P. Török

S. H. Wiersma

T. D. Visser

Török

Wiersma

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Focusing through an interface:?scanning and localizing the intensity

Wiersma, S.H.

Visser, T.D.

Torok, P.

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March 15, 1998 / Vol. 23, No. 6 / OPTICS LETTERS 415Focusing through an interface: scanning and
localizing the intensity
S. H. Wiersma and T. D. Visser
Department of Physics and Astronomy, Free University, De Boelelaan 1081, 1081 HV Amsterdam, The Netherlands
P. To¨ro¨k
Department of Engineering Science, University of Oxford, Parks Road, Oxford OX1 3PJ, UKReceived December 16, 1997
Focusing light through an interface leads to an aberrated intensity distribution that is highly extended with a
relatively low peak intensity. We present a method, using a well-chosen annular aperture, that can greatly
improve the localization of the intensity about a prescribed point on the axis. Also, the intensity at that
point can be increased significantly. By continuously varying the annulus radii, we can scan the intensity
peak through the second medium. This localization and scanning method has possible applications in three-
dimensional imaging and lithography.  1998 Optical Society of America
OCIS codes: 140.5960, 230.3670, 230.5590, 260.3800.The present authors and others have recently shown
that focusing through a dielectric interface produces
strongly aberrated wave fronts.1 – 4 An example of this
process is shown in Fig. 1. Compared with the case
when the two refractive indices are equal, here the
axial intensity distribution is elongated and exhibits
many secondary peaks.
Although the form of the intensity distribution can
be calculated accurately with the results of the studies
mentioned above, it is highly desirable to have a means
to compensate for the aberrations to decrease the width
of the distribution and also to increase the intensity
locally. Methods suggested thus far, such as phase
masks and adaptive-optics solutions, are difficult to
implement in practical optical systems. Although it is
relatively easy to realize an annular aperture to reduce
on-axis aberrations, this method appears to lower the
intensity because only a limited fraction of the incident
beam passes the aperture. It is our purpose in this
Letter to demonstrate that the use of a well-chosen
annular aperture can lead to a simultaneous increase
in the intensity and substantial decrease in the width
of the axial intensity distribution. In particular, the
peak intensity can be increased in this manner. In
addition, we present a new scanning mechanism.
In Ref. 2 it was derived that, for the case of an
unobscured lens, the electric-f ield distribution on the
z axis (the axis of rotational symmetry, perpendicular
to the interface) in the second medium is given by
Exszd ­ Cszd
Z V1
0
expsik2s 2 ik1tdgsu1, zddu1 , (1)
with
Cszd ­
f
2
s f 2 dd2
µ
z
f 2 d
2 1
¶
expsik1f d , (2)
gsu1, zd ­
µ
1
s3
2
ik2
s2
¶
shs 1 hp cos u2dtan u1 . (3)0146-9592/98/060415-03$15.00/0The subscript x in Eq. (1) indicates that the incident
plane wave is linearly polarized along the x direction.
V1 is the semiaperture angle of the lens, f is the focal
length, and u2 follows from u1 through Snell’s law.
Also, f 2 d is the distance between the focus of the
lens and the interface, ki si ­ 1, 2d is the wave number
in medium i, and hs and hp are Fresnel transmission
coefficients. The origin of the z axis coincides with the
focal point, and the direction of increasing z is toward
the lens. The functions s and t are defined as
tsu1d ­
f 2 d
cos u1
, (4)
ssu1, zd ­ ft2 1 z2 2 2zs f 2 ddg1/2 . (5)
A stationary phase analysis of this integral shows
that the main contribution to the intensity at the axial
Fig. 1. Intensity distribution (in arbitrary units) along
the z axis (in micrometers) for an unobscured lens with
a semiaperture angle V1 ­ 45– and a focal length of
1022 m. The lens is positioned in air (n1 ­ 1d. The
interface with the second medium sn2 ­ 1.51d is located
300 mm in front of the focus of the lens. The wavelength
in air is l0 ­ 632.8 nm. 1998 Optical Society of America
416 OPTICS LETTERS / Vol. 23, No. 6 / March 15, 1998Fig. 2. Intensity I sz ­ 2203 mmd as a function of ulow and
uhigh. All other parameters are as in Fig. 1.
position z is due to those rays for which the phase
factor is an extremum. It is possible to show that this
condition occurs precisely for that ray that is refracted
according to Snell’s law such that it crosses the axis
at z. The angle of incidence in the first medium of
this ray is denoted ustatszd. By choosing, instead of an
unobscured lens, an annular aperture that corresponds
to the angular interval fulow szd, uhighszdg that contains
ustatszd, we can modify the axial intensity distribution
significantly. By numerically optimizing the size of
this angular interval, we can achieve both of our aims
(viz., the localization of the intensity around z and
the increased intensity at z). Note that this can be
done only for points that lie between the so-called
geometrical shadow boundaries.2 As an illustration of
our method we show how, e.g., the secondary peak at
z ­ 2203 mm in Fig. 1 can be selected as the center of
the improved intensity distribution. The dependence
of the intensity at this particular point on the interval
fulow , uhighg is depicted in Fig. 2. The optimal annulus
(i.e., the one that give the highest intensity) in this
case was found to be f28.04–, 36.71–g. Note that this
interval indeed includes ustatsz ­ 2203d ­ 32.7–. For
a lens that obeys the sine condition,5 the relation
between the radius r of the annulus (placed in the
back focal plane or the exit pupil) and u is given by
r ­ f sin u.
The axial intensity distribution for the optimized
annulus is shown in Fig. 3. The intensity is now
much more localized, and the number of secondary
maxima and their heights are both strongly reduced.
The intensity profile is indeed centered about the
prescribed axial position. Also, the intensity at z ­
2203 mm has changed from 0.47 to 0.70, an increase of
more than 48%.
In the same way, the highest peak (at
z ­ 2166.7 mm) in Fig. 1 can be enhanced. For
a u1 interval of f0–, 25.51–g the intensity at that point is
increased by 25%. At the same time, a much narrower
intensity profile is obtained. It is striking that weobtain this increased peak intensity by blocking 63% of
the total incident power.
In Fig. 4 the value of ustat as a function of the axial
position is shown. Also, the optimized values (i.e.,
those that lead to the highest intensity) of ulow and uhigh
for all axial points z between the geometrical shadow
boundaries are depicted. Note that ulow suddenly
becomes nonzero near the position of the original
maximum. This result is related to the fact that the
paraxial rays, which together make up the maximum
peak for the case of an unobscured lens, gradually get
out of phase with the rays near ustat as z becomes more
negative. Therefore, from a certain z value onward,
these paraxial rays are no longer part of the optimized
annulus. Also, it is seen that from certain z values on,
uhigh ­ 45–. This is so because uhigh cannot exceed V1.
When the annular illumination interval is chosen as
fulow szd, uhighszdg the intensity distribution will be cen-
tered about z. Not only ustat but also ulow and uhigh
are monotonic functions of z. Therefore we can, by
varying the radii of the annulus in a continuous man-
ner, scan a single intensity peak through the second
medium between the geometrical shadow boundaries.
This new scanning mechanism may be useful in, e.g.,
three-dimensional imaging and lithography.
Fig. 3. Intensity distribution along the z axis (in micro-
meters) for an optimized annular aperture with 28.04– ,
u1 , 36.71–. All other parameters and the normalization
are as in Fig. 1.
Fig. 4. Stationary phase ustat (dashed curve) and the two
interval limits ulow (lower curve) and uhigh (upper curve)
that give optimal intensity as a function of axial position z
(in micrometers). All parameters are as in Fig. 1.
March 15, 1998 / Vol. 23, No. 6 / OPTICS LETTERS 417In conclusion, we have presented a method to im-
prove the width of the axial intensity distribution that
results when light is focused through an interface.
This method, which can be implemented easily, em-
ploys an annular aperture with optimized radii. It
results in a higher localized intensity at a prescribed
point while reducing the secondary maxima strongly.
The intensity of the original peak can also be increased
in this manner. In addition, this method provides a
novel scanning mechanism.
Address correspondence to T. D. Visser: E-mail
address tvisser@nat.vu.nl.References
1. P. To¨ro¨k, P. Varga, Z. Laczik, and G. R. Booker, J. Opt.
Soc. Am. A 12, 325 (1995).
2. S. H. Wiersma and T. D. Visser, J. Opt. Soc. Am. A 13,
320 (1996).
3. S. H. Wiersma, P. To¨ro¨k, T. D. Visser, and P. Varga, J.
Opt. Soc. Am. A 14, 1482 (1997).
4. V. Dhayalan, ‘‘Focusing on Electromagnetic waves,’’
Ph.D. dissertation (University of Bergen, Bergen, Nor-
way, 1996).
5. M. Born and E. Wolf, Principles of Optics, 6th ed.
(Cambridge U. Press, Cambridge, 1997).


Focusing through an Interface: Scanning and Localizing the Intensity.

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VU Research PortalFocusing through an Interface: Scanning and Localizing the Intensity.Wiersma, S.H.; Visser, T.D.; Torok, P.published inOptics Letters1998DOI (link to publisher)10.1364/OL.23.000415document versionPublisher's PDF, also known as Version of recordLink to publication in VU Research Portalcitation for published version (APA)Wiersma, S. H., Visser, T. D., & Torok, P. (1998). Focusing through an Interface: Scanning and Localizing theIntensity. Optics Letters, 23(6), 415-417. https://doi.org/10.1364/OL.23.000415General rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.            • Users may download and print one copy of any publication from the public portal for the purpose of private study or research.            • You may not further distribute the material or use it for any profit-making activity or commercial gain            • You may freely distribute the URL identifying the publication in the public portal ?Take down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.E-mail address:vuresearchportal.ub@vu.nlDownload date: 23. May. 2021March 15, 1998 / Vol. 23, No. 6 / OPTICS LETTERS 415Focusing through an interface: scanning andlocalizing the intensityS. H. Wiersma and T. D. VisserDepartment of Physics and Astronomy, Free University, De Boelelaan 1081, 1081 HV Amsterdam, The NetherlandsP. TörökDepartment of Engineering Science, University of Oxford, Parks Road, Oxford OX1 3PJ, UKReceived December 16, 1997Focusing light through an interface leads to an aberrated intensity distribution that is highly extended with arelatively low peak intensity. We present a method, using a well-chosen annular aperture, that can greatlyimprove the localization of the intensity about a prescribed point on the axis. Also, the intensity at thatpoint can be increased significantly. By continuously varying the annulus radii, we can scan the intensitypeak through the second medium. This localization and scanning method has possible applications in three-dimensional imaging and lithography.  1998 Optical Society of AmericaOCIS codes: 140.5960, 230.3670, 230.5590, 260.3800.The present authors and others have recently shownthat focusing through a dielectric interface producesstrongly aberrated wave fronts.1 – 4 An example of thisprocess is shown in Fig. 1. Compared with the casewhen the two refractive indices are equal, here theaxial intensity distribution is elongated and exhibitsmany secondary peaks.Although the form of the intensity distribution canbe calculated accurately with the results of the studiesmentioned above, it is highly desirable to have a meansto compensate for the aberrations to decrease the widthof the distribution and also to increase the intensitylocally. Methods suggested thus far, such as phasemasks and adaptive-optics solutions, are difficult toimplement in practical optical systems. Although it isrelatively easy to realize an annular aperture to reduceon-axis aberrations, this method appears to lower theintensity because only a limited fraction of the incidentbeam passes the aperture. It is our purpose in thisLetter to demonstrate that the use of a well-chosenannular aperture can lead to a simultaneous increasein the intensity and substantial decrease in the widthof the axial intensity distribution. In particular, thepeak intensity can be increased in this manner. Inaddition, we present a new scanning mechanism.In Ref. 2 it was derived that, for the case of anunobscured lens, the electric-f ield distribution on thez axis (the axis of rotational symmetry, perpendicularto the interface) in the second medium is given byExszd ­ CszdZ V10expsik2s 2 ik1tdgsu1, zddu1 , (1)withCszd ­f2s f 2 dd2µzf 2 d2 1∂expsik1f d , (2)gsu1, zd ­µ1s32ik2s2∂shs 1 hp cos u2dtan u1 . (3)0146-9592/98/060415-03$15.00/0The subscript x in Eq. (1) indicates that the incidentplane wave is linearly polarized along the x direction.V1 is the semiaperture angle of the lens, f is the focallength, and u2 follows from u1 through Snell’s law.Also, f 2 d is the distance between the focus of thelens and the interface, ki si ­ 1, 2d is the wave numberin medium i, and hs and hp are Fresnel transmissioncoefficients. The origin of the z axis coincides with thefocal point, and the direction of increasing z is towardthe lens. The functions s and t are defined astsu1d ­f 2 dcos u1, (4)ssu1, zd ­ ft2 1 z2 2 2zs f 2 ddg1/2 . (5)A stationary phase analysis of this integral showsthat the main contribution to the intensity at the axialFig. 1. Intensity distribution (in arbitrary units) alongthe z axis (in micrometers) for an unobscured lens witha semiaperture angle V1 ­ 45± and a focal length of1022 m. The lens is positioned in air (n1 ­ 1d. Theinterface with the second medium sn2 ­ 1.51d is located300 mm in front of the focus of the lens. The wavelengthin air is l0 ­ 632.8 nm. 1998 Optical Society of America416 OPTICS LETTERS / Vol. 23, No. 6 / March 15, 1998Fig. 2. Intensity I sz ­ 2203 mmd as a function of ulow anduhigh. All other parameters are as in Fig. 1.position z is due to those rays for which the phasefactor is an extremum. It is possible to show that thiscondition occurs precisely for that ray that is refractedaccording to Snell’s law such that it crosses the axisat z. The angle of incidence in the first medium ofthis ray is denoted ustatszd. By choosing, instead of anunobscured lens, an annular aperture that correspondsto the angular interval fulow szd, uhighszdg that containsustatszd, we can modify the axial intensity distributionsignificantly. By numerically optimizing the size ofthis angular interval, we can achieve both of our aims(viz., the localization of the intensity around z andthe increased intensity at z). Note that this can bedone only for points that lie between the so-calledgeometrical shadow boundaries.2 As an illustration ofour method we show how, e.g., the secondary peak atz ­ 2203 mm in Fig. 1 can be selected as the center ofthe improved intensity distribution. The dependenceof the intensity at this particular point on the intervalfulow , uhighg is depicted in Fig. 2. The optimal annulus(i.e., the one that give the highest intensity) in thiscase was found to be f28.04±, 36.71±g. Note that thisinterval indeed includes ustatsz ­ 2203d ­ 32.7±. Fora lens that obeys the sine condition,5 the relationbetween the radius r of the annulus (placed in theback focal plane or the exit pupil) and u is given byr ­ f sin u.The axial intensity distribution for the optimizedannulus is shown in Fig. 3. The intensity is nowmuch more localized, and the number of secondarymaxima and their heights are both strongly reduced.The intensity profile is indeed centered about theprescribed axial position. Also, the intensity at z ­2203 mm has changed from 0.47 to 0.70, an increase ofmore than 48%.In the same way, the highest peak (atz ­ 2166.7 mm) in Fig. 1 can be enhanced. Fora u1 interval of f0±, 25.51±g the intensity at that point isincreased by 25%. At the same time, a much narrowerintensity profile is obtained. It is striking that weobtain this increased peak intensity by blocking 63% ofthe total incident power.In Fig. 4 the value of ustat as a function of the axialposition is shown. Also, the optimized values (i.e.,those that lead to the highest intensity) of ulow and uhighfor all axial points z between the geometrical shadowboundaries are depicted. Note that ulow suddenlybecomes nonzero near the position of the originalmaximum. This result is related to the fact that theparaxial rays, which together make up the maximumpeak for the case of an unobscured lens, gradually getout of phase with the rays near ustat as z becomes morenegative. Therefore, from a certain z value onward,these paraxial rays are no longer part of the optimizedannulus. Also, it is seen that from certain z values on,uhigh ­ 45±. This is so because uhigh cannot exceed V1.When the annular illumination interval is chosen asfulow szd, uhighszdg the intensity distribution will be cen-tered about z. Not only ustat but also ulow and uhighare monotonic functions of z. Therefore we can, byvarying the radii of the annulus in a continuous man-ner, scan a single intensity peak through the secondmedium between the geometrical shadow boundaries.This new scanning mechanism may be useful in, e.g.,three-dimensional imaging and lithography.Fig. 3. Intensity distribution along the z axis (in micro-meters) for an optimized annular aperture with 28.04± ,u1 , 36.71±. All other parameters and the normalizationare as in Fig. 1.Fig. 4. Stationary phase ustat (dashed curve) and the twointerval limits ulow (lower curve) and uhigh (upper curve)that give optimal intensity as a function of axial position z(in micrometers). All parameters are as in Fig. 1.March 15, 1998 / Vol. 23, No. 6 / OPTICS LETTERS 417In conclusion, we have presented a method to im-prove the width of the axial intensity distribution thatresults when light is focused through an interface.This method, which can be implemented easily, em-ploys an annular aperture with optimized radii. Itresults in a higher localized intensity at a prescribedpoint while reducing the secondary maxima strongly.The intensity of the original peak can also be increasedin this manner. In addition, this method provides anovel scanning mechanism.Address correspondence to T. D. Visser: E-mailaddress tvisser@nat.vu.nl.References1. P. Török, P. Varga, Z. Laczik, and G. R. Booker, J. Opt.Soc. Am. A 12, 325 (1995).2. S. H. Wiersma and T. D. Visser, J. Opt. Soc. Am. A 13,320 (1996).3. S. H. Wiersma, P. Török, T. D. Visser, and P. Varga, J.Opt. Soc. Am. A 14, 1482 (1997).4. V. Dhayalan, ‘‘Focusing on Electromagnetic waves,’’Ph.D. dissertation (University of Bergen, Bergen, Nor-way, 1996).5. M. Born and E. Wolf, Principles of Optics, 6th ed.(Cambridge U. Press, Cambridge, 1997).

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Focusing through an Interface: Scanning and Localizing the Intensity.

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