Novel concepts of negative- optics in master’s level educational courses
The novel ideas of negative refraction index () optics suitable for teaching special courses in universities at master’s level are systematized and analyzed. The most important innovative ideas in this field are recounted in the logical order necessary for achieving the best understanding of the mat...
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| Опубліковано в: : | Semiconductor Physics Quantum Electronics & Optoelectronics |
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
2017
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| Цитувати: | Novel concepts of negative- optics in master’s level educational courses / G.Yu. Rudko // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2017. — Т. 20, № 2. — С. 235-239. — Бібліогр.: 7 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860292457192751104 |
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| author | Rudko, G.Yu. |
| author_facet | Rudko, G.Yu. |
| citation_txt | Novel concepts of negative- optics in master’s level educational courses / G.Yu. Rudko // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2017. — Т. 20, № 2. — С. 235-239. — Бібліогр.: 7 назв. — англ. |
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| description | The novel ideas of negative refraction index () optics suitable for teaching special courses in universities at master’s level are systematized and analyzed. The most important innovative ideas in this field are recounted in the logical order necessary for achieving the best understanding of the material. They are: the opposite signs of phase and group velocities of light; the change of the ordered right-hand triad of vectors E, B and V from the right-handed to the left-handed one; the change of the sign of the Doppler effect; the bent of the incident light, when entering the negative n material, in the “wrong” direction; the emergence of new class of materials – artificial metamaterials that have negative n; current state of the search for possibilities to achieve invisibility.
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| first_indexed | 2026-03-20T09:54:25Z |
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Semiconductor Physics, Quantum Electronics & Optoelectronics, 2017. V. 20, N 2. P. 235-239.
doi: https://doi.org/10.15407/spqeo20.02.235
© 2017, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
235
PACS 42.15.-i, 42.25.-p, 78.67.Pt, 01.40.gb
Novel concepts of negative-n optics
in master’s level educational courses
G.Yu. Rudko1,2
1V. Lashkaryov Institute of Semiconductors Physics, NAS of Ukraine
45, prospect Nauky, 03028 Kyiv, Ukraine; e-mail: g.yu.rudko@gmail.com
2National University of Kyiv-Mohyla Academy
2, Skovorody str., 04070 Kyiv, Ukraine
Abstract. The novel ideas of negative refraction index (n) optics suitable for teaching
special courses in universities at master’s level are systematized and analyzed. The most
important innovative ideas in this field are recounted in the logical order necessary for
achieving the best understanding of the material. They are: the opposite signs of phase
and group velocities of light; the change of the ordered right-hand triad of vectors E, B
and V from the right-handed to the left-handed one; the change of the sign of the Doppler
effect; the bent of the incident light, when entering the negative n material, in the
“wrong” direction; the emergence of new class of materials – artificial metamaterials that
have negative n; current state of the search for possibilities to achieve invisibility.
Keywords: negative refraction index, metamaterial, invisibility..
Manuscript received 11.01.17; revised version received 25.04.17; accepted for
publication 14.06.17; published online 18.07.17.
1. Introduction
Nano-physics and nanomaterials are a rapidly develo-
ping area of knowledge. With the emergence of this vast
field many conventional concepts are revised and new
approaches in various fields of physics appear. Opening
new fascinating horizons in physics leads to the necessi-
ty of changing the tuition programs, and, especially, the
special courses at graduate students level. Among the
impressive new effects that have already gained recogni-
tion of public, the most spectacular are the phenomena
observed in the newly-constructed class of meta-
materials. These materials were purposely developed to
achieve the naturally non-existing property of refractive
media – negative refraction index. After the first reports
on fabrication of the materials with negative refraction
index, predictions of the theory of negative-n media
(which was seemingly far from reality) were verified.
The coincidence of the experimental results obtained
with the predictions of the theory that at the time of its
appearance looked like a game of mind proves the power
of purely mathematical approaches and is favorable for
the development of student’s thinking flexibility.
The development of new revolutionary concepts in
optics goes back to 1967 when Viktor Veselago
published the paper [1] with the detailed analysis of
electrodynamics of light propagation in hypothetic
media that are characterized by negative n. At that time,
this paper sounded as a mere theoretical curiosity. In the
assumption of the existence of negative-n materials
Veselago predicted that light would behave in ways not
found in nature and many of generally assumed and
convenient properties of optical systems would be
altered. In the 70-s of the previous century, there were
no ways for experimental verification of this theory,
because none of known natural materials has the
required properties. Moreover, at that time physicists
were not sure whether the existence of materials with
both negative ε and μ is forbidden because of first
principles or not.
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2017. V. 20, N 2. P. 235-239.
doi: https://doi.org/10.15407/spqeo20.02.235
© 2017, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
236
This obstacle has been overcome almost 30 years
later when a group of scientists reported on the
fabrication of a hybrid system, a metamaterial, that met
the demands [2].
Thus, the new field of optics had proven its
viability and got being known far beyond the society of
physicists, therefore, at present there is an urgent need to
develop new instructional approaches that would prepare
the students for perception of altered physical concepts.
In what follows, we propose the sketch of graduate–level
instruction in introducing novel concepts based on
negative index of refraction.
2. Basic concepts of wave optics in negative-n media
Originally the source for the revision of basic concepts
in optics lies in the relation between the refractive index
of a medium and its fundamental properties – electrical
permittivity and magnetic permeability. As this relation
is n2 = εμ or n = (εμ)1/2, the value of refractive index
must not change after reverse of signs for both ε and μ.
Veselago started to explore possible changes of
electrodynamical consequences with the analysis of
Maxwell equations. In these equations, the values ε and
μ are present not as a product but separately, thus their
sign is of importance. In the assumption that ε < 0 and
μ < 0, Veselago got the following conclusions.
For propagation of a monochromatic plane wave,
he found that, on the contrary to the conventional case of
positive n when the triad of vectors k, E and H is right-
handed, the corresponding triad in the negative-n media
has to be a left-handed one. However, the direction of
the Pointing vector S = c/4π [E, H] turned out to be
independent of the sign of n. Physically, it means that
the phase and group velocities of light have opposite
signs. In other words, the wave fronts of a wave travel in
one direction, while the energy of this wave propagates
in the opposite direction.
This consequence immediately leads to another
exotic conclusion: the sign of Doppler shift depends on
sign of refractive index. In negative-n materials, the
waves of a source that moves away from the observer
are registered as the ones having shorter wavelengths,
while in the materials with n > 0 the wavelengths
become longer. Similarly, in the materials with n > 0 the
Vavilov–Cherenkov effect will also be inversed.
3. Basic concepts of ray optics in negative-n media
One of the most difficult tasks that has to be addressed
while preparing a modern ray optics course narrative is
to achieve understanding of rays propagation in the
negative-n materials. This concept in particular leads to
a cognitive dissonance in light of unexpected
visualization of Snell’s law as compared to that known
by students since high school. Detailed explanation of
each effect caused by newly learned concept of negative
index of refraction necessitates thorough re-analysis of
many well-known effects and devices.
Snell’s law for negative-n materials [1, 4]. The
very first concepts of optics that are studied in the school
are the concepts of geometrical optics or ray optics,
which describe propagation of light in terms of rays. The
simplicity and obviousness of the ray optics laws, such
as law of reflection and law of refraction (Snell’s law)
makes them easy for understanding and memorizing.
While studying the latter law, the value named the
refractive index (or index of refraction), n, has been
introduced. It determines how much the light beam is
bent, or refracted, when crossing an interface between
two different materials. The mathematical formulation of
Snell’s law is n1 sinθ1 = n2 sinθ2, where θ1 and θ2 are the
angles of incidence and refraction, respectively, of a ray
crossing the interface between two materials with
refractive indices n1 and n2. The refractive index is used
to represent the factor by which the velocity of light is
reduced when traveling through a refractive medium
with respect to its value in vacuum: the velocity of light
in a medium is V = c/n, where c is the speed of light in
vacuum. The traditional illustration of Snell’s law
known from school is shown in Fig. 1a. Note that at
school level, the students are taught to use only the
positive value n which is, as it was shown by Veselago,
not obvious a priori.
More strict approach to refraction index interpreta-
tion that implies to the hypothetic possibility of negative
sign of the refraction index (in accordance with the
possibility of the negative sign of square root (εμ)1/2)
leads to refracting light in a way that light is not nor-
mally refracted in nature. A beam incident on a material
with negative n from a material with positive n refracts
to the same side of the normal as the incident ray. This
situation is illustrated by Fig. 1b. It shows that the light
beam in the medium with negative n is deflected in the
“wrong” direction (from the conventional point of view).
Thus, the unshakable conviction of students about the
direction of light that was gained during previous studies
must be revised. However, it should be stressed that
Snell’s law is not violated. Indeed, the change of the sign
of refraction index leads merely to the change of sign of
the angle of refraction.
Fig. 1. The beam path at the refraction on the boundary of
vacuum and medium with the refractive index n. a) The
refractive index of the medium is positive. b) The refractive
index of the medium is negative. 1 – incident beam; 2 –
reflected beam, 3 – refracted beam at n > 0; 4 – refracted beam
at n < 0.
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2017. V. 20, N 2. P. 235-239.
doi: https://doi.org/10.15407/spqeo20.02.235
© 2017, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
237
Changes in lenses properties. One more conviction
of students has to be ruined when teaching negative-n
optics. While the biconvex lens (or plano-convex) made
of positive-n material is positive or converging lens, the
same type of lens made of negative-n material is
negative or diverging. The collimated parallel beam
passing through this lens will be diverged (spread). And
vice versa, the light passing through the biconcave or
plano-concave lens made of negative-n material
converges to a spot (a focus). Thus, the lens is a positive
or converging lens.
Flat quasi-lens [5]. Due to different refraction, a
flat parallel slab of the material with negative n can
exhibit “focusing” properties, which also contradicts to
everyday experience of students. It can be shown that the
rays from a point source impinging on a flat, parallel
slab of negative-n material would be refocused to a point
on the opposite side of the material. The condition of
this refocusing is that the width of the slab is larger than
the distance between the source and the slab. Fig. 2
illustrates the beams passing the parallel slabs of
positive-n (a) and negative-n (b) materials. Despite
certain similarity with action of the lens (the possibility
to focus light from a point source) the slab of negative n
material is not able to focus parallel beam of light into
point, thus it is named quasi-lens. Another name of this
device is a super-lens or perfect lens, because it
transforms 3D object into 3D image without distortions
and overcomes the diffraction limit (i.e. there are no
losses of evanescent waves).
Fig. 2. The beams passing through the parallel slabs of posi-
tive-n (a) and negative-n (b) materials.
4. Metamaterials – artificial materials
with negative n: basic ideas and realization
As it was mentioned above, none of the naturally
existing materials has negative refraction index. In the
Table, signs of the electrical permittivity and magnetic
permeability of all natural materials are systematized
(rows 1-3). The row 4 of this Table corresponds to the
artificial materials that will be discussed below. All
these materials can be distinguished in accordance with
specific features of their reflectance and refraction.
Fig. 3 demonstrates interaction of the incident light
beam with the materials with varied signs of ε and μ.
In view of the absence of natural materials with
negative n (in other words, with simultaneously negative
electrical permittivity and magnetic permeability) and in
order to bring to life the promising predictions of
negative-n optics, the new class of materials has been
developed. These materials have diverse notifications.
Mostly, they are named metamaterials because they are
not the convenient materials built of atoms and
molecules, instead, they are constructed of tiny elements
fabricated by modern nano-technologies. Another name
of these materials is negative refraction index materials
in accordance with the main property of interest. One
can also find the name left-handed materials because of
different orientation of E, H and k vectors in the triad.
One more name is backward wave media, which is
caused by the unusual direction of phase velocity – in
the opposite direction to the energy propagation. Thus,
all these new terms and concepts should be elucidated
for students and deserve including to the special courses
of master’s level, keeping in mind that all these names
are used in accordance with the focus of corresponding
discussion. Hereafter, we will use the name
metamaterials.
The fundamental idea of creating negative-n
materials was the construction of such a hybrid system
that would combine the elements responsible for
negative electrical permittivity and negative magnetic
permeability. The first success was achieved by the
authors of [2], who have assembled arrays of tiny
components.
Table.
Name of material Electrical
permittivity ε
Magnetic
permeability μ
Naturally existing materials
Right-handed material (the name appeared
after the introduction of the term left-handed
materials)
ε > 0 μ > 0, Usual dielectrics,
ferroelectrics
No special name ε < 0 μ > 0, Plasmas:
– electronic plasma in metals,
– ionic plasma in ionosphere
No special name ε > 0 μ < 0 No isotropic material exist with μ<0
Anisotropic ferromagnets in magnetic field
can exhibitμ<0
Left-handed material ε < 0 μ < 0 Only artificial metamaterials
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2017. V. 20, N 2. P. 235-239.
doi: https://doi.org/10.15407/spqeo20.02.235
© 2017, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
238
They were the metallic rods, which were
responsible for ε < 0, and split-ring resonators, which
provided μ < 0. These components must be of
subwavelength size, therefore, the first metamaterial
worked at microwave frequencies. More modern
metamaterials are the nanostructured artificial media
with specific building blocks. Fabrication of these
structures poses a big challenge because of rather high
resolution demands: the shorter the wavelength, the
higher spatial resolution [6]. High reproducibility of the
building blocks geometry and composition is also of
great importance.
5. Invisibility or electromagnetic cloaking
One of the most thrilling perspectives of negative-n
optics is the possibility to achieve invisibility, thus it
deserves a separate paragraph. Invisibility that for
centuries and centuries has been a subject of fairy tales
comes to the scene due to the emergence of
nanotechnologies that make it possible to produce the
materials with negative n. Up to now, real invisibility
was associated either with absolute transparency or
with mimicry (like that observed for chameleon or
certain species of bottom-dwelling fishes that are
champions of camouflage). The fictitious artifacts, like
Harry Potter’s cloak, were supposed just to provide
visual disappearance of an object. Regardless fairy
tales, what would be the demands to a possible device
that could ensure invisibility of a macroscopic object?
This device must have the following characteristics: it
must neither reflect nor scatter the incident light, it
must not produce a shadow, and, naturally, it must not
absorb the light. That is, this device must not perturb
the electromagnetic field around the object. Obviously,
all these demands can be fulfilled only in a limited
spectral range. And, naturally, the most appealing is the
visible range. The devices for gaining invisibility are
built of novel metamaterials. In them, an object is
hidden inside a hollow surrounded by a metamaterial (a
metamaterial cloak). This coverage works by bending
the light around an object in the way shown in Fig. 4b.
Scattering of light by the same object without coverage
is shown for comparison in Fig. 4a. It is obvious that
this type of invisibility differs from that used in Stealth
technology, because its aim is not to prevent reflection
of electromagnetic waves. Review of current
experimental state-of-the-art of simple core–shell
invisibility cloaking can be found in [7]. It should be
noted that modern technologies provide the possibility
to construct artificial materials with regulated values of
ε and μ (positive as well as negative). However, the
metamaterials are not ideal media, they absorb light
thus the presence of an object can cast weak shadows;
thus, the cloaking performance degrades with in-
creasing optical losses.
Fig. 3. Light reflection and refraction in the materials
with notified ε and μ signs at the boundary with air.
Fig. 4. Light scattering by an object (a) and invisibility
(b) gained due to propagation of light inside the co-
verage made of metamaterial.
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2017. V. 20, N 2. P. 235-239.
doi: https://doi.org/10.15407/spqeo20.02.235
© 2017, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
239
6. Conclusions
In summary, the present paper is a brief overview of new
fundamental concepts and ideas that appeared in physics
with the emergence of the new vast field – the optics of
negative-n materials. The most important innovative
ideas in this field are intuitively controversial to the
well-established concepts of conventional physics. By
mastering these new non-trivial phenomena, the students
get deeper insight into the nature of light and its
interaction with refracting media. New tuition material
has to be communicated in a such way that it increases
elucidation and is built on simple concepts, steadily
increasing the degree of difficulty when letting new
concepts settle in students memory. Students need to
grasp that the group velocity direction is opposite to
phase velocity direction, that E, H and k-vectors follow
left-hand screw rule instead of right-hand, that Doppler
effect acquires negative sign, etc. The special appeal of
the course is achieving of understanding of invisibility
(or electromagnetic cloaking) that causes acute interest
and promises attractive applications. This interest
stimulates learning the basics of nanotechnologies that
are the only known interest to produce the materials with
negative n (metamaterials). The revolutionary ideas of
building the materials that never existed before facilitate
broadening of the horizons of comprehension of physics
and optics, in particular.
References
1. Veselago V.G. The electrodynamics of substances
with simultaneously negative values of ε and μ.
Physics – Uspekhi. 1968. 10, No. 4. P. 509–514.
2. Shelby R.A., Smith D.R., Schultz S. Experimental
verification of a negative index of refraction.
Science. 2001. 292(5514). P. 77–79.
3. Smith D.R., Padilla W.J., Vier D.C., Nemat-Nasser
S.C., Schultz S. Composite medium with
simultaneously negative permeability and
permittivity. Phys. Rev. Lett. 2000. 84. P. 4184.
4. Rautian S.G. Reflection and refraction at the
boundary of medium with a negative group
velocity. Physics – Uspekhi. 2008. 51, No. 10. P.
981–988.
5. Pendry J.B. Negative refraction makes a perfect
lens. Phys. Rev. Lett. 2000. 85, No. 18. P. 3966–
3969.
6. Shadrivov I.V., Sukhorukov A.A., and Kivshar
Yu.S. Beam shaping by a periodic structure with
negative refraction. Appl. Phys. Lett. 2003. 82. P.
3820–3822.
7. Schittny R., Niemeyer A., Mayer F., Naber A.,
Kadic M., Wegener M. Invisibility cloaking in
light-scattering media. Laser & Photon. Rev. 2016.
10, No. 3. P. 382–408.
|
| id | nasplib_isofts_kiev_ua-123456789-214926 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1560-8034 |
| language | English |
| last_indexed | 2026-03-20T09:54:25Z |
| publishDate | 2017 |
| publisher | Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
| record_format | dspace |
| spelling | Rudko, G.Yu. 2026-03-04T12:49:20Z 2017 Novel concepts of negative- optics in master’s level educational courses / G.Yu. Rudko // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2017. — Т. 20, № 2. — С. 235-239. — Бібліогр.: 7 назв. — англ. 1560-8034 PACS: 42.15.-i, 42.25.-p, 78.67.Pt, 01.40.gb https://nasplib.isofts.kiev.ua/handle/123456789/214926 https://doi.org/10.15407/spqeo20.02.235 The novel ideas of negative refraction index () optics suitable for teaching special courses in universities at master’s level are systematized and analyzed. The most important innovative ideas in this field are recounted in the logical order necessary for achieving the best understanding of the material. They are: the opposite signs of phase and group velocities of light; the change of the ordered right-hand triad of vectors E, B and V from the right-handed to the left-handed one; the change of the sign of the Doppler effect; the bent of the incident light, when entering the negative n material, in the “wrong” direction; the emergence of new class of materials – artificial metamaterials that have negative n; current state of the search for possibilities to achieve invisibility. en Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України Semiconductor Physics Quantum Electronics & Optoelectronics Novel concepts of negative- optics in master’s level educational courses Article published earlier |
| spellingShingle | Novel concepts of negative- optics in master’s level educational courses Rudko, G.Yu. |
| title | Novel concepts of negative- optics in master’s level educational courses |
| title_full | Novel concepts of negative- optics in master’s level educational courses |
| title_fullStr | Novel concepts of negative- optics in master’s level educational courses |
| title_full_unstemmed | Novel concepts of negative- optics in master’s level educational courses |
| title_short | Novel concepts of negative- optics in master’s level educational courses |
| title_sort | novel concepts of negative- optics in master’s level educational courses |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/214926 |
| work_keys_str_mv | AT rudkogyu novelconceptsofnegativeopticsinmastersleveleducationalcourses |