- Email: [email protected]
Errata to: “nominal model for structure-property relations of chiral dielectric sculptured thin films”
COMPUTER MODELLING
PERGAMON
Mathematical
and Computer
Modelling 35 (2002)
1355-1363 www.elsevier.com/locate/mcm
Errata To: “Nominal Model for Structure-Property Relations of Chiral Dielectric Sculptured Thin Films” Mathl.
Comput. Modelling, Vol. 34, No. 12/13, pp. 1499-1514, 2001
J. A. SHERWIN AND A. LAKHTAKIA CATMAS-Computational and Theoretical Materials Science Group Department of Engineering Science and Mechanics Pennsylvania State University University Park, PA 16802-6812, U.S.A. [email protected]
(Received January 2002; accepted February 200.2) Abstract-In the subject paper, calculationsreportedfor the stated angular valuea of x were actually made for n/2 - x. Consequently, Figures l-7 are incorrect, and a few quantitative, but not qualitative, conclusions drawn therefrom are also incorrect. The error in the computer program came to light very recently, after the printing of the journal issue. We deeply regret the error. Corrected versions of Sections 5 and 6 of the paper are presented here,and will also be availablein the forthcomingPh.D. Thesis of the first author. @ 2002 Elsevier Science Ltd. All rights reserved.
5.
NUMERICAL
RESULTS
AND DISCUSSION
Spectrums of optical rotation, transmittance ellipticity, linear and circular dichroisms, and apparent linear and circular dichroisms were calculated as functions of the constitutive and the geometric parameters Ed, 7~~73, x, Sz, f, b, and the frwspace wavelength X0 = 27r/ko. Maximum magnitudes in the computed spectrums were determined in wavelength-regimes that cover the relevant Bragg regimes. All spectral maximums are identified here by the superscript max; thus, LD:-; is the maximum value of LD,,, over a prescribed range of X0. The variations of these maximums were then examined with respect to any one of the constitutive and the geometric parameters, while the other parameters were held fixed. The relative permittivity e, was chosen to obey a Lorentz resonance model as per E,
Ps
(A,) = 1+ 1+
(N,-l -
ixsx,‘)2’
We set p, = 0.8, N, = 50, and X, = 180 nm for calculations to.ensure that absorption is moderate at visible wavelengths. All calculations were performed for structurally right-handed TFHBMs. 5.1. The Slenderness
Aspect
Ratio
73
Let us begin with the effect of 7s. We chose 72 = 1, R = 200nm, x = 20”, and the flm thickness L = 20 R. The inclusion volume fraction f was chosen to take on the values 0.2, 0.5, 0.8; 0895-7177/02/i! - see front matter@ 2002 Elsevier Science Ltd. AN rights reserved. PII: SO895-7177(02)00088-2
Typeset
by &@-w
J. A. SHERWINAND A. LAKHTAKIA
1356
accordingly, the Bragg regimes are roughly 411.3 < XO < 425.7 nm for f = 0.2,434 < X0 < 448 nm for f = 0.5, and 457.1 < X0 < 471.4nm for f = 0.8. Plots of r$g” and \krx versus 73 for p-polarized incidence are shown in Figures la and lb. The behaviors of the maximums of LCD,
CD,,,,CD, and CD,,
with respect to 73 are similar, and are, therefore, not shown here. Clearly,
each observable property maximum becomes virtually independent of 73 when the ratio y3/y2 exceeds about 10, and the same conclusion emerged from the calculations made for s-polarized incidence. Hence, we chose 73 = 20 and varied 1 5 72 < 3 for most of the remaining calculations.
(4
(b)
Figure 1. (a) Optical rotation maximumand (b) transmittanceellipticity maximum for p-polarizedincidenceplotted as functionsof 73. The maximums were determined over the spectral range X0 E [400,600] nm; -yz = 1, n = 2OOnm, L = 20R, x = 20°, and f = 0.2, 0.5, 0.8.
5.2. The Transverse Aspect
Ratio --ye
Next, we come to the role played by 72. Spectral maximums of optical rotation and transmittance ellipticity over the range X0 E [400,600] nm are plotted in Figures 2a and 2b as functions of 72 for ppolarized incidence, when 73 = 20, f = 0.5, R = 200nm, L = 2OQ and X = 20°, 40’) 60”.
Similar plots also emerged for s-polarized incidence, as well as for all four
dichroisms. The latter are exemplified by the plot of CD”*”
versus 72 in Figure 2c.
As Figures 2a-2c clearly indicate, for each chosen value of X, there exists a critical value of 72 such that the optical activity reduces to zero. We denote thii value of 72 by 7:. After curve-fitting exercises on plots for all observable optical property maximums, we found that ‘y!j G%1.485X-l.ll~9, where X is in radian. As +$ depends essentially on the inverse of X, it is quite sensitive to changes in this variable. Let us, however, observe here that X and E,,b,c are intimately related in a directional vapor deposition process [14]. The maximum values of all the observable properties were also found to obey simple empirical relations with respect to 72 and X. Thus,
Pmax = A(X) (72 - 7:)4 + B(X) (72 - 7:)3 + C(X) (72 - 7;)2+ D(x) (72 - 7;),
(2)
where A(X), B(x), C(x), and D(x) qre coefficent functions of X, while p stands for a particular observable property. The coefficient functions were found to have a quadratic dependence on in Figure 2b wm found tb obey equation (2) with coefficients A(X) = X. For example, @” -0.12 + 0.57X - 0.16X2, B(X) = 0.43 + 0.80X - 2.43X2, C(X) = 4.44 - 13.55X + 10.95X2, and * D(X) = 2.19 - 5.10X+2.78X2, where X is in radian. The previous paragraph implies that a high degree of optical activity is possible by employing large values of 72 when X is very large. ‘Conversely, the optical activity can be maximized by employing small values of X when 72 2 1. Indeed, Venugopal et al. [ll] have shown that the optical activity of locally uniaxial TFHBM slabs decreases in intensity as X increases. This
1357
Errata
---- x=2oo ,- -,- x=4oo
0.25. 0.20 :*\ ‘.
0.020
-
x=60’
‘*
.-..
x=2-jo
'-1-1x=40° r=60°
0.016 ‘*,, ‘. 0.012
'***, '. b '. l. '. o0.006 8. I-l._ l. 8
l.
0.004 *
"*,*..,
O.OOtp-
,...1..-' 2.0
1.5
2.5
3
Figure 2. (a) Optical rotation maximum and (b) transmittance elipticity maximum for p-polarized incidence, and (c) circular dichroism maximum plotted as functions of yz. The maximums were determined over the spectral range X0 E [400,600] nm; ~s=20,SZ=200nm,L=20R,f=0.5,and~=20°,400,600.
becomes width
apparent
from the center-wavelength
( AXc)Bragg of the Bragg regime,
Xtragg and the full-width-at-half-maximum
estimated
band-
as follows [12]:
(3)
where Ed =
&a&b
(5)
& COs2X -+ &bsin2 X ’
As x increases, sc -+ sa for real CTFs, while Ed + E, from equation (5). Accordingly, (AXe)Bragg + 0. Indeed, to maximize optical activity, one should either keep x as small as possible and 72 2 1, or keep x very large and 72 as large as possible, while keeping 7s > 72. We repeated our calculations, but keeping x = 40” fixed while varying f = 0.2, 0.5, 0.8. The counterparts of Figures 2a-2c are presented as Figures 3a-3c. From these plots and related ones not shown here, we determined 7: = 3.10 - 2.14f + 1.06f2. When x GZ!90”, -y$ is virtually independent of mid-range values of f. All observable properties were found to obey relations of the form P max = A(f)
(~2 - Y;)~
+ B(f)
(-12 - Y;)~
+ CC.0
(~2 - T$)~
+ W)
(~2 -
7;)
,
(6)
1358
J. A. SHERWINAND A. LAKHTAKIA
(4
(b)
Figure 3. The same ss Figure 2 except x = 40’ and the inclusion volume fraction f = 0.2, 0.5, 0.8.
where A(f),
etc., are quadratic functions of f.
As an example, the coefficients for @”
calculated from the data for Figure 3a as A(f) = 2.08 - 7.18f + 6.84f2, B(f) 1.74f2, C(f) = -0.13 + 1.80f - 1.82f2, and D(f) = 0.003 - 0.06.f + 0.08f2.
were
= 0.37 - 2.01f +
5.3. The Inclusion Volume Fraction f We now delineate the role played by f. Representative plots of optical rotation maximum and transmittance ellipticity maximum for ppolarized incidence, as well as maximums of CD, CD,,,, CD, and CD,,,, over the range Xs E [400,600] nm and 7s = 1,2,3 are provided in Figures 4a-4f. We note from these figures that any observable property maximum pmax + 0 as f + 0,l. This is because the TFHBM is isotropic (and homogeneous) when either f = 0 or f = 1. Furthermore, for all observable properties, there exists a single maximum value of pmBXfor each value of 7s. The value of f at which the maximum occurs is hereafter denoted by fs. After curve-fitting exercises for all pm=-f plots, we found that fc obeys the functional form fs = a + by2 + cyg. The corresponding value of p”““(fs) E pf;” varies as a quadratic function of 7s. For example, 40”“” N 7.90 - 5.0772 + 0.877;. Unlike $, fs varies from property to property even when all parameters are the same. For example, the maximum values of dr and q:” occur in Figures 4a and 4b at f’ = 0.41, 0.38, 0.32 for ys = 1,2,3, respectively. The maximums of CD”= in Figure 4d are located at fo = 0.62, 0.59, 0.52, which are all higher than the fs values for 4:“” and QE=. The values offs for the maximums of LDrpy are nearly the same as for those of LD”“. The maximums of CD”= in Figure 4e occur at fo = 0.30, 0.24, 0.20, which are lower than for the maximums of 4:” and
1359
.Errata
“‘0
0.2
0.4
0.6
0.6
1
f
f
(b)
(4 0.03
----
0.05
y,=l
..,.,. yzz2 0.02.
.8 ,-’ l.0.\.. -.
z 9
.’ ,
-0’
-
‘-
0.04
r&J
0.03 ‘.
l.
---.
y,=l
,.,.,. yzz2
*
-
..-•-•..
y*=3
.’ 0’
%
,.’
,. l*
‘-.
‘. \
\ ‘\,
f
f (4
(cl 0.03
----
y2=l
,-,.,- y*=2 ,,‘-.-.,
0.02~
:’
..
-
8,. ‘.
L o
: O.Olr
/
‘. \\,, .
I:
r,*
‘8.
0.005.
,,,* .
0
0.2
0.6
0.4
,..---__
I= @ 0.001 0
0.6
,: l’
#’
-.
‘.
‘,
‘.
‘.
:’
‘.
‘\
1
f
(e) Figure 4. (a) Optical rotation maximum and (b) transmittance ellipticity maximum for ppolarized incidence, (c) linear dichroism maximum, (d) apparent linear dichroism maximum, (e) circular dichroism maximum, and (f) apparent circular dichroism maximum plotted as functions of f. The maximums were determined over the spectral range Xo E [400,600] nm; -ya = 20, R = 200 nm, L = 20S2, x = 20°, and ys = 1,2,3.
QK”“. The CD::; maximums of 4:“”
maximum values, however, are located at nearly the same fo values as the and QF”.
Unlike the geometric ratios 72 and 73, the wavelength at which the maximum values of observable properties occur in the spectral signatures depends on the inclusion volume fraction f.
1360
J. A. SHERWINAND A. LAKHTAKIA
4000
0.2
0.4
0.6
0.8
f Figure 5. The wavelength Xo of optical rotation maximum for p-polarized incidence plotted ss a function off. All other parameters are the same se for Figure 4.
The wavelength of positive maximum values of the optical rotation for ppolarized incidence is plotted against f in Figure 5. The dependence of this wavelength on f is clearly linear. The endpoints of the plots in Figure 5 need an explanation. Although optical activity may appear to occur at f = 0,1, it cannot. While the vertical axis of the plots can be directly related to Xfragg, we must also observe that (AXo)B’agg= 0 when f = 0,l. In other words, the circular Bragg phenomenon vanishes at the endpoints of the plots. 5.4. The Angle of Rise x
The variations of q$” and QK” with respect to the angle of rise x are presented for p polarized incidence in Figures 6a and 6b. The ratios “(3 = 20 and “yz= 2, R = 200 nm, L = 20 R, and f takes on the values 0.2, 0.5, 0.8. Clearly, there is a value of x at which there is no optical activity. This value of x is denoted by XQ. From Figures 6a and 6b, we get xc = 42’, 47O, 49” when f = 0.2, 0.5, 0.8. The value of xc is the same for all observable property maximums. An analytical explanation for the existence of ~0 is as follows [13]. For axial excitation, the local linear birefringence of a TFHBM is the difference between & and fi [13]. When x = xc, cc = .& and the optical activity vanishes.
(4
W
Figure 6. (a) Optical rotation maximum and (b) transmittance ellipticity maximum for ppolarted incidence plotted ss functions of x. The maximums were determined over the spectral range Xo E [400,600] nm; 7s = 20, m = 2, f = 0.2, 0.5, 0.8, n=2OOnm,andL=200.
1361
Errata
5.5. The Structural
Half-Period
52
The role of R has been amply investigated by Venugopal et al. [11,13], who determined that Brass and ~7 increases linearly with s2, when all other parameters are fixed. Our numer(Ax01 increases virtually linearly ical studies also showed that the wavelength corresponding to $y with 0; see also, [12]. 5.6. Orientational
Averaging
The presented Bruggeman formalism is predicated on the assumption that all the helicoidal columns are locally tilted at the same angle x with respect to any zy plane. In practice, vapor deposition is not perfectly exact, and some distribution of x must be accounted for. In the absence of any relevant quantitative morphological studies, we confine ourselves here to a simple illustrative example. Let us assume that x E [xa - 6x, xa + 6x1, where xa is the center-value and 6, is the maximum deviation. We define the orientational averages
which lead to the orientationally
averaged Bruggeman equation f
If the distribution resulting $,
(cl,)+(1-
f)
(gv)=p.
(8)
F(v)
is symmetric about v = 0, then (a,) and (a,) are diagonal and the = is also diagonal in the primed coordinate system. The=form of the distribution
chosen here is F(v) =
COP @v/26,) s_“;, cosm (7rva/26x) duo ’
(9)
where m is an integer. The remainder of the analysis to determine EHCM remains the same as = in Section 3. In Figure 7, $JY for ppolarized incidence is plotted as a function of 6,. Here, yz = 3, ys = 20, 0 = 200nm, f = 0.5, and L = 200, while xa = 20” and m = 1. The dependence of ~~ on 6, is given by +Eax = 0.3754 - 0.0706, - 1.0566:, where 6, is in radian. Similar quadratic relations hold for all other observable property maximums. We note that the larger the width 26, of the distribution, the smaller is c$F. Obviously, when 6, = 90”, all orientations in a vertical plane become possible, which would seriously impair the degree of anisotropy of the TFHBM. It should also be clear that a larger value of m corresponds to a distribution that decays more rapidly about v = 0 than a distribution corresponding to a lower value of m. This implies that smaller values of m accentuate the effects of 6,. We examined m = 0,1,2 and the results followed the trend previously outlined. However, the differences between the results for different values of m were negligibly small over the range of 6, examined. More importantly, we conclude that optical activity is only mildly perturbed when 6, is small. For instance, Figure 7 shows that 4:” changes by x 1% when S, increases from lo to Y. This finding suggests that x in directional vapor deposition does not have to be very precisely controlled-at least, for optical applications of TFHBMs.
6. CONCLUDING
REMARKS
We presented a simple but richly endowed model to account for the permittivity dyadic and the optical response properties of dielectric TFHBMs. In this nominal model, the helicoidal columnar structure of a TFHBM is emulated by a distribution of ellipsoidal inclusions. The
1362
J. A. SHERWIN AND A. LAKHTAKIA
Figure 7. Optical rotation maximum for gpolarised incidence plotted as a function of 6x. The maximums were determined over the spectral range Ao E [400,600] nm; 72 = $73 = 20, n = 200nm, L = 20Q, x0 = 20°, and m = 1.
shape, the volume fraction, and the orientation of the inclusions are significant variables in our model, which must be calibrated against measured optical data. We examined the optical response of an axially excited TFHBM slab embedded in vacuum with reference to variations in the slenderness and transverse aspect ratios 73 and 72, the inclusion volume fraction f, the angle of rise x, and the distribution width 6,. Concentrating on various measures of optical activity, we arrived at the following conclusions.
(4
All observable properties strongly depend on the transverse aspect ratio 72, 1 5 ^I;?< 7s. There exists a specific value of 7s denoted by $j such that all optical activity disappears. The value of 7: can be parameterized in terms of other geometric factors and sg. All observable property maximums are best-fitted to fourth-order polynomials of 72. (b) All observable property maximums have similar dependencies on f; furthermore, pmax --+ 0 (cl as f + 0,l. (4 There exists an fe for each value of 7s such that pmax(fc) > pm=(f), f # fo. The value of fc depends both on the values of other geometric and constitutive parameters and on the property p. (4 The quantities +5:“, %gax, and CD&y reach their respective maximum values with respect to f E [0, l] at the same value of fc. The values of fc for LDma” and LDFpy are higher than for +KBx, !I$“, and CDrpy and are approximately the same for each other; while the values of fc for CDma” are lower. Except for their magnitudes, &!“, XQ”, and CDEpy have precisely the same dependencies on f and 72. This result does not hold for CDma”, LDFpy, and CDm” Small distributions of the angle of rise x do not significantly affect the observable property maximums explored here. Validation of the proposed model and verification of the aforementioned perimental data shall be pursued in the near future.
findings against ex-
REFERENCES R. Messier and A. Lakhtakia, Sculptured thin films-II. Experiments and applications, Mat. Res. Innouat. 2, 217-222 (1999). A. Lakhtalda, R.. Messier, M.J. Brett and K. Robbie, Sculptured thin films (STFS) for optical, chemical and biological applications, Znnou. Mat. Res. 1, 165-176 (1996). A. Lakhtakia and W.S. Weiglhofer, On light propagation in helicoidal bianisotropic mediums, Ptwc. R. Sot. Land. A 448, 419-437 (1995); Erratum 454, 3275 (1998). E. Ertekin and A. Lakhtalda, Sculptured thin films for optical sensing of gas concentration, Eur. Phys. J. Appl. P/q/s. 5, 45-50 (1999).
Errata
1363
5. P.D. Sunal, A. Lakhtakia and R. Messier, Simple model for dielectric thin-film helicoidal bianisotropic media, Opt. Commun. 158, 119-126 (1998). 6. W.S. Weiglhofer and A. Lakhtakia, On electromagnetic waves in biaxial media, Electromagnetics 19, 351-362 7. a. 9. 10. 11.
12. 13. 14.
(1999). Q.h. Wu, I.J. Hodgkinson and A. Lakhtakia, Circular polarization filters made of chiral sculptured thin films: Experimental and simulation results, Opt. Engg. 39, 1863-1868 (2000). B. Michel, A. Lakhtakia and W.S. Weiglhofer, Homogenization of linear bianisotropic particulate composite media-Numerical studies, Int. J. Appl. Electromagn. Mech. 9, 167-178 (1998); 10, 537-538 (1999). D. Stroud, Generalized effective-medium approach to the conductivity of an inhomogeneous material, Phys. Rev. B 12, 3368-3373 (1975). J.L. Buchanan and P.R. Turner, Numerical Methods and Analysis, McGraw-Hill, New York, (1992). V.C. Venugopal and A. Lakhtakia, Second harmonic emission from an axially excited slab of a dielectric thin-film helicoidal bianisotropic medium, Proc. R. Sot. Lond. A 464, 1535-1571 (1998); Erratum 455, 4383 (1999). A. Lakhtakia, Spectral signatures of axially excited slabs of dielectric thin-film helicoidal bianisotropic mediums, Eur. Phys. J. Appl. Phys. 8, 129-137 (1999). V.C. Venugopal and A. Lakhtakia, Electromagnetic plane-wave response characteristics of non-axially excited slabs of dielectric thin-film helicoidal bianisotropic mediums, Proc. R. Sot. Lond. A 456, 125-161 (2000). I.J. Hodgkinson and Q.h. Wu, Birefringent Thin Films and Polarizing Elements, World Scientific, Singapore, (1997).
PERGAMON
Mathematical
and Computer
Modelling 35 (2002)
1355-1363 www.elsevier.com/locate/mcm
Errata To: “Nominal Model for Structure-Property Relations of Chiral Dielectric Sculptured Thin Films” Mathl.
Comput. Modelling, Vol. 34, No. 12/13, pp. 1499-1514, 2001
J. A. SHERWIN AND A. LAKHTAKIA CATMAS-Computational and Theoretical Materials Science Group Department of Engineering Science and Mechanics Pennsylvania State University University Park, PA 16802-6812, U.S.A. [email protected]
(Received January 2002; accepted February 200.2) Abstract-In the subject paper, calculationsreportedfor the stated angular valuea of x were actually made for n/2 - x. Consequently, Figures l-7 are incorrect, and a few quantitative, but not qualitative, conclusions drawn therefrom are also incorrect. The error in the computer program came to light very recently, after the printing of the journal issue. We deeply regret the error. Corrected versions of Sections 5 and 6 of the paper are presented here,and will also be availablein the forthcomingPh.D. Thesis of the first author. @ 2002 Elsevier Science Ltd. All rights reserved.
5.
NUMERICAL
RESULTS
AND DISCUSSION
Spectrums of optical rotation, transmittance ellipticity, linear and circular dichroisms, and apparent linear and circular dichroisms were calculated as functions of the constitutive and the geometric parameters Ed, 7~~73, x, Sz, f, b, and the frwspace wavelength X0 = 27r/ko. Maximum magnitudes in the computed spectrums were determined in wavelength-regimes that cover the relevant Bragg regimes. All spectral maximums are identified here by the superscript max; thus, LD:-; is the maximum value of LD,,, over a prescribed range of X0. The variations of these maximums were then examined with respect to any one of the constitutive and the geometric parameters, while the other parameters were held fixed. The relative permittivity e, was chosen to obey a Lorentz resonance model as per E,
Ps
(A,) = 1+ 1+
(N,-l -
ixsx,‘)2’
We set p, = 0.8, N, = 50, and X, = 180 nm for calculations to.ensure that absorption is moderate at visible wavelengths. All calculations were performed for structurally right-handed TFHBMs. 5.1. The Slenderness
Aspect
Ratio
73
Let us begin with the effect of 7s. We chose 72 = 1, R = 200nm, x = 20”, and the flm thickness L = 20 R. The inclusion volume fraction f was chosen to take on the values 0.2, 0.5, 0.8; 0895-7177/02/i! - see front matter@ 2002 Elsevier Science Ltd. AN rights reserved. PII: SO895-7177(02)00088-2
Typeset
by &@-w
J. A. SHERWINAND A. LAKHTAKIA
1356
accordingly, the Bragg regimes are roughly 411.3 < XO < 425.7 nm for f = 0.2,434 < X0 < 448 nm for f = 0.5, and 457.1 < X0 < 471.4nm for f = 0.8. Plots of r$g” and \krx versus 73 for p-polarized incidence are shown in Figures la and lb. The behaviors of the maximums of LCD,
CD,,,,CD, and CD,,
with respect to 73 are similar, and are, therefore, not shown here. Clearly,
each observable property maximum becomes virtually independent of 73 when the ratio y3/y2 exceeds about 10, and the same conclusion emerged from the calculations made for s-polarized incidence. Hence, we chose 73 = 20 and varied 1 5 72 < 3 for most of the remaining calculations.
(4
(b)
Figure 1. (a) Optical rotation maximumand (b) transmittanceellipticity maximum for p-polarizedincidenceplotted as functionsof 73. The maximums were determined over the spectral range X0 E [400,600] nm; -yz = 1, n = 2OOnm, L = 20R, x = 20°, and f = 0.2, 0.5, 0.8.
5.2. The Transverse Aspect
Ratio --ye
Next, we come to the role played by 72. Spectral maximums of optical rotation and transmittance ellipticity over the range X0 E [400,600] nm are plotted in Figures 2a and 2b as functions of 72 for ppolarized incidence, when 73 = 20, f = 0.5, R = 200nm, L = 2OQ and X = 20°, 40’) 60”.
Similar plots also emerged for s-polarized incidence, as well as for all four
dichroisms. The latter are exemplified by the plot of CD”*”
versus 72 in Figure 2c.
As Figures 2a-2c clearly indicate, for each chosen value of X, there exists a critical value of 72 such that the optical activity reduces to zero. We denote thii value of 72 by 7:. After curve-fitting exercises on plots for all observable optical property maximums, we found that ‘y!j G%1.485X-l.ll~9, where X is in radian. As +$ depends essentially on the inverse of X, it is quite sensitive to changes in this variable. Let us, however, observe here that X and E,,b,c are intimately related in a directional vapor deposition process [14]. The maximum values of all the observable properties were also found to obey simple empirical relations with respect to 72 and X. Thus,
Pmax = A(X) (72 - 7:)4 + B(X) (72 - 7:)3 + C(X) (72 - 7;)2+ D(x) (72 - 7;),
(2)
where A(X), B(x), C(x), and D(x) qre coefficent functions of X, while p stands for a particular observable property. The coefficient functions were found to have a quadratic dependence on in Figure 2b wm found tb obey equation (2) with coefficients A(X) = X. For example, @” -0.12 + 0.57X - 0.16X2, B(X) = 0.43 + 0.80X - 2.43X2, C(X) = 4.44 - 13.55X + 10.95X2, and * D(X) = 2.19 - 5.10X+2.78X2, where X is in radian. The previous paragraph implies that a high degree of optical activity is possible by employing large values of 72 when X is very large. ‘Conversely, the optical activity can be maximized by employing small values of X when 72 2 1. Indeed, Venugopal et al. [ll] have shown that the optical activity of locally uniaxial TFHBM slabs decreases in intensity as X increases. This
1357
Errata
---- x=2oo ,- -,- x=4oo
0.25. 0.20 :*\ ‘.
0.020
-
x=60’
‘*
.-..
x=2-jo
'-1-1x=40° r=60°
0.016 ‘*,, ‘. 0.012
'***, '. b '. l. '. o0.006 8. I-l._ l. 8
l.
0.004 *
"*,*..,
O.OOtp-
,...1..-' 2.0
1.5
2.5
3
Figure 2. (a) Optical rotation maximum and (b) transmittance elipticity maximum for p-polarized incidence, and (c) circular dichroism maximum plotted as functions of yz. The maximums were determined over the spectral range X0 E [400,600] nm; ~s=20,SZ=200nm,L=20R,f=0.5,and~=20°,400,600.
becomes width
apparent
from the center-wavelength
( AXc)Bragg of the Bragg regime,
Xtragg and the full-width-at-half-maximum
estimated
band-
as follows [12]:
(3)
where Ed =
&a&b
(5)
& COs2X -+ &bsin2 X ’
As x increases, sc -+ sa for real CTFs, while Ed + E, from equation (5). Accordingly, (AXe)Bragg + 0. Indeed, to maximize optical activity, one should either keep x as small as possible and 72 2 1, or keep x very large and 72 as large as possible, while keeping 7s > 72. We repeated our calculations, but keeping x = 40” fixed while varying f = 0.2, 0.5, 0.8. The counterparts of Figures 2a-2c are presented as Figures 3a-3c. From these plots and related ones not shown here, we determined 7: = 3.10 - 2.14f + 1.06f2. When x GZ!90”, -y$ is virtually independent of mid-range values of f. All observable properties were found to obey relations of the form P max = A(f)
(~2 - Y;)~
+ B(f)
(-12 - Y;)~
+ CC.0
(~2 - T$)~
+ W)
(~2 -
7;)
,
(6)
1358
J. A. SHERWINAND A. LAKHTAKIA
(4
(b)
Figure 3. The same ss Figure 2 except x = 40’ and the inclusion volume fraction f = 0.2, 0.5, 0.8.
where A(f),
etc., are quadratic functions of f.
As an example, the coefficients for @”
calculated from the data for Figure 3a as A(f) = 2.08 - 7.18f + 6.84f2, B(f) 1.74f2, C(f) = -0.13 + 1.80f - 1.82f2, and D(f) = 0.003 - 0.06.f + 0.08f2.
were
= 0.37 - 2.01f +
5.3. The Inclusion Volume Fraction f We now delineate the role played by f. Representative plots of optical rotation maximum and transmittance ellipticity maximum for ppolarized incidence, as well as maximums of CD, CD,,,, CD, and CD,,,, over the range Xs E [400,600] nm and 7s = 1,2,3 are provided in Figures 4a-4f. We note from these figures that any observable property maximum pmax + 0 as f + 0,l. This is because the TFHBM is isotropic (and homogeneous) when either f = 0 or f = 1. Furthermore, for all observable properties, there exists a single maximum value of pmBXfor each value of 7s. The value of f at which the maximum occurs is hereafter denoted by fs. After curve-fitting exercises for all pm=-f plots, we found that fc obeys the functional form fs = a + by2 + cyg. The corresponding value of p”““(fs) E pf;” varies as a quadratic function of 7s. For example, 40”“” N 7.90 - 5.0772 + 0.877;. Unlike $, fs varies from property to property even when all parameters are the same. For example, the maximum values of dr and q:” occur in Figures 4a and 4b at f’ = 0.41, 0.38, 0.32 for ys = 1,2,3, respectively. The maximums of CD”= in Figure 4d are located at fo = 0.62, 0.59, 0.52, which are all higher than the fs values for 4:“” and QE=. The values offs for the maximums of LDrpy are nearly the same as for those of LD”“. The maximums of CD”= in Figure 4e occur at fo = 0.30, 0.24, 0.20, which are lower than for the maximums of 4:” and
1359
.Errata
“‘0
0.2
0.4
0.6
0.6
1
f
f
(b)
(4 0.03
----
0.05
y,=l
..,.,. yzz2 0.02.
.8 ,-’ l.0.\.. -.
z 9
.’ ,
-0’
-
‘-
0.04
r&J
0.03 ‘.
l.
---.
y,=l
,.,.,. yzz2
*
-
..-•-•..
y*=3
.’ 0’
%
,.’
,. l*
‘-.
‘. \
\ ‘\,
f
f (4
(cl 0.03
----
y2=l
,-,.,- y*=2 ,,‘-.-.,
0.02~
:’
..
-
8,. ‘.
L o
: O.Olr
/
‘. \\,, .
I:
r,*
‘8.
0.005.
,,,* .
0
0.2
0.6
0.4
,..---__
I= @ 0.001 0
0.6
,: l’
#’
-.
‘.
‘,
‘.
‘.
:’
‘.
‘\
1
f
(e) Figure 4. (a) Optical rotation maximum and (b) transmittance ellipticity maximum for ppolarized incidence, (c) linear dichroism maximum, (d) apparent linear dichroism maximum, (e) circular dichroism maximum, and (f) apparent circular dichroism maximum plotted as functions of f. The maximums were determined over the spectral range Xo E [400,600] nm; -ya = 20, R = 200 nm, L = 20S2, x = 20°, and ys = 1,2,3.
QK”“. The CD::; maximums of 4:“”
maximum values, however, are located at nearly the same fo values as the and QF”.
Unlike the geometric ratios 72 and 73, the wavelength at which the maximum values of observable properties occur in the spectral signatures depends on the inclusion volume fraction f.
1360
J. A. SHERWINAND A. LAKHTAKIA
4000
0.2
0.4
0.6
0.8
f Figure 5. The wavelength Xo of optical rotation maximum for p-polarized incidence plotted ss a function off. All other parameters are the same se for Figure 4.
The wavelength of positive maximum values of the optical rotation for ppolarized incidence is plotted against f in Figure 5. The dependence of this wavelength on f is clearly linear. The endpoints of the plots in Figure 5 need an explanation. Although optical activity may appear to occur at f = 0,1, it cannot. While the vertical axis of the plots can be directly related to Xfragg, we must also observe that (AXo)B’agg= 0 when f = 0,l. In other words, the circular Bragg phenomenon vanishes at the endpoints of the plots. 5.4. The Angle of Rise x
The variations of q$” and QK” with respect to the angle of rise x are presented for p polarized incidence in Figures 6a and 6b. The ratios “(3 = 20 and “yz= 2, R = 200 nm, L = 20 R, and f takes on the values 0.2, 0.5, 0.8. Clearly, there is a value of x at which there is no optical activity. This value of x is denoted by XQ. From Figures 6a and 6b, we get xc = 42’, 47O, 49” when f = 0.2, 0.5, 0.8. The value of xc is the same for all observable property maximums. An analytical explanation for the existence of ~0 is as follows [13]. For axial excitation, the local linear birefringence of a TFHBM is the difference between & and fi [13]. When x = xc, cc = .& and the optical activity vanishes.
(4
W
Figure 6. (a) Optical rotation maximum and (b) transmittance ellipticity maximum for ppolarted incidence plotted ss functions of x. The maximums were determined over the spectral range Xo E [400,600] nm; 7s = 20, m = 2, f = 0.2, 0.5, 0.8, n=2OOnm,andL=200.
1361
Errata
5.5. The Structural
Half-Period
52
The role of R has been amply investigated by Venugopal et al. [11,13], who determined that Brass and ~7 increases linearly with s2, when all other parameters are fixed. Our numer(Ax01 increases virtually linearly ical studies also showed that the wavelength corresponding to $y with 0; see also, [12]. 5.6. Orientational
Averaging
The presented Bruggeman formalism is predicated on the assumption that all the helicoidal columns are locally tilted at the same angle x with respect to any zy plane. In practice, vapor deposition is not perfectly exact, and some distribution of x must be accounted for. In the absence of any relevant quantitative morphological studies, we confine ourselves here to a simple illustrative example. Let us assume that x E [xa - 6x, xa + 6x1, where xa is the center-value and 6, is the maximum deviation. We define the orientational averages
which lead to the orientationally
averaged Bruggeman equation f
If the distribution resulting $,
(cl,)+(1-
f)
(gv)=p.
(8)
F(v)
is symmetric about v = 0, then (a,) and (a,) are diagonal and the = is also diagonal in the primed coordinate system. The=form of the distribution
chosen here is F(v) =
COP @v/26,) s_“;, cosm (7rva/26x) duo ’
(9)
where m is an integer. The remainder of the analysis to determine EHCM remains the same as = in Section 3. In Figure 7, $JY for ppolarized incidence is plotted as a function of 6,. Here, yz = 3, ys = 20, 0 = 200nm, f = 0.5, and L = 200, while xa = 20” and m = 1. The dependence of ~~ on 6, is given by +Eax = 0.3754 - 0.0706, - 1.0566:, where 6, is in radian. Similar quadratic relations hold for all other observable property maximums. We note that the larger the width 26, of the distribution, the smaller is c$F. Obviously, when 6, = 90”, all orientations in a vertical plane become possible, which would seriously impair the degree of anisotropy of the TFHBM. It should also be clear that a larger value of m corresponds to a distribution that decays more rapidly about v = 0 than a distribution corresponding to a lower value of m. This implies that smaller values of m accentuate the effects of 6,. We examined m = 0,1,2 and the results followed the trend previously outlined. However, the differences between the results for different values of m were negligibly small over the range of 6, examined. More importantly, we conclude that optical activity is only mildly perturbed when 6, is small. For instance, Figure 7 shows that 4:” changes by x 1% when S, increases from lo to Y. This finding suggests that x in directional vapor deposition does not have to be very precisely controlled-at least, for optical applications of TFHBMs.
6. CONCLUDING
REMARKS
We presented a simple but richly endowed model to account for the permittivity dyadic and the optical response properties of dielectric TFHBMs. In this nominal model, the helicoidal columnar structure of a TFHBM is emulated by a distribution of ellipsoidal inclusions. The
1362
J. A. SHERWIN AND A. LAKHTAKIA
Figure 7. Optical rotation maximum for gpolarised incidence plotted as a function of 6x. The maximums were determined over the spectral range Ao E [400,600] nm; 72 = $73 = 20, n = 200nm, L = 20Q, x0 = 20°, and m = 1.
shape, the volume fraction, and the orientation of the inclusions are significant variables in our model, which must be calibrated against measured optical data. We examined the optical response of an axially excited TFHBM slab embedded in vacuum with reference to variations in the slenderness and transverse aspect ratios 73 and 72, the inclusion volume fraction f, the angle of rise x, and the distribution width 6,. Concentrating on various measures of optical activity, we arrived at the following conclusions.
(4
All observable properties strongly depend on the transverse aspect ratio 72, 1 5 ^I;?< 7s. There exists a specific value of 7s denoted by $j such that all optical activity disappears. The value of 7: can be parameterized in terms of other geometric factors and sg. All observable property maximums are best-fitted to fourth-order polynomials of 72. (b) All observable property maximums have similar dependencies on f; furthermore, pmax --+ 0 (cl as f + 0,l. (4 There exists an fe for each value of 7s such that pmax(fc) > pm=(f), f # fo. The value of fc depends both on the values of other geometric and constitutive parameters and on the property p. (4 The quantities +5:“, %gax, and CD&y reach their respective maximum values with respect to f E [0, l] at the same value of fc. The values of fc for LDma” and LDFpy are higher than for +KBx, !I$“, and CDrpy and are approximately the same for each other; while the values of fc for CDma” are lower. Except for their magnitudes, &!“, XQ”, and CDEpy have precisely the same dependencies on f and 72. This result does not hold for CDma”, LDFpy, and CDm” Small distributions of the angle of rise x do not significantly affect the observable property maximums explored here. Validation of the proposed model and verification of the aforementioned perimental data shall be pursued in the near future.
findings against ex-
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Errata
1363
5. P.D. Sunal, A. Lakhtakia and R. Messier, Simple model for dielectric thin-film helicoidal bianisotropic media, Opt. Commun. 158, 119-126 (1998). 6. W.S. Weiglhofer and A. Lakhtakia, On electromagnetic waves in biaxial media, Electromagnetics 19, 351-362 7. a. 9. 10. 11.
12. 13. 14.
(1999). Q.h. Wu, I.J. Hodgkinson and A. Lakhtakia, Circular polarization filters made of chiral sculptured thin films: Experimental and simulation results, Opt. Engg. 39, 1863-1868 (2000). B. Michel, A. Lakhtakia and W.S. Weiglhofer, Homogenization of linear bianisotropic particulate composite media-Numerical studies, Int. J. Appl. Electromagn. Mech. 9, 167-178 (1998); 10, 537-538 (1999). D. Stroud, Generalized effective-medium approach to the conductivity of an inhomogeneous material, Phys. Rev. B 12, 3368-3373 (1975). J.L. Buchanan and P.R. Turner, Numerical Methods and Analysis, McGraw-Hill, New York, (1992). V.C. Venugopal and A. Lakhtakia, Second harmonic emission from an axially excited slab of a dielectric thin-film helicoidal bianisotropic medium, Proc. R. Sot. Lond. A 464, 1535-1571 (1998); Erratum 455, 4383 (1999). A. Lakhtakia, Spectral signatures of axially excited slabs of dielectric thin-film helicoidal bianisotropic mediums, Eur. Phys. J. Appl. Phys. 8, 129-137 (1999). V.C. Venugopal and A. Lakhtakia, Electromagnetic plane-wave response characteristics of non-axially excited slabs of dielectric thin-film helicoidal bianisotropic mediums, Proc. R. Sot. Lond. A 456, 125-161 (2000). I.J. Hodgkinson and Q.h. Wu, Birefringent Thin Films and Polarizing Elements, World Scientific, Singapore, (1997).