Approximate magnifying perfect lens with positive refraction

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by  Tomas Tyc

Approximate magnifying perfect lens with positive refraction
Tom´aˇs Tyc and Martin Sˇarbort
Institute of Theoretical Physics and Astrophysics,
Masaryk University, Kotla´ˇrska´ 2, 61137 Brno, Czech Republic
(Dated: January 13, 2012)
We propose a device with a positive isotropic refractive index that creates an approximate
magnified perfect real image of an optically homogeneous three-dimensional region of space within
geometrical optics. Its key ingredient is a new refractive index profile that can work as an
approximate perfect lens on its own, having a very moderate index range.
PACSnumbers: 42.15.-i,42.15.Eq,42.30.Va
2
1
Inperfectimaging, lightraysemergingfromanypoint
0
2 P of some three-dimensional region are perfectly (stig-
matically) reassembled at another point P’, the image
n
of P. Perfect imaging has been one of the hot topics in
a
J modern optics since 2000 when J. Pendry showed that a
slab of a material with negative refractive index [1] can
2
1 work as a perfect imaging device (perfect lens). Much
effort has then been put into designing and constructing
] perfect lenses based on materials with negative refrac- FIG.1: Mediumwithaspherically-symmetricrefractiveindex
s thatfocusesparallelraysinsideanopticallyhomogeneousunit
c tive index [2], which led e.g. to demonstrations of sub-
sphere (blue) to a point at distance R from its center (here
i wavelength resolution [3].
t R=7).
p
On the other hand, as early as in 1854 J. C. Maxwell
o
found a device with an isotropic and positive refractive
.
s index that images the whole space perfectly, which he Thekeyingredientofourlensisasphericallysymmet-
c
i called fish eye. More than 150 years later, U. Leonhardt ric refractive index profile n(r) that focuses all parallel
s and T. Philbin showed that Maxwell’s fish eye provides rays within a sphere of, say, radius 1 and a constant re-
y
h perfect imaging not only in terms of geometrical optics fractive index to a point at a distance R > 1 from the
p alsointhefullframeworkofwaveoptics[4,5],andthere- center of the sphere (Fig. 1). To see why this can be
[ fore enables sub-wavelength resolution similarly as per- useful, imagine that a spherical mirror is placed at the

fect lenses based on negative refraction. Recent experi- radius R (Fig. 2). A ray that emerges from a point P
2
v ments have confirmed this [6]. Only a few other perfect insidetheunitspherereachesthemirror,isreflectedand
8 lenses with an isotropic positive refractive index were enters the sphere again. Because of the law of reflection
7 known until recently. Even less was known about de- andtheabovefocusingproperty,therayafterre-entering
1 vicesthatwouldimageperfectlyhomogeneousregionsof the sphere will be parallel with the original ray and lie
3 space, i.e., regions with a uniform refractive index. In- opposite to it from the center of the sphere. Therefore
0. deed,eveninthelastissueofBornandWolf’sPrinciples it will pass approximately through the point P’ that is
1 of Optics [7] we read that the only known example of oppositeofP,viewedfromthecenterofthesphere. This
0 such a device is a plane mirror or a combination thereof. shows that an approximate perfect real image of point P
1 This has been changed by a recent excellent work of J. is formed at point P’.
:
v C. Min˜ano [8] who proposed several new perfect lenses Tofindthecorrespondingrefractiveindexn(r),weuse
i imaginghomogeneousregionsandalsoshowedthatsome theexpressionforthepolaranglesweptbytherayduring
X
well-known optical devices such as Eaton lens or Luneb- propagation from r =1 to r =R [10] (see Fig. 3):
r urg lens [9] are in fact perfect lenses as well. All of them
a
haveunitmagnification,givinganimageofthesamesize (cid:90) R dr
as the original object. ∆φ=L √ . (1)
r n2r2−L2
1
Here we present a lens that provides an approximate
perfect real image of a homogeneous region of 3D space Here L = nrsinα and α denotes the angle between the
with an arbitrary magnification. Our device is a non- ray and the radius vector. L corresponds to the angu-
trivial combination of Maxwell’s fish eye and a new re- larmomentumintheequivalentmechanicalproblem[11]
fractiveindexprofile. Thisprofileequippedwithaspher- and is conserved in any spherically-symmetric refractive
ical mirror can work as an approximate perfect lens on index profile. Assuming that the refractive index inside
itsown,givingarealimageofahomogeneoussphereand theunitsphereisequaltoone,Lisequaltothedistance
using just a moderate refractive index range. Our lens of the ray from the center and ∆φ = arcsinL (Fig. 3).
employsisotropicmaterialwithpositiverefractiveindex. Inserting this into Eq. (1) and making the substitutions
2
gration, we obtain
(cid:88)
A g (L)=arcsinL, (4)
i i
i
whereg (L)=L(cid:82)N1ϕ (N)(N2−L2)−1/2dN. Foracho-
i 1 i
sen set of functions ϕ (N) we need to calculate the un-
i
knowncoefficientsA . Forthispurposewedefineanother
i
set of functions ψ (L) on the interval (0,1). Multiplying
j
both sides of Eq. (4) by ψ (L), integrating over L from
j
0 to 1 and interchanging the order of summation and
integration, we obtain the matrix equation
(cid:88)
σ A =B , (5)
ij i j
i
(cid:82)1
where σ = ψ (L)g (L)dL and B =
ij 0 j i j
(cid:82)1
ψ (L)arcsinLdL. The unknown coefficients A
0 j i
FIG.2: Approximateperfectlensformedbysurroundingthe are then solutions of the system of linear equations (5).
mediumfromFig.1withasphericalmirrorofradiusR(shown Using the calculated coefficients A , the approximate
i
in black). solution of function u(N) is finally obtained. In our cal-
culation, we have chosen polynomials as basis functions,
ϕ (N) = Ni,ψ (L) = Lj, with i,j running from 0 to
i j
some maximum value M.
The second, less sophisticated but equally efficient
L ��
method,wasbasedonnumericalminimizationofthelens
1 R aberration. We have represented the function N(x) in
n = 1 Eq. (2) by a polynomial N(x) = 1+(cid:80)ka xi with coef-
1 i
ficients a and calculated the aberration
i
(cid:32) (cid:33)2
(cid:88) (cid:90) X dx
A= L −arcsinL (6)
i (cid:112) i
FIG.3: Notationusedforderivingtherefractiveindexprofile N(x)2−L2
n (r). {Li} 0 i
R
as a function of a ,...,a . Here L denotes a chosen set
1 k i
of representative values of L from the interval (0,1]. To
r =ex, N =nr, we obtain an integral equation find the minimum aberration, we have employed the nu-
merical function NMinimize of the program Mathemat-
(cid:90) X dx ica. It turned out that using polynomial of degree k =5
L =arcsinL, (2)
(cid:112) and ten uniformly distributed values of L gave a refrac-
N(x)2−L2 i
0
tive index with a negligible aberration; for example, for
X = 3 the mean difference of the angle ∆φ from the
where X = lnR. As has been shown [12], this equation
correct value was 10−6 radians. It turned out that the
does not have an exact solution; we therefore employed
methodworksverywellforX ≥2,butdoesnotworkfor
two different methods to find an approximate solution
X ≤1, which would correspond to R≤e.
N(x) numerically.
Both methods give similar results for n(r). Fig. 4
In the first method, we changed the integration vari-
shows the refractive index n(r) obtained by the second
able in Eq. (2) from x to N to obtain
method for several values of R. The ray trajectories are
shown in Fig. 1 for R = 7. Of course, the whole func-
(cid:90) N1 dN tion n(r) (including the region r < 1) can be multiplied
L u(N)√ =arcsinL, (3)
N2−L2 by any real constant C without changing the focusing
1
properties of the lens. In addition, the size of the lens
where u(N) = dx/dN and N = Rn(R). This equation can be scaled by an arbitrary factor D. If we denote the
1
canbesolvedbyGalerkin’smethodforthelinearintegral original refractive index distribution by n (r) instead of
R
equations of the first kind [13] as follows. The unknown n(r) to emphasize that it depends on the parameter R,
(cid:80)
function u(N) is first expanded as u(N)= A ϕ (N), then the most general refractive index of our lens be-
i i i
where A are real coefficients and ϕ (N) is a set of func- comes n(r)=Cn (r/D). This lens focuses parallel rays
i i R
tionsontheinterval(1,N ). Substitutingthisexpression withinthesphereofradiusDtothepointatthedistance
1
into Eq. (3) and interchanging the summation and inte- DRfromitscenter. UsingforinstanceR=7andsetting
3
FIG. 4: Refractive index n (r) calculated by the method of
R
minimizing aberration for R=5 (red full line), R=10 (blue
dashed line) and R=15 (green dash-dotted line). The verti-
callinemarkstheborderofthehomogeneousregionatr=1.
z - axis
FIG. 6: Ray tracing in the approximate magnifying perfect
lens with m = 3/2. The parts of the rays in regions I, II
Region III
and III are shown in blue, red and green, respectively. The
numbers help to match the two parts of the same ray. The
absorber Region I
primesmarktheraysthatarereflectedfromthemirrorbefore
m mR entering region III while unprimed rays are reflected after
1 R leaving region III.
mirror
Region II
in region III is described by Maxwell’s fish eye profile
n (1+1/m)
n = 1 , (7)
III 1+r2/(mR2)
with the fish eye radius, i.e., the radius of ray that runs
√
atafixeddistancefromthecenterO,equaltoa=R m,
and n = n (R). It is easy to check that the refractive
FIG. 5: Regions in the approximate magnifying perfect lens, 1 R
index is continuous at the hemispherical interfaces of re-
the z axis (θ =0) being vertical. The radii of the individual
spheres are marked alon√g horizontal axis. The dashed line gion III with both regions I and II, i.e., nI(R)=nIII(R)
shows the radius a = R m of Maxwell’s fish eye profile lo- and nII(mR)=nIII(mR).
catedinRegionIII.Theobjectandimagespacesaremarked TheradiusaofMaxwell’sfisheyewaschosensuchthat
by a slightly darker color. it images the hemispheric interface of regions I and III
tothehemisphericinterfaceofregionsIIandIII.Indeed,
the relation of radial coordinates r,r(cid:48) of a point and its
C = 2.74 to bring n(r) above one in the whole lens, we image in the fish eye is rr(cid:48) =a2 [7], which is satisfied in
get 1≤n≤3.11, which is a very moderate range. our case.
Having described the lens that approximately focuses We will show now that this device approximately im-
parallel rays to a single point, we proceed now to con- ages the object space, which is the homogeneous hemi-
struction of the magnifying lens. For this purpose, we sphereofradiusr =1andrefractiveindexn=1inregion
divide the whole Euclidean 3D space into three regions I, to the image space, which is the homogeneous hemi-
denotedI,IIandIII(seeFig.5). Usingsphericalcoordi- sphere with radius r = m and refractive index n = 1/m
nates(r,θ,φ)centeredatapointO,wedefinetheregions in region II (see Fig. 5). Consider a ray emerging from
as follows. RegionI isgiven bythe conditions0≤r ≤R some point P placed at the radius vector r in the ob-
P
and 0 ≤ θ < π/2; region II is given by the conditions ject space in the direction described by a unit vector ν
0 ≤ r ≤ mR and π/2 < θ ≤ π, where m ≥ 1 is going that has a positive component in z-direction. Due to
to be the lens magnification; region III occupies the rest the focusing properties of the profile n (r) in region I,
R
of the space. Refractive index in region I is given by the it will hit the interface of regions I and III at the point
above described profile, i.e., n = n (r). Refractive in- A=Rν. The Maxwell’s fish eye profile in region III will
I R
dex in region II is n = n (r/m)/m. Refractive index ensure that the ray will then propagate along a circle
II R
4
and hit the interface of regions III and II at the point In summary, we have proposed a lens that makes an
B= −mRν. Moreover, since the ray is a segment of a approximate magnified perfect real image of a homoge-
circle, clearly the angle between the ray and the straight neous3Dregion. Performanceofthislensinthefullwave
lineAOBatpointAisthesameastheanglebetweenthe opticsregimeisasubjectofinvestigation,butwebelieve
ray and the line AOB at point B. The ray hence enters that it may provide sub-wavelength resolution similarly
region II with the same impact angle with which it left as Maxwell’s fish eye. Manufacturing this lens would be
region I. Now since the index profiles in regions I and II difficult because the refractive index of Maxwell’s fish
differ only by a spatial scaling by m and a multiplica- eye profile in region III goes to zero for r →∞, and rays
tive factor, it is clear that the shape of the ray in region forming the image do get very far from the origin. Mul-
II will be the same as its shape in region I, up to scal- tiplying the whole index profile by a large number will
ing by m. In particular, when the ray enters the image not help much because then the index in the object and
space, itwillresumeitsoriginalpropagationdirectionν, the image spaces then becomes very large. An option
but the straight line on which it lies will be m× more how to reduce refractive index range significantly would
distant from the origin O than the straight line of the be to position regions I and II differently and use some
first segment of the ray in region I. This means that the other, not spherically symmetric index profile in region
ray will be directed towards the point P’ placed at mr . III to image the surface of one sphere to the other. On
P
Now if many rays emerge from point P, they will form the other hand, the index profile n (r) that is a part of
R
an image at P’; however, this point lies outside of region thedevicecanworkasanapproximateperfectlensonits
II, so the image would be virtual. Because of this, we own,requiringjustamoderaterefractiveindexrangeand
place a mirror at the interface of regions I and II that therefore having potential to become a practical device.
changes the virtual image at P’ into a real image at a
After we published this paper on the arXiv, our col-
point P” at position Zˆmr , with Zˆ meaning the oper-
P leagueKlausBeringprovedanalyticallythatthereexists
ation of inverting the z-coordinate. Furthermore, if we
no function N(x) that solves the integral equation (2)
make the mirror double-sided, then we can take advan-
exactly and therefore the refractive index n with the
tage also of the rays that emerge “down” from P (those R
focusingpropertiesdoesnotexisteither. Beforethat,we
with negative z-component of ν). These rays will be re-
believedthatEq.(2)doeshaveanexactsolutionbutthat
flected from the mirror while still propagating in object
we just could not find it. A detailed proof of the non-
space and then reach the image P” without a further re-
existence of the solution is given in Ref. [12]. In spite
flection, as marked in Fig. 6 by primes. It turns out,
of that, we believe that even though our device works
however, that some rays with positive z-component of ν
only approximately, it still may have value for further
willhittheflatinterfacebetweenregionsIIIandII.Since
research; therefore we have replaced the paper on arXiv
it is not possible to use these rays for imaging, we block
instead of withdrawing it. In the meantime, we have
them by placing an absorber at this flat interface on the
found another method for designing magnifying perfect
sideofregionIII,keepingthemirroronthesideofregion
lenses (or absolute instruments) [14] that works exactly
II. This way, not all rays emerging from P reach P”, but
and is much more flexible than the method presented in
itisanegligibleminorityofraysthatarelostinthisway,
this paper.
especially if the magnification m is not too large. Still,
it is not necessary for a device to capture all rays to be WethankKlausBering,UlfLeonhardt,MichaelKrbek
called a perfect lens within geometrical optics [7]. and Aaron Danner for very useful discussions.
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[6] Y.G.Ma,C.Ong,S.Sahebdivan,T.Tyc,andU.Leon- New J. Phys. 13, 115004 (2011).
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