Content uploaded by Andrzej Kolodziejczyk
Author content
All content in this area was uploaded by Andrzej Kolodziejczyk on Apr 07, 2015
Content may be subject to copyright.
Imaging with extended focal depth by means of
lenses with radial and angular modulation
G. Mikuła1, Z. Jaroszewicz2, 3*, A. Kolodziejczyk1, K.Petelczyc1, and M. Sypek1
1Faculty of Physics, Warsaw University of Technology, Koszykowa 75, 00-662 Warsaw, Poland
2 Institute of Applied Optics, Kamionkowska 18, 03-805 Warsaw, Poland
3 National Institute of Telecommunications, Szachowa 1, 04-894 Warsaw, Poland
*Corresponding author: mmtzjaroszewicz@post.pl
Abstract: The paper presents imaging properties of modified lenses with
the radial and the angular modulation. We analyze three following optical
elements with moderate numerical apertures: the forward logarithmic axicon
and the axilens representing the radial modulation as well as the light sword
optical element being a counterpart of the axilens with the angular
modulation. The abilities of the elements for imaging with extended depth
of focus are discussed in detail with the help of structures of output images
and modulation transfer functions corresponding to them. According to the
obtained results only the angular modulation of the lens makes possible to
maintain the acceptable resolution, contrast and brightness of the output
images for a wide range of defocusing. Therefore optical elements with
angular modulations and moderate numerical apertures seem to be
especially suitable for imaging with extended focal depth.
©2007 Optical Society of America
OCIS codes: (110.2990) Image formation theory; (110.4100) Modulation transfer function;
(080.2740) Geometrical optics, optical design; (050.1970) Diffractive optics
References and links
1. M. Mino and Y. Okano, “Improvement in the optical transfer function of a defocused optical system through
the use of shaded apertures,” Appl. Opt. 10, 2219-2225 (1971).
2. J. Ojeda-Castañeda, P. Andres, and A. Diaz, “Annular apodizers for low sensitivity to defocus and to
spherical aberration,” Opt. Lett. 11, 487-489 (1986).
3. J. Ojeda-Castañeda, E. Tepichin, and A. Diaz, “Arbitrary high focal depth with a quasioptimum real and
positive transmittance apodizer,” Appl. Opt. 28, 2666-2670 (1989).
4. J. Ojeda-Castañeda and L. R. Berriel-Valdos, “Zone plate for arbitrarily high focal depth,” Appl. Opt. 29,
994-997 (1990).
5. E. R. Dowski, Jr. and W. T. Cathey, “Extended depth of field through wave-front coding,” Appl. Opt. 34,
1859-1866 (1995).
6. S. Bradburn, W. T. Cathey, and E. R. Dowski, Jr., “Realizations of focus invariance in optical-digital
systems with wave-front coding,” Appl. Opt. 36, 9157-9166 (1997).
7. H. B. Wach, E. R. Dowski, Jr., and W. T. Cathey, “Control of chromatic focal shift through wave-front
coding,” Appl. Opt. 37, 5359-5367 (1998).
8. S. C. Tucker, W. T. Cathey, and E. R. Dowski, Jr., “Extended depth of field and aberration control for
inexpensive digital microscope systems,” Opt. Express 4, 467-474 (1999).
9. J. Sochacki, A. Kołodziejczyk, Z. Jaroszewicz, and S. Bara, “Nonparaxial design of generalized axicons,”
Appl. Opt. 31, 5326-5330 (1992).
10. J. Sochacki, Z. Jaroszewicz, L.R. Staro ski, and A. Kołodziejczyk, “Annular-aperture logarithmic axicon,” J.
Opt. Soc. Am. A 10, 1765-1768 (1993).
11. W. Chi, and N. George, “Electronic imaging using a logarithmic asphere,” Opt. Lett. 26, 875-877 (2001).
12. M. A. Golub, V. Shurman, and I. Grossinger, “Extended focus diffractive optical element for Gaussian laser
beams,” Appl. Opt. 45, 144-150 (2006).
13. J. Ares, R. Flores, S. Bara, and Z. Jaroszewicz, “Presbyopia compensation with a quartic axicon,” Optom.
Vis. Sci. 82, 1071-1078 (2005).
14. G.-m. Dai, “Optical surface optimization for the correction of presbyopia,” Appl. Opt. 45, 4184-4195 (2006).
15. A. Flores, M. R. Wang, and J. J. Yang, “Achromatic hybrid refractive-diffractive lens with extended depth of
focus,” Appl. Opt. 43, 5618-5630 (2004).
16. Z. Liu, A. Flores, M. R. Wang, and J. J. Yang, “Diffractive infrared lens with extended depth of focus,” Opt.
Eng. 46, 018002 (1-9) (2007).
#83838 - $15.00 USD Received 6 Jun 2007; revised 2 Jul 2007; accepted 4 Jul 2007; published 11 Jul 2007
(C) 2007 OSA 23 July 2007 / Vol. 15, No. 15 / OPTICS EXPRESS 9184
17. E. E. Garcia-Guerrero, E. R. Mendez, H. M. Escamilla, T. A. Leskova, and A. A. Maradudin, “Design and
fabrication of random phase diffusers for extending the depth of focus,” Opt. Express 15, 910-923 (2007).
18. B.-Z. Dong, J. Liu, B.-Y. Gu, and G.-Z. Yang, “Rigorous electromagnetic analysis of a microcylindrical
axilens with long focal depth and high transverse resolution,” J. Opt. Soc. Am. A 18, 1465-1470 (2001).
19. J.-S. Ye, B.-Z. Dong, B.-Y. Gu, G.-Z. Yang, and S.-T. Liu, “Analysis of a closed-boundary axilens with long
focal depth and high transverse resolution based on a rigorous electromagnetic theory,” J. Opt. Soc. Am. A
19, 2030-2035 (2002).
20. F. Di, Y. Yingbai, J. Guofan, and W. Minxian, “Rigorous concept for the analysis of diffractive lenses with
different axial resolution and high lateral resolution,” Opt. Express 17, 1987-1994 (2003).
21. J. Lin, J. Liu, J. Ye, and S. Liu, “Design of microlenses with long focal depth based on general focal length
function,” J. Opt. Soc. Am. A 24, 1747-1751 (2007).
22. N. Davidson, A. A. Friesem, and E. Hasman, “Holographic axilens: high resolution and long focal depth,”
Opt. Lett. 16, 523-525 (1991).
23. A. Kołodziejczyk, S. Bara, Z. Jaroszewicz, and M. Sypek, “The light sword optical element – a new
diffraction structure with extended depth of focus,” J. Mod. Opt. 37, 1283-1286 (1990).
24. J. Sochacki, S. Bara, Z. Jaroszewicz, and A. Kołodziejczyk, “Phase retardation of the uniform-intensity
axilens,” Opt. Lett. 17, 7-9 (1992).
25. G. Mikuła, A. Kolodziejczyk, M. Makowski, C. Prokopowicz, and M. Sypek, “Diffractive elements for
imaging with extended depth of focus,” Opt. Eng. 44, 058001(2005).
26. S. Bara, C. Frere, Z. Jaroszewicz, A. Kołodziejczyk, and D. Leseberg, “Modulated on-axis circular zone
plates for a generation of three-dimensional focal curves,” J. Mod. Opt. 37, 1287-1295 (1990).
27. H. H. Emsley, Visual Optics (London: Hatton Press, 1952).
28. M. Sypek, “Light propagation in the Fresnel region. New numerical approach,” Opt. Commun. 116, 43-48
(1995).
1. Introduction
The limited range of focus is a significant disadvantage of incoherent imaging systems. The
depth of focus is especially important when three-dimensional scenes or three-dimensional
objects are imaged. Extending depth of field of optical imaging systems has been a subject of
intensive investigations. Application of optical power-absorbing apodizers can increase the
depth of focus [1-4]. This method leads to some disadvantages. Apodization limits an
effective aperture of the imaging set-up what causes a substantial loss of an incident light
energy and reduces significantly a resolution of imaging. The elements of this kind can be
only used in a case of sufficiently strong illumination what limits their practical applications.
Another possible method of extending depth of field is based on the two-step process [5-8].
The optical imaging system is designed in such a way that the point-spread function (PSF) is
insensitive to misfocus while the optical transfer function (OTF) has no regions of zero values
within its passband. Then the electronic processing of inverse filtration is used to restore the
image formed by the optical system. The same electronic processing restores the image for all
values of misfocus because the OTF is insensitive to misfocus. The electronic processing of
an optical image seriously limits application of this method. The electronic stage makes
impossible imaging in a real time and complicates a construction of an imaging set-up.
The most promising optical elements for imaging with extended depth of focus in real-
time seem to be optical elements focusing an incident plane wave into a focal line segment.
These elements can be regarded as modified lenses with controlled aberrations. The
modification should lead to output images characterized by the possible highest contrast,
brightness and sharpness. A fixed point of the focal segment is connected to a proper input
plane or an output plane in an imaging process. The optical elements of this kind were
intensively studied lately in many papers. The authors of them attempted to solve the problem
of extended focal depth by different methods leading to different optical structures as for
example axicons [9-13], elements defined by a numerical iterative approach [14-16] or optical
diffusers [17]. Some works published in recent years demonstrated usefulness of the axilens in
optical systems with a long focal depth [18-21]. The analyzed axilens was based on design
proposed by Davidson at al. [22].
All the above mentioned optical elements exhibit the radial symmetry. According to the
results presented for imaging set-ups with such elements, it is very difficult to maintain high
quality of output images when defocus becomes large [13, 14, 16, 17]. This difficulty lies in a
nature of the image formation process. According to the geometrical optics light focused
#83838 - $15.00 USD Received 6 Jun 2007; revised 2 Jul 2007; accepted 4 Jul 2007; published 11 Jul 2007
(C) 2007 OSA 23 July 2007 / Vol. 15, No. 15 / OPTICS EXPRESS 9185
around a point of an optical axis diverges quickly substantially spoiling neighboring focal
points of the segment. Therefore a quality of the focal depth is limited, especially when a
numerical aperture of an imaging system is large. Some disadvantages of elements with the
symmetry of revolution can be overcome by an application of the angular modulation. The
angular modulation offers an additional degree of freedom during element’s design and
modifies harmful diffraction effects corresponding to focusing.
The aim of this paper is to illustrate usefulness and advantages of optical elements with
angular modulation for imaging with extended depth of focus. For this purpose we study and
compare imaging properties of three following optical elements: the forward logarithmic
axicon [9] (FLA), the axilens [22] (AXL) and the light sword optical element [23] (LSOE).
The LSOE is a modified convergent lens with an angular modulation while two others
elements represent lenses with radial modulation. The terms lens modulation or lens
modification used in this paper means modulation or modification of a phase transmittance of
a lens. Particularly, we analyze in detail quality of output images by means of calculated
modulation transfer functions (MTFs) corresponding to all elements and different object
planes in the imaging set-up. We have intentionally chosen the parameters of an imaging
arrangement similar to those described in works dealing with a presbyopia correction [13, 14],
since it enables to evaluate usefulness of investigated elements for ophthalmologic
applications.
2. Analyzed elements and an imaging set-up.
An assumed imaging set-up includes a thin optical imaging element. An imaging plane is
placed in a fixed distance q=20 mm behind the element. The assumed distances between the
input objects and the element vary and belong to the range
)
∞∈ ,cm 25p. According to
geometrical optics, the thin optical element focusing incident plane wave into a proper light
segment makes possible to realize imaging with extended depth of field in the above set-up.
Points of the focal segment should be situated from a distance mm 18.5
1=f up to a distance
mm 20
12 =Δ+= fff behind the element in order to cover the assumed range of the object
distances. Hence f=1.5 mm denotes a length of the focal segment. We designed the FLA, the
AXL and the LSOE fulfilling the above conditions. In all cases we have assumed circular
apertures of the elements with a radius R=2 mm and the wavelength =632.8 nm of a
monochromatic illumination corresponding to a He-Ne laser. Because of moderate numerical
apertures corresponding to focusing we have used the Fresnel paraxial approximation during
our design and numerical calculations.
2.1 Forward logarithmic axicon (FLA)
The element is designed by means of geometrical optics and the principle of energy
conservation [9]. According to the ray tracing the FLA exactly focuses light in an assumed
focal segment with a uniform intensity distribution. The phase transmittance of the element
has the following form:
()
+−=Φ 1
2
1ln
2f
ar
a
k
r, (1)
where 2
/Rfa Δ= ,
λ
π
/2=k; is a wavelength of light and r denotes a radial
coordinate in an element plane. Potential abilities of the FLA or its simplified version for
imaging with extended depth of focus were confirmed numerically and experimentally [11-
13].
#83838 - $15.00 USD Received 6 Jun 2007; revised 2 Jul 2007; accepted 4 Jul 2007; published 11 Jul 2007
(C) 2007 OSA 23 July 2007 / Vol. 15, No. 15 / OPTICS EXPRESS 9186
2.2 Axilens (AXL)
The AXL was proposed by Davidson at al. [22]. The phase transmittance of the element is
defined by the following phase:
()
()
[]
22
1
2
/2 Rfrf
kr
rΔ+
−=Φ . (2)
Design leading to the above phase function violates the law of the energy conservation [24]
therefore the AXL does not focus exactly incident light in an assumed fragment of an optical
axis. Nevertheless, latest intensive investigations demonstrated that the AXL can be
successfully used as a lens with a large focal depth [18-21]. According to the published results
of these investigations the AXL seem to be especially suitable for imaging with extended
depth of focus. Therefore we have decided to analyze imaging properties of the AXL and to
compare them with those corresponding to the optical element with the angular modulation.
2.3 Light sword optical element (LSOE)
The LSOE is a counterpart of the AXL where the radial modulation of the lens was
substituted by an angular one. Preliminary results illustrating abilities of the LSOE with a
small numerical aperture for imaging with extended depth of focus were reported elsewhere
[25]. The phase defining transmittance of the LSOE is given as follows:
() ()
[]
πθ
2/2 1
2
ff kr
rΔ+
−=Φ , (3)
Fig. 1. Geometry of focusing by the LSOE. The infinitesimal angular sector of the element
focuses an incident plane wave into a segment PP1 oriented perpendicularly to the sector.
where is an azimuthal coordinate in an element’s plane. The LSOE corresponds to a limiting
case of the element focusing light into a curve lying on a lateral surface of a cylinder with its
radius going to zero [26]. The element described by Eq. (3) forms approximately an assumed
focal segment even within the geometrical optics. According to the paraxial ray tracing
implemented to polar coordinates points (r, ) of the infinitesimal angular sector =const of
the element are connected to the following points ( , ) of the output plane
πθΔ
2/
1ffz += :
()
/2 , 24/ 1
πθϕθΔπΔρ
+=+= fffr . (4)
#83838 - $15.00 USD Received 6 Jun 2007; revised 2 Jul 2007; accepted 4 Jul 2007; published 11 Jul 2007
(C) 2007 OSA 23 July 2007 / Vol. 15, No. 15 / OPTICS EXPRESS 9187
The geometry of light focusing by the LSOE according to the ray tracing method is shown in
Fig. 1. The infinitesimal angular sector of the element corresponding to the angular coordinate
focuses light into a small line segment PP1 instead into an assumed point P with coordinates
()
πθ
2/,0,0 1ff Δ+ in the Cartesian coordinate system OXYZ. The length L of the segment is
defined by the following relation:
()
θΔπΔ
fffRL 24/ 1+= , (5)
where R denotes a radius of the LSOE’s aperture. The segment PP1 is oriented perpendicularly
to the angular sector, i.e. the segment’s direction is defined by a semi-line corresponding to an
angular coordinate +/2. Taking into account this geometrical approach, the assumed
parameters R=2 mm, mm 18.5
1=f, f=1.5 mm, =632.8 nm,
[
)
πθ
2,0∈ and Eq. (5) the
lengths of the segments PP1 belong to the range [11.94 m , 12.90 m]. For comparison,
diameters of the central Airy spots formed by lenses with the same aperture and focal
distances correspond to the range [7.14 m , 7.72 m].
3. Numerical and experimental results
The assumed optical set-up resembles that used in an model of the human eye [27]. Hence we
have analyzed the images of optotypes of Snellen with an angular dimension 5 minutes of arc
and the smallest details equal to 1 minute of arc. The Snellen optotypes are commonly used in
the ophthalmology for vision acuity examinations. The satisfying recognition of the Snellen
optotypes with an angular dimension 5’ corresponds to the standard 20/20 vision acuity. We
have selected for our analysis the input objects in a form shown in Fig. 2. The singular object
was consisted of four optotypes in a shape of a capital E oriented in different directions. Three
parallel strips of the letter E form a fragment of the Ronchi grating. According to the
parameters of the assumed optical set-up an ideal image of grating in the output plane
corresponds to the fundamental spatial frequency =86 lines/mm.
Fig. 2. The form of input object used in numerical simulations and experiments. Each letter E
has the same angular dimension 5 minutes of arc. Singular strips of the letter have an angular
width 1 minute of arc.
In order to analyze in detail imaging abilities of the studied optical elements we have
prepared numerical simulations corresponding to the following 12 different object distances p
given in millimeters: 250, 300, 350, 400, 450, 500, 600, 700, 800, 1000, 1500, 2000.
Numerical simulations were conducted using a diffractive modeling package working
according to the modified convolution approach [28] on a matrix of 4096x4096 points.
Making calculations we have assumed that the input object has been illuminated by a
monochromatic spatially incoherent light of a He-Ne laser with a wavelength =632.8 nm.
The columns AXL-s, FLA-s and LSOE-s of the Fig. 3 show the intensity distributions of
output images of the Snellen optotypes created by the AXL, the FLA and the LSOE and
obtained in numerical simulations. The intensities are presented in a gray scale with 256
different levels. All these images have the same maximal intensity defined by the highest
level.
The transfer functions are useful tools for an analysis of imaging properties. Therefore we
computed MTFs corresponding to the numerical results presented in Fig. 3. The MTFs shown
in Figs. 4-5 were calculated as moduli of Fourier transforms of incoherent point spread
functions. The LSOE does not exhibit a symmetry of revolution then in a case of this element
we present two cross-sections of MTFs corresponding to perpendicular directions x and y in
#83838 - $15.00 USD Received 6 Jun 2007; revised 2 Jul 2007; accepted 4 Jul 2007; published 11 Jul 2007
(C) 2007 OSA 23 July 2007 / Vol. 15, No. 15 / OPTICS EXPRESS 9188
p AXL-s AXL-e FLA-s FLA-e LSOE-s LSOE-e
250
300
350
400
450
500
600
700
800
1000
1500
2000
Fig. 3. Intensity distributions of the output images formed by the AXL, FLA and LSOE for
different object distances p given in milimeters. The columns AXL-s, FLA-s and LSOE-s
includes results of numerical simulations. The remaining distributions correspond to
experimental verifications.
#83838 - $15.00 USD Received 6 Jun 2007; revised 2 Jul 2007; accepted 4 Jul 2007; published 11 Jul 2007
(C) 2007 OSA 23 July 2007 / Vol. 15, No. 15 / OPTICS EXPRESS 9189
p AXL FLA
250
300
350
400
450
500
600
700
800
1000
1500
2000
Fig. 4. MTFs calculated for the AXL and the FLA for different object distances p given in
millimeters.
#83838 - $15.00 USD Received 6 Jun 2007; revised 2 Jul 2007; accepted 4 Jul 2007; published 11 Jul 2007
(C) 2007 OSA 23 July 2007 / Vol. 15, No. 15 / OPTICS EXPRESS 9190
p LSOE H LSOE V
250
300
350
400
450
500
600
700
800
1000
1500
2000
Fig. 5. MTFs calculated for the LSOE and different object distances p given in millimeters. The
column LSOE H corresponds to a direction x in the spatial frequencies domain and the column
LSOE V to a direction y.
#83838 - $15.00 USD Received 6 Jun 2007; revised 2 Jul 2007; accepted 4 Jul 2007; published 11 Jul 2007
(C) 2007 OSA 23 July 2007 / Vol. 15, No. 15 / OPTICS EXPRESS 9191
a frequency domain ( x, y). The cross-sections of MTFs showed in Fig. 5 are representative.
The plots for other cross-sections have a similar character.
Then we have verified experimentally results of our numerical simulations. The elements
assumed in the simulations were fabricated as binary-phase diffractive structures by electron
beam lithography at the Institute of Electronic Materials in Warsaw by the technique
described below in more detail.
The glass plate with a conductive layer (ITO) was covered with electron resist by its
pulverization on the rotating substrate. The thickness of the resist was controlled by the
velocity of rotation and should result in a phase shift equal to (in our case, compared with
the light propagating in air). A substrate prepared in this way was exposed with the help of an
electron beam lithography device with variable shaped e-beam system (ZBA-20 by Jenoptik
GmbH). After developing, the exposed areas were removed to achieve the binary phase
element. The described process enables us to fabricate 0.5- m-spot-sized structures with
accuracy of 0.1 m in both the x and y directions.
The produced binary-phase diffractive structures have limited diffraction efficiency
theoretically equal to 40.5%. The output images are formed only by the (+1) orders of
structures. Nevertheless in our imaging system the other orders have negligible influence for
output intensity distributions. The (-1) order produces divergent wave-fronts and higher orders
correspond to very small diffraction efficiencies. The columns AXL-e, FLA-e and LSOE-e of
the Fig. 3 present output images formed by the fabricated elements in an optical set-up and
captured by a CCD camera. The object was illuminated by He-Ne laser. In order to obtain
spatially incoherent illumination, a rotated ground glass was inserted between the object and a
He-Ne laser.
4. Discussion of the obtained results
According to the results shown in Fig. 3, the experiment confirmed numerical simulations.
Generally the elements form images with different brightness and contrast. We registered the
output images in the experiment using the same illumination intensities making possible to
compare a brightness and a contrast of the output images in an optical set-up. Therefore
otherwise to simulations the experimental results are counterparts of non-normalized intensity
images. Numerical and experimental imaging results shown in Fig. 3 present satisfied
coincidence. The structures of images are almost the same. The slight difference is probably
caused by imperfections in the fabrication of diffractive structures and their limited diffractive
efficiencies. Good agreement between numerical simulations and experiments justifies our
assumption about the used paraxial approach.
Characteristic features of output images can be explained using the MTFs shown in Fig. 4-
5. Only the LSOE forms recognizable images of Snellen optotypes for all object distances.
MTFs corresponding to the LSOE have not zeros for spatial frequencies smaller than 100
lines/mm. The nonzero ranges of the MTFs are substantially wider than these corresponding
to the FLA and the AXL. Some MTFs of the FLA and the AXL have zeros around the
characteristic spatial frequency of the optotypes =86 lines/mm. Therefore the relative output
images are completely blurred for object distances 400 mm, 600mm in a case of the FLA and
500mm, 1000mm in a case of the AXL. Additionally MTFs corresponding to the FLA and the
AXL exhibit narrow maxima around the zero spatial frequency. Moreover these MTFs have
smaller values for higher frequencies than in a case of the LSOE. Hence generally the images
created by the FLA and the AXL demonstrate lower contrasts. This effect is especially
recognizable in a case of the AXL where the MTFs decrease the most rapidly from the central
maximum. The images created by the FLA and the AXL are considerably blurred when object
distances become longer than 500 mm. The blurs correspond to rapid oscillations appearing in
MTFs plots. These oscillations may cause substantial contrast inversions of output images.
The obtained numerical and experimental results prove superiority of the LSOE for
imaging with extended depth of focus in the analyzed set-up. Generally the LSOE forms
images with better resolution, contrast and higher brightness than those created by the FLA
#83838 - $15.00 USD Received 6 Jun 2007; revised 2 Jul 2007; accepted 4 Jul 2007; published 11 Jul 2007
(C) 2007 OSA 23 July 2007 / Vol. 15, No. 15 / OPTICS EXPRESS 9192
and the AXL. Surprisingly good abilities of the LSOE for imaging with extended depth of
field are probably connected with the flow of energy during focusing. According to ray
tracing shown in Fig. 1 the focusing process has an off-axis character. The maximum intensity
of the focal spot lies outside the optical axis OZ and waves around it [25]. This phenomenon
causes a displacement of the output images. Taking into account the assumed parameters
corresponding to our simulations and experiments, the angular displacement is equal
approximately to 2 minutes of arc. The flow of energy changes its main direction during
focusing and positions of output images rotate around the optical axis. Then the effect of
mutual disturbance between neighboring images corresponding to different focal lengths or
different object distances is less harmful than in cases of the FLA and the AXL where the
flow of energy during focusing has the same main direction along the optical axis.
5. Conclusion
We have analyzed in detail abilities of lenses with the radial and angular modulation for
imaging with extended depth of focus. We have chosen for our analysis the imaging set-up
similar to that representing a model of the human eye and three following optical elements:
the forward logarithmic axicon and the axilens exhibiting radial modulation of the lens
transmittance as well as the light sword optical element representing a lens with angular
modulation. The structures of output images in numerical simulations and experiments were
discussed by means of the calculated MTFs. The LSOE contradictory to the FLA and the
AXL forms well recognizable output images for a wide range of object distances. The angular
modulation of the lens transmittance used in a case of the LSOE modifies the flow of light
energy during focusing what improves quality of imaging.
Nevertheless the LSOE exhibits some disadvantages. It forms slightly stretched focal spots
without radial symmetry [25] what generates characteristic blur of output images. These
images waves around the optical axis. The above disadvantages do not seriously limit
usefulness of the LSOE for many imaging applications. Moreover the LSOE is only one
relatively simple example of the lens with an angular modulation of a transmittance. Imaging
abilities of the LSOE can be probably substantially improved by optimization of its design.
Analytical modification can be realized by a substitution of the linear function of in a
denominator of the phase transmittance given in Eq. (3) by an another, properly selected
angular function. According to the lately published works especially promising seem to be
iterative optimization methods [14-16].
The presented results give evidence that modified lenses with angular modulation of
phase transmittances can be very useful tools for imaging with extended focal depth.
Generally the angular modulation offers the additional degree of freedom in a design process.
This modulation changes the flow of energy during focusing what can improve an imaging
quality. According to the presented results angularly modified lenses of moderate numerical
apertures can be at least used successfully in machine vision and ophthalmologic applications.
Acknowledgments
We thank Salvador Bará from Santiago de Compostela University in Galiza for many fruitful
discussions, encouragement to write the present study and last but not least, for careful
reading of the manuscript. This work was supported by Warsaw University of Technology
and the Network of Excellence in Micro-Optics (NEMO).
#83838 - $15.00 USD Received 6 Jun 2007; revised 2 Jul 2007; accepted 4 Jul 2007; published 11 Jul 2007
(C) 2007 OSA 23 July 2007 / Vol. 15, No. 15 / OPTICS EXPRESS 9193
