Pis'ma v ZhETF, vol. 102, iss. 4, pp. 240-244
© 2015 August 25
Two-dimensional dynamic photonic crystal creation by means of three non-coplanar laser beams interference in colloidal CdSe/ZnS quantum
dots solution
A. M. SmirnovI. V. Tikhonov, V. N. M&ntsevich, V. S. Dneprovskii Department of Physics, Lornonosov MSU, 119991 Moscow, Russia Submitted 14 July 2015
We demonstrated a simple way to create dynamic photonic crystals with different lattice symmetry by interference of three non-coplanar laser beams in colloidal solution of CdSe/ZnS quantum dots. Two-dimensional dynamic photonic crystal was formed due to the periodical changing of refraction and/or absorbtion of resonantly excited excitons in CdSe/ZnS quantum dots. The formation of dynamic photonic crystal was confirmed by the observed diffraction of the beams that have excited photonic crystal at the angles equal to that calculated for the corresponding two-dimensional lattice (self-diffraction regime).
DOI: 10.7868/S0370274X15160043
1. Introduction. Materials in which the refractive index (dielectric constant) is modulated on a length scale close to the wavelength necessary for operation are called photonic crystals. Multiple interference between waves scattered from each unit cell of the crystal results in photonic bandgap formation - a set of frequencies within which no propagating electromagnetic modes exist [1, 2].
Photonic crystals have demonstrated an attractive potential for numerous devices fabrication [3-5], which, for example, deal with high-capacity data storing and other related applications [6]. Considerable progress has been made in constructing two-dimensional structures with the use of conventional lithography technique [2], but this method does not provide the possibility to produce dynamic photonic crystals. Colloidal crystals may be used as templates to make submicrometer structures [7-12] but the use of closed-packed spheres severely restricts the range of lattices that may be produced [13] and allows almost no freedom to alter the structure of a unit cell. That's why formation of photonic crystals based on the quantum dots colloidal solution is an actual problem of great interest, because one can change the parameters of dynamic photonic crystal (spacial symmetry, unit cell shape and size, crystal dimension) by means of quantum dots concentration and dimensions changing from the one hand and by means of experimental geometry changing from the other hand. Moreover, obtained dynamic structure opens new horizons for careful analysis of relaxation processes in the QDs.
^e-mail: hieroglifics@mail.ru
In the present paper we describe a technique, which gives possibility to form two-dimensional tunable dynamic photonic crystals with different lattice symmetry. With this technique we have made micro-periodic dynamic semiconductor structure with strong nonlinear changing of refraction and absorbtion and analyzed the self-diffraction processes of three non-coplanar laser beams at the dynamic photonic crystal (diffraction grating). To our knowledge, we demonstrated for the first time that two-dimensional tunable dynamic photonic crystal (diffraction grating) can be formed in the colloidal quantum dots solution due to the periodical changing of refraction and/or absorbtion caused by three non-coplanar laser beams interference. To reach the best uniform contrast of the structure under investigation and for better understanding of the problems, specially raised by the interference of multiple non-coplanar beams we have also performed theoretical calculation of the periodic intensity field in the quantum dots solution.
2. Experimental setup. We used 35 picosecond pulses train from a mode-locked frequency-doubled Nd+3:YAG-laser (A = 532 nm) to irradiate the cell with colloidal solution of CdSe/ZnS quantum dots. Three laser beams were created by splitting the laser output twice with dielectric beam-splitters. Three coherent non-coplanar laser beams with the same linear polarization and equal intensities h = h = h = I attain the cell with colloidal solution of quantum dots simultaneously. The beams geometry is shown schematically in Fig-1.
240
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Fig. 1. Beams geometry. Wave vectors of the three laser beams are drawn as arrows
wave, respectively, and % is the angle between the polarization directions of the -¿th and jth waves. Due to experimental geometry in our case linear polarization state of each beam was the same, consequently, Oij = 0 and ipoi — <poj = 0. Fig. 2 shows calculated plane (see Fig. 2a) and three-dimensional (see Fig. 2b) intensity distribution, which induces hexagonal two-dimensional photonic crystal in colloidal solution of quantum dots.
Evaluated intensity distribution demonstrates that by means of three interacting coherent non-coplanar plane waves with equal amplitudes one can obtain two-dimensional dynamic photonic crystal with the periodicity in two directions, which is determined by the vectors ai and a2 (Fig. 2a). Vectors ai and a2 have equal length, which can be evaluated from the obtained intensity distribution (see Fig. 2):
The intensity and linear polarization state of each beam were controlled by means of a half-wave plate and a dielectric polarizing beam-splitter; the polarizer is the last optical element before the cell with colloidal solution of CdSe/ZnS quantum dots [14, 15]. The diffraction pattern formed by transmitted and self-diffracted beams was registered with the help of Nikon D70 camera. Colloidal solution was based on the CdSe/ZnS quantum dots with radii about 2.4 nm and dimensions dispersion 20% dissolved in hexane with concentration 1017sm~3. Linear absorbtion coefficient of the colloidal solution was 35 sm-1. Peculiarities of three laser beams interaction in the nonlinear optical medium were analyzed in the vicinity of resonant excitation of main exciton transition in CdSe/ZnS quantum dots [16, 17]. The CdSe/ZnS quantum dots colloidal solution was utilized due to the possibility of control under physical properties, which determine nonlinear system response on the resonance laser excitation.
3. Results and discussion. Non-stationary two-dimensional photonic crystal can be formed due to the interaction of three coherent non-coplanar plane waves (Ei = Eio cos(cat — kiX + ipoi), « = 1,2,3) in the colloidal solution of CdSe/ZnS quantum dots due to the spatial modulation of medium optical properties. The intensity distribution of the interference field of three plane waves of the same wavelength A:
I Elj + ^ lEoi.Eoj
COS tiii X
j i<j x cos[(Kj - Kj)r + <fioi - tpoj], hj = 1-3,
(1)
where K^), and <fioi(j) are the amplitude, the
wave vector and the initial phase of the «(j)th plane
|ai| = |a2| =
A
! sin 9
(2)
One can easily prove the presence of two-dimensional dynamic photonic crystal analyzing the results of self-diffraction of the waves, which form non-stationary two-dimensional photonic crystal. Diffraction patterns were registered with the help of Nikon D70 camera and demonstrate three transmitted beams holding the direction of incident beams and 21 additional beams. The presence of additional beams can be explained by the self-diffraction of three incident beams at the two-dimensional dynamic photonic crystal, caused by the periodical spatial changing of absorbtion of colloidal quantum dots, which takes place in the interference light field (see Fig. 2). Periodical changing of absorbtion during the resonant single-photon excitation of main exciton transition in colloidal solution of quantum dots by means of second harmonic picosecond pulses can be explained by the presence of competing and coexisting effects of states filling and Stark shift of the exciton absorbtion [18, 19]. Nonlinear absorbtion changing can also be accompanied by the nonlinear changing of refraction [20]. Small red shift of second harmonic wavelength from the resonant wavelength of colloidal quantum dots absorbtion may result in formation of two-dimensional dynamic photonic crystal (phase diffraction grating). It was estimated that induced changing of refractive index in the regions with maximum value of intensity can reach An ~ 10~3 - enough for formation of two-dimensional non-stationary photonic crystal. The presence of diffraction rings is the direct manifestation of self-diffraction at the induced transparency channel [19].
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max
min
Fig. 2. (a) - Calculated plane surface of three-beam laser interference pattern, (b) - Calculated three-dimensional intensity distribution
\ /Wi -2 -1 0 1 2 3
H X 61
H 46.3 X 53.3 X
-3 >< 46.3 X 32 X 61
-2 61 x 23.6 X 32 X
-1 X 32 X 11.6 X 46.3
0 53.3 X 11.6 X 23.6 X
1 X 32 X 11.6 X 46.3
2 6i X 23.6 X 32 X
3 46.3 X 32 X 6i
4 46.3 X 53.3 X
5 61 X^
Fig. 3. (Color online) Propagation angles of the laser beams self-diffracted on the induced two-dimensional dynamical photonic crystal for different diffraction maxima (mi;m2)
To determine the propagation angles of self-diffracted laser beams the Laue method for the two-dimensional diffraction grating was applied:
ci(œsce — cos ceo) = mi A,
£-2 (COS ß — COS ßo) = 7712 A,
cos2 ceo + cos2 ßo + cos2 70 = 1, cos2 ce + cos2 ß + cos2 7 = 1,
(3)
where ceo, A)? and 70 are the angles, which incident beams form with the axis x, y and z correspondingly; ce, /5, and 7 are angles of diffracted beams, rn^eZ, c\ and C2 are the translation vectors for rectangular unit cell, which can be expressed through the translation vectors ai and a2 for the hexagonal unit cell in the following way:
|Cl| = lai I
|c2| = 2|a2| cos
6
(4)
One can also find from Fig. 1 the following relations between the angles:
ßo =
2'
= 7o +
2'
sm 70 = — cos ceo =
suit
(5)
Consequently, one can obtain the following system of equations for angles a, /3, and 7:
cos ß = m2 sin#, sin 7 = д/со52се + cos2 ß.
(6)
According to obtained equations the possible value
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