ 2m + 1 (15)
The peak of function sI 0 (ρ i , 0), shown in Fig. 6, has approximately the same magnitude as u(ρ ), shown in Fig. 2. Some integrals of Bessel products like those in (15) are shown graphically as functions of (ρ i B) in Chapter 2 of [3]. From these it seems likely that the major contribution to sI 0, in the range 0 < ρ i < 0.5, is from the term for m = 0. For ρ i < 0.5, this term is approximately proportional to ρ i so when combined with the variation as cos φ i its contribution to g is approximately proportional to (ri cos φ i ) or distance (in the image plane) perpendicular to the plate edge. This term, produced by the component of the plate transmission varying as cos θ , appears to describe the shading of intensity perpendicular to the plate edge that is observed with straight-edge plates [1]. 3.2.3. Choice of α for weak-phase object In (12), (13) and (14), the terms copying the object are multiplied by the sine of α , the phase introduced by the plate. These
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C.J. Edgcombe / Ultramicroscopy 182 (2017) 124–130
Fig. 4. Function W0 I (q0 ri ) (independent of azimuthal angle φ i ) for B = 0.5, 1, 2, 5, integrated over Q from B to ∞.
Fig. 5. Approximate forms of φ i -dependent functions sR and sI for φ i = 0 and B = 1. Both functions were integrated over Q from B to 10 0 0. The function sR was summed over 2m from 2 to 20. Function sI was summed over (2m + 1) from 1 to 21. At φ i = π /2, sI = 0.
4. Discussion
Fig. 6. Approximate form of function sI 0 for B = 0 and at φ i = 0; summed for (2m + 1) = 1–21 and integrated over Q from 0 to 10 0 0. When B = 0, then sR 0 = 0.
results show that the usual choice of α = π minimises the linear copy of the phase distribution of the object (constant with φ i ). If it is desired that the image contain this linear copy, the phase change of the plate should be chosen as some other value than π . With α = π /2 and q0 ≥ 0,
|g| = u1 − 2 γ u1 [w0I − sR + sI ] + O γ 2
2
2
This choice of α ensures that the weak-phase response is linear in the object phase γ , but the shading contribution from components of S is still present. For an object of strong phase, the optimum value of α is likely to differ from π /2, as discussed by Beleggia [10].
For the configuration analysed in Section 3.1, the most suitable value of α depends on how important it is to minimise the shading contribution from = 1. If its presence is unimportant or even desired, then a value of α = π /2 or 3π /2 will maximise the contribution from the object shape. However, as α is reduced from π /2, the magnitude of (1 − cos α )/sin α falls and the relative contribution from = 1 decreases. This form of plate is simple to make in comparison with some other types but, as usual, care will be needed to minimise possible charging at the edge of the plate, from the direct beam. A recent study by Koeck [11], which takes account of defocus and spherical aberration, shows that a phase shift of π /2 has the further benefit of extending the first passband, when operation is at Scherzer defocus. 5. Conclusion The image of a disc phase object produced by a round lens with a Hilbert phase plate has been calculated by Abbe’s theory of imaging with the 2D transformation expressed in cylindrical coordinates. The conventional choice of π for the phase added by the plate leaves the image of this object out of phase with the direct beam, and as the plate edge approaches the axis of the lens system, the linear contribution of the object’s phase to the image intensity is cancelled. However, if the phase added by the plate is
C.J. Edgcombe / Ultramicroscopy 182 (2017) 124–130
reduced to π /2 (for a weak-phase object), then the image intensity contains a copy of the object that is linear in the phase of the object. In addition to this copy of the object, the image contains a series of azimuthal harmonics including one varying approximately as (ri cos φ i ) which appears to be the source of the observed gradation of intensity perpendicular to the plate edge.
A.3. Conversion from Cartesian form The Cartesian 2D transform of f can be specified by (for example, Section 2.1.1 of [8] with 2π fX replaced by qx and similarly for fy )
F= Acknowledgements The calculations of Figs. 4–6 were made with Mathematica 11.1.1. This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors. Appendix A. The 2D transform in cylindrical coordinates The transmission of plates that have some azimuthal structure can in general be specified as a function of both q, the magnitude of radial frequency in angular measure, and an azimuthal angle θ . Any such function can in principle be expressed as a Fourier series in θ , with components of azimuthal periodicity and coefficients that are functions of and q. The spectrum for each can be inverted to give the image function for the same periodicity in the image angle φ i . Before deriving the full 2D transform, we state first the Fourier series and Hankel transform to be used here. Section (A.3) shows that the transform in cylindrical coordinates is obtained by direct change of coordinates from a standard 2D transform in Cartesian coordinates.
A given function f(r, φ ) that is periodic in the angle φ with period 2π can be represented by a Fourier series in exp iφ as ∞
eiφ f (r )
(A.1)
=−∞
where is integral and the (complex) f are defined by
π 1 f (r ) = e−iφ f (r, ϕ ) dφ 2 π −π
(A.2)
f (r ) =
0
P (q ) J (rq ) q dq
(A.3)
0
∞
f (r ) J (qr ) r dr
∞ −∞
f exp −i(qx x + qy y ) dx dy
(A.5)
Here the initial constant has been chosen for later convenience. Relation (A.5) can be expressed in cylindrical coordinates (r, φ ), (q, θ ) by converting the Cartesian coordinates to their cylindrical equivalents:
qx = q cos θ , qy = q sin θ , x = r cos φ , y = r sin φ In applications, θ is the azimuthal angle of q = (q,θ ), measured from the same axis and with the same origin as φ . After substituting into (A.5) with suitable changes to the ranges of integration, and specifying f and F as functions of cylindrical coordinates at given transverse planes,
F (q, θ ) =
π
1 2π
−π
∞
0
f (r, φ ) exp −i[qr cos (φ − θ )] r dr dφ (A.6)
An object function specified as f(r, φ ) is necessarily periodic in φ with period 2π and so can be expressed as a Fourier series in exp iφ . To obtain the harmonic dependence on θ , we first substitute in (A.6) from (A.1). We then define a new angle ψ by
ψ =φ−θ
F (q, θ ) =
∞ 1
π
eiθ
∞ 0
l=−∞
f (r )
π 0
cos ψ
× exp − i(qr cos ψ ) dψ r dr and after using Bessel’s integral (10.9.2 of [12]) for ψ with θ constant, F can be written as ∞
F (q, θ ) =
i− eiθ
∞ 0
=−∞
f (r ) J (qr ) r dr
(A.7)
(A.4)
and J (qr) is a Bessel function of the first kind and integral order . The pair described as a Hankel transform in [12] and some other tables is an alternative form of the same relations but uses different weighting functions, so before any given table is applied, the definitions it uses should be checked. In applications, it may be useful to think of q as a radial phase constant in angular measure, while bearing in mind that the local wavelength of a Bessel function J (qr) differs from 2π /q. If f is a constant independent of r, the integral in (A.4) diverges, so some constraint on f is needed. In practice f0 is limited since the wave has zero amplitude outside some maximum radius.
∞
i− eiθ P (q )
(A.8)
=−∞
On substituting (A.2) in (A.7), the first relation of the 2D transform as a function of the original object function is:
F (q, θ ) =
where P is defined by
P (q ) =
∞ −∞
F (q, θ ) =
A given function f (r) can be represented (subject to certain conditions) by a function P (q) as ∞
or, with the definition (A.4),
A.2. Hankel transform
1 2π
Then, with θ held constant as φ varies, F can be expressed as
A.1. Fourier series
f (r, φ ) =
129
π 1 − iθ ∞ i e e−iφ f (r, φ ) dφ J (qr )r dr 2π 0 −π
A.4. Inverse transform An application such as Section 3 may provide the spectrum in the form of (A.8), from which the P can be identified. If not, the P can be found from F(q, θ ) by the Fourier inversion of (A.8):
P (q ) =
1 2π
π
−π
i e−iθ F (q, θ ) dθ
(A.9)
When the P are known, f can be found by combining (A.1) and (A.3) as
f (r, φ ) =
∞ =−∞
eiφ
∞ 0
P (q ) J (qr ) q dq
(A.10)
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The relation (A.10) is used in Section 3 to find the image function from a known Fourier series for the spectrum. The inverse relation of the 2D transform, for f as a function of F, can be found by inserting (A.9) in (A.10):
f (r, ϕ ) =
∞ 1 iφ ∞ π −iθ ie e F (q, θ )dθ J (qr ) q dq 2π =−∞ 0 −π
v (ri , b) = i 1 − eiα W , > 0 v0 (ri , b) = p0 I0 (ρi , B ) + W0 (ρi , B ) Then from (8), Eq. (10) is obtained:
+ i w0I sinα + 1 − eiα S
(A.11) Alternatively, on inserting (A.9) in (A.3) and substituting the resulting expression for f into (A.1), the complete inversion agrees with (A.11).
g(ri , φi ) = p0 u1 + eiγ − 1 [ p0 I0 (ρi , B ) + w0R
where
S ( ρ i , φi , B ) = s R + i s I = 2
∞
i W cos φi
=1
Appendix B. The general image function
For convenience in using Mathematica, S can be rearranged as an integral of a sum, giving (11):
We seek to evaluate the coefficients v given by (9) above:
v ( ri , b ) = i b
∞ 0
t (q ) J1 (qb) J (qri )dq
S ( ρ i , φi , B ) = s R + i s I =
t (q ≤ |q0 | ) =
0, p0 ,
=
l = 0 l=0 l = 0 l=0
where
∞
σSI =
t0R + i t0I = [cos α + (θ0 /π )(1 − cos α )] + i{1 − (θ0 /π )} sin α The integrals must be evaluated by summing over two ranges of q, less than and greater than |q0 |. Here B = q0 b may be negative, zero or positive. 1) 0 < q < |q0 |:
t (q ) J1 (qb) J (qri )dq = 0, > 0
and for = 0,
0
b
t0 (q ) J1 (qb ) J0 (qri )dq = p0 I0 (ρi , B )
2) |q0 | < q: Using (6c), for > 0: ∞
|q 0 |
t (q ) J1 (qb ) J (qri )dq = 1 − eiα W , > 0
where
W (ρi , B ) =
∞ 1 sin θ0 (q ) J1 (Q ) J (ρi Q )dQ, > 0 π |B|
While the W above are real, when = 0 then W0 is complex:
W0 = w0R + i w0I sinα =
π
∞
|B |
(−1 )m
∞
(−1 )m
m=0
|q 0 |
2
m=1
B = q0 b, Q = qb
b
|B| =1
i
1 sin θ0 (q ) J (ρi Q )
σS (ρi , φi , Q, B)J1 (Q ) dQ
σS = σSR + i σSI σSR =
θ0 = arccos (−q0 /q ) = arccos (−B/Q )
0
∞ ∞
where
1 − eiα sin θ0 (q )/π , t (|q0 | < q ) = t0R + i t0I ,
|q 0 |
π
× cos φi J1 (Q ) dQ
by using the coefficients for the disc object and Hilbert plate as in (6a-d) above
2
∞
|B|
(t0R + i t0I ) J1 (qb) J0 (qri )d (qb)
On summing over both ranges of q to find v as in (9):
1 sin [2mθ0 (Q, B )] cos 2mφi J2m (ρi Q ) 2m
1 sin [(2m + 1 )θ0 (Q, B )] 2m + 1
× cos (2m + 1 )φi J2m+1 (ρi Q ) References [1] R. Danev, H. Okawara, N. Usuda, K. Kametani, K. Nagayama, A novel phasecontrast transmission electron microscopy producing high-contrast topographic images of weak objects, J. Biol. Phys. 28 (2002) 627–635, doi:10.1023/ A:1021234621466. [2] Y. Kaneko, R. Danev, K. Nitta, K. Nagayama, In vivo subcellular ultrastructures recognized with Hilbert differential contrast transmission electron microscopy, J. Electron Microsc. (Tokyo). 54 (2005) 79–84, doi:10.1093/jmicro/dfh105. [3] C.J. Edgcombe, Phase plates for transmission electron microscopy, Ch. 2 of Vol. 200, in: P.W. Hawkes (Ed.), Advances in Imaging and Electron Physics, Academic Press, Burlington, 2017, doi:10.1016/bs.aiep.2017.01.007. [4] C.J. Edgcombe, Imaging by Zernike phase plates in the TEM, Ultramicroscopy 167 (2016) 57–63, doi:10.1016/j.ultramic.2016.05.003. [5] S. Lowenthal, Y. Belvaux, Observation of phase objects by optically processed Hilbert transform, Appl. Phys. Lett. 11 (1967) 49, doi:10.1063/1.1755023. [6] M. Born, E. Wolf, Principles of Optics, sixth ed., Pergamon Press, Oxford, 1993. [7] P.W. Hawkes, E. Kasper, Wave Optics (Principles of Electron Optics Vol. 3), Academic Press, London, 1994. [8] J.W. Goodman, Introduction to Fourier Optics, second ed., McGraw-Hill, London, 1996. [9] M.J. Lighthill, Introduction to Fourier analysis and generalised functions, Cambridge University Press, Cambridge, 1959. [10] M. Beleggia, A formula for the image intensity of phase objects in Zernike mode, Ultramicroscopy 108 (2008) 953–958, doi:10.1016/j.ultramic.2008.03. 003. [11] P.J.B. Koeck, Improved Hilbert phase contrast for transmission electron microscopy, Ultramicroscopy 154 (2015) 37–41, doi:10.1016/j.ultramic.2015.03. 002. [12] DLMF, in: F.W.J. Olver, A.B. Olde Daalhuis, D.W. Lozier, B.I. Schneider, R.F. Boisvert, C.W. Clark, B.R. Miller, B.V. Saunders (Eds.), 2016 DLMF: NIST Digital Library of Mathematical Functions, Release 1.0.13 http://dlmf.nist.gov/.
Contents lists available at ScienceDirect
Ultramicroscopy journal homepage: www.elsevier.com/locate/ultramic
Imaging with straight-edge phase plates in the TEM C.J. Edgcombe TFM Group, Department of Physics, University of Cambridge, CB3 0HE Cambridge, UK
a r t i c l e
i n f o
Article history: Received 24 March 2017 Revised 15 June 2017 Accepted 19 June 2017 Available online 22 June 2017 Keywords: Phase plate Foucault plate Hilbert plate Fourier–Bessel transform 2D transform
a b s t r a c t The image of a simple phase object produced by a round lens with a Foucault or Hilbert phase plate can be determined with Abbe imaging theory and a 2D transform expressed in cylindrical coordinates. The contributions to the image amplitude from a uniform disc object and an azimuthally varying plate can then be distinguished and their phases relative to the incident wave can be compared. It appears that the usual choice of added phase for a Hilbert plate causes the image of a weak disc object to vanish as the plate edge approaches the axis, but a different choice of plate thickness can enable a weak phase object to provide a linear contribution to image intensity. © 2017 Elsevier B.V. All rights reserved.
1. Introduction Phase plates are currently of great interest for cryomicroscopy of biomedical materials since they offer the possibility of reducing the electron dose to objects that produce a weak phase shift. The two major types that have been widely investigated are the Zernike plate, which has rotational symmetry, and the Hilbert plate, consisting of a half-plane of dielectric material with a straight edge (Fig. 1), [1,2]. A recent survey [3] reviews the many forms in which these plates have been realized, and also gives some analysis of the operation of both types by applying Abbe’s wave theory to a rotationally symmetric lens and object. The method was also used in a separate paper [4] to analyse the effect of a Zernike phase plate in the TEM. Here we extend the previous analysis to include azimuthal variation and so to describe the behaviour of straight-edge (Foucault and Hilbert) plates. To do this, relations for the 2D transform have been written using cylindrical coordinates, after which the images produced by certain combinations of a simple phase object and a phase plate can be readily deduced. This process yields information about the components of the image (both the copy of the object and artefacts due to the plate), their amplitudes, and in addition their relative phases. The detailed analysis suggests how the performance of the Hilbert plate can be improved. This plate was so named by Lowenthal and Belvaux [5] because the Hilbert transform was used in processing the image. The phase added by the plate was chosen in [5] as π , without discussion of other choice of angle. The detailed
E-mail address: [email protected] http://dx.doi.org/10.1016/j.ultramic.2017.06.020 0304-3991/© 2017 Elsevier B.V. All rights reserved.
analysis given here shows that a different choice of thickness of the Hilbert plate may lead to substantially improved performance. 2. Method for post-specimen phase plates Analysis of imaging in the TEM often considers the phase contrast transfer function (PCTF), which defines the relative phase introduced by the lens system as a function of spatial frequency. It is well known (for example, Section 8.6.3(c) of [6]) that, if the argument of the PCTF can be changed by π /2, then the contrast at low frequencies can be increased. The established transfer functions show how the system responds to a single spatial frequency (of infinite transverse extent), but they do not show the image produced by a bounded object that contains many sources of scattering and many spatial frequencies. However, theory for the transfer of a spectrum of frequency components has been defined for optical systems by Abbe (Section 8.6.3(b) of [6]), and for TEM by Hawkes and Kasper (Section 65.2 of [7]). This theory, known elsewhere as Fourier optics [8], shows how the image can be obtained from the object exit wave and the frequency response of intermediate parts of the system. Here we apply this transfer theory to a system in which the object and the lens each have rotational symmetry and there is a Foucault or Hilbert phase plate at the standard post-lens location. Since the behaviour is periodic in azimuth, we use cylindrical coordinates: (r, φ ) at exit from the object, (q, θ ) at the phase plate and (ri , φ i ) at the image plane. According to Abbe’s theory, the object wave can be specified by its spectrum in the spatial frequency q. The lens redistribute the wave so that all parts that have the same magnitude of q before reaching the lens converge after the lens to one physical radius that is proportional to q, on a specific plane, the ‘diffraction’ plane, transverse to the cylindrical axis. If the illumination is par-
C.J. Edgcombe / Ultramicroscopy 182 (2017) 124–130
125
Fig. 2. Unit step, used to describe illumination and phase of disc object.
to object space. The inversion of the modified spectrum then gives the image functions directly in these scaled coordinates. In the 2D transforms as defined in Appendix A, each function f (r) is both a coefficient of a Fourier series and a source for a Hankel transform; similarly for each function t used below. In previous work [4] the name ‘Fourier–Bessel’ was used for the radial transform with no azimuthal variation; either this name, or possibly ‘Fourier-Hankel’, now seems more suitable for the general 2D transform in cylindrical coordinates. 2.1. Illumination Fig. 1. Location and form of a conventional Hilbert plate (from Fig. 1a,b, Principle of difference-contrast TEM, in reference [1], © 2002 Kluwer Academic Publishers, with permission of Springer.)
allel, this plane is the back focal plane. It is the relation between q and physical radius at the diffraction plane that allows a phase plate to modify particular ranges of q (Eqs. (65.15)–(65.17) of [7]). The modified spectrum then propagates further, and its inversion defines the focused image. The complex amplitude of the image can be converted to intensity and then shows the contributions of the component terms to the detectable intensity. The spectrum of the simple object considered here can be found analytically. The present work describes the transmission of a Foucault or Hilbert plate as a series of azimuthal harmonics. Here we use the 2D transform expressed in cylindrical coordinates; many functional relations can be deduced and a consistent set, based on harmonics of exp iθ , is given in Appendix A. A benefit of Abbe’s method is that it shows not only the spatial distribution but also the relative phases of components of the image amplitude. Analysis shows that the image intensity obtained with the Foucault plate contains a copy of the object’s spatial distribution, whose magnitude is proportional to the square of the phase change due to the object. Objects with phase change less than one radian (‘weak phase objects’) then contribute less to the image intensity than if it were linear in the object phase. The Foucault intensity also contains an approximately linear gradation perpendicular to the plate edge, with magnitude proportional to the phase of a weak uniform object. The Hilbert plate produces a similar gradation of intensity, but can generate a copy of the object shape that varies linearly with the object phase, as is desired. Analysis shows how the image depends on the thickness of the plate, from which some new deductions can be made. The theory given here omits known contributions to the PCTF of a TEM such as defocus, aberrations, energy spread, signal-to-noise ratio and other effects. The object is assumed to produce zero absorption and so to be a ‘pure phase object’. The plate is assumed to be located at the axial plane at which each value of q occurs at a single radius. The coordinates (ri , φ i ) of the image are assumed to be scaled by reduction and rotation as in Section 65.2 of [7]; or, as an alternative statement, the image coordinates are referred
An easily defined form of illumination is a transversely infinite plane wave, but a question arises about its transform. When Cartesian coordinates are used, the Fourier transform in each coordinate direction is represented by a Dirac delta function and justified by distribution theory [9]. When we use cylindrical coordinates, it is not immediately clear how to determine the Hankel transform of a radially uniform wave of infinite extent. We can avoid using distribution theory by noting that in practice the illumination will always be limited at some outer radius. We therefore assume that at the plane of the object exit wave, the illumination is uniform within a radius a much greater than the specimen radius, and is zero outside radius a. Here we omit the factor exp i(kz − ωt), and represent the illuminating wave at the specimen by
ψ = u1 (r/a ) where u1 is a unit step function (Fig. 2):
u1 ( ρ ) =
0<ρ<1 1<ρ
1, 0,
The Hankel transform P0 of this rotationally symmetric wave is (using Eq. (A.4) with = 0)
P0 (q, a ) =
0 2
∞
u1 (r/a ) J0 (qr )r dr
= a J1 (qa )/qa Then when a is not infinite, P0 is well-behaved as q → 0. 2.2. Object The object is assumed to advance the phase of the wave (relative to free-space propagation) by γ h(r, φ ), where γ is a constant angle, h is a specified function of approximately unit magnitude and is zero for r greater than some value b which is less than a, and r and φ are the cylindrical coordinates transverse to the axis of the system. Then, with the incident wave as in Section 2.1, the exit wave is described by
ψ = u1 (r/a ) exp iγ h(r, φ ) (With this definition, the phase change γ h is defined only modulo 2π .) For convenience in discussing weak-phase behaviour, ψ is written as the sum of the incident wave and a scattered wave f:
ψ = u1 (r/a ) + f (r, φ )
126
C.J. Edgcombe / Ultramicroscopy 182 (2017) 124–130
intersect the material of the plate. Then as q0 becomes more positive, more of the beam is clear of the plate. For a given (q, θ ), transmission outside the plate occurs when
q cos θ > − q0 The magnitude q is non-negative, but q0 may be positive, zero or negative.
Fig. 3. Geometry for straight-edge plates, looking along the axis of the system.
from which
f = u1 (r/a ) eiγ h − 1 = eiγ h − 1 = iγ h(r, φ ) + O
γ2
(1)
The second equality follows because u1 is unity over the range r < a, and h is zero for r > b. Here we consider only a notional disc object. Although this is not typical of most biological objects, it can be used to analyse many types of plate. 2.2.1. Disc object This object is assumed to be a circular disc that produces uniform phase change γ (relative to free-space propagation) over its diameter, 2b, while outside this diameter the phase change is zero (Fig. 2). The scattered wave at this object can be written with another unit step function:
h = u (ρ ) where u is as defined in Section 2.1 but now a function of ρ , defined by
ρ = r/b
2.3.2. Relation of q to physical radius at plate By equating the scattering angle for a wave with phase constant k, acquiring transverse wave vector q, to the angle for a particle scattered to radius x at the focal distance f from a thin lens, we obtain the relation q/k ≈ x/f. This is only approximate since the objective usually has thick-lens properties. The simple relation suggests that the value of q0 corresponding to a physical spacing x0 of the plate edge from the axis is q0 ∼ 2π x0 /λf, so for 200 kV electrons, a focal length of 3 mm and a spacing x0 of 1 micrometre, the corresponding value of q0 is about 0.84 nm−1 . 2.3.3. Intercepting plates The Foucault plate is opaque to the beam and blocks part of the spectrum. The transmission function T then has the value 0 or 1 as a function of q and θ . These plates, like apertures and beam stops, are found to produce weak-phase image amplitudes that differ in phase by π /2 from the direct beam. As described in Section 3.1, the intensities of desired images are then proportional only to the square of the object phase γ and so are undesirably small for weak-phase objects. 2.3.4. Hilbert plate The Hilbert plate is also a half-plane plate but instead of being opaque, it consists of dielectric material that produces a phase change, additional to that of free space, denoted here by α . Any absorption by the plate is ignored here. The transmission T at (q, θ ) depends on the relative magnitudes of q and q0 , as shown in Fig. 3. In the following theory, it will be convenient to use the symbol p0 defined by
p0 ( q0 ) =
The wave at the plane of exit from the object is then, from (1),
f d (ρ ) = e
iγ h
−1= e
iγ
− 1 u (ρ )
(2)
The object is rotationally symmetric so its azimuthal periodicity is zero. The Hankel transform of (2) according to (A.4) for = 0 can be written as
Fd (q ) = eiγ − 1 b2 J1 (Q )/Q
(3)
where
Q = qb and J1 () is a Bessel function of the first kind.
1, exp iα ,
0 < q0 q0 < 0
(4)
Those elements of the electron wave that have q ≤ |q0 | are transmitted with the same phase change for all θ . For q0 > 0, the phase change is zero, while for q0 < 0, all transmitted electrons suffer phase change α . Thus
T (q ≤ |q0 |, θ ) = p0 For parts of the wave with q > |q0 |, the phase change depends on θ :
T (q > |q0 |, θ ) =
1, exp iα ,
−θ0 < θ < θ0 −π < θ < −θ0 ,
θ0 < θ < π
(5)
where θ 0 is defined as in Fig. 3 by
θ0 (q ) = arccos (−q0 /q ) 2.3. Plate The phase added by the Hilbert plate is represented (as for the Zernike plate) by the argument of a complex number. 2.3.1. Location of a straight-edge plate The plate lies in a plane perpendicular to the axis of the lens system. The intersection of the axis with this plane is taken as the origin of transverse coordinates (q, θ ) (Fig. 3). The direction of the perpendicular outward from the plate edge is chosen to define θ = 0. The perpendicular spacing of the plate edge from the axis is denoted by q0 and is defined as positive when the axis does not
These T are either constant or periodic in θ with period 2π , so they can be expressed as Fourier series of the form
T (q, θ ) =
∞
t (q ) exp iθ
=−∞
By the usual method for Fourier series, the t can be written as
t (q ≤ |q0 | ) =
0, p0 ,
l = 0 l=0
1 − eiα sin θ0 (q )/π , t (q > |q0 | ) = t0R + i t0I ,
(6a,b) l = 0 l=0
(6c,d)
C.J. Edgcombe / Ultramicroscopy 182 (2017) 124–130
× cos φi J (ρi Q ) J1 (Q )
where
t0R + i t0I = [θ0 /π + {1 − (θ0 /π )} cos α ] + i{1 − (θ0 /π )} sin α
3. Images with Hilbert plates
The terms in the image function will be shown to have complex amplitudes which define their relative phases. Real parts of these amplitudes have the phase of the incident beam. If the images of the incident and scattered waves can be represented as u and (iη), with phases differing by π /2, then their total image intensity is |u + iη|2 = u2 + η2 and the intensity is modulated as η2 . However, if part of the image of the scattered wave is in phase with the incident wave and has the real amplitude η, then the total intensity |u + η|2 = u2 + 2 ηu + η2 , and the intensity is modulated by η linearly. 3.2. A disc object with a Hilbert plate The spectrum of a disc object, Fd (q) as in (3), is independent of azimuthal angle θ . The transmission of the Hilbert plate was defined by relations (5) and (6). The spectrum at exit from this plate then has the form ∞
eiθ t (q )
=−∞
This spectrum corresponds to the form of (A.8), so by equating terms
P (q ) = i t (q ) Fd (q )
(7)
We can now use (A.10) to find the image amplitude. After substituting in (7) for Fd from (3), inserting P in (A.10), writing g(ri , φ i ) for the image amplitude f in (A.10) including the unscattered part u1 (ri /a) (with the phase shift (4) that occurs when q0 < 0), and noting that t and (i J ) are independent of the sign of ,
g ( r i , φi ) = p 0 u 1 + e iγ − 1
v0 ( ri , b ) + 2
∞
v (ri , b) cos φi (8)
v ( r i , b ) = i b
∞
0
t (q ) J1 (qb) J (qri )dq
(9)
To evaluate the integrals in (9), it is necessary to sum over the two ranges 0 < q < |q0 | and |q0 | < q < ∞, using the specific value of t for each range. The tedious calculation for the general case is relocated to Appendix B. The result is
g(ri , φi ) = p0 u1 + eiγ − 1 [ p0 I0 (ρi , B ) + w0R + i w0I sinα
+ 1 − e iα S
I0 (ρi , B ) =
|B| 0
(10)
J1 (Q ) J0 (ρi Q ) dQ; B = q0 b, Q = qb
w0R + i w0I sinα =
∞
|B|
θ 0 iα θ0 + 1− e J1 (Q ) J0 (ρi Q )dQ π π
and, with S arranged as an integral of a sum,
S ( ρ i , φi , B ) = s R + i s I =
2
π
∞ ∞ B
=1
i
1 sin θ0 (Q, B )
|g|2 = u1 2 − 2 γ u1 [(w0I − sR ) sin α + sI (1 − cos α )] + O(γ 2 ) (12) and for q0 < 0 (including the effect of the phase change at q = 0 produced by the plate)
|g|2 = u1 2 − 2 γ u1 [(−w0R + w0I cos α − sR ) sin α − sI (1 − cos α )] + O(γ 2 )
(13)
The image intensity thus contains the intensity of the incident beam (u1 2 , equal to u1 ) and a term linear in the object phase, γ . This linear contribution includes w0 I , which is an integral of the object spectrum independent of φ i , together with φ i -dependent artefacts sR and sI . The forms of w0 I for B = 0.5, 1, 2 and 5 are shown in Fig. 4 as functions of ρ i B = q0 ri . The central region of the disc object is imaged only if B = q0 b is less than about 1, that is if the object radius is less than a limit set by system parameters. Larger disc objects show up in the image by the discontinuities at their edges. Approximate calculations of sR and sI as functions of ρ i = ri /b for B = 1 are shown in Fig. 5. Like w0 I , sI produces a response mainly at the edge of a disc, but sI varies with φ i . Eq. (12) shows that the relative sizes of these two contributions to the intensity depend on the plate phase change α (see Section 3.2.3). 3.2.2. Special case q0 b = ε (0 < ε << 1) When the edge of the plate is close to the axis, θ 0 becomes independent of q and equal to π /2. Then B, I0 and sR become negligible, and the image amplitude and intensity simplify to
g ( r i , φi ) = u 1 + e iγ − 1 q0 = 0+
|g|2 = u1 2 − 2 γ u1
=1
where
(11)
3.2.1. Weak-phase disc object and Hilbert plate On approximating (10) to first order in the object phase γ and forming the intensity, the result for q0 > 0 is
3.1. Linear and quadratic components of intensity
T (q, θ )Fd (q ) = Fd (q )
dQ
The function I0 appeared in earlier work on Zernike plates with no azimuthal variation [4], where it was named Id . It has no relation to the modified Bessel function In ().
These coefficients t are independent of the sign of .
where
127
1 2
1 + eiα u(ρi )/2 + i 1 − eiα sI0 ,
u(ρi ) sin α + sI0 (1 − cos α ) + O(γ 2 ) (14)
where
sI0 (ρi , φi ) =
2
π
∞ ∞ 0
m=0
cos (2m + 1 )φi J2m+1 (ρi Q )J1 (Q ) dQ 2m + 1 (15)
The peak of function sI 0 (ρ i , 0), shown in Fig. 6, has approximately the same magnitude as u(ρ ), shown in Fig. 2. Some integrals of Bessel products like those in (15) are shown graphically as functions of (ρ i B) in Chapter 2 of [3]. From these it seems likely that the major contribution to sI 0, in the range 0 < ρ i < 0.5, is from the term for m = 0. For ρ i < 0.5, this term is approximately proportional to ρ i so when combined with the variation as cos φ i its contribution to g is approximately proportional to (ri cos φ i ) or distance (in the image plane) perpendicular to the plate edge. This term, produced by the component of the plate transmission varying as cos θ , appears to describe the shading of intensity perpendicular to the plate edge that is observed with straight-edge plates [1]. 3.2.3. Choice of α for weak-phase object In (12), (13) and (14), the terms copying the object are multiplied by the sine of α , the phase introduced by the plate. These
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C.J. Edgcombe / Ultramicroscopy 182 (2017) 124–130
Fig. 4. Function W0 I (q0 ri ) (independent of azimuthal angle φ i ) for B = 0.5, 1, 2, 5, integrated over Q from B to ∞.
Fig. 5. Approximate forms of φ i -dependent functions sR and sI for φ i = 0 and B = 1. Both functions were integrated over Q from B to 10 0 0. The function sR was summed over 2m from 2 to 20. Function sI was summed over (2m + 1) from 1 to 21. At φ i = π /2, sI = 0.
4. Discussion
Fig. 6. Approximate form of function sI 0 for B = 0 and at φ i = 0; summed for (2m + 1) = 1–21 and integrated over Q from 0 to 10 0 0. When B = 0, then sR 0 = 0.
results show that the usual choice of α = π minimises the linear copy of the phase distribution of the object (constant with φ i ). If it is desired that the image contain this linear copy, the phase change of the plate should be chosen as some other value than π . With α = π /2 and q0 ≥ 0,
|g| = u1 − 2 γ u1 [w0I − sR + sI ] + O γ 2
2
2
This choice of α ensures that the weak-phase response is linear in the object phase γ , but the shading contribution from components of S is still present. For an object of strong phase, the optimum value of α is likely to differ from π /2, as discussed by Beleggia [10].
For the configuration analysed in Section 3.1, the most suitable value of α depends on how important it is to minimise the shading contribution from = 1. If its presence is unimportant or even desired, then a value of α = π /2 or 3π /2 will maximise the contribution from the object shape. However, as α is reduced from π /2, the magnitude of (1 − cos α )/sin α falls and the relative contribution from = 1 decreases. This form of plate is simple to make in comparison with some other types but, as usual, care will be needed to minimise possible charging at the edge of the plate, from the direct beam. A recent study by Koeck [11], which takes account of defocus and spherical aberration, shows that a phase shift of π /2 has the further benefit of extending the first passband, when operation is at Scherzer defocus. 5. Conclusion The image of a disc phase object produced by a round lens with a Hilbert phase plate has been calculated by Abbe’s theory of imaging with the 2D transformation expressed in cylindrical coordinates. The conventional choice of π for the phase added by the plate leaves the image of this object out of phase with the direct beam, and as the plate edge approaches the axis of the lens system, the linear contribution of the object’s phase to the image intensity is cancelled. However, if the phase added by the plate is
C.J. Edgcombe / Ultramicroscopy 182 (2017) 124–130
reduced to π /2 (for a weak-phase object), then the image intensity contains a copy of the object that is linear in the phase of the object. In addition to this copy of the object, the image contains a series of azimuthal harmonics including one varying approximately as (ri cos φ i ) which appears to be the source of the observed gradation of intensity perpendicular to the plate edge.
A.3. Conversion from Cartesian form The Cartesian 2D transform of f can be specified by (for example, Section 2.1.1 of [8] with 2π fX replaced by qx and similarly for fy )
F= Acknowledgements The calculations of Figs. 4–6 were made with Mathematica 11.1.1. This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors. Appendix A. The 2D transform in cylindrical coordinates The transmission of plates that have some azimuthal structure can in general be specified as a function of both q, the magnitude of radial frequency in angular measure, and an azimuthal angle θ . Any such function can in principle be expressed as a Fourier series in θ , with components of azimuthal periodicity and coefficients that are functions of and q. The spectrum for each can be inverted to give the image function for the same periodicity in the image angle φ i . Before deriving the full 2D transform, we state first the Fourier series and Hankel transform to be used here. Section (A.3) shows that the transform in cylindrical coordinates is obtained by direct change of coordinates from a standard 2D transform in Cartesian coordinates.
A given function f(r, φ ) that is periodic in the angle φ with period 2π can be represented by a Fourier series in exp iφ as ∞
eiφ f (r )
(A.1)
=−∞
where is integral and the (complex) f are defined by
π 1 f (r ) = e−iφ f (r, ϕ ) dφ 2 π −π
(A.2)
f (r ) =
0
P (q ) J (rq ) q dq
(A.3)
0
∞
f (r ) J (qr ) r dr
∞ −∞
f exp −i(qx x + qy y ) dx dy
(A.5)
Here the initial constant has been chosen for later convenience. Relation (A.5) can be expressed in cylindrical coordinates (r, φ ), (q, θ ) by converting the Cartesian coordinates to their cylindrical equivalents:
qx = q cos θ , qy = q sin θ , x = r cos φ , y = r sin φ In applications, θ is the azimuthal angle of q = (q,θ ), measured from the same axis and with the same origin as φ . After substituting into (A.5) with suitable changes to the ranges of integration, and specifying f and F as functions of cylindrical coordinates at given transverse planes,
F (q, θ ) =
π
1 2π
−π
∞
0
f (r, φ ) exp −i[qr cos (φ − θ )] r dr dφ (A.6)
An object function specified as f(r, φ ) is necessarily periodic in φ with period 2π and so can be expressed as a Fourier series in exp iφ . To obtain the harmonic dependence on θ , we first substitute in (A.6) from (A.1). We then define a new angle ψ by
ψ =φ−θ
F (q, θ ) =
∞ 1
π
eiθ
∞ 0
l=−∞
f (r )
π 0
cos ψ
× exp − i(qr cos ψ ) dψ r dr and after using Bessel’s integral (10.9.2 of [12]) for ψ with θ constant, F can be written as ∞
F (q, θ ) =
i− eiθ
∞ 0
=−∞
f (r ) J (qr ) r dr
(A.7)
(A.4)
and J (qr) is a Bessel function of the first kind and integral order . The pair described as a Hankel transform in [12] and some other tables is an alternative form of the same relations but uses different weighting functions, so before any given table is applied, the definitions it uses should be checked. In applications, it may be useful to think of q as a radial phase constant in angular measure, while bearing in mind that the local wavelength of a Bessel function J (qr) differs from 2π /q. If f is a constant independent of r, the integral in (A.4) diverges, so some constraint on f is needed. In practice f0 is limited since the wave has zero amplitude outside some maximum radius.
∞
i− eiθ P (q )
(A.8)
=−∞
On substituting (A.2) in (A.7), the first relation of the 2D transform as a function of the original object function is:
F (q, θ ) =
where P is defined by
P (q ) =
∞ −∞
F (q, θ ) =
A given function f (r) can be represented (subject to certain conditions) by a function P (q) as ∞
or, with the definition (A.4),
A.2. Hankel transform
1 2π
Then, with θ held constant as φ varies, F can be expressed as
A.1. Fourier series
f (r, φ ) =
129
π 1 − iθ ∞ i e e−iφ f (r, φ ) dφ J (qr )r dr 2π 0 −π
A.4. Inverse transform An application such as Section 3 may provide the spectrum in the form of (A.8), from which the P can be identified. If not, the P can be found from F(q, θ ) by the Fourier inversion of (A.8):
P (q ) =
1 2π
π
−π
i e−iθ F (q, θ ) dθ
(A.9)
When the P are known, f can be found by combining (A.1) and (A.3) as
f (r, φ ) =
∞ =−∞
eiφ
∞ 0
P (q ) J (qr ) q dq
(A.10)
130
C.J. Edgcombe / Ultramicroscopy 182 (2017) 124–130
The relation (A.10) is used in Section 3 to find the image function from a known Fourier series for the spectrum. The inverse relation of the 2D transform, for f as a function of F, can be found by inserting (A.9) in (A.10):
f (r, ϕ ) =
∞ 1 iφ ∞ π −iθ ie e F (q, θ )dθ J (qr ) q dq 2π =−∞ 0 −π
v (ri , b) = i 1 − eiα W , > 0 v0 (ri , b) = p0 I0 (ρi , B ) + W0 (ρi , B ) Then from (8), Eq. (10) is obtained:
+ i w0I sinα + 1 − eiα S
(A.11) Alternatively, on inserting (A.9) in (A.3) and substituting the resulting expression for f into (A.1), the complete inversion agrees with (A.11).
g(ri , φi ) = p0 u1 + eiγ − 1 [ p0 I0 (ρi , B ) + w0R
where
S ( ρ i , φi , B ) = s R + i s I = 2
∞
i W cos φi
=1
Appendix B. The general image function
For convenience in using Mathematica, S can be rearranged as an integral of a sum, giving (11):
We seek to evaluate the coefficients v given by (9) above:
v ( ri , b ) = i b
∞ 0
t (q ) J1 (qb) J (qri )dq
S ( ρ i , φi , B ) = s R + i s I =
t (q ≤ |q0 | ) =
0, p0 ,
=
l = 0 l=0 l = 0 l=0
where
∞
σSI =
t0R + i t0I = [cos α + (θ0 /π )(1 − cos α )] + i{1 − (θ0 /π )} sin α The integrals must be evaluated by summing over two ranges of q, less than and greater than |q0 |. Here B = q0 b may be negative, zero or positive. 1) 0 < q < |q0 |:
t (q ) J1 (qb) J (qri )dq = 0, > 0
and for = 0,
0
b
t0 (q ) J1 (qb ) J0 (qri )dq = p0 I0 (ρi , B )
2) |q0 | < q: Using (6c), for > 0: ∞
|q 0 |
t (q ) J1 (qb ) J (qri )dq = 1 − eiα W , > 0
where
W (ρi , B ) =
∞ 1 sin θ0 (q ) J1 (Q ) J (ρi Q )dQ, > 0 π |B|
While the W above are real, when = 0 then W0 is complex:
W0 = w0R + i w0I sinα =
π
∞
|B |
(−1 )m
∞
(−1 )m
m=0
|q 0 |
2
m=1
B = q0 b, Q = qb
b
|B| =1
i
1 sin θ0 (q ) J (ρi Q )
σS (ρi , φi , Q, B)J1 (Q ) dQ
σS = σSR + i σSI σSR =
θ0 = arccos (−q0 /q ) = arccos (−B/Q )
0
∞ ∞
where
1 − eiα sin θ0 (q )/π , t (|q0 | < q ) = t0R + i t0I ,
|q 0 |
π
× cos φi J1 (Q ) dQ
by using the coefficients for the disc object and Hilbert plate as in (6a-d) above
2
∞
|B|
(t0R + i t0I ) J1 (qb) J0 (qri )d (qb)
On summing over both ranges of q to find v as in (9):
1 sin [2mθ0 (Q, B )] cos 2mφi J2m (ρi Q ) 2m
1 sin [(2m + 1 )θ0 (Q, B )] 2m + 1
× cos (2m + 1 )φi J2m+1 (ρi Q ) References [1] R. Danev, H. Okawara, N. Usuda, K. Kametani, K. Nagayama, A novel phasecontrast transmission electron microscopy producing high-contrast topographic images of weak objects, J. Biol. Phys. 28 (2002) 627–635, doi:10.1023/ A:1021234621466. [2] Y. Kaneko, R. Danev, K. Nitta, K. Nagayama, In vivo subcellular ultrastructures recognized with Hilbert differential contrast transmission electron microscopy, J. Electron Microsc. (Tokyo). 54 (2005) 79–84, doi:10.1093/jmicro/dfh105. [3] C.J. Edgcombe, Phase plates for transmission electron microscopy, Ch. 2 of Vol. 200, in: P.W. Hawkes (Ed.), Advances in Imaging and Electron Physics, Academic Press, Burlington, 2017, doi:10.1016/bs.aiep.2017.01.007. [4] C.J. Edgcombe, Imaging by Zernike phase plates in the TEM, Ultramicroscopy 167 (2016) 57–63, doi:10.1016/j.ultramic.2016.05.003. [5] S. Lowenthal, Y. Belvaux, Observation of phase objects by optically processed Hilbert transform, Appl. Phys. Lett. 11 (1967) 49, doi:10.1063/1.1755023. [6] M. Born, E. Wolf, Principles of Optics, sixth ed., Pergamon Press, Oxford, 1993. [7] P.W. Hawkes, E. Kasper, Wave Optics (Principles of Electron Optics Vol. 3), Academic Press, London, 1994. [8] J.W. Goodman, Introduction to Fourier Optics, second ed., McGraw-Hill, London, 1996. [9] M.J. Lighthill, Introduction to Fourier analysis and generalised functions, Cambridge University Press, Cambridge, 1959. [10] M. Beleggia, A formula for the image intensity of phase objects in Zernike mode, Ultramicroscopy 108 (2008) 953–958, doi:10.1016/j.ultramic.2008.03. 003. [11] P.J.B. Koeck, Improved Hilbert phase contrast for transmission electron microscopy, Ultramicroscopy 154 (2015) 37–41, doi:10.1016/j.ultramic.2015.03. 002. [12] DLMF, in: F.W.J. Olver, A.B. Olde Daalhuis, D.W. Lozier, B.I. Schneider, R.F. Boisvert, C.W. Clark, B.R. Miller, B.V. Saunders (Eds.), 2016 DLMF: NIST Digital Library of Mathematical Functions, Release 1.0.13 http://dlmf.nist.gov/.