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PHYSICS
УДК 546.722+546.271 doi:10.21685/2072-3040-2022-2-4
Subwavelength optical microscopy of nanostructured objects on the surface of an optical metamaterial with a zero refractive index
O.N. Gadomsky1, N.M. Ushakov2, V.E. Katnov3, D.O. Musich4, M.M. Gebeydullov5, A.H. Adam6
i,4,5,6uiyanovsk State University, Ulyanovsk, Russia 2Kotel'nikov Institute of Radio Engineering and Electronics of the Russian Academy of Sciences, Saratov Branch, Saratov, Russia 3Kazan Federal University, Kazan, Russia
1gadomsky@mail.ru, 2nmu@bk.ru, 3vkatnov@yandex.ru, 4zokeraf@mail.ru, 5marat735654@gmail.com, 6adamali43434@gmail.com
Abstract. It has been proven theoretically that light reflected from the surface of an optical metamaterial with a zero refractive index is focused in the area of a focal spot with a radius of I / 16, where I is the wavelength of light. It is shown that at a zero refractive index, the laws of refraction and reflection of light are violated, the angles of refraction and reflection become random quantities. The reflectivity of the surface in the region of the focal spot is calculated in the presence of a diatomic nanostructured object on the surface. It is shown that the size of the focal spot increases with the distance from the center of the focal spot from the surface. In this case, an image of a nanostructured object can be obtained.
Keywords: nanostructures, optical metamaterial, zero refractive index, focal spot For citation: Gadomsky O.N., Ushakov N.M., Katnov V.E., Musich D.O., Gebeydullov M.M., Adam A.H. Subwavelength optical microscopy of nanostructured objects on the surface of an optical metamaterial with a zero refractive index. Izvestiya vysshikh uchebnykh zavedeniy. Povolzhskiy region. Fiziko-matematicheskie nauki = University proceedings. Volga region. Physical and mathematical sciences. 2022;(2):40-53. (In Russ.). doi:10.21685/2072-3040-2022-2-4
Субволновая оптическая микроскопия наноструктурированных объектов на поверхности оптического метаматериала с нулевым показателем преломления
О. Н. Гадомский1, Н. М. Ушаков2, В. Е. Катнов3, Д. О. Мусич4, М. М. Губейдуллов5, А. Х. Адам6
1Д5,6Ульяновский государственный университет, Ульяновск, Россия 2Институт радиотехники и электроники имени В. А. Котельникова РАН, Саратовский филиал, Саратов, Россия 3Казанский (Приволжский) федеральный университет, Казань, Россия
1gadomsky@mail.ru, 2nmu@bk.ru, 3vkatnov@yandex.ru, 4zokeraf@mail.ru, 5marat735654@gmail.com, 6adamali43434@gmail.com
© Gadomsky O.N., Ushakov N.M., Katnov V.E., Musich D.O., Gebeydullov M.M., Adam A.H., 2022. Контент доступен по лицензии Creative Commons Attribution 4.0 License / This work is licensed under a Creative Commons Attribution 4.0 License.
Аннотация. Теоретически доказано, что свет, отраженный от поверхности оптического метаматериала с нулевым показателем преломления, фокусируется в области фокального пятна с радиусом X/16, где X - длина волны света. Показано, что при нулевом показателе преломления законы преломления и отражения света нарушаются, углы преломления и отражения становятся случайными величинами. Рассчитана отражательная способность поверхности в области фокального пятна при наличии на поверхности двухатомного наноструктурированного объекта. Показано, что размер фокального пятна увеличивается по мере удаления от центра фокального пятна от поверхности. В этом случае можно получить изображение наноструктурированного объекта.
Ключевые слова: наноструктуры, оптический метаматериал, равный нулю показатель преломления, фокальное пятно
Для цитирования: Gadomsky O.N., Ushakov N.M., Katnov V.E., Musich D.O., Gebeydullov M.M., Adam A.H. Subwavelength optical microscopy of nanostructured objects on the surface of an optical metamaterial with a zero refractive index // Известия высших учебных заведений. Поволжский регион. Физико-математические науки. 2022. № 2. С. 40-53. doi:10.21685/2072-3040-2022-2-4
1. Introduction
A large number of studies have been devoted to various methods of designing an optical superlens, and among them, several methods that have already been implemented experimentally can be distinguished.
Silver is used in many optical devices, including devices in which subwavelength focusing of light is achieved [1-4], for example, an optical resolution of 170 nm is achieved with a one-dimensional grating with a period of 500 nm on the silver surface. In our paper [5], it is shown that a record size of the focal spot can be achieved using a one-dimensional grating on the silver surface, which is determined only by the grating period.
The review [6] presents some examples of the implementation of subwavelength light focusing based on thin silver layers. Thus, in [7], using a 35 nm thick silver film with a periodic structure of slits, light was focused in the focusing region, the linear dimensions of which are 270 nm. Approximately the same result was achieved in works [8-11].
In [12-14], more complex silver-based lens designs are used for subwavelength focusing of light.
In [15], experimental results of subwavelength focusing of light using a thin chromium Fresnel plate are presented. In this work, focusing of laser radiation with a wavelength of X = 532 nm into an elliptical spot with dimensions of 0.64X and 0.42X is achieved.
Review [16] presents examples of experimental implementation of superlens based on metasurfaces with nanoatoms of various shapes and gold films with na-noparticles. Thus, using V-shaped nanoparticles at a wavelength of 676 nm, focusing of radiation into a focal spot 630 nm in diameter with an efficiency of 10% was achieved.
Several problems can be identified in the implementation of subwavelength focusing of light. One of them is to achieve the smallest possible focusing area of light. The second key problem is to achieve focusing of light away from the super-lens in the wave zone. The third key problem is to achieve subwavelength focusing of light regardless of the wavelength of external radiation. This article will investigate the possibility of subwavelength focusing of light reflected from the surface of
an optical metamaterial with a zero refractive index and the possibility of designing a subwavelength optical microscope based on this effect.
In works [17-20] it is shown that when in "fishnet" structures and in structures from a set of layers, the real part of the permittivity and permeability can be close to zero in a narrow wavelength range.
Our research group has developed a technology for the synthesis of a metamaterial (PMMA + Ag) with silver nanoparticles (see references to our patents in [21, 22]) and, on the basis of experimental reflection and transmission spectra of layers of this metamaterial, it has been established that the metamaterial we synthesize has a random close to zero refractive index. Unique optical phenomena have been discovered in this metamaterial. Thus, in [21], experimental data are presented showing a violation of the principle of reversibility of light fluxes in (PMMA + Ag) / glass samples on a glass substrate in a wide wavelength range from 450 to 1200 nm. In [22], a significant decrease in the shadow of a highly reflective sphere covered with a layer of metamaterial (PMMA + Ag) was found experimentally, which indicates the possibility of masking bodies using layers of different thicknesses from this metamaterial.
In [23], the effect of photon localization in the (PMMA + Ag) layer was experimentally discovered. We hypothesize that the use of this metamaterial could play an important role in solving key problems and in the development of a super-lens. In this theoretical article, the boundary problem of reflection and refraction of light with different wavelengths in the range from 450 to 1200 nm from the surface of a semi-infinite medium with a zero refractive index is solved. It is shown that the surface of an optical metamaterial with a zero refractive index can be the basis for the implementation of subwavelength optical microscopy of nanostructured objects located on the surface of this metamaterial.
2. Generalized laws of refraction and reflection of light
We write the generalized law of refraction in the following form:
«1 sin 01 = «2 sin 02, (1)
Where «1 is the refractive index of the input medium 1, 01 is the angle of incidence of light, «2 is the refractive index of a semi-infinite medium 2, equal to zero, 02 is the angle of refraction in medium 2. Since «2 = 0, then the angle of refraction 02 is indeterminate, that is, it is random.We will assume that the random angle 02 distributed with equal probability in the range of values [-n /2, n /2]. Then, at normal incidence of light on the surface of medium 2, we have 01 = 0 and the possible directions of refraction are shown in fig. 1a.
We represent the generalized law of reflection as
0r =n-02, (2)
where 0r is the random angle of reflection (Fig. 1b). Possible angles of reflection are related to possible angles of refraction in accordance with the law of conservation of momentum of photons at the interface 1-2 of two media. In this case, the zero-vector k2 determines the direction of refraction and reflection of light.
The angles of refraction can be both complex and real. For complex angles of refraction, we have the following relations:
i ii n I 2 ii i ✓
02 =02 - i02 , cos 02 = ish%2 = i\x2 -1, x2 = ch02 , sin 02 = X2, 02 = % . (3)
a)
b)
Fig. 1. Possible directions of refraction (a) and reflection (b) of light on the surface of an optical metamaterial with a zero refractive index
In this case, the angle 02 determines the direction of wave attenuation, and i
the angle 02 - the direction of wave propagation. For real angles of refraction, we have the following relations:
sin 02 = y2, cos 02 = д/l—
у2.
(4)
3. Refraction of light at the boundary of a vacuum-semi-infinite medium with zero refractive index
The amplitude of the refracted wave Tq is determined using the extinction theorem [24, 25], which takes into account the extinction of the external wave at the boundary and instead of it, a wave with a zero wave vector propagates in the medium. Calculating the surface integral in the extinction theorem under the condition
k0r » 1,
(5)
where k0 = 2n / X, X is the light wavelength, r is the depth of the observation point inside the environment, we obtain the following equality:
A
(( , = 1 зш^+М , (,.r0 )-To ]
2 cos 92sin 02 L J
(6)
Here A( ^ is the amplitude of the external plane wave, the angle 92 is defined as sin 92 = «2 sin 02 and the unit vector s has the following components:
= sin 92, Sy = 0, sz = -cos 92. (7)
Since «2 = 0, then the angle 92 = 0, therefore, formula (6) is greatly simplified. The phase A in equality (6) has the form:
A = —0 (rs1 ) + k0 (rs), (8)
where S1 is the unit vector along the direction of propagation of the external wave, Ix = -sin 01, = 0, s^ = -cos 01. (9)
In the generalized law of refraction (1), the angle 01 = 0, therefore the phase A = 0.
As a result, for s-polarized waves with amplitudes Al and we obtain from (6) the following formula:
tl= 2a_l . (10)
For p-polarized waves with amplitudes A| and Tj| we obtain that
T = cok A|. (11)
4. Focusing the reflected light
Let us now consider the observation points over the surface of the medium with a zero refractive index, taking into account the generalized law of reflection (2). The calculations in this case are similar, only now in the corresponding formulas the coordinates inside the medium z' = -r should be replaced by z = r. The electric field strength of the reflected wave takes the following form:
Er =-
1 sin (( -02 )
• [sr (Sr • To)-To ] e~ik°(rSR), (12)
2 cos 92 sin 02
where the random unit vector sr along the direction of propagation of the reflected wave has the following components
sxr =-sin 0R, Syr = 0, Szr = - cos 0R . (13)
Denoting by Rl and R| the corresponding projections of Er in directions perpendicular and parallel to the plane of incidence, we obtain
R±= A±exp[zkQ (rsR)] , (14)
R| = Aiexp[ik0 (rsR )]. (15)
Transform the random phase in these formulas using the polar angle 9 for the vector r . Then we get that
(rsR ) = r cos 0, (16)
where the angle varies within 0 < 0<n .
Let's average the values R^ and R| over the random angles of refraction 02 . Squaring the resulting expression in absolute value, we get the following function:
|F (r, z) = ii-^.
2 (V)
(17)
where we represent the variable r as r = x2 + (RQ + z)2 taking into account the condition k0Rq »1 . For k0r = 0 and Rq = 0 function (17) is equal to
|F (x, z )) = 1, for 2kor = n/4 the function is |F|2 = 0.95, that is, this function is slowly decreasing with increasing argument (2kor) near the maximum of this
function. This makes it possible to estimate the radius of the brightest part of the focal spot Ar using the following equality
Ar = (/16). (18)
Function (17) vanishes at 2kor =±2n, ±4n etc. (fig. 2). Function (17) determines the brightness of the focal spot and is characterized by a Strehl number close to unity, since the side maxima are much less than the main maximum. The brightness of the focal spot area corresponding to the classical diffraction limit is approximately five times less than the brightness of the central area of the focal spot.
|F(x,z)|
0.75
2
/ 0.5
/ 0.25
-4tt
-2tt
2tt
4tt 2k0-r
I \|2 1 1 - cos (2k0r) Fig. 2. Function graph IF (x, z ) =----
1,2 (V)2
When focusing natural light, the reflectivity of the surface of the medium with a zero refractive index is determined using the equality
R
A=1F z r
(19)
In fig. 3 shows the structure of the focal spot. At the edge of the focal spot at 2k0r = n/4 at Rq = 0 we get R / A, equal to 0.95. The integral value of the surface reflectivity over the area of the focal spot is defined as
R
A
i
Ax(0)
О (0 )]2 V
J
1 - cos X
dx,
(20)
where the variable of integration x = 2k0r, Ax(0) = 2k0Ar(0), Ar(0) - is the radius of the focal spot at Rq = 0 . Calculation of this integral gives the value (R / A) = 1.
Fig. 3. Reflectivity of the surface of a metamaterial with zero refractive index in the region of the focal spot. The wavelength is X = 600 nm, the coordinate of the center of the focal spot is r(0, 0, Ar), where Ar = 37.5 nm. In the center of the focal spot, the reflectivity of the surface of an optical metamaterial with a zero refractive index R/A = 1
Let us now consider the properties of reflectivity (R / A) when changing the distance Rq from the center of the focal spot to the surface z = 0. Let us determine the radius of the focal spot Ar at an arbitrary distance of the focal center from the surface z = 0 as follows:
2koAr = 4 + 2nm , m = 0,1,2...
(21)
At m = 1 the reflectivity of surface (19) at the edge of the focal spot is 41.3 times less than the corresponding value (R / A) at m = 0 . This means that with increasing distance Rq the local brightness of the focal spot decreases, however, the area of the focal spot increases. The integral value of the reflectivity of the surface of the medium with a zero refractive index has the form:
A = ~~2 f (P), (22)
A P
where p = 2ko Ar (Rq )sin P, Ar (Rq ) is the radius of the focal spot, the center of which is at a distance Rq from the surface z = 0 , the angle P is defined as tg (P) = Ar / Rq. The graph of the function f (p) is shown in Fig. 4. As seen from fig. 4, under the condition
2k0Ar (RQ) sin p« 1 (23)
we obtain from (22) the value (R / A)«1.
f(p)
f (P ) =
Fig. 4. Function graph f (p)
P . + . P
2•2! 4•4! 6•6!
, where p = 2k0 Ar (R0) sin P
Let us also determine the relationship between Rq and the radius of the focal spot Ar (Rq ), taking into account that
2
Ar2 = xB +(zB - Rq )2
(24)
where xb , zb are coordinates at the border of the focal spot. We also take into ac-
1
count that xb = zb = —j=Ar . Then from relation (24) we obtain that
v2
R0 = ArV2 .
(25)
In fig. 5 shows the focal spots at different distances of the centers of the focal spots from the surface of an optical metamaterial with a zero refractive index. It can be seen that the size of the focal spots increases with increasing this distance. The integral value of the reflectivity of the boundary z = 0 does not depend on the distance, but in this case the local values of the reflectivity decrease.
Fig. 5. Dependence of the area of the focal spot on the distance between the center of the focal spot and the surface of an optical metamaterial with a zero refractive index
5. Subwavelength optical microscopy of nanostructured objects
Let us consider the image of a diatomic nanostructured object located on the surface z = 0 of the medium, in the region of the focal spot, the center of which is at a distance Rq from the surface. The reflectivity of the surface, taking into account a diatomic nanostructured object, has the form:
'2 <\F(x,z)|2, (26)
where
R
m
1 + f (x,z)e~ik°Xl + f2 (x,z)e
—iko Xi
Ю
f1 =—T a
(Rf R3
(27a)
ю
f2 =--t a
(R f
R23
(27b)
с
с
e is the unit vector of polarization of external radiation, R = |r - rj |, R2 = \r - , r - is the radius vector of the observation point in the focal spot region, r (-xj, 0, 0), r2 (xj, 0, 0) -are the radius vectors of atoms on the surface of the optical metamaterial, and a is the polarizability of atoms. In expressions (27.a) and (27.b), only the retarded parts of the atomic dipole field are presented, since the observation point is in the wave zone with respect to the surface z = 0.
In fig. 6 shows images of atoms of a nanostructured object at the edges of a focal spot, the diameter of which is greater than the wavelength. This means that a nanostructured object can be detected with an optical microscope.
Fig. 6. Image of a diatomic nanostructured object of two Na atoms in the region
of the focal spot. a = (2^ / h\-1-r, d0 - the dipole moment of the 3S-3P
v ;œ0-œ-i / T2
transition of the sodium atom, 1/72, - natural resonance width, œ0 - transition frequency, d0 = 5.110-18 un. CGSE, 1/7, = 10 MHz, œ0 = 3.197 1015 rad/sec
So, this article provides a theoretical justification for the fact that a subwave-length optical microscope can be built on the basis of an optical metamaterial with a zero refractive index to study nanostructured objects on the surface of this metamaterial. We assume that such a subwavelength optical microscope can be implemented on the basis of an optical metamaterial PMMA + Ag with silver nano-particles synthesized using our nanotechnology in a wide wavelength range from 450 to 1200 nm.
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Список литературы
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Информация об авторах / Information about the authors
Олег Николаевич Гадомский доктор физико-математических наук, профессор, профессор кафедры радиофизики и электроники, Ульяновский государственный университет (Россия, г. Ульяновск, ул. Льва Толстого, 42)
E-mail: gadomsky@mail.ru
Николай Михайлович Ушаков
доктор физико-математических наук, профессор, главный научный сотрудник лаборатории субмикронной электроники, Саратовский филиал Института радиотехники и электроники имени В. А. Котельникова Российской академии наук (Россия, г. Саратов, ул. Зеленая, 38)
E-mail: nmu@bk.ru
Владимир Евгеньевич Катнов кандидат химических наук, доцент, старший научный сотрудник, Институт геологии и нефтегазовых технологий, Казанский (Приволжский) федеральный университет (Россия, г. Казань, ул. Кремлевская, 18)
E-mail: vkatnov@yandex.ru
Oleg N. Gadomskiy
Doctor of physical and mathematical
sciences, professor, professor
of the sub-depatment of radiophysics
and electronics, Ulyanovsk State University
(42 L'va Tolstogo street, Ulyanovsk,
Russia)
Nikolay M. Ushakov Doctor of physical and mathematical sciences, professor, principal researcher of the Laboratory of submicron electronics, Kotel'nikov Institute of Radio Engineering and Electronics of the Russian Academy of Sciences, Saratov Branch (38 Zelenaya street, Saratov, Russia)
Vladimir E. Katnov
Candidate of chemical sciences, associate professor, principal researcher, Institute of Geology and Petroleum Technologies, Kazan Federal University (18 Kremlevskaya street, Kazan, Russia)
Дмитрий Олегович Мусич программист службы протектората по научной работе, Научно-исследовательский технологичный институт имени С. П. Капицы, Ульяновский государственный университет (Россия, г. Ульяновск, ул. Л. Толстого, 42)
E-mail: zokeraf@mail.ru
Dmitriy O. Musich Protectorate service programmer for scientific work, Research Institute of Technology named after S.P. Kapitsa, Ulyanovsk State University (42 L'va Tolstogo street, Ulyanovsk, Russia)
Марат Маратович Губейдуллов аспирант, Ульяновский государств университет (Россия, г. Ульяновск, ул. Льва Толстого, 42)
E-mail: marat735654@gmail.com
Marat M. Gebeydullov Postgraduate student, Ulyanovsk State University (42 L'va Tolstogo street, Ulyanovsk, Russia)
Али Хусейн Адам аспирант, Ульяновский государств университет (Россия, г. Ульяновск, ул. Льва Толстого, 42)
E-mail: adamali43434@gmail.com
Ali Hussein Adam Postgraduate student, Ulyanovsk State University (42 L'va Tolstogo street, Ulyanovsk, Russia)
Авторы заявляют об отсутствии конфликта интересов / The authors declare no conflicts of interests.
Поступила в редакцию / Received 23.03.2022
Поступила после рецензирования и доработки / Revised 19.05.2022 Принята к публикации / Accepted 22.06.2022
