Lateral Resolution Enhancement with Standing Evanescent Waves
by
George E. Cragg
M.S., Physics (1997)
University of Rochester
Submitted to the Department of Mechanical Engineering
in Partial Fulfillment of the Requirements for the Degree of
Master of Science in Mechanical Engineering
at the
Massachusetts Institute of Technology
June 2000
MASSACHUSETTS INSTITUTE
OF TECHNOLOGY
2000 Massachusetts Institute of Technology
All rights reserved
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Signature of Author .............................. ......
Department of Mechanical Engineering
May 23, 2000
Certified by.............................
..............
Peter T. C. So
Assistant Professor of Mechanical Engineering
Thesis Supervisor
A ccepted by.............................................
Am A. aunim
Chairman, Department Committee on Graduate Students
2
Lateral Resolution Enhancement with Standing Evanescent Waves
by
George E. Cragg
Submitted to the Department of Mechanical Engineering
on May 23, 2000 in Partial Fulfillment of the
Requirements for the Degree of Master of Science in
Mechanical Engineering
ABSTRACT
We have developed a novel fluorescence microscopy technique that achieves a lateral
resolution of better than one-eighth of the emission wavelength of the sample. With a
total internal reflection geometry, standing evanescent waves generate a high frequency
intensity modulation across the sample. At each of three different standing wave phases,
an image is recorded. These three images are then weighted and summed in such a way
that enhances resolution in the given direction. Two dimensional resolution enhancement
is attained from a linear combination of several such scans performed symmetrically
across the sample plane. The proposed imaging system is analyzed through theoretical
calculations of its point spread function and its optical transfer function.
Thesis Supervisor: Peter T. C. So
Title: Assistant Professor of Mechanical Engineering
3
1. Introduction
Although optical microscopy is the most widely used high-resolution technique, it
has failed to address problems with length scales below 100 nm. Atomic scale resolution
has been achieved in semiconductor samples with scanning probe techniques such as near
field scanning optical microscopy, scanning tunneling microscopy, and atomic-force
microscopy.
However, in biological specimens, the applications of scanning probe
techniques suffer from slow imaging speed, resolution degradation, and from the need to
make mechanical contact with the specimen. Clearly, there is a need for a complete
optical approach that can be adapted to study a wide variety of specimens.
Standing wave total internal reflection fluorescence (SWTIRF) microscopy can
achieve a lateral resolution better than one-eighth of the emission wavelength. This
technique is based on the formation of a high frequency standing wave excitation in a
total internal reflection geometry. This high frequency information can be extracted to
produce high-resolution images from a series of images taken at different phases and
directions of the standing wave. Of particular promise is the ability to perform noncontact imaging of biological specimens at a resolution that is comparable to or better
than that of scanning probe microscopy.
2. Overview
Before proceeding with the development of SWTIRF microscopy, some preliminary
background is in order. To understand the fundamentals of imaging systems, coherent
and incoherent imaging is introduced in the next section. With this background, lateral
resolution is quantified in section four. The physics behind this technique involves the
4
interaction of light at an interface, which is reviewed in section five. In section six this
discussion is extended to evanescent fields, which are the mode of excitation in all total
internal reflection (TIR) microscopy. Section seven then describes standing evanescent
waves as a means of providing a high frequency intensity modulation to the sample.
Drawing on these results, sections eight and nine describe the resolution enhancement
achieved by SWTIRF microscopy.
3. Coherent and Incoherent Optical Imaging Systems
An imaging system can be defined as any device that reproduces the geometry of
an object. An ideal optical imaging system is a system free from all aberrations.
Therefore, an ideal system's representation of an object is limited only by the diffraction
effect. We shall restrict our attention to the ideal class of systems throughout. Consider
the simple imaging system of the lens shown below in Fig. 3.1.
h-
Object
(.,
T )
z2
Lens
Focal
plane
Image
(u, v)
Fig. 3.1 shows a lens as a simple imaging system where z I = object distance and z 2= image distance.
5
In this case, the light from the object is the input, which is processed by the lens and then
converted to an image.
If the object is coherently illuminated, then the complex amplitude of the electric
field at the image plane, Ui (u, v), is given by the following expression:
Ui(u,v)=
Jh(u-
,v- i)U 0 ( , i)didi,
(3.1)
where U0 (4, rl) is the complex electric field amplitude at the object and (u,v) are the
image coordinates. The reduced object coordinates, ((,ij), are given by
=
M and
= Mil where M is the magnification defined by - z 2 / z 1 . The amplitude impulse
response, h(u, v), is the electric field produced at the image plane given a Dirac delta
function electric field amplitude at the object plane. Since Eq. (3.1) is a convolution, it
sums the impulse responses of each of the points of the object. Hence, we say that
coherent imaging is linear in complex amplitude. Typically, the measured quantity at the
image plane is the intensity or the squared modulus of the complex amplitude. For
coherent imaging, the intensity is given as
2
Ui (u, v)f 2 = Jh(u -
=
,v - fi)U
0 ((, ij)didij
h((, ij)0U0 ((,ii),
where 0 denotes the convolution operation. Thus, we see that although a coherent
system is linear in complex amplitude, it is not linear in intensity.
(3.2)
6
In an incoherent imaging system on the other hand, each illuminated point on the
object differs from every other point of illumination by a random phase. In this case, the
image, I( ,ij), is given as
I(,j=h( , ii)J ( U.(4, j)1
=
P(, ij) 0 ( ,ij),
(3.3)
where P(,, i)= h(, ij)I is the point spread function (PSF) of the imaging system and
O((,j) =
jU0 (p,j)
is the intensity distribution on the object plane, scaled by the
magnification factor. We refer to O(,, i) as the object function, since it represents the
"exact" geometry of the object.
4. Resolution of Fluorescence Microscopy
In fluorescence microscopy, a sample is usually illuminated by coherent laser
light thereby causing fluorescence emission from the stained portions of the sample.
Upon being collected by the objective, these emissions are relayed to optics that project
an image of the labeled sections of the sample on the image plane (Fig. 4.1). Typically, a
CCD or some other device is used to record the image.
7
Image plane
(CCD)
Objective
Sample plane
Fig. 4.1 shows a simple microscope.
These fluorescence emissions occur spontaneously and randomly; hence the
imaging light is incoherent. The PSF of the objective lens is a measure of the extent to
which the system "smears out" or distorts the points of the object. Therefore, it is natural
to seek a criterion for lateral resolution directly from the PSF. Resolution can be
quantified as the minimum distance between two point particles such that they still can be
seen as distinct. This distance is given by the Rayleigh criterion as the full width at half
maximum (FWHM) of the PSF. For microscope objectives, the PSF is usually given as
P(r)=
_____
27NAr/Xe
I
2
(4.1)
where J1 is the first order bessel function of the first kind, ke is the free space emission
wavelength of the source, and r = (r, 0) is the radial vector on the image plane. Without
loss of generality, a magnification of unity is assumed in Eq. (4.1) and throughout.
Additionally, the numerical aperture, NA, is a number, typically ranging from 1 to 1.3,
that is dependent upon the properties of the objective. With Eq. (4.1) as the PSF, the
Rayleigh criterion is given by
FWHM =
O.6l1e
*
NA
.
(4.2)
8
Hence we see that for an emission wavelength of 560 nm and an NA of 1.3, the
resolution of conventional fluorescence microscopy is about 263 nm.
An alternative way of describing the resolution capabilities of an optical system is
to examine its optical transfer function (OTF), which is defined as the Fourier transform
of the PSF. Recall that the two-dimensional Fourier transform, g(k ,
y
of a function
g(x, y) is given as
&X
, k Y)=
g(x, y)e-i(kxx+kyy)dxdy,
(4.3)
where k X, k Y are referred to as spatial frequencies. The inverse Fourier transform is
known as the Fourier integral representation of the original function, g(x, y):
g(x, y)=
(27r)
2
j(kX, ky)ei(kxx+kyy)dkdky.
Hence, the Fourier transform of g(x, y),
(4.4)
(k Xk y ), is the frequency spectrum of the
original function, g(x, y). In this way, the OTF reveals both the spatial frequencies that
are present in the image as well as the relative weight of these frequency components.
Because the OTF can be used to describe the low-pass filtering characteristics of the
imaging system, the resolution is determined from the cutoff frequency of the OTF. For
the case of conventional fluorescence microscopy, the OTF corresponding to Eq. (4.1),
P(k), is given as
P(k) =
2 {arccos(k / 2a)-(k / 2a)[1-(k / 2a)2 1/2
0,
0< k
k
k >2a
2a
(4.5)
9
where k = (k x2 +kY 2)1/
2
is the radial spatial frequency coordinate. Thus, the
conventional cutoff frequency is k max= 4ixNA / Xe. This cutoff frequency corresponds
to a resolution limit of about
-kmax
27
NA
which is in good agreement with
the Rayleigh result.
5. Boundary Equations
Before describing SWTIRF imaging, it will be helpful to review some of the
physics behind its implementation. As this technique is implemented using total internal
reflection (TIR), it will be useful to review the equations that describe the propagation of
light at a surface. Consider the refraction of light at a dielectric boundary between
medium 1 and medium 2 as shown in Fig. 5.1.
a)
b)
z
z
H
E
t
2
.
kt
Et
2
16
kt
-
Ht
r x
14/O
k.
k
r
H
r
Ei
E.k
k
E
/Hi
H).
Fig. 5.1 depicts the reflection and refraction of a) s-polarized and b) p-polarized incident light, where X
points into the page and 9 points out of the page.
10
The equations governing the behavior of light at the interface are the well-known
Maxwell equations and the associated boundary conditions. Assuming no free charges or
currents, then Maxwell's equations, in MKS units, are
aR
(i)
VxE=-p-
(iii)
V- E = 0
at
(ii)
V xH=
(5.1)
(iv)
--
at
V -pH = 0.
Here, E is the electric field, R is the magnetic field and Pt and s are the permeability and
permittivity, respectively, of the medium in which the light wave is propagating. The
associated boundary conditions are given as
(i)
E11 = E211
(iii)
sjEj I = 62E2_L
(5.2)
(ii)
HIII=H211
(iv)
pHij_=p2H2J_,
where the subscripts 1 and 2 denote medium 1 and medium 2, respectively, and 11and
I represent the components that are parallel to the boundary and those perpendicular to
the boundary, respectively. Given Eq. (5.1) and Eq. (5.2), it can be deduced that the
incident, reflected and transmitted wave vectors (k , k r , and k t, respectively) all lie in
the same plane (the plane of incidence). Figure 5.1 (a) depicts the situation in which the
light is s-polarized meaning that the electric field is perpendicular to the plane of
incidence. Additionally, Fig. 5.1 (b) depicts p-polarization, or polarization in the plane of
incidence. By splitting the incident beam into s and p components, it is possible to
deduce the behavior of an arbitrarily polarized wave. For any incident polarization, it can
be shown that the angle of incidence is equal to the angle of reflection, or O; = 0,
Moreover, Snell's law, nI sin(
1 )=
n 2 sin(0 2 ), follows directly from Eqs. (5.1) and
II
(5.2), where n, 2 , the index of refraction of the material, is given by the ratio of the
velocity of light in free space, c, to the velocity of light in the given medium, V 1 2 , or
n1
2
-
g1,1,2 61, 2
P 00OF0
_C
-
--
vCO
1,2
If A is the complex amplitude of the electric vector of the incident field, then
resolving the incident electric field into components that are parallel to (p-polarized) and
perpendicular to (s-polarized) the plane of incidence gives the following:
(a)
Eix = -All cos(Oj)ei(ki-r-wt)
(b)
E iy = A Ie i(k -r-ot)
(c)
Ei z = All sin(Oi )eitki-r-oet)
(5.3)
where o is the angular frequency of the light. Referring to Fig. 5.1, the incident wave
vector, ki, is given as
ki =k 1[sin(O )i + cos(0;)Z],
(5.4)
where k i= o / v1 , and i, y, i are unit vectors along the three axes. Substituting Eqs.
(5.3) into Eq. [5.1 (i)] yields the components of the incident magnetic field:
(a)
Hix = -A_
(b)
Hiy= -Al
I cos(j)eikr-ot)
A Ivi
1 ei(ki-r-ot)
AIV
1
S1 1
Similarly, if T and R are the respective complex amplitudes of the transmitted and
reflected fields, then we have
(5.5)
12
(a)
Et x = -TI cos(Ot)ei(kt -r-wt)
(d) Ht x
cos(6tei(ktr-t)
= -T
11 2 V2
(b)
Ety = Tei(Ci(r-r-ot)
(e)
(c)
Et z = TI, sin(Otle i(kt-r-cot)
(f) Ht z = TL
t=
t
fie+COS(
[sm
in(te
1
Hty = -T,
ei(kt-r-(ot)
(5.6)
PI 2V2
I sin(Otle i(k t -r-wt)
9 2V2
with
(5.7)
for the transmitted field, and
(a)
Er x = -RI cos(Orei(krr-t)
(d)
Hrx
cos(Orei(kr-r-ot)
= -
9 Ivi
(b)
Ery = Rei(kr-r-ot)
(e)
1 ei(kr-r-ct)
Hry = -R
(5.8)
9 Ivi
(c)
Erz = R11 sin(Orei(kr r-wt)
(f) Hrz
=
I
in(Orei(kr-r-wt)
9 Ivi
with
k r = kl[sin(9 )X^-cos(65 )Z],
(5.9)
for the reflected field, where we have used the result that Or = Oi. From the boundary
conditions, Eqs. (5.2), the relationships between the incident, reflected, and refracted
waves can be derived. These relationships are known as the Fresnel formulae:
TI
- 2sin(Ot )cos(Q1)
sin(O; +Ot )cos(Oi -0t)
(5.10)
2sin(Ot )cos(
sin(Oi +Ot )
A
13
(a)
R
1
=tan(O; -0t) Al
tan(Oi +Ot )
(5.11)
(b)
R
=
sin(
sin(Oi +Ot )
A
.0t)A
6. Evanescent Fields
Consider the case in which light, incident from an optically dense medium,
impinges on a less optically dense medium, i.e. for n 2 < n I in Fig. 5.1. From Snell's
law, if the incident angle exceeds the critical value of 0 crit = sin~ 1 (n 1 /n 2 ), then the
transmitted angle becomes imaginary. With sin(Ot =(n 1/ n 2 ) sin(0i), we have
cos(t)=i i(n I /n 2 ) 2 sin 2 (0i)-1 .
(6.1)
Using Eqs. [5.6 (a-c)], (5.7), and (6.1) the transmitted field, Et, is given as
(n 1I/n 2 )2 sin 2(85)-1Jz
= ik2sin(t~xk2l
T
E t = Teik2sin(ot)xek2
= Teiki sin(Oi)x e-z/ 2d
(6.2)
,
where the time dependence e-ict has been omitted and the decay constant, d, is given as
2d =
ko 2
4ir
[n 1
2
sin (0i) -n 2 2 F 1/2
.
(6.3)
Thus, the field is exponentially decreasing in z while propagating with wavenumber kI in
x, thereby forming an evanescent surface wave. Inserting typical parameters in Eq. (6.3),
one finds that the energy of the evanescent field is confined to within 100 - 200 nm of the
surface of the prism. For this reason, TIR microscopy is predominantly used in the study
14
of surface phenomena. To find out what happens to the reflected wave in this case, Eq.
(6.1) must be substituted into the Fresnel equations (5.11):
(a)
R11
n2 cos(O5)-i sin2(Oi)-n2
2 (ei)-n
n 2 cos(0 1)+i
11
(6.4)
2(0
(b)
where n=n
R
2
cos(O)-i sin (O1)-n
cos(Oi)+i sin 2 (0)-n
/nI.
2
A
FromEqs.(6.4)itisseenthatR l=A1
and jR I=1A
fI,from
which we conclude that all of the incident light is reflected. From Eqs. (6.4) we also have
that
(a)
All
e
(b)
A1
= e''
,
(6.5)
where
(a)
51= -2 tan
1 sin2(01)-n2
n2O) n2]
(6.6)
(b)
8- = -2 tan[
Ssin2(
)-n~
.s(O)
Thus, at the interface the reflection suffers a phase shift that depends on the ratio of
refractive indices, n = n 2 / n .
At first, the total reflection conclusion seems to be contradictory since there must
be an energy transfer in order to establish an evanescent field in the second, less optically
dense medium. The answer to this apparent paradox comes with the realization that the
above analysis is based on the assumption that the boundary surface and wave fronts are
15
of infinite extent. In any actual experiment, the incident beam will be bounded in both
space and in time. Moreover, at the beginning of the reflection process, there will be
transient effects in which some of the incident wave's energy does indeed penetrate into
the second medium. Therefore, it is more accurate to say that in the steady state no net
energy crosses the boundary. However, there are ways of extracting a steady flow of
energy from the incident beam through the evanescent field. For example, in the case of
frustrated total internal reflection (FTIR) the evanescent region is sandwiched between
two high index regions (Fig. 6.1).
Evanescent region n2 <n
Fig. 6.1 shows FTIR.
In FTIR, the photons incident on the first surface tunnel through the low index
evanescent region and appear in the high index region on top. Another instance in which
energy can be forced to flow across the boundary is in the case where fluorescent probes
are inserted into the evanescent medium. This scenario is typically employed in TIR
microscopy, where an evanescent field is used to excite a fluorescently labeled specimen.
7. Standing Evanescent Waves
In order to extend the resolution of standard TIR microscopy, a means is sought
which will impart a high frequency intensity modulation to the evanescent field. One
16
way of accomplishing this task is by creating a standing evanescent wave at the interface
between the sample and the prism (Fig. 7.1). Provided that the refractive index of the
sample does not change appreciably with distance along the sample plane (as is the case
for most biological specimens) then the phase shifts predicted by Eqs. (6.6) will remain
constant over the front of the reflected beam. Therefore, it will be possible to generate a
uniform standing wave by retroreflecting the totally reflected beam back onto itself. To
have the freedom of shifting the fringes of the standing wave, the retroreflection is
accomplished through a piezo driven mirror as shown in Fig. 7.1.
CCD
Objective
Standing wave
4 ~Sample
(n)2)
iror
Incoming
beam
Piezo
Fig. 7.1 is a schematic of the single direction SWTIRF setup.
In considering the effect of polarization on the evanescent field, Eqs. (5.10) and
(6.1) are substituted into Eq. (6.2):
17
(a)
2
Aei[kl sin(Oj)x-4 1+/2] e-z/ d
2cos(Oi)[sin 2(0i)-n2 1 /2
Et x =
[n4 cos 2(
= 2cos(Oi)
S- 2(1n)1/2
(b)
Et y
(c)
Et z ={
)sin2
(O)-n 2 1/2
2
AI ei[kl sin(O0)x-4 1 ] e-z/ d
2 cos(Oi ) sin(Oi)
[n 4
cos 2 (0, )+sin 2 (0,)-n2]1/ 2}
(7.3)
2
d
A 1ei[k, sin(0)x-4j] e-z/
where
ta-l [sin2(i)n2
(a)
1/2
n 2 Cos(6;)
and
(b)
(7.4)
[sin2(0i)-n2
4 _ =tan-1
= tanO
11/2
If the incident light is p-polarized, i.e. if A I = 0, then the evanescent electric field
cartwheels along the surface with a spatial period of
.X
n, sin(0i)
(where X is the free
space wavelength of the excitation beam) as shown in Fig. 7.2.
X
I _-
n1 si(e 1)
r
E field
n2
ni
e
oi
Fig. 7.2 illustrates the cartwheeling electric field produced by a p-polarized incident beam.
18
This cartwheeling effect precludes the possibility of forming a standing wave simply
from the retroreflection of the totally reflected beam. However, if the incident beam is spolarized (A I = 0 ) then the incident and retroreflected beams do combine to give a
standing wave. Using Eq. [7.3 (b)] the standing wave is given as
=
where
A1
[
2cos(0i)
_ I-n2 )1/2
(1-n) )
I
i[k sin(Oi)+4/2] e-z/ 2 d
4 is some arbitrary phase.
(7.5)
The time averaged square of the real part of Eq. (7.5)
yields the intensity modulation, I(x,z), produced by the standing wave:
2
I(x2y)=(A I ) 4cos (0)-l
[1+cos(Kx+)]ez
(1- n )
,
(7.6)
in which K is given by
K =4cn I sin(O; )/X.
(7.7)
In the further discussions on resolution enhancement, only the modulation factor
1+ cos(Kx + $) will be of concern, hence the other factors will be dropped in the
expression for the intensity modulation.
8. Resolution Enhancement in one Direction
Suppose that the sample shown in Fig. 7.1 has some fluorophore distribution
given by O(r). The goal here is to try to represent O(r) as accurately as possible from
images taken with the setup shown. The intensity on the sample plane, ISP (r), is given
by the product of the modulated evanescent intensity (excluding the amplitude and the z
dependent factors) with the object function O(r):
19
(8.1)
I , (r) = O(r)[1 + cos(Kx + j)].
Assuming a magnification of unity throughout, the image, 10 (r), of the sample plane
intensity pattern of Eq. (8.1) is given as
10 (r) = 0(r)[1 + cos(Kx + + ')] 0 P(r),
(8.2)
where 0 denotes the convolution operation, P(r), is the conventional PSF given by Eq.
(4.1), and 0' is a variable phase allowed by the movement of the piezo driven mirror
(Fig. 7.1).
Resolution enhancement requires three standing wave images, each taken at a
different phase from the other two. Choosing the phases of 0' =0, )'= n /2, and
n' ,
the three images may be expressed, with the help of Eq. (8.2), as
I e i(K++jx/ 2 ) O(r)
Ij (r) = O(r) 0 P(r) +
0 P(r)
L2
+[
2
(8.3)
e -(Kx +4
/ 2) O(r)
P(r);
j = 0, 1, 2;
where the cosine has been expressed in terms of complex exponentials and the
convolution operation has been expanded. Equations (8.3) can be solved for
O(r)0P(r), [(1/ 2)ei(Kx++ji/
2)
O(r)]OP(r), and [(1/ 2)e i(x+++jn/ 2) O(r)])P(r)
in terms of the measured images 10 (r), II(r), and 12 (r):
(a)
0(r)OP(r)= (1/2)[IO(r)+12(r)]
(b) [(1/2)e1(Kx+*)O(r)] 0P(r)= (1/4)[(1-i)Io(r)+2i11(r)-(1+i)12 (r)]
(c)
[(1/2)e~(Kx+)O(r)]OP(r)=(1/4)[(1+i)Io(r)-2ill(r)+(-1+i)1 2 (r)].
By writing out the convolution operation in Eqs. [8.4 (b), (c)] we have
(8.4)
20
[(1/ 2)e "x
+ )O(r)]0 P(r) = (1 / 2)J e ti(Kx'+4)O(r')P(r - r')d2 r'
=
(1/ 2)e±i(Kx+) J0(r')e±iK(x x)P(r - r')d2 r'
=
(1/ 2)e ±iK
0(r')e iK(x-x)P(r - r')d2 r'
= (1/2)e{0x+ ){O(r)0[e:
(8.5)
P(r)]}.
Using this result, Eq. [8.4 (b)] becomes
(r) [eiKxP(r)]} = (1/4)[(1 -i)IO(r) + 2ill(r)- (1+i)12 (r)].
(1/2)e(Kx0{0
(8.6)
Similarly, the complex conjugate, Eq. [8.4 (c)], becomes
(1/2)e~(
'KP(r)]}
+**){O(r)®[e
=
(1/4)[(1+i)10 (r) - 211 (r) + (-1+i)12 (r)].
(8.7)
An enhanced image, I(r), is obtained by first multiplying Eq. (8.6) and (8.7) by
e
i(x+a) , respectively, and then summing the result with Eq. [8.4 (a)]:
I(r)= O(r) 0 {P(r)[1 + cos(Kx - 4 + a)] }.
(8.8)
Likewise, I(r) can be expressed as the sum of the three images of Eqs. (8.3):
1(r)
= I(r)(l/ 2)[1+ Jicos(Kx +a+7 / 4)]+1 1 (r)sin(Kx +a)
+ 12 (r)(1 / 2)[1 - v cos(Kx + a
-
(8.9)
n / 4)],
where each image receives its own sinusoidal weighting factor with arbitrary phase a.
Equation (8.9) reveals how to construct the enhanced image from three standing wave
images, whereas Eq. (8.8) shows that the enhancement is achieved by transferring the
modulation from the object function to the conventional PSF.
A few comments are in order concerning the phases
4
and a. Note that
4
is an
arbitrary phase that specifies the position of the standing wave relative to the object,
O(r), whereas we are completely free to specify a in the weighting factors. From Eq.
21
(8.8), it can be seen that the conventional PSF is symmetrically attenuated for a = 0 .
Thus, a condition is sought which will determine this value of x. This phase matching
condition will be satisfied by extremizing the total intensity, I tot(a):
'tot (a) = fI(r)d 2 r = ff0(r')P(r -r'){l+ cos[K(x - x')-
+a]}d 2 r' d 2 r,
(8.10)
where the integral of I(r) is taken over the entire image plane, E. Upon interchanging the
integrations in Eq. (8.10), the maximization condition is
tot=-0(r')
P(r -r')sin[K(x - x')-
+ ]d2r
d2r'= 0.
(8.11)
Since P(r - r') is an even function, condition (8.11) is fulfilled only when a =4.
Therefore, the value of a which maximizes Eq. (8.10) is substituted into Eq. (8.9)
thereby obtaining the enhanced image.
With the phase determined, the PSF of the enhanced image, Pswx (r), is given by
Eq. (8.8) upon substitution a Dirac delta function in as the object, O(r):
Pswx (r) = [1+ cos(Kx)]P(r),
(8.12)
Since the diffraction limited PSF, P(r), is modulated by the standing wave excitation,
the width of the central peak is narrowed along the x-axis while its conventional width is
retained along the y-axis (Fig. 8.1). The degree to which this narrowing takes place
depends on the spatial frequency K, which, by Eq. (7.7), depends linearly on the
refractive index of the prism used. Thus, materials with a high refractive index will
provide the greatest narrowing of the PSF. Quantitatively, the FWHM of the central peak
along the x-axis is given by
22
(8.13)
.
FWHM = 7-=
K 4nnsin(9i)
For an angle of incidence of 750, the FWHM is approximately Xe /6 and Xe /9 for
quartz (nI=1.46 at 532 nm) and lithium niobate (LiNbO 3 , nI= ordinary index
=
2.32 at
532 nm) prisms, respectively. Note that Xe denotes the emission wavelength whereas X
represents the excitation wavelength in free space. As Fig. 8.1 shows, the sidebands in
the standing wave PSF will increase as nI increases. Fortunately, it may be possible to
remove the sidebands by use of simple deconvolution algorithms. Additionally, SWTIRF
imaging may be combined with two photon or pump-probe imaging techniques for
further resolution enhancement as well as sideband suppression.
The OTF of the single direction SWTIRF application, Pswx (k), is given as the
normalized Fourier transform of Eq. (8.12):
Pswx (k) = [2+P(-Ke ) +P(Kex )]~1 [2P(k)+P(k -Kex)+P(k+Kex)],
(8.14)
where P(k) is defined in Eq. (4.5) and K is defined in Eq. (7.7). Thus, Pswx (k) is a sum
of three conventional OTF functions, one at the center, one shifted by K along k X, and
the third shifted by -K along k X , where the center function has twice the weight of the
outer two (Fig. 8.2). As was previously pointed out, the sidebands in the point spread
function increase as K increases. This situation manifests itself in frequency space as the
amount of overlap of the three P(k) functions. If all three overlap perfectly (K = 0), there
is no ringing, but there is no resolution enhancement. As K increases, the cutoff
frequency along the k x axis increases at the cost of decreasing the overlap of the P(k)
functions. If K is large enough to create frequency gaps along k x (K > 8nNA / A e), the
23
outer functions will separate from the center one thereby causing severe ringing in the
PSF. For cases in which there are no frequency gaps, the cutoff frequency of the single
direction SWTIRF application is 4nNA / X e + K along the k x axis but retains the
conventional value of 4nNA / X e along the k Y axis.
24
a)
1
&--a Conventional
A-A Confocal
+--
0.81
i:n
0.6
N
0.41
z
0.2
0
0
50
100
ID SWTIRF Quartz (n=1.46)
ID SWTIRF LiNbO 3 (n=2.32)
150
250
200
Distance Along x-axis (nm)
c)
b)
...........
A
..........
CA
0.5.
0
0
020
-200
-200
0
-20
-200
-0
-200
-200
Fig. 8.1. For the representative parameters of Oi = 75*, NA= 1.3, X = 532 nm., and ke = 560nm, plots
are generated that show a) the PSF's of conventional, confocal, and SWTIRF microscopies, the PSF's of a
single direction (ID) SWTIRF scan for b) quartz and c) lithium niobate prisms.
25
Conventional
ID SWTIRF Quartz
ID SWTIRF LiNbO 3
13B-1
0.81
"o
0.6
0.41
0.2
ft -- M-M,
om--
- -
a an
-. 9
-. A-
-0.08 -0.06 -0.04 -0.02
0
0.02
0.04
0.06
Spatial Frequency Along kg-axis (cycles/nm)
b)
c)
1.5
......
I
L&
0.5-
0.5
0
0
-0.05
-
0.05
0
.05
-0.05
Fig. 8.2 Frequency space counterparts (OTF's) of Figs. 8.1 a), b), and c), respectively.
-005
26
9. Two Dimensional Resolution Enhancement
Resolution enhancement in two dimensions will require multiple scans performed
along different directions across the sample plane. To this end, Eq. (8.12) is generalized
to an arbitrary direction along the sample plane:
Pswt (r) = {l + cos[Kr cos(O - t)]}P(r),
(9.1)
where r = (r, 0) and 4 is the angle that the standing wave makes with the x-axis.
At this point, we can identify two main criteria that the final image must satisfy.
For consistency with incoherent imaging, we require the enhanced image to be linear in
intensity. Therefore, the final image must be formed from a sum of images. As there is
no justification for precluding any image from this sum, we must allow for both the
single direction SWTIRF images of Eq. (8.9) as well as the diffraction limited image of
Eq. [8.4 (a)]. Secondly, the PSF and the OTF must be symmetric since we would like to
have a uniform resolution in all directions along the sample plane. Hence, N>2 scans are
performed, one at every n / N radians in the sample plane. With the help of Eq. (9.1), the
general expression for a PSF satisfying these criteria is given by
Psw (r, N)=
i~~'
1 (1 N-i1
-+
-
I 1 + Cos Kr cos0
-7
+
P(r)
(9.2)
-
1
1+p
S(r, N)P(r),
where S(r, N) is the sum in parenthesis, 1/(1+ ) is a normalization factor and P is a
constant that determines the weight of the diffraction limited image in the sum.
The constant 0 is chosen to maximize resolution, which, by the Rayleigh
criterion, is synonymous with minimizing the width of the symmetric modulation
27
S(r, N). It is therefore required that S(r, N) and its first derivative with respect to r
vanish for some distance rmin < Xe /2:
S(rmin ,0, N) =0 ,
(9.3)
(9.4)
=0.
-S(r,0, N)
]r=rmi
27t
With the help of the formula J0 (w) = (l/2i) Jcos[w cos(0+O)]dO, Eqs. (9.3) and (9.4)
0
are first integrated from 0 to 2n in 0 and then divided through by n giving
Jo(Krmin )+l+2P =0,
(9.5)
-J(Kr)
(9.6)
dr
=0,
r=rm
where J0 is the zero order Bessel function of the first kind. Equation (9.6) is used to find
the minimum value of J0 , J 0 (Krmin )
obtain the constant P = -0.3.
Psw (r, N) = 1.43 ±
Nm
-0.4, which is then substituted into Eq. (9.5) to
Upon inserting this value into Eq. (9.2) we have
1 + cos Kr cos
-N
-
J
-
0.3 P(r),
(9.7)
which is the formula showing how to construct the enhanced two-dimensional image.
Figure 8.3 shows line plots and mesh plots of the enhanced two-dimensional PSF
for both quarts and LiNbO 3 prisms. In this case, quartz produces a FWHM of
approximately X e /5 whereas the LiNbO 3 FWHM is k e / 8. Although adding scans
together in this way produces a symmetric PSF with reduced sidebands, its width is
28
somewhat broader than its single direction counterpart, Pswx (r). The frequency space
representation of Eq. (9.7), Psw (k, N) (Fig. 8.4), is given as
(i
2N-1
+ K[cos(mn / N)ex + sin(mir / N)ey ]}+1.7P(k)
Psw (k, N) =Y r2NP{k
(
where y is a normalization factor.
M=)
(9.8)
29
a) 1
Conventional
Confocal
+-ID SWTIRF Quartz (n=1.z 46)
e--o 2D SWTIRF Quartz
--- ID SWTIRF LiNbO 3 (n=2. 32)
2D SWTIRF LiNbO 3
6-E
A-,
0.81
-I,
0.61
0.41
0.2
0
150
200
Distance Along x-axis (nm)
0
50
100
250
c)
1.5-Y.U
1
'0
U
05
0
Z
0
-200
0
-200
Fig. 8.3. For the representative parameters of N = 4,
020
ei
-200
-200
0
= 750, NA = 1.3, X = 532 nm, and Xe = 560nm ,
plots are generated that show a) the PSF's of conventional, confocal, and SWTIRF microscopies, the PSF's
of the symmetric two-dimensional (2D) SWTIRF implementation for b) quartz and c) lithium niobate
prisms.
30
a)
1
Conventional
8--B
e-o 2D SWTIRF Quartz
2D SWTIRF LiNbO 3
0.8
~0
0.6-
0.4-
0
0.2 -
I
0
-
-0.c 8 -0.06 -0.04 -0.02
0
0.02
0.04
0.06
Spatial Frequency Along k.-axis (cycles/nm)
c)
b)
1
0.5 -
.0)
00
0.05
0.05
-0.05
-0.05
0
0.06
0.05
0-0. 05
Fig. 8.4. Frequency space counterparts (OTF's) of Figs. 8.3 a), b), and c), respectively.
-0.050
31
10. Conclusion
In assessing how SWTIRF is to be used, its range of applicability must be
recognized. Due to the shallow depth of field, the technique will primarily be limited to
surface studies, such as in single molecules or cell membranes. However, as the critical
angle is approached, the evanescent field penetrates deeper into the sample, thereby
allowing for the possibility of extending the technique into three dimensions. Moreover,
it may be possible to develop a version of the technique that will achieve high resolution
in coherent imaging as well. This would be a great advantage in cases where fluorescent
dyes would be impractical or impossible to use, such as in semiconductors. In these
cases the coherent laser light could be used to illuminate the sample, thus making
labeling unnecessary. For specimens in which the refractive index changes appreciably,
it may be necessary to employ two separate counter propagating beams instead of the
retroreflection geometry proposed in Fig. 5.1. Because the evanescent wave itself acts as
the probing mechanism, no mechanical contact is required with the specimen thereby
allowing high fidelity imaging of soft biological samples. Finally, video rate image
acquisition times may be possible as not more than three images are required per standing
wave direction.
32
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