Stefan Bernet et al- Quantitative imaging of complex samples by spiral phase contrast microscopy

8/3/2019 Stefan Bernet et al- Quantitative imaging of complex samples by spiral phase contrast microscopy

http://slidepdf.com/reader/full/stefan-bernet-et-al-quantitative-imaging-of-complex-samples-by-spiral-phase 1/14

Quantitative imaging of complex

samples by spiral phase contrast

microscopy

Stefan Bernet, Alexander Jesacher, Severin Furhapter,

Christian Maurer, and Monika Ritsch-Marte

Division for Biomedical Physics, Innsbruck Medical University, M¨ ullerstr. 44

A-6020 Innsbruck, Austria

[email protected]

Abstract: Recently a spatial spiral phase filter in a Fourier plane of

a microscopic imaging setup has been demonstrated to produce edge

enhancement and relief-like shadow formation of amplitude and phase

samples. Here we demonstrate that a sequence of at least 3 spatially filtered

images, which are recorded with different rotational orientations of the

spiral phase plate, can be used to obtain a quantitative reconstruction of

both, amplitude and phase information of a complex microscopic sample,

i.e. an object consisting of mixed absorptive and refractive components. The

method is demonstrated using a calibrated phase sample, and an epithelial

cheek cell.

© 2006 Optical Society of America

OCIS codes: (070.6110) Spatial filtering, (090.1970) Diffractive optics, (100.5090) Phase-only

filters.

References and links

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(C) 2006 OSA 1 May 2006 / Vol. 14, No. 9 / OPTICS EXPRESS 3792

#68805 - $15.00 USD Received 8 March 2006; revised 25 April 2006; accepted 25 April 2006

mailto:[email protected]

mailto:[email protected]



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1. Introduction

The use of a spiral phase plate as a spatial filter in a Fourier plane of an imaging setup has been

proposed [1, 2, 3] and demonstrated [4, 5, 6] as an isotropic edge detection method providing

strong contrast enhancement of microscopic amplitude and phase samples. A similar imaging

procedure applied to samples with a larger optical thickness (on the order of a few wavelength)was shown to result in a novel kind of spiral shaped interferograms, which have the unique

property that a complete sample phase topography can be unambiguously reconstructed from

only one single interferogram [7, 8].

Recently [9], the experimental significance of the central singularity of the spiral phase

plate has been pointed out. It was shown that the effect of a transmissive central pixel in a spiral

phase plate leads to a violation of the otherwise isotropic edge enhancement, resulting in useful

relief-like shadow images of the sample topography. The shadow orientations can be rotated

continuously by shifting the phase of this central pixel with respect to the remaining spiral

phase plate. For optically thin samples it was shown [9] that the shadow effect can be used to

obtain a high contrast image of a phase sample by numerical post-processing of a sequence of

at least three spiral-filtered images recorded with different shadow orientations.

Here we demonstrate that this method can be even used for the imaging of a complex sample,

i.e. a sample consisting of both, amplitude and refractive index modulations. In principle, themethod provides a quantitative relative measurement of the amplitude transmission of a sample

(normalized to its maximum transmission), and even an absolute measurement of the phase

topography without the need of a previous calibration or comparison with a reference sam-

ple. Such a quantitative measurement is hard to achieve with other microscopic methods like

standard phase-contrast or differential interference contrast (Nomarski-) methods [10], which

deliver just qualitative data. The spiral phase method fora quantitative measurement of both, ab-

solute optical thickness and transmission of complex samples has various practical applications,

like e.g., lithography mask inspection in semiconductor industry, or quantitative measurements

of biological objects.

2. Basics of spiral phase filtering

The significance of the spiral phase transform, which is also known as the Riesz transform, vor-tex transform, or two-dimensional isotropic Hilbert-transform has been pointed out in different

publications. As a purely numerical tool, the method is used for example in fringe analysis of

interferograms [11, 12], or, very recently, as a tool for the analysis of speckle patterns [13].

There are also applications, where the transformation is performed with optical methods, by

introducing a spiral phase filter into a Fourier plane of an imaging setup. These experiments





have developed rapidly with the availability of high resolution spatial light modulators (SLMs)

which can act as two-dimensional arrays of individually addressable pixels, acting as program-

mable phase shifters. There, the spiral phase transform can be applied with an on-axis element -

a so-called spiral phase plate [14], or by diffraction from a specially designed off-axis hologram

[15].

A sketch of a so-called 4f-system as one possible setup for implementing a spatial Fourier

filter is shown in Fig. 1.

Fig. 1. Basic principle of a spiral phase plate spatial Fourier filter. A transmissive input

image is illuminated by a plane wave. The illumination beam is scattered into the directions

of amplitude or phase gradients within the input image (two directions indicated by red

and blue rays). The largest part of the illumination light passes without being scattered

(green rays). A first lens (L1) located at a focal distance after the input image creates

a Fourier transform of the image in its right focal plane, where the spiral phase plate is

located. The design of the spiral phase plate is shown below (grey-values correspond tophase values in a range between 0 and 2π ). The undiffracted part of the illumination beam

(green) corresponds to the zero-order Fourier component of the image field and focuses in

the center of the phase plate. The diffracted parts of the input field (red and blue) focus at

different positions at the spiral phase plate (indicated below), which are determined by their

propagation directions in front of L1, and thus by the gradient directions within the input

image. The spiral phase plate adds a phase offset to each off-axis beam. A second lens L2

placed at a focal distance behind the spiral phase plate performs a reverse Fourier transform

and creates the output image in its right focal plane. There, the zero-order component of the

incident light field (green) is again a plane wave, superposing coherently with the remaining

light field. This remaining light field now carries a spatially dependent phase-offset with

respect to the input image, which corresponds to the geometrical angle into which the

(amplitude- or phase) gradient of the input image is directed.

Typically, the spiral phase transformation is defined as a multiplication of the Fourier trans-form of an input image with a vortex phase profile, i.e. with exp(iφ ), where φ is the polar angle

in a plane transverse to the light propagation direction measured from the center of the spiral

phase plate. This definition excludes information about one point, i.e. the center of the spiral

phase element, where a phase singularity exists. However, if a real spiral phase element is used





as a Fourier spatial filter, this central position becomes of utmost importance, since it coincides

with the zero-order Fourier component of the input image which typically contains the major

amount of the total light intensity.

In practice, real spiral phase elements have a central point which is no singularity, but has a

well-defined amplitude- and phase transmission property. For example, in our case the phase

shifting element is a pixelated spatial light modulator with individually addressable phase

values for its 1920 x 1200 pixels. Only in the case where a central region (or pixel) with a size

on the order of the zero-order Fourier component of the input image has no transmission, the

resulting spiral phase transform is really isotropic, resulting in an isotropically edge enhanced

output image.

However, if the central region acts as a transmissive phase shifter, then the rotational symme-

try of the spiral phase filter is broken. This can be seen by the fact that an absolute orientation

of the plate can be (for example) defined by the radial direction where the phase plate values

correspond to the phase value of the central pixel. If such a non-isotropic spiral phase filter with

transmissive center is used as a spatial Fourier filter, then the output image shows a relief-like

shadow profile, similar to a topographic surface which is illuminated from an oblique direction.

The reason for this behavior is that each amplitude or phase gradient within the original input

image diffracts an incoming illumination beam into a well-defined direction, corresponding to

the gradient direction (see Fig. 1). In the Fourier plane of the imaging setup, each of these

well-defined scattered beams is focused at a certain position, at a polar angle corresponding to

the direction of the gradient. The effect of the spiral phase plate is then, to add a certain phase

value to this beam, which also corresponds to the polar angle of the beam position in the Fourier

plane, i.e. to the gradient direction in the original image. Afterwards, the light field is Fourier

back-transformed by an additional lens into an output image. Compared to the input image, the

output image has therefore an additional phase offset at the positions where the input image has

an amplitude or phase gradient. These additional phase offsets equal the geometric direction

angles into which the gradients within the input image are pointing.

If such an output image is interferometrically superposed with a plane wave, the ”interfero-

gram” will differ from the input image at all positions where the sample has an amplitude or

phase gradient. There will always be one gradient direction showing maximum constructive

interference, i.e. edge amplification, whereas the opposite gradient direction shows maximal

destructive interference, i.e. an edge ”shadow”. Image regions where the gradients have otherdirections show a smooth transition between constructive and destructive interference. This be-

havior creates useful pseudo-relief shadow images, where elevations and depressions within a

phase topography can be distinguished at a glance.

In the case of an non-isotropic spiral phase plate with transmissive center the plane wave re-

quired for the interferometric superposition is automatically delivered by the zero-order Fourier

component of the input image field, focussing in the center of the spiral phase plate, since such

a focal point is automatically transformed into a plane wave by the reverse Fourier transform

performed by the following lens. Thus, a spiral phase plate with a transmissive center acts effec-

tively as a self-referenced (or common-path) interferometer [16, 17, 18], using the unmodulated

zero-order component of the original input light field as a reference wave for interferometric

superposition with the remaining, modulated image field. Therefore, changing the phase of the

central pixel of the spiral phase plate results in a corresponding rotation of the apparent shadow

direction. The same effect can be observed, if the phase of the central pixel is kept constant, butthe whole spiral phase plate is rotated by a certain angle around its center.

In the following, it will be shown how this rotating shadow effect can be used to reconstruct

the exact phase and amplitude transmission of a complex sample. Basically, the possibility to

distinguish amplitude from phase modulations results from a π /2-phase offset between the scat-





tering phases of amplitude and phase structures, resulting in a corresponding rotation angle of

π /2 between the respective shadow orientations [9]. The feature that an image can be uniquely

reconstructed is based on the fact that there is no information loss in non-isotropically spiral

phase filtered light fields (using a transmissive center of the spiral phase element) as compared

to the original light fields, due to the reversibility of the spiral phase transform. For exam-

ple, a second spiral phase transform with a complimentary spiral phase element (consisting of

complex conjugate phase pixels) can reverse the whole transform without any loss in phase or

amplitude information. Note that this would not be the case, for example, if a spiral phase filter

of the form ρ exp(iφ ) (where ρ is the radial polar coordinate) was used. Although such a filter

would produce the true two-dimensional gradient of a sample [19], it simultaneously erases

the information about the zero-order Fourier component of a filtered input image. Therefore,

the image information could only be restored up to this zero-order information, consisting of

an unclear plane wave offset in the output image. This missing information would not just re-

sult in an insignificant intensity offset, but in a strong corruption of the image, since the plane

wave offset coherently superposes with the remaining image field, leading to an amplification

or suppression of different components.

3. Numerical post-processing of a series of rotated shadow images

The relief-like shadow images obtained from the non-isotropic spiral phase filter give a niceimpression of the sample topography. In contrast to the Nomarski or differential interference

contrast method [10] - which creates similar shadow images - the spiral phase method works

also for birefringent samples. For many applications quantitative data about the absolute phase

and transmission topography of a sample are desired. Here we show that such quantitative data

can be obtained by post processing a series of at least three shadow images (even better results

of real samples are obtained by a higher number of images), recorded at evenly distributed

shadow rotation angles in an interval between 0 and 2π .The intensity distribution of a series of three images I out 1,2,3 = | E out 1,2,3 |

2 in the output plane

of a spiral phase filtering setup can be written as:

I out 1,2,3 = |( E in− E in0)⊗Φexp(iα 1,2,3) + E in0

|2 (1)

There, E in = | E in( x, y)|exp[iθ in( x, y)] is the complex amplitude of the input light field, E in0= | E in0

|exp(iθ in0) is the constant zero-order Fourier component (including the complex

phase) of the input light field, and α 1,2,3 are three constant rotation angles of the spiral phase

plate which are adjusted during recording of the three images, and which are evenly distrib-

uted in the interval between 0 and 2π , e.g. α 1,2,3 = 0,2π /3,4π /3. The symbol ⊗Φ denotes

a convolution process with the Fourier transform of the spiral phase plate (i.e. Φ(ρ ,φ ) =F −1{exp[iφ ( x, y)]} = i exp[iφ ( x, y)]/ρ 2, where F −1 symbolizes the reverse Fourier transform

[11]).

Thus the three equations (1) mean that the input image field without its zero-order Fourier

component ( E in− E in0) is convoluted with the reverse Fourier transform of the spiral phase plate

(this process corresponds to the actually performed multiplication of the Fourier transform of

the image field with the spiral phase function [20]), which is rotated during the three expo-

sures to three rotational angles α 1,2,3. Then the unmodulated zero-order Fourier component

E in0 which has passed through the center of the spiral phase plate is added as a constant planewave. The squared absolute value of these three ”interferograms” corresponds to the intensity

images which are actually recorded.

The three equations (1) can be rewritten as:





I out 1,2,3 = |( E in − E in0)⊗Φ|2 + | E in0

|2

+[( E in− E in0)⊗Φ] E ∗in0

exp(iα 1,2,3)

+[( E in− E in0)⊗Φ]∗ E in0

exp(−iα 1,2,3) (2)

Here, a∗ symbolizes the complex conjugate of a number a. In order to reconstruct the orig-

inal image information E in( x, y), a kind of complex average I C is formed by a numerical mul-

tiplication of the three real output images I out 1,2,3 with the three known complex phase factors

exp(−iα 1,2,3), and a subsequent averaging:

I C =1

3[ I out 1 exp(−iα 1) + I out 2 exp(−iα 2) + I out 3 exp(−iα 3)] (3)

Analyzing this operation, it is obvious that the multiplication with the complex phase factors

exp(−iα 1,2,3) supplies the first and the third lines of Eq. (2) with a complex phase angle of

exp(−iα 1,2,3), and exp(−2iα 1,2,3), respectively, but it cancels the phase term behind the second

line. The subsequent summation over the three complex images leads to a vanishing of all terms

with phase factors, since the three angles are evenly distributed within the interval between 0

and 2π . Thus the result is:

I C = [( E in − E in0)⊗Φ] E ∗in0

(4)

Since E in0is a (still unknown) constant, the convolution in Eq. (4) can be reversed by numer-

ically performing the deconvolution with the inverse convolution functionΦ −1, i.e.:

( E in− E in0) E ∗in0

= I C ⊗Φ−1 (5)

This deconvolution corresponds to a numerical spiral-back transformation, which can be un-

ambiguously performed due to the reversibility of the spiral phase transform. In practice, it is

done by a numerical Fourier transform of I C , then a subsequent multiplication with a spiral

phase function with the opposite helicity as compared to the experimentally used spiral phase

plate, i.e. with exp[−iφ ( x, y)], followed by a reverse Fourier transform. Note that for this nu-

merical back-transform it is not necessary to consider the phase value of the central point in thespiral phase kernel, since the zero-order Fourier component of I C is always zero.

Equation (5) suggests that the original image information E in( x, y) can be restored from the

”spiral-back-transformed” complex average I C ⊗Φ−1 by:

| E in( x, y)|exp[i(θ in( x, y)−θ in0)] = ( I C ⊗Φ

−1 + | E in0|2)/| E in0

| (6)

There, the complex image information E in( x, y) has been split into its absolute value and its

phase. Therefore, if the intensity | E in0|2 of the constant zero-order Fourier component of the

input image is known, it is possible to reconstruct the complete original image information E in

up to an insignificant phase offset θ in0, which corresponds to the spatially constant phase of the

zero-order Fourier component.

Thus, the final task is to calculate the intensity of the zero-order Fourier component of the

input image | E

in0|

2

from the three spiral transformed images. For this purpose, we first calculatethe ”normal” average I Av of the three recorded images, which is an image consisting of real,

positive values, i.e.:

I Av =1

3( I out 1 + I out 2 + I out 3 ) (7)





Again, all terms within Eq. (2) which contain a complex phase factor exp (±iα 1,2,3) will

vanish after the averaging, due to the fact that the three angles α 1,2,3 are evenly distributed

within the interval 0 and 2π . The result is:

I Av = |( E in− E in0)⊗Φ|2 + | E in0

|2 (8)

Comparing Eq. (8) with Eq. (4) one obtains

| E in0|4− I Av| E in0

|2 + | I C |2 = 0, (9)

from which one can calculate the desired value for | E in0|2 as:

| E in0|2 =

1

2 I Av±

1

2

I 2 Av−4| I C |2 (10)

Note, that using this equation, | E in0|2 can be calculated for each image pixel individually,

although it should be a constant. For the ideal case of numerically simulated samples, | E in0|2

in fact delivers the same value at each image pixel. In practice there can be some jitter due to

image noise around a mean value of | E in0|2, which is an indicator for the noise of the imaging

system and delivers a useful consistency check. In numerical tests and real experimentsit turned

out that in this case the best results are obtained by searching the most frequently occurringvalue (rather than the mean value) of | E in0|2 in a histogram, and to insert this value for further

processing of Eq. (6).

Interestingly, there are two possible solutions for the intensity | E in0|2 of the zero-order

Fourier component, which differ by the sign in front of the square root. In the two cases, the

constant intensity of the zero-order Fourier component at each image pixel exceeds or falls

below one half of the average image intensity. This means, that in the case of a positive sign,

most of the total intensity at an image pixel is due to the plane wave contribution of the zero-

order Fourier component of the input image, and the actual image information is contained in a

spatially dependent modulation of the plane ”carrier-wave” by the higher order Fourier compo-

nents. This always applies for pure amplitude samples, and for samples with a sufficiently small

phase modulation, which is typically the case when imaging thin phase objects in microscopy.

On the other hand, if the sample has a deep phase modulation (on the order of π or larger) with

a high spatial frequency then the solution with the negative sign is appropriate. This happens,for example, for strongly scattering samples like ground glass, where the zero-order Fourier

component is mainly suppressed. In practice, our thin microscopic samples investigated to date

have all been members of the ”low-scattering” group, where the positive sign in front of the

square root in equation (10) has to be used.

After inserting | E in0|2 from equation (10) into Eq. (6), the absolute phase topography of the

sample (up to an insignificant offset) is obtained by calculating the complex phase angle of the

right hand side of Eq. (6). Note that the sample phase profile is obtained in absolute phase units,

i.e. there is no undetermined scaling factor which would have to be determined by a previous

calibration. Furthermore, the transmission image of the sample object can be computed by cal-

culating the square of the absolute value of the right hand side of Eq. (6). The result corresponds

to a bright-field image of the object, which could be also recorded with a standard microscope.

However, the transmission image of the spiral phase filtering method has a strongly reduced

background noise as compared to a standard bright-field image, due to the coherent averagingof I C (see Eq. (3)) over a selectable number of shadow images. There, all image disturbances

which are not influenced by the phase shifting during the different exposures(like readout-noise

of the image sensor, stray light, or noise emerging from contaminated optics behind the Fourier

plane) are completely suppressed.





In practice, noise reduction and image quality can be even enhanced by a straightforward

generalization of the described method to the imaging of more than three shadow images. The

only condition for this generalization to multiple exposures is that the rotation angles of the

spiral phase plate are evenly spread over the interval between 0 and 2π .

4. Experimental results

Our actual experimental setup for demonstrating the features of the spiral phase transform is

sketched in Fig. 2, and explained in the figure caption.

Fig. 2. Sketch of the experimental setup: The sample is illuminated with a collimated white-

light beam. The transmitted light passes the objective (NA 0.95, 63x), then a first folding

mirror M1, and a set of two lenses L1 and L2, which project the Fourier transform of the

image at the upper part of a reflective SLM. There, a spiral phase creating hologram with

a typical fork-like dislocation in its center is displayed (as sketched in the upper part of

the SLM image). If the zero-order Fourier component of the incident light field coincides

with the central grating dislocation, the first order diffracted light field is the desired spiral

phase filtered image, however, with an undesired dispersion due to the bandwidth of the

illumination light (indicated as red/green/blue rays in the figure). In order to compensate for

the dispersion, the diffracted light field passes through a further Fourier-transforming lens

L3, which creates a real image in its focal plane where a mirror M2 is located. The mirror

is adjusted such that the back-reflected light passes again through the Fourier-transforming

lens L3 and focuses at another position on the SLM. There, a ”normal” grating with the

same spatial frequency as that of the spiral phase hologram is displayed (lower image at the

right side), from where another first-order diffraction process compensates the dispersion

induced by the first one. Finally, the diffracted light field is reflected by a further folding

mirror M3 to a camera objective lens L4, which projects the spatially filtered image at a

CCD chip.





It differs from the principle setup sketched in Fig. 1 in two main points: First, spiral filtering

is not performed by an on-axis transmissive spiral phase plate, but instead by diffraction from

an off-axis vortex-creating hologram displayed at a high resolution liquid crystal SLM (1920

x 1200 pixels, each pixel is 10x10 μ m 2). Such holographic gratings with a characteristic fork-

like dislocation in their center (see upper image at the right side of Fig. 2) are typically used

to create so-called doughnut beams (Laguerre-Gauss modes) from an incident Gaussian beam

[15]. The main reason to use diffraction from such a hologram is that we cannot generate

a sufficiently accurate on-axis spiral phase plate with our spatial light modulator, due to its

limited phase-modulation capabilities. Therefore we use off-axis diffraction, where the phase

of the diffracted light field is encoded with high precision within the spatial arrangement of

the hologram structures, rather than in the phase shifts of the individual SLM pixels. Thus, the

limited phase-modulation capabilities of the SLM are only influencingthe diffraction efficiency,

but not the phase accuracy of the spiral filtered image.

The second difference to the simple principle setup of Fig. 1 is a further diffraction step of

the filtered light field at a second ”normal” grating with the same spatial frequency as used

for the first one, in order to compensate for the dispersion due to the white light illumination.

Basically, the setup uses one more Fourier-transforming lens L3, and a back-reflection mirror

M2, which are arranged such that a copy of the light field in the upper part of the SLM plane

is produced in the lower part of the SLM plane. There, diffraction at a ”normal” second grating

compensates for the dispersion introduced by the first one, before recording the spiral-phase

filtered image at a CCD camera. This dispersion control would not be necessary, if an on-axis

spiral phase plate was used, or in the case of monochromatic illumination.

In order to produce spiral phase filtered images with an adjustable and controlled shadow

effect, a circular area in the central part of the vortex creating hologram with a diameter on the

order of the size of the zero-order Fourier spot of the incident light field (typically 100 microns

diameter, depending on the collimation of the illumination light, and on the focal lengths of

the objective, and the lens set L1 and L2) is substituted by a ”normal” grating. There, the zero-

order Fourier component is just deflected (without being filtered) into the same direction as the

remaining, spiral-filtered light field. A controlled rotation of the shadow images can then be

performed by shifting the phase of the central grating, or - preferably - by keeping the phase of

the central grating constant, but rotating the remaining part of the spiral phase hologram (before

calculating its superposition with a plane grating in order to produce the off-axis hologram).This second method is the holographic off-axis analogue to a simple rotation of an on-axis

spiral phase plate around its center.

This setup was then used for the imaging of a commercially available phase test pattern (so-

called ”Richardson slide”), which consists of a micro-pattern with a depth on the order of h =240 nm etchedinto a transmissive silica sample with a refractive index of n=1.56, corresponding

to an optical path difference of (n−1)h≈ 135 nm. Like any pure phase object which is imaged

with an optical system with a limited numerical aperture, there is also some intensity contrast

in the image. The mechanism is based on the fact that small phase structures within the object

scatter the transmitted light at diffraction angles which can be larger than the maximal aperture

angle of the microscope objective. As a result, sharp changes of the phase structures in an object

appear darker than their unmodulated surroundings. Thus, this ”spurious” intensity contrast

may be reduced by using objectives with a higher numerical aperture, however, for our test

experiment this effect is desired since it provides us with a quasi complex sample (note that theeffect of Fourier filtering does not depend on the mechanism of the intensity modulation, i.e.

there is no difference whether a local intensity reduction is due to an absorber in the sample,

or to an intensity loss due to scattering). Thus, the Richardson slide can be used as a model for

a quantitative complex sample, since it possesses a structured phase topography as well as an





intensity modulation.

Fig. 3. Imaging of a Richardson phase pattern. The total length of the two scale bars at

the right and lower parts of the image are 80 μ m, each divided into 8 major intervals with

a length of 10 μ m. All images are displayed as negatives (i.e. dark areas correspond to

bright structures in the real images) for better image contrast. (A), (B), and (C) are three

shadow-effect images recorded at spiral phase plate angles of 0, 2π /3 and 4π /3, respec-

tively. For comparison, (D) is a brightfield image recorded with our setup by substituting

the spiral phase hologram at the SLM by a ”normal” grating. (E) and (F) are the corre-

sponding intensity and phase images, respectively, obtained by numerical processing of the

shadow-images (A)-(C) according to the method described in the text.

Fig. 3(A-C) show three shadow-effectimages of the Richardson slide, recorded at three spiral

phase plate rotation angles of 0, 2π /3 and 4π /3, respectively. For display purposes, all images

of Fig. 3 are printed as negatives, i.e. dark areas correspond to bright ones in the actual images.

Each image is assembled by 4× 2 individual images, since the field of view of the setup was

limited by the diffraction angle of the SLM holograms such that the whole test sample could

not be recorded at once. Note that no further image processing (like background subtraction

etc.) was used. Obviously, the three shadow-effect images produce a relief-like impression of

the sample topography, similar to Nomarski- or differential interference contrast methods. For

comparison, Fig. 3(D) shows a bright-field image of the sample, recorded by substituting the

spiral phase pattern displayed at the SLM with a ”normal” grating. As mentioned above, in-

tensity variations are visible, although the sample is a pure phase-object. The results for the

intensity transmission and phase from the numerical processing of the three shadow-effect im-ages are shown in Fig. 3(E) and (F), respectively. As expected, the measured bright-field image

in (D) is in good agreement with its numerically obtained counterpart in (E). Obviously, the best

contrast of the sample is obtained from the phase of the processed image, displayed in image

(F). In order to check the accuracy of the method, a section of this phase image is compared in

Fig. 4 with an accurate surface map of the sample, recorded with an atomic force microscope





(AFM, Nanoscope, by courtesy of Michael Helgert, Research Center Carl Zeiss, Jena).

Fig. 4. Comparison of the spiral phase contrast method with data obtained from an atomic

force microscope (AFM). (A) shows a section from the sample displayed in Figure 3(F),

which corresponds to the section (B) scanned with the AFM. In (C), the phase topography

of the selected section as measured with the spiral phase method is displayed as a surface

plot, with the calculated depth of the etched pattern scaled in absolute units. It turns out

that the pattern depth measured with the spiral phase method seems to be (150 ± 20) nm as

compared to the AFM reference measurement, where a depth of (240 ± 10) nm is obtained.

Fig. 4(A) and (B) show the same section of the sample phase profile, as recorded with the

spiral phase method (A) and the AFM (B), respectively. The size of the measured area is 25 ×25μ m2. The two images show again a good qualitative agreement, i.e. even details like the

partial damage of the sample in the lower right quadrant of the sample are well reproduced

by the spiral phase method. However, the resolution of the AFM is obviously much better, i.e.

the actual spiral phase image becomes blurred in the third ring (measured from the outside)





of the Richardson spiral. Since the thickness of the bars in the outer rings of the star are 3.5

μ m (outmost ring), 1.75 μ m, 1.1 μ m, 0.7 μ m, and 0.35 μ m,respectively, the actual transverse

resolution of the spiral filtering method turns out to be on the order of 1 μ m. The effect of this

limited spatial resolution becomes clearer in Fig. 4(C), where the phase profile is plotted as a

surface plot with absolute etching depth values in nm as obtained by the numerical processing.

Obviously, the contrast of the method (i.e. the height of the inner parts of the sample) decreases

if the spatial resolution comes to its limit, which is the case for structure sizes below 1 μ m.

However, for larger structures, the contrast and the measured phase profile are independent

from the shape or the location of the structure, i.e. in Figure 3(F), different etched objects (like

the maple leaves, circles and square, and the Richardson star) show the same depth of the phase

profile within a range of 10%.

However, there is one drawback,concerning the measured absolute depth of the phase profile.

The comparison of the groove depths measured by the AFM (240 ± 10 nm) and the spiral

contrast method (150 ± 20nm) reveals that our method underestimates the optical path length

difference by almost 40 %, which disagrees with the theoretical assumption that the spiral

phase method should measure absolute phase values even without calibration. More detailed

investigations show that this underestimation is due to the limited spatial coherence of the

white-light illumination (emerging from a fiber with a core diameter of 0.4 mm) which is not

considered in the theoretical investigation of the previous section. Briefly, the limited spatial

coherence of the illumination results in a zero-order Fourier spot in the SLM plane, which is

not diffraction limited but has an extended size, i.e. it is a ”diffraction disc”. All the other Fourier

components of the image field in the SLM plane are thus convolved with this disc, resulting in

a smeared Fourier transform of the image field in the SLM plane. Such a smearing results in

a decrease of the ideal anticipated edge-enhancement (or shadow-) effect. However, since this

edge-enhancement effect encodes the height of a phase profile in the ideal spiral phase method,

its reduction due to the limited coherence of the illumination seems to result from an apparently

smaller profile depth, which is actually computed.

We tested this assumption by repeating similar measurements with coherent TEM 00 illumi-

nation from a laser diode. There, in fact the structure depth was measured correctly within ±10% accuracy. However, for practical imaging purposes the longitudinal coherence of the laser

illumination is disturbing, since it results in laser speckles. In contrast to the suppression of in-

coherent image noise (like background light) by the spiral phase method, the coherent specklenoise is not filtered out.

Nevertheless, the spiral phase method can be used for a quantitative measurement of phase

structures. This is due to the fact that the apparent decrease of the phase profile does not depend

on details of the sample, but just on the setup. Therefore, the setup can be calibrated with

a reference test sample like the Richardson slide. In our actual setup the results of further

measurements can be corrected by considering the 40 % underestimation of the phase depth.

An example for such a measurement is shown in Figure 5. A bright-field image of the cheek

cell as recorded by substituting the spiral phase grating at the SLM with a normal grating

is displayed in (A). Image size is 25×25μ m2. (B-D) show three corresponding shadow-effect

images recorded with the same settings as used for Fig. 3. Numerical processing of these images

results in the calculated intensity transmission (E) and phase profile (F) images. There, details

of structures within the cell are visible with a high contrast. The phase profile is plotted again

in (G) as a surface plot, where the z-axis corresponds to the computed phase profile depth inradians (without consideration of the calibration factor). In order to find the actual optical path

length difference of structures within the sample, the phase shift at a certain position has to

be multiplied by the corresponding calibration factor of 1.6, and divided by the wavenumber

(2π /λ , λ ≈ 570 nm) of the illumination light. For example, the maximal optical path length





Fig. 5. Spiral phase imaging of a cheek cell. (A) is a bright-field image of the cell. (B-D)

are three shadow-effect images with apparent illumination directions of 0, 2π /3 and 4π /3,

respectively. (E) and (F) are numerically processed intensity transmission and phase profile

images of the sample. (G) is a surface plot of the phase profile, displaying the absolute

calculated phase shift (without calibration correction) in radians.

difference between the lower part of the cell (red in Fig. 5) and its surrounding is approximately

70 nm. In order to calculate the absolute height of the cell, the difference of the refractive indices

between the cell and its surrounding (water) is required. On the other hand, if the actual height

of the cell was measured by another method (e.g. an AFM), then the refractive index of the cell

contents could be determined.

5. Discussion

In this paper we demonstrated a spiral phase contrast method for quantitative imaging of the

amplitude transmission and the phase profile of thin, complex samples. The method is based on

the numerical post-processing of a sequence of at least three shadow-effect images, recorded

with different phase offsets between the zero-order Fourier spot, and the remaining, spiral fil-

tered part of the image field. After a straightforward numerical algorithm, a complex image is

obtained, whose amplitude and phase correspond to the amplitude and phase transmission of

the imaged object. In principle, the method is supposed to give quantitative phase profiles of

samples with a height in the sub-wavelength regime, even without requiring a preceding cali-

bration. However, measurements performed at a phase sample calibrated with an AFM revealed

that the method underestimates the height of the phase profile. This is due to the limited spatial

(transverse) coherenceof the illuminationsystem, and could be avoided by using TEM 00 illumi-

nation from a laser diode. On the other hand, there is no requirement for longitudinal (temporal)

coherence for performing spiral phase filtering, with the exception of dispersion control if dis-persive elements (like gratings) are in the beam path. For practical imaging, broadband light

illumination is advantageous, since disturbing speckles are suppressed. In this case, the method

can still be used for quantitative phase measurements, if it is calibrated with a reference phase

sample.

If an on-axis spiral phase plate [14] would be used as a transmissive spiral phase filter, un-





desired dispersion effects could be avoided without further dispersion control. If such a plate

would be implemented in the back aperture plane of a microscope objective, then a rotation

of the shadow direction would just require a corresponding rotation of the spiral phase plate.

The actual feature of such a setup to measure sub-wavelength optical path differences is based

on the fact that the setup acts as a self-referenced interferometer, comparing the zero-order

Fourier component of a light field with its remainder. Similar self-referenced phase measure-

ments can be in principle performed with a ”normal” phase contrast method by stepping the

phase of the zero-order Fourier component with respect to the remaining, non-filtered image

field [16, 17, 18]. However, using a ”normal” phase contrast method, the interference contrast

depends strongly on the phase difference, whereas the spiral phase method ”automatically” de-

livers images with a maximal (but spatially rotating) contrast, since the phase of a spiral phase

filtered image always covers the whole range between 0 and 2π . Advantageously, the phase

shifting in the case of spiral phase filtering just requires a rotation of an inserted spiral phase

plate by the desired phase angle, which cannot be achieved as easily with a ”normal” phase

contrast method. In principle, the method can be also used in reflection mode for measuring

surface phase profiles with an expected resolution on the order of 10 nm or better, which can

have applications in material research and semiconductor inspection.

Acknowledgments

The authors want to thank Michael Helgert (Research Center Carl Zeiss, Jena) for the supply

of the Richardson slide, and for the AFM measurements in Fig. 4. This work was supported

by the Austrian Academy of Sciences (A.J.), and by the Austrian Science Foundation (FWF)

Project No. P18051-N02.



Stefan Bernet et al- Quantitative imaging of complex samples by spiral phase contrast microscopy

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Transcript of Stefan Bernet et al- Quantitative imaging of complex samples by spiral phase contrast microscopy