Optics and Spectroscopy, Vol. 95, No. 1, 2003, pp. 70–79. Translated from Optika i Spektroskopiya, Vol. 95, No. 1, 2003, pp. 77–86. Original Russian Text Copyright © 2003 by Gevorgyan.
SOLID-STATE SPECTROSCOPY
Optics of Helical Periodic Media in the Presence of a Wave Modulating Parameters of the Medium: I. New Regions of Diffraction Reflection A. H. Gevorgyan Department of Physics, Yerevan State University, Yerevan, 375025 Armenia e-mail:
[email protected] Received September 13, 2002
Abstract—Reflection and transmission of light through a layer of a helical periodic medium in the presence of a longitudinal hypersonic field are studied. The cases of periodic and aperiodic media are considered in the perturbation theory approximation. Regions of diffraction reflection (RDRs) of different character are shown to appear, including RDRs caused by the helicity of the medium, by its stratification, and simultaneously by the medium helicity and stratification. The results are compared with the exact numerical solution of the problem. © 2003 MAIK “Nauka/Interperiodica”.
INTRODUCTION In connection with the wide application of helical periodic media (HPMs) in electro- (magneto- and acousto-) optical arrangements, the study of the effect of external fields on the optical properties of HPMs (cholesteric liquid crystals (CLCs), chiral smectics, etc.) is of considerable interest. This effect has its origins in the anisotropy of the local dielectric (magnetic) susceptibility of HPMs, which, upon application of an external field, gives rise to forces distorting the structure of an HPM. This change in the structural properties, in its turn, has an effect on the optical characteristics of the HPM. Manifestations of the action of external fields on the structure (and, hence, on the optical properties) are very diverse. The generally accepted classification of these effects [1] is as follows: (1) effects caused by a change in the pitch of the helix and (2) effects caused by a change in textures and by conduction. In addition to these fundamental changes in the structure, which lead to the most interesting and practically important effects, in the case of CLCs, also significant are effects in thin layers, when it is impossible to neglect the influence of the surface on the parameters, in particular, on the pitch of helix. Among other external influences, mechanical fields strongly affect the optical properties of CLCs. These fields can be produced by mechanical deformations and can also arise during the propagation of ultrasonic waves through CLCs. The interaction of light and ultrasonic waves in crystals is used for controlling the parameters of light beams and investigating physical properties of matter. The effect of external fields on optical properties of CLCs was considered in [1–10] (see also references cited therein). In [11], the orienting action of a light wave on the cholesteric mesophase was
studied. It was shown, in particular, that a light wave also gives rise to modulation of parameters of a medium. Below, we present the results of the study of the effect of a longitudinal hypersonic wave on the optical properties of CLCs: The results obtained in [12] are generalized and compared with the results of the numerical calculations performed according to the exact theory of propagation of light through an inhomogeneous layer. A hypersonic wave induces a change in the principal values of the local tensor of permittivity (photoelastic effect) and in the pitch of helix. The medium also becomes stratified, with the period of stratification equal to the length of the hypersonic wave. This periodic perturbation varies both in space and in time. In particular, if the hypersound represents a traveling wave, the periodic perturbation travels with the velocity of the hypersound. Since the speed of hypersound is smaller than the speed of light by many orders, the periodic perturbation caused by the hypersound can be considered stationary and we can neglect the time dependence of the parameters of the medium in the wave equation, i.e., in calculating (1/c2)∂2D/∂t 2, we can omit differentiation of the medium parameters with respect to time and take their time dependence into account only in the final results. As is known [13], this can be done if Ω/ω Ⰶ 1 ,
(1)
where Ω and ω are the frequencies of the hypersonic and light waves, respectively. This condition can easily be fulfilled. Note that the distinction of the present problem from those considered in [1–11] formally consists in the presence of two periods of nonuniformity: the undis-
0030-400X/03/9501-0070$24.00 © 2003 MAIK “Nauka/Interperiodica”
OPTICS OF HELICAL PERIODIC MEDIA IN THE PRESENCE OF A WAVE
torted pitch of the helix and the hypersound wavelength. With these periods properly chosen, the consideration can be reduced to [7, 8]. In the general case, this is impossible. This is connected with the fact that, generally speaking, the medium under consideration can also be aperiodic, which takes place when the ratio of the two periods mentioned cannot be represented as a ratio of two integers.
71
mittivity tensor to the form εˆ (z) = Rˆ (az) εˆ 0 R–1(az), where ε 0 0 1 εˆ 0 = 0 ε 2 0 0 0 ε2 is the local permittivity tensor and
NEW REGIONS OF DIFFRACTION REFLECTION We will assume that the modulation is produced by a plane hypersonic wave and that the Bragg diffraction regime is realized in the medium. Let the plane hypersonic wave propagate along the z axis. Then, the principal values of the local permittivity tensor ε1 and ε2 and a = 2π/σ (σ is the pitch of the helix) are no longer constant and vary with z and t. Let the laws of their variation be as follows: a ( z, t ) = a 0 + a 1c ( t ) cos bz + a 1s ( t ) sin bz, ε 1 ( z, t ) = ε 10 + ε 1c ( t ) cos bz + ε 1s ( t ) sin bz,
(2)
ε 2 ( z, t ) = ε 20 + ε 2c ( t ) cos bz + ε 2s ( t ) sin bz, where a0 = 2π/σ0 and σ0, ε10, and ε20 are the corresponding parameters of the unperturbed helix. These relationships were obtained under the assumption that two waves propagate in the medium that modulate its parameters: forward and backward waves. From relationships (2), for the angle between the directions of propagation and the x axis of the laboratory coordinate system, we have ϕ ( z, t ) = ϕ 0 + a 1c ( t ) cos bz/b + a 1s ( t ) sin bz/b.
(3)
cos az – sin az 0 Rˆ ( az ) = sin az cos az 0 0 1 0 is the rotation matrix; (2) assuming that the permittivity tensor is invariable in the local system, one can transform the vectors of the electric and magnetic fields and of the induction according to the equation E(z) = –1 Rˆ (az)E(0). Although these methods are equivalent and ultimately give the same result, here, we will use the second method, which has certain advantages; in particular, the periodic dependence of the permittivity on z in the absence of hypersound is excluded. This makes it possible to reduce the presence of the two periodicities (the helicity caused by the torsion and the stratification created by the hypersound) to the stratification only and allows one to apply the same approach both in the case when the ratio of two periods can be represented as a ratio of two integers (the medium is periodic) and in the case when this cannot be done (the medium is aperiodic). Let us substitute relationships (2) into the wave equation and pass in it to the components related to the x' and y' axes. We will seek the field in the form (0)
It follows from Eqs. (2) and (3) that, in this case, the permittivity, generally speaking, is not a periodic function of the coordinates. From the requirement of periodicity of the medium, we obtain (4)
ba 0 /2 = m 1 /m 2 ,
where m1 and m2 are integers. We will consider the general case when relationship (4) can break down. In study of the optical properties of HPMs, two coordinate systems are commonly used: the laboratory system (x, y, z), whose z axis is parallel to the axis of the HPM helix, and the rotating system (x', y', z'), whose z' axis coincides with the z axis, while the x' and y' axes are parallel to the principal directions of the permittivity tensor for a given value of z. In solving Maxwell’s equations, two approaches can be applied: (1) assuming that the vectors of the electric and magnetic fields and the induction are invariable, one can transform the perOPTICS AND SPECTROSCOPY
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(1)
Ᏹ ( z, t ) = Ᏹ ( z, t ) + Ᏹ ( z, t ) 4
=
+∞
∑∑
Ᏹ mn exp [ i ( k mn z – ωt ) ].
(5)
m = 1 n = –∞
We then obtain 2 2 ω2 a 1c a 1s 2 2 -----2- ε 10 – k mn – a 0 – ------- – ------- Ᏹ mnx 2 2 c 2 ω ε 1c - ( Ᏹ mn – 1 x + Ᏹ mn + 1 x ) + 2ia 0 k mn Ᏹ mny – a 0 a 1c – -----2- ---- c 2
a 1c a 1s - ( Ᏹ mn – 2 x – Ᏹ mn + 2 x ) + i ------------2 ω ε 1s - ( Ᏹ mn – 1 x – Ᏹ mn + 1 x ) + i a 0 a 1s – -----2- ---- c 2 2
72
GEVORGYAN 2
except for Ᏹm0x and Ᏹm0y , we reduce Eqs. (6) (in the components related to the x' and y' axes) to the form
2
a 1c a 1s + – ------ + ------- ( Ᏹ mn – 2 x + Ᏹ mn + 2 x ) 4 4
2 2 ω ------ ε – k – a 0 Ᏹ m0 x + 2ia 0 k m0 Ᏹ m0 y = 0, c 2 10 m0 2
+ ia 1c ( k mn – 1 Ᏹ mn – 1 y + k mn + 1 Ᏹ mn + 1 y ) a 1s b - ( Ᏹ mn – 1 y + Ᏹ mn + 1 y ) + --------2
ω 2 2 – 2ia 0 k m0 Ᏹ m0 x + -----2- ε 20 – k m0 – a 0 Ᏹ m0 y = 0. c 2
+ a 1s ( k mn – 1 Ᏹ mn – 1 y – k mn + 1 Ᏹ mn + 1 y )
The condition for existence of a nontrivial solution of set (7) expressed by the equality ∆m0 = 0, where ∆m0 is the determinant of set (7),
ia 1c b - ( Ᏹ mn – 1 y – Ᏹ mn + 1 y ) = 0, + ----------2 2 2 ω2 a 1c a 1s 2 2 ---- 2 ε 20 – k mn – a 0 – ------- – ------- Ᏹ mny – 2ia 0 k mn Ᏹ mnx 2 2 c
(7)
(6)
ω ω 2 2 2 2 ∆ m0 = -----2- ε 10 – k m0 – a 0 -----2- ε 20 – k m0 – a 0 c c 2
2
– ω ε 2c - ( Ᏹ mn – 1 y + Ᏹ mn + 1 y ) – a 0 a 1c – -----2- ---- c 2
(8)
2 2 4a 0 k m0 ,
2
leads, as one would expect, to the values of km0 determined by the formula km0 = 2π ε m b±/λ, where b± = 0
a 1c a 1s - ( Ᏹ mn – 2 y – Ᏹ mn + 2 y ) + i ------------2
1 + χ – δ 0 ± γ (γ = 2
4χ + δ 0 , δ0 = (ε10 – ε20)/(ε10 +
2
2
2
a 1c a 1s + – ------ + ------- ( Ᏹ mn – 2 y + Ᏹ mn + 2 y ) 4 4 – ia 1c ( k mn – 1 Ᏹ mn – 1 x + k mn + 1 Ᏹ mn + 1 x ) a 1s b - ( Ᏹ mn – 1 x + Ᏹ mn + 1 x ) + --------2 – a 1s ( k mn – 1 Ᏹ mn – 1 x – k mn + 1 Ᏹ mn + 1 x ) ia 1c b - ( Ᏹ mn – 1 x – Ᏹ mn + 1 x ) = 0 – ----------2 for any kmn. Here, kmn = km0 + nb (n = ±1, ±2, ±3, …). Let us apply the perturbation theory to set (6) thus obtained. As small parameters, we will consider the deviations of the permittivity tensor components and of the quantity a from the values that these quantities had in the absence of modulation of parameters of the medium. In addition, for applicability of the perturbation theory, it is necessary that the amplitudes Ᏹmnx and Ᏹmny be much smaller than Ᏹm0x and Ᏹm0y . In the zero-order approximation, neglecting all the quantities that are proportional to the modulation parameters (a1c, s , ε1c, s, and ε2c, s) and all the amplitudes
2
ε20), χ = λ/(σ0 ε m ), and ε m = (ε10 + ε20)/2). In the zeroorder approximation, the field in the medium can be written as 0
ω ε 2s - ( Ᏹ mn – 1 y – Ᏹ mn + 1 y ) + i a 0 a 1s – -----2- ---- c 2
2
0
4
(0)
Ᏹ ( z, t ) =
∑Ᏹ
m0 exp [ i ( k m0 z
– ωt ) ].
(9)
m=1
The set of equations (6) represents an infinite homogeneous set of equations in the unknown amplitudes Ᏹmnx and Ᏹmny . However, as is known [14], not all the amplitudes Ᏹmnx and Ᏹmny are related to one another. It turns out that only the amplitudes Ᏹm0x , Ᏹm0y and Ᏹmnx , Ᏹmny with various values of n are most strongly coupled. Physically, this means that there is a resonance coupling between the components Ᏹm0x, y and Ᏹmnx, y and that, simultaneously with the amplitude Ᏹm0x, y, the amplitude Ᏹmnx, y differs significantly from zero. In the first-order approximation, assuming that, along with Ᏹm0x, y the amplitudes Ᏹm1x, y differ from zero, neglecting all the other amplitudes, and restricting ourselves to terms that are linear in the small parameters a1c, s , ε1c, s , and ε2c, s , we obtain from Eqs. (6) that ω 2 2 ----- ε – k – a 0 Ᏹ m1 x + 2ia 0 k m1 Ᏹ m1 y c 2 10 m1 2
2 2 ω ε 1c ω ε 1s - – i a 0 a 1s – -----2- ----- Ᏹ m0 x = a 0 a 1c – -----2- ---- c 2 c 2
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b – i k m0 + --- ( a 1c – ia 1s ) Ᏹ m0 y , 2 (10)
ω 2 2 – 2ia 0 k m1 Ᏹ m1 x + -----2- ε 20 – k m1 – a 0 Ᏹ m1 y c 2
0
2
m' ≠ m
2
k m'1 = k m0 ,
From set (10), we find the expressions for the amplitudes Ᏹm1x, y ω ω ε 1, 2c 2 2 Ᏹ m1 x, y = -----2- ε 2, 10 – k m1 – a 0 a 0 a 1c – -----2- --------c 2 c 2
(16)
is fulfilled (m' and m run from 1 to 4), then, according to Eq. (8), the right-hand side of expression (12) will vanish again. Condition (16) specifies one more region of diffraction reflection, whose wavelength is approximately determined by the expression λ 3 = 2Λ ε m . 0
(17)
Conditions (14) and (16) give three values of the frequency at which ∆m, 1 = 0. The frequency corresponding to the diffraction caused by the helicity is contained in the zero-order equations, and the boundaries of this region of diffraction reflection are determined by the formula
ω ε 1, 2s - Ᏹ m0 x, y – i a 0 a 1s – -----2- -------- c 2 2
b − + i k m0 + --- ( a 1c – ia 1s ) ( Ᏹ m0 y, x ) 2
(11)
ω ε 2, 1c − + 2ia 0 k m0 a 0 a 1c – -----2- ---------- c 2 2
1, 2
λ0
= σ 0 ε m ( 1 ± δ 0 ). 0
(18)
Thus, we have four diffraction regions. One of them, with the mean wavelength λ0 = σ0 ε m (its boundaries are determined by formula (18)), corresponds to the diffraction reflection caused by the helicity. The diffraction region with the wavelength λ3 corresponds to the light diffraction from the stratification. Two other regions of diffraction reflection, with the wavelengths λ1, 2, are caused by both periods of the inhomogeneity of medium and correspond to the diffraction from periodic perturbations of the helicity. When deriving Eqs. (15) and (17), we assumed that resonance coupling between the amplitudes Ᏹm0x, y and Ᏹm1x, y takes place. It is clear that, in the general case, there may exist regions of diffraction reflection caused by the existence of a resonance coupling between Ᏹm0x, y and Ᏹmnx, y with n also differing from unity. In this case, in a similar way, instead of Eqs. (15) and (17), we obtain 0
2 ω ε 1, 2s - Ᏹ m0 y, x + i a 0 a 1s – -----2- -------- c 2
b ± i k m0 + --- ( a 1c – ia 1s ) Ᏹ m0 x, y /∆ m, 1 , 2 where ∆ m, 1
(15)
Here, we took into account that b = 2π/Λ, where Λ is the hypersound wavelength. If the condition
b + i k m0 + --- ( a 1c – ia 1s ) Ᏹ m0 x . 2
ω ω 2 2 2 2 = -----2- ε 10 – k m1 – a 0 -----2- ε 20 – k m1 – a 0 c c 2
From condition (14), two regions of diffraction reflection are determined; approximate expressions for their wavelengths have the form 2Λσ 0 ε m -. λ 1, 2 = ----------------------2Λ − + σ0
2 2 ω ε 2c ω ε 2s ------------= a 0 a 1c – 2 – i a 0 a 1s – 2 ------ Ᏹ m0 y c 2 c 2
2
73
2
(12)
2 2
– 4a 0 k m, 1 . For applicability of the perturbation theory, we must satisfy the conditions
2Λσ 0 ε m -, λ 1, 2 = -----------------------2Λ − + nσ 0
(19)
2Λ 0 λ 3 = ------- ε m . n
(20)
0
Ᏹ m1 x, y Ⰶ Ᏹ m0 x, y .
(13)
These conditions are violated close to the frequencies at which ∆m, 1 = 0. According to expressions (8) and (12), ∆m, 1 = 0 if 2
2
(14)
k m1 = k m0 . OPTICS AND SPECTROSCOPY
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If n ≠ ±1, these regions of diffraction reflection are called higher-order regions, because, according to
74
GEVORGYAN
expressions (19) and (20), they usually take place at higher frequencies. Now, we consider the exact solution of the problem. Let a layer of the medium occupy the space between the planes z = 0 and z = d (d is the layer thickness). The axis of the medium coincides with the z axis. A plane hypersonic wave also propagates along the z axis. The plane of incidence of light coincides with the (x, z) plane, and the wave is incident normally to the surface, i.e., along the z axis. Let us expand the components of the field amplitudes of the incident, reflected, and transmitted waves in the basis circular polarizations according to +
+
–
E i, r, t = E i, r, t n + + E i, r, t n – =
E i, r, t – E i, r, t
,
(21)
where n+ and n– are the unit vectors of the basis circular polarizations, and represent the solution of the problem in the form E r = Rˆ E i ,
E t = Tˆ E i .
(22)
To construct the Jones matrices Rˆ and Tˆ , let us break the layer of the medium under consideration into a great number of thin layers with the thicknesses d1, d2, d3, …, dN. If their maximum thickness is small enough (much smaller than σ0 and Λ), the parameters of the medium can be considered constant in each layer. Then, according to [15], the problem of determining Rˆ and Tˆ is reduced to solving the following set of difference matrix equations: –1 Rˆ j = rˆ j + ˆt j Rˆ j – 1 ( ˆI – rˆ j Rˆ j – 1 ) ˆt j , –1 Tˆ j = Tˆ j – 1 ( ˆI – rˆ j Rˆ j – 1 ) ˆt j ,
(23)
ˆ and Tˆ 0 = ˆI ; here, Rˆ j , Tˆ j , Rˆ j – 1 , and Tˆ j – 1 with Rˆ 0 = O are the Jones matrices for the media with the homogeneous layers j and j – 1, respectively; rˆ j and ˆt j are the ˆ is a Jones matrices of the jth homogeneous layer; O zero matrix; and ˆI is the unit matrix. By using expressions for the elements of the Jones matrices of a homogeneous layer of an HPM [16], one can calculate the elements of the matrices of an HPM located in a longitudinal hypersonic field and, then, calculate also the reflectance R = |Er |2/|Ei |2 and the transmittance T = |Et |2/|Ei |2, as well as other optical characteristics of the system.
NUMERICAL CALCULATIONS AND CONCLUSIONS Numerical calculations according to the exact theory were performed for the case when the dependences of the pitch of the helix and the principal values of the permittivity tensor on the coordinate z are approximated by the expressions a ( z, t ) = a 0 + a 1s ( t ) sin bz,
(24)
ε 1, 2 ( z, t ) = ε 01, 2 + ε 1, 2s ( t ) sin bz,
(25)
and a1s and ε1, 2s are equal to zero. The calculations were carried out for the three following situations: (1) Λ ~ σ0, (2) Λ Ⰶ σ0, and (3) Λ Ⰷ σ0. (1) Λ ~ σ0. Figure 1a demonstrates the dependence of the reflectance R and Fig. 1b, the dependence of the variation in the reflectance ∆R caused by the stratification (∆R = R' – R, where R' and R are the reflectances in the presence of ultrasound (R') and in its absence (R)) on the wavelength λ. As follows from the figures, the stratification can lead both to an increase and to a decrease in the reflectance, with substantial changes occurring close to the zero-order region of diffraction reflection (in the 0.585–0.605 µm region). As one can see from the figures, the reflectance undergoes a sharp change (∆R passes through its peak) at the wavelength λ = 0.4264 µm. This wavelength corresponds to λ2, which is determined by expression (19) with n = 1. Thus, one can see the appearance of the firstorder diffraction reflection, caused by the simultaneous presence of helicity and stratification. One can see in the inset in Fig. 1a that the reflectance also undergoes a sharp change at the wavelength λ = 0.985 µm. This wavelength corresponds to λ1, which is also determined by expression (19) with n = 1. The results obtained show that the diffraction reflection is stronger at the wavelength λ2 than at the wavelength λ1. This is caused by the large distance between this diffraction region and the zero-order diffraction region; the nearer a region of diffraction reflection to the main region of diffraction reflection, the more intense the diffraction reflection in this region. As the calculations show, the diffraction reflection at the wavelength λ3, which corresponds to the region of diffraction reflection caused by the stratification, is virtually not observed with the adopted parameters of the problem. (2) Λ Ⰶ σ0. Figure 2a demonstrates the dependence of the reflectance R and Fig. 2b, the dependence of the variation in the reflectance ∆R caused by the stratification on the wavelength λ. As can be seen from the figures, in this case, new first-order regions of diffraction reflection caused by the simultaneous presence of helicity and stratification are also excited. However, if, at the wavelength λ2 = 0.1202 µm, the light with the left-hand circular polarization undergoes diffraction reflection (as in the zeroOPTICS AND SPECTROSCOPY
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OPTICS OF HELICAL PERIODIC MEDIA IN THE PRESENCE OF A WAVE R 1.0
R 0.04
75
(‡)
0.02
1 1
0.5 0 0.9
1.0
1.1 λ, µm 2
0 0.375
0.475
0.575
∆R
0.675 λ, µm
(b)
0.1
1
2 0
400
500
600
700 λ, µm
Fig. 1. Dependence (a) of the reflectance R and (b) of the variation in the reflectance ∆R on the wavelength λ. a1s = 0.1 µm–1, Λ = 0.5 µm, σ0 = 0.4 µm, ε01 = 2.29, ε02 = 2.143, ε1s = 0.0001, ε2s = 0.00005, and the layer thickness d = 10 µm. The incident wave has (1) left- and (2) right-hand circular polarization. The helix is left-hand.
order region of diffraction reflection), then, at the wavelength λ1 = 0.2018 µm, the light with the right-hand circular polarization undergoes diffraction reflection. In addition, one can see that, although λ2 is closer to the main region of diffraction reflection than λ1, here, the diffraction reflection is here weaker than at the wavelength λ1. These and a number of other specific features noted above have explanations. For example, it is known that the sign of the helicity is determined by the sign of the off-diagonal element in the HPM permittivity tensor in the laboratory coordinate system. It is also known that light with one type of circular polarization undergoes diffraction from the helical structure of the medium, whereas light with the reverse circular polarization does not. At the same time, it is known from the theory of diffraction that, by expanding the periodic permittivity tensor in a series in terms of the reciprocal lattice vectors, one can determine the possible set of waves excited and, under certain conditions, diffracting in the OPTICS AND SPECTROSCOPY
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medium. Considering a1s and ε1, 2s to be small quantities; expanding the diagonal and off-diagonal elements of the permittivity tensor in a series; and retaining the terms linear in these small parameters, we obtain ε xx, yy =
ε m ibz i2a 0 z – i2a 0 z – ibz + ----( e + e ) ± δ0 [ ( e +e ) 2
0 εm 1
s
a 1s i ( 2a + b )z – i ( 2a 0 + b )z -[(e 0 +e ) + -------0 bε m – (e
i ( 2a 0 – b )z
+e
– i ( 2a 0 – b )z
(26)
)]]
δ i(2a 0 + b)z – i(2a 0 + b)z i(2a 0 – b)z – i(2a 0 – b)z ± ----s [( e +e ) + (e +e )], 4 i2a 0 z – i2a 0 z 0 –e ) ε xy, yx = ε m [iδ 0 ( e
76
GEVORGYAN R 1.0
R (a)
0.02
1
0.01 0.5
2 0 0.11
0.15
λ, µm
0.19
0.2
0
1
0.4
0.6
λ, µm
∆R (b) 0.02 1
2 0
0
0.2
0.6
0.4
–0.02
λ, µm
Fig. 2. Dependence (a) of the reflectance R and (b) of the variation in the reflectance ∆R on the wavelength λ. Λ = 0.05 µm. The remaining parameters and designations are the same as in Fig. 1.
a 1s i ( 2a + b )z – i ( 2a 0 + b )z -[(e 0 –e ) + -------0 bε m – (e
i ( 2a 0 – b )z
–e
– i ( 2a 0 – b )z
)]]
the Bragg condition, which has the form k = (ω/c) ε m = g/2, where g is the corresponding reciprocal lattice vector, is fulfilled. The wavelength of the diffraction reflection caused by light diffraction from the 0
helicity of the medium (λ0 = σ0 ε m ) is determined δ s i(2a0 + b)z –i(2a0 + b)z i(2a 0 – b)z – i(2a 0 – b)z + i ---- [ ( e –e ) + (e –e ) ], from the condition g = 2a0. The conditions g = 2b and 4 g = (2a0 ± b) give three other wavelengths, determined by expressions (17) and (15), respectively. It is also s where ε m = (ε1s + ε2s)/(ε10 + ε20) and δs = (ε1s – ε2s)/(ε10 – known that the widths and the intensities of diffraction ε20). Thus, in the linear approximation, the modes with reflection in these regions are proportional to the ampli0 0 s 0 the wave vectors 2a0, b, and (2a0 ± b) and with the tudes ε m δ 0 , ε m ε m /2 , and δ0a1s /b + ε m δ s /4 , respec0 0 s amplitudes proportional to ε m δ 0 , ε m ε m /2 and δ0a1s /b + tively. It can be seen from Eq. (26) that, for 2a0 – b > 0, the sign of the diffracting circular polarization of the 0 ε m δ s /4 , respectively, are excited in the medium. corresponding mode coincides with the sign of the cirAccording to [14], diffraction reflection occurs when cular polarization for the mode determined by the helic0
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0.5
0 0.118
(a)
(b)
(c)
0.02
0.8
0.01
1
0 0.145
0.121
77
1
0.4
2
0.150 λ, µm
2
0 0.155 0.198
0.203
0.208
Fig. 3. Dependence of the reflectance R on the wavelength λ at new wavelengths of diffraction reflection (a) λ2, (b) λ3, and (c) λ1. Λ = 0.05 µm and a1s = 1.5 µm–1. The remaining parameters and designations are the same as in Fig. 2.
ity of the medium. For 2a0 – b < 0, the sign of the diffracting circular polarization of the corresponding mode changes. If Λ ~ σ0, the sign of 2a0 – b coincides with the sign of 2a0, while for Λ Ⰶ σ0, the quantity 2a0 − b is negative. This circumstance explains the fact that, when Λ Ⰶ σ0, the light with the reverse circular polarization undergoes diffraction reflection at the wavelength λ1 = 0.2018 µm (in contrast to the case of light diffraction in the zero-order region of diffraction reflection). The calculations also show that, with the adopted parameters of the problem, in this case (i.e., for Λ Ⰶ σ0), the diffraction reflection at the wavelength λ3 (which corresponds to the region of diffraction reflection caused by the stratification) is virtually not observed, which can be explained by small values of the depths of modulation ε1, 2s . The calculations show that, on an increase in the depths of modulation a1s and ε1, 2s, the diffraction reflection at the wavelengths λ1 and λ2 occurs more strongly and, in addition, diffraction reflection at the wavelength λ3 is also observed. Figure 3 presents the dependence of the reflectance on the wavelength for the depth of modulation a1s = 1.5 µm–1. Note another specific feature: At the wavelength λ3, the light of both circular polarizations undergoes diffraction reflection. This is not surprising, since this region of diffraction reflection is caused only by stratification of the medium. Figure 4 illustrates the dependence of the reflectance R on the depth of modulation a1s at the wavelengths λ1 (1), λ2 (2), and λ3 (3). The depth of modulation a1s (as well as ε1s and ε2s) changes with time. Then, according to Fig. 4, the intensity of the reflected (transmitted) light will also change. Consequently, such a layer can be used for modulation of light. It can be shown that, with a change in a1s(t), the ellipticity of the transmitted wave, the rotation of the OPTICS AND SPECTROSCOPY
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plane of its polarization, and so on, will also change, i.e., other parameters of light will also be modulated. (3) Λ Ⰷ σ0. Figure 5a demonstrates the dependence of the reflectance R, and Fig. 5b, the dependence of the variation in the reflectance ∆R caused by the stratification on the wavelength λ. As can be seen from the figures, in this case, many regions of diffraction reflection appear. In addition, virtually total suppression of the diffraction reflection at the wavelength λ0 = σ0 ε m , caused by the light diffraction from the unperturbed helicity, is observed. These specific features can also be easily understood if one keeps in mind that, in this case, the value of a1s /b is substantially larger and, as we have 0
R 1.0 2 1
0.5
3 0
2
4
a1c
Fig. 4. The reflectance R as a function of the depth of modulation a1s at the wavelengths (1) λ1, (2) λ2, and (3) λ3. The solid curves correspond to the case when the wave with the left-hand circular polarization is incident on the layer and the dashed curves correspond to the right-hand circular polarization. The remaining parameters are the same as in Fig. 2.
78
GEVORGYAN (a)
R 1.0
1 0.5
0 0.45
0.50
0.55
0.60 (b)
∆R 1.0
0.65
0.70
0.75 λ, µm
1 0.5
0 0.50
0.60
λ, µm
0.70
–0.5
–1.0 Fig. 5. Dependences of (a) the reflectance R and (b) the variation in the reflectance ∆R on the wavelength λ. Λ = 5 µm. The remaining parameters and designations are the same as in Fig. 1.
already noted above, the amplitudes of the modes with the wave vectors 2a0 ± b are proportional to δ0a1s /b + ε m δ s /4 . Therefore, higher order modes with the wavelengths determined by formula (19) with n = ±2, ±3, … are also excited. It is clear that, in this case, both the perturbation theory and the two-wave theory of diffraction are inapplicable for quantitative description of the diffraction phenomena; however, as in other cases, formulas (18)–(20) determine the spectral location of the new regions of diffraction reflection virtually exactly. Varying the hypersound wavelength and the depth of modulation, one can change both the width of these regions and the spectral spacing between them. Consequently, such systems can be used as controllable narrow-band polarization filters and mirrors. 0
In conclusion, note that, as the calculations show, the perturbation theory is also applicable in the case when the medium is aperiodic (i.e., condition (4) is not fulfilled) and the sample is assumed to be thick: dδ0 > σ0, Λ. REFERENCES 1. L. M. Blinov, Electro-Optics and Magneto-Optics of Liquid Crystals (Nauka, Moscow, 1978; Wiley, New York, 1983). 2. P. G. de Gennes, Solid State Commun. 6, 163 (1968). 3. R. Meyer, Appl. Phys. Lett. 14, 208 (1969). 4. S. C. Chou, L. Cheung, and R. B. Meyer, Solid State Commun. 11, 977 (1972). OPTICS AND SPECTROSCOPY
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OPTICS OF HELICAL PERIODIC MEDIA IN THE PRESENCE OF A WAVE 5. R. Dreher, Solid State Commun. 12, 519 (1973). 6. S. Shtrikman and M. Tur, J. Opt. Soc. Am. 64, 1178 (1974). 7. V. A. Belyakov and V. E. Dmitrienko, Fiz. Tverd. Tela (Leningrad) 17, 491 (1975) [Sov. Phys. Solid State 17, 307 (1975)]. 8. V. A. Belyakov and A. S. Sonin, Optics of Cholesteric Liquid Crystals (Nauka, Moscow, 1982). 9. S. M. Osadchiœ, Kristallografiya 29, 976 (1984) [Sov. Phys. Crystallogr. 29, 573 (1984)]. 10. D. G. Khoshtariya, S. M. Osadchiœ, and G. S. Chilaya, Kristallografiya 30, 755 (1985) [Sov. Phys. Crystallogr. 30, 440 (1985)]. 11. B. Ya. Zel’dovich and N. V. Tabiryan, Zh. Éksp. Teor. Fiz. 82, 167 (1982) [Sov. Phys. JETP 55, 99 (1982)].
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12. A. H. Gevorgyan and O. S. Eritsyan, Izv. Akad. Nauk Arm. SSR, Fiz. 19, 135 (1984). 13. L. A. Ostrovskiœ and N. B. Stepanov, Izv. Vyssh. Uchebn. Zaved., Radiofiz. 14, 489 (1971). 14. A. Yariv and P. Yeh, Optical Waves in Crystals: Propagation and Control of Laser Radiation (Wiley, New York, 1984; Mir, Moscow, 1987). 15. A. H. Gevorgyan, K. V. Papoyan, and O. V. Pikichyan, Opt. Spektrosk. 88, 586 (2000) [Opt. Spectrosc. 88, 586 (2000)]. 16. A. H. Gevorgyan, Opt. Spektrosk. 89, 685 (2000) [Opt. Spectrosc. 89, 631 (2000)].
Translated by N. Reutova
