Specular and nonspecular, thickness-dependent, spectral holes in a slanted chiral sculptured thin film with a central twist defect

Optics Communications 215 (2003) 79–92 www.elsevier.com/locate/optcom

Specular and nonspecular, thickness-dependent, spectral holes in a slanted chiral sculptured thin film with a central twist defect Fei Wang 1, Akhlesh Lakhtakia *,1 CATMAS – Computational and Theoretical Materials Sciences Group, Department of Engineering Science and Mechanics, Pennsylvania State University, 212 EES Building, University Park, PA 16802–6812, USA Received 20 August 2002; accepted 18 November 2002

Abstract We demonstrate analytically that two types of spectral holes located approximately in the center of the Bragg regime could be exhibited by a slanted chiral sculptured thin film (STF) containing a 90° twist defect midway through its thickness. One is a nonspecular spectral reflection hole excited by an incident circularly polarized plane wave of the same handedness as the chiral slanted STF, and the other a specular transmission hole excited by an incident circularly polarized plane wave of the opposite handedness. The occurrences of these holes depend on the device thickness, and a crossover thickness can be defined. The existence of both types of spectral holes is sensitive to the dual-periodicity of the slanted chiral STF, and can be completely subverted by the Rayleigh–Wood phenomenon. Ó 2002 Elsevier Science B.V. All rights reserved. PACS: 42.25.Fx; 42.40.Eq; 42.79.Dj; 77.55.+f; 78.20.Fm Keywords: Chirality; Circular Bragg phenomenon; Rayleigh–Wood anomaly; Sculptured thin films; Spectral reflection hole; Specular transmission hole; Structural handedness

1. Introduction Volume gratings are commonly used as dielectric mirrors in optics. Such gratings are either

*

Corresponding author. Tel.: +1-814-863-4319; fax: +1-814865-9974. E-mail addresses: [email protected] (F. Wang), [email protected] (A. Lakhtakia). 1 Fax: +1-814-863-7967.

. . .HLHLHL. . . multilayers [1] (where H stands for a high-permittivity layer and L for a low-permittivity layer) or the permittivity varies sinusoidally in the thickness direction [2]. Being periodic systems, volume gratings exhibit the Bragg phenomenon – which manifests itself as a high-reflectance wavelength-regime, provided the grating thickness spans a sufficient number of permittivity periods. Haus and Shank [3] proposed the creation of a spectral hole in the reflection spectrum by inserting

0030-4018/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 ( 0 2 ) 0 2 2 0 4 - 6

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a phase defect half-way along a volume grating. The spectral hole is a narrow transmission feature in the center of the Bragg wavelength-regime, and is useful for narrowband filtering. The proposal is implemented chiefly in quarter-wavelength-shifted distributed feedback lasers [4]. When normally illuminated, so that the wave propagation vector inside a volume grating is parallel to the thickness direction, the Bragg phenomenon is insensitive to the polarization state of the incident light. This is because the described volume gratings are made of isotropic dielectric substances. But periodically nonhomogeneous materials can possess structural handedness; and if they do, the Bragg phenomenon turns to be polarization-sensitive. Cholesteric liquid crystals (CLCs) [5,6] as well as chiral sculptured thin films (STFs) [7,8] are periodically nonhomogeneous in the thickness direction, as their permittivity dyadics vary helicoidally in that direction. Accordingly, the high-reflectance characteristic of the Bragg regime is observed only when the circular polarization state of incident light matches the structural handedness of the material; otherwise, low reflectance is observed. Liquid-crystalline as well as thin-film chiral mirrors are available for circular-polarization-sensitive reflection [9,10]. The introduction of a phase defect half-way inside a structurally chiral structure also gives rise to a spectral hole – but only when the circular polarization state of the incident light matches the structural handedness [11–13]. The polarization-sensitivity of structurally chiral spectral-hole filters cannot, of course, be exhibited by isotropic spectral-hole filters. Very recently, Kopp and Genack [14] reported an even more interesting feature. As a CLC with a phase defect becomes thicker, but with its period fixed, the spectral hole in the co-handed reflectance spectrum diminishes steadily and eventually vanishes. Simultaneously, a spectral hole appears and grows in the transmittance spectrum for incident light of the other circular polarization state. The bandwidth of the second spectral hole is a few thousandths of the bandwidth of the initial spectral hole, according to theoretical calculations. Although these conclusions were reported for CLCs, in this paper we have verified

the results to qualitatively hold for chiral STFs as well. Reflection and transmission for all structures discussed heretofore are purely specular [15,16]. Spectral-hole filters implemented with these structures, therefore, cannot be used for nonspecular applications [17]. If, however, the direction of nonhomogeneity were inclined with respect to the thickness direction, then the structure would be periodic in the thickness direction as well as in a direction perpendicular to it [18]. The specular nature of the Bragg phenomenon and the nonspecular nature of the Rayleigh–Wood phenomenon (generally observed with surface-relief gratings [19,20]) can be expected to interact with each other. With this motivation, slanted chiral STFs were recently proposed by us [21]. Several physical vapor deposition techniques have emerged for manufacturing STFs [10,22–26]. Normally, the helical microcolumns of a chiral STF grow upright on the substrate. However, it is possible to grow the microcolumns slanted at angle a 6¼ 0° with respect to the substrate normal [25,27], which should lead to the fabrication of slanted chiral STFs. The interactions of the two orthogonal periodicities of slanted chiral STFs have been researched [21]. In particular, the circular Bragg phenomenon for slanted chiral STFs ða 6¼ 0°Þ is partially nonspecular, in contrast to that for an unslanted chiral STF ða ¼ 0°Þ, and is highly affected or even totally subverted at large jaj by the Rayleigh–Wood anomalies that occur due to the transverse periodicity. In this paper, we examine the optical response of a slanted chiral STF with a phase defect inserted midway through its thickness. The chosen defect is a twist defect characterized by an angle /t 6¼ 0° [12–14]. The plan of this paper is as follows: Section 2 provides a detailed description of the theoretical treatment of the boundary value problem to be solved. First, the geometry of the boundary value problem, the constitutive relations of the two halves of the slanted chiral STF, and the field representations above and below the central twist defect are presented. Then, coupled wave theory [28,29] is used to formulate a matrix ordinary differential equation [16] for the fields excited in

F. Wang, A. Lakhtakia / Optics Communications 215 (2003) 79–92

both the halves. Finally, the numerical solution procedure adopted for the boundary value problem is briefly described, the details being available elsewhere [21]. Section 3 is devoted to the presentation and discussion of the numerical results obtained. An expðixtÞ time-dependence is implicit, with x as the angular frequency of the incident plane wave, and t as the time. Vectors are in boldface, while dyadics are double-underlined.

2. Theoretical analysis Let the region 0 6 z 6 2L be occupied by the slanted chiral STF with a twist of angle /t between the upper and the lower halves about their common axis of nonhomogeneity, as shown in Fig. 1, while the half-spaces z 6 0 and z P 2L are vacuous. A plane wave is incident from the lower half-space z 6 0 on to the plane z ¼ 0. As a result, reflection and transmission into the two half-spaces occur. 2.1. Constitutive relations The relative permittivity dyadic
ðr; k0 Þ of the slanted chiral STF of thickness 2L is factorable as
ðr; k0 Þ ¼ S y ðaÞ  S z ðrÞ  S y ðvs Þ 
ref ðk0 Þ  S Ty ðvs Þ  S Tz ðrÞ  S Ty ðaÞ;

0 6 z 6 2L;

ð1Þ

where the position vector r ¼ xux þ yuy þ zuz ; k0 is the free-space wavelength, and the superscript T

81

denotes the transpose. As most STFs are locally biaxial [24,30], the reference relative permitivity dyadic is given by [31]
ref ðk0 Þ ¼
a ðk0 Þuz uz þ
b ðk0 Þux ux þ
c ðk0 Þuy uy :

ð2Þ

The wavelength-dependences of the scalars
a;b;c are assumed to emerge from a single-resonance Lorentzian model [32,33] as pa;b;c
a;b;c ðk0 Þ ¼ 1 þ  ð3Þ
2  ; 1 1 þ Na;b;c  ika;b;c k1 0 where pa;b;c are the oscillator strengths. The parameters ka;b;c and Na;b;c determine the resonance wavelengths and absorption linewidths. Based on the local columnarity of the STFs, we surmise that j
b ðk0 Þj P j
c ðk0 Þj P j
a ðk0 Þj away from the resonance wavelengths of the bulk material deposited as the thin film [31,34]. The rotational nonhomogeneity of the chosen thin film is captured by the dyadic hp i   S z ðrÞ ¼ ux ux þ uy uy cos ð r  u‘ Þ þ / X hp i   þ h uy ux  ux uy sin ðr  u‘ Þ þ / X þ uz uz ; ð4Þ the axis of rotational nonhomogeneity (i.e., the helical axis) being parallel to the unit vector u‘ ¼ sin aux þ cos auz . The structural period along the helical axis is denoted by 2X. The parameter h ¼ 1 for structural right-handedness, while h ¼ 1 for structural left-handedness. The central twist defect is described through the angle 0°; 0 6 z 6 L; /¼ ð5Þ L 6 z 6 2L: /t ; Finally, the dyadic S y ðcÞ ¼ ðux ux þ uz uz Þ cos c þ ðuz ux  ux uz Þ sin c þ uy uy

Fig. 1. Schematic of the boundary value problem involving the slanted chiral sculptured thin film with a twist angle of /t introduced between the upper and lower halves about the axis of nonhomogeneity. Nonspecular reflection and transmission can occur because a 6¼ 0°.

ð6Þ

serves two different roles. Whereas S y ðvs Þ delineates the role of the growth process with 90°  vs being the angle of declination from the helical axis, S y ðaÞ represents the slanted orientation of the helical axis. Since STFs are generally fabricated using physical vapor deposition [22,26], the

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growing microcolumns of an STF must be oriented at some angle to the substrate plane such that the so-called angle of rise v ¼ vs  jaj. Hence, v 2 ð0°; 90° for the ideal helical microcolumns of a slanted chiral STF to always grow upwards in relation to the substrate plane, while 0° < vs < 90° is mandated by the equation of a helix. Thus, a is restricted to the range ðvs ; vs Þ. When a ¼ 0°, the slant is absent and the usual chiral STFs are represented by
ðr; k0 Þ 
ðz; k0 Þ. 2.2. Incident, reflected and transmitted plane waves Clearly,
ðr; k0 Þ 
ðx; z; k0 Þ is periodic along the x and the z axes, but is independent of y. Therefore, the field representation in the two half-spaces must be periodic with respect to x. It must also comprise both specular and nonspecular planewave components, in accordance with the Floquet–Bloch theorem [28,35,36]. Let the incident plane wave propagate at an angle hinc to the þz axis and at an angle winc to the þx axis (in the xy plane), in the lower half-space z 6 0. Hence, the incident, the reflected and the transmitted electromagnetic field phasors are best expressed in a set of Floquet harmonics [29] respectively, as follows:      X  isn  pþ isn þ pþ ðnÞ ðnÞ n n pffiffiffi pffiffiffi Ei ¼ aL  aR 2 2 n¼0; 1; 2;...    exp i kxn x þ ky0 y þ kzn z ; z 6 0; ð7Þ     isn  pþ isn þ pþ ðnÞ ðnÞ n n pffiffiffi pffiffiffi Hi ¼ aL þ aR 2 2 n¼0; 1; 2;...    exp i kxn x þ ky0 y þ kzn z ; z 6 0; ð8Þ X

i g0



    isn  p isn þ p ðnÞ ðnÞ n n pffiffiffi pffiffiffi Er ¼  rL þ rR 2 2 n¼0; 1; 2;...    exp i kxn x þ ky0 y  kzn z ; z 6 0; ð9Þ X





      isn  p isn þ p i ðnÞ ðnÞ n n pffiffiffi pffiffiffi Hr ¼  rL  rR g 2 2 n¼0; 1; 2;... 0    exp i kxn x þ ky0 y  kzn z ; z 6 0; ð10Þ X

    isn  pþ isn þ pþ ðnÞ ðnÞ n n pffiffiffi pffiffiffi tL  tR Et ¼ 2 2 n¼0; 1; 2;... 

  exp i kxn x þ ky0 y þ kzn ðz  2LÞ ; z P 2L; ð11Þ X



    isn  pþ isn þ pþ ðnÞ ðnÞ n n pffiffiffi pffiffiffi Ht ¼ tL þ tR 2 2 n¼0; 1; 2;... 

  exp i kxn x þ ky0 y þ kzn ðz  2LÞ ; z P 2L: ð12Þ X

i g0



pffiffiffiffiffiffiffiffiffiffiffi In (7)–(12) and thereafter, g0 ¼ l0 =
0 is the intrinsic impedance, while l0 and
0 are the permittivity and the permeability, of free space. The transverse wavenumbers kxn ¼ k0 sin hinc cos winc þ njx ; ky0 ¼ k0 sin hinc sin winc ;

ð13Þ n ¼ 0; 1; 2; . . . ; pffiffiffiffiffiffiffiffiffi where k0 ¼ x l0
0 ¼ 2p=k0 is the free-space wavenumber, and jx ¼ ðp=XÞj sin aj

ð14Þ

because the slanted chiral STF has a period Lx ¼ 2X=j sin aj along the x axis. The vertical wavenumbers, 8
1=2 > 2 2 < þ k2  k2  k2 if k02 P kxn þ ky0 ; 0 xn y0 kzn ¼
1=2 > 2 2 : þi kxn þ ky0  k02 otherwise; ð15Þ are either real-valued (for propagating harmonics) or imaginary (for evanescent harmonics). The nth-order Floquet harmonic in any of the fields (7)–(12) involves left and right circularly polarized (LCP and RCP) components with amðnÞ ðnÞ plitudes aL and aR for the incident plane wave, ðnÞ ðnÞ ðnÞ ðnÞ rL and rR for the reflected field, and tL and tR for the transmitted field. As the incident plane wave is a Floquet harmonic of order n ¼ 0, the ðnÞ ðnÞ coefficients aL ¼ aR ¼ 0 8n 6¼ 0. The vectors, ky0 ux þ kxn uy sn ¼ ;  kxyn  kxn ux þ ky0 uy kzn kxyn ð16Þ p ¼  þ uz ; n kxyn k0 k0 n ¼ 0; 1; 2; . . . ;

F. Wang, A. Lakhtakia / Optics Communications 215 (2003) 79–92

individually denote linear polarization of the perpendicular and the parallel types in electromagnetics literature [37], respectively, with respect to  the direction vectors d ¼ k u þ ky0 uy kzn uz = xn x n k0 ; whilst qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 þ k2 ð17Þ kxyn ¼ þ kxn y0 is defined for convenience. ðnÞ ðnÞ Since faL g and faR g are supposedly known, ðnÞ ðnÞ ðnÞ ðnÞ the amplitude sets frL g, frR g, ftL g and ftR g, (n ¼ 0; 1; 2; . . .), have to be determined by solving a boundary value problem. 2.3. Coupled wave equations Each of the nine components
pq ðx; zÞ; ðp; q ¼ x; y; zÞ, of the relative permittivity dyadic
ðrÞ of (1) is represented as X ðnÞ
pq ðzÞ exp ðinjx xÞ;
pq ðx; zÞ ¼ n¼0; 1; 2 ð18Þ p; q ¼ x; y; z; 0 6 z 6 2L; ðnÞ where
pq ðzÞ are the Fourier amplitudes; we have dropped explicitly mention of wavelength-dependences at this point. The Floquet–Bloch theorem entails the decompositions i X h EðrÞ ¼ ExðnÞ ð zÞux þ EyðnÞ ð zÞuy þ EzðnÞ ð zÞuz

83

r  EðrÞ ¼ ixl0 HðrÞ; r  HðrÞ ¼ ix
0
ðrÞ  EðrÞ; 0 < z < 2L;

ð21Þ

and exploiting the orthogonalities of the functions exp½iðkxn x þ ky0 yÞ across any plane z ¼ constant, we derive the following set of coupled wave equations: d ðnÞ E ð zÞ  ikxn EzðnÞ ð zÞ ¼ ik0 g0 HyðnÞ ð zÞ; dz x

ð22Þ

d ðnÞ E ð zÞ  iky0 EzðnÞ ð zÞ ¼ ik0 g0 HxðnÞ ð zÞ; dz y

ð23Þ

ky0 ExðnÞ ð zÞ  kxn EyðnÞ ð zÞ ¼ k0 g0 HzðnÞ ð zÞ;

ð24Þ

d ðnÞ H ð zÞ  ikxn HzðnÞ ð zÞ dz x X h ik0
ðn~nÞ ðzÞExð~nÞ ð zÞ ¼ g0 n~¼0; 1; 2;... yx

i nÞ ð~ nÞ ðn~ nÞ ð~ nÞ þ
ðn~ ðzÞE ð z Þ þ
ðzÞE ð z Þ ; yy y yz z

ð25Þ

d ðnÞ H ð zÞ  iky0 HzðnÞ ð zÞ dz y X h ik0
ðn~nÞ ðzÞExð~nÞ ð zÞ ¼ g0 n~¼0; 1; 2;... xx

i nÞ ð~ nÞ ðn~ nÞ ð~ nÞ þ
ðn~ ð z ÞE ð z Þ þ
ðzÞE ð z Þ ; xy y xz z

ð26Þ

n¼0; 1; 2;...

   exp i kxn x þ ky0 y ; H ðrÞ ¼

X

h

0 6 z 6 2L;

ð19Þ

HxðnÞ ð zÞux þ HyðnÞ ð zÞuy þ HzðnÞ ð zÞuz

i

n¼0; 1; 2;...

   exp i kxn x þ ky0 y ;

0 6 z 6 2L;

ð20Þ

for the electromagnetic field phasors within the chosen thin-film device. Parenthetically, we note that Rokushima and Yamakita [18] also used the Floquet–Bloch theorem for locally uniaxial chiral liquid crystals; however, their representation is periodic with respect to the helical axis. On substituting (18)–(20) in the frequency-domain Maxwell curl postulates

ky0 HxðnÞ ð zÞ  kxn HyðnÞ ð zÞ X h k0 ðn~ nÞ
ðn~nÞ ðzÞExð~nÞ ð zÞ þ
zy ðzÞEyð~nÞ ð zÞ ¼ g0 n~¼0; 1; 2;... zx i nÞ ð~ nÞ ð z ÞE ð z Þ : ð27Þ þ
ðn~ zz z These six equations hold for all n ¼ 0; 1; 2; . . . The solution procedure for (22)–(27) is described in detail in a predecessor paper [21], to which we refer the interested readers. It suffices to mention here that both n and n~ are restricted to the range ½NT ; NT ; NT > 0, and a system of 4ð2NT þ1Þ ordinary differential equations is solved numerically, with (7)–(12) providing the necessary boundary conditions.

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F. Wang, A. Lakhtakia / Optics Communications 215 (2003) 79–92 ðnÞ

Finally, reflectances (RLL , etc.) and the transðnÞ mittances (TLL , etc.) are defined as 2 ðnÞ 2 ðnÞ 3 " # jrL j jrL j2 ðnÞ ðnÞ 2 Imðnn fn Þ 6 ja0L j ja0R j2 7 RLL RLR ¼ 4 5; ðnÞ ðnÞ Imðn0 f0 Þ jrRðnÞ j2 jrRðnÞ j2 RRL RRR "

ðnÞ TLL ðnÞ TRL

ðnÞ TLR ðnÞ TRR

2

# ¼

Imðnn fn Þ 6 4 Imðn0 f0 Þ

ja0L j2

ðnÞ jtL j2 ja0L j2 ðnÞ jtR j2 ja0L j2

ja0R j2 ðnÞ

jtL j2 ja0R j2 ðnÞ jtR j2 ja0R j2

3

ð28Þ

7 5;

n 2 ½NT ; NT ; where ImðÞ_ denotes the imaginary part, and   1 kxn kzn ky0 nn ¼ pffiffiffi i ; kxyn 2 k0 kxyn   ð29Þ 1 ky0 kzn kxn i fn ¼ pffiffiffi ; kxyn 2 k0 kxyn n 2 ½NT ; NT : Co-polarized remittances (i.e., reflectances and transmittances) have both subscripts identical, while cross-polarized remittances have both subscripts different. The principle of conservation of energy mandates the inequalities i X h ðnÞ ðnÞ ðnÞ ðnÞ RLL þ RRL þ TLL þ TRL 6 1; ð30Þ n2½NT ;NT

X

h

i ðnÞ ðnÞ ðnÞ ðnÞ RLR þ RRR þ TLR þ TRR 6 1;

ð31Þ

n2½NT ;NT

which reduce to equalities for nondissipative thin films. The sums on the left sides of (30) and (31) are denoted by 1  AL and 1  AR , where AL and AR are the absorbances for incident LCP and RCP plane waves, respectively. 3. Results and discussion 3.1. Preliminaries The solution procedure was implemented using double-precision arithmetic in Fortran 90 on a Solaris computer. The following constitutive and structural parameters were chosen: pa ¼ 2:0; pb ¼ 2:6 and pc ¼ 2:1; ka ¼ kc ¼ 140 nm and kb ¼ 150 nm; Na ¼ Nb ¼ Nc ¼ 40,000; X ¼ 300 nm; vs ¼ 30°; h ¼ 1. These constitutive parameters are po-

tentially realizable using silicon dioxide, and are thus likely to be compatible with semiconductor and optical technologies. The half-thickness L varied from 25X to 100X in our study, the minimum value of L chosen to let the circular Bragg phenomenon develop fully [16,21]. The slant angle a was set to be positive and less than vs . The twist angle /t was generally assumed equal to 90°, in order to produce a spectral hole roughly in the center of the Bragg regime. For compatibility with the commonplace planar structures in electronics and optics, we examined only the normal-incidence case (i.e., hinc ¼ 0°) with winc ¼ 0°; hence, sin wn ¼ 0 8n. We focused on the wavelength-regime k0 2 ½1000; 1110 nm. The maximum order parameter for the Floquet harmonics was fixed at NT ¼ 3 for the chosen wavelength-regime, after ensuring that every reflectance and transmittance greater than 0.001 converged to 1% accuracy. All propagating harmonics and some evanescent harmonics were thereby covered. Moreover, we ensured that the left sides of both (30) and (31) converged, and that neither condition was violated by more than 1 ppm. 3.2. Unslanted chiral STF device Let us begin with a ¼ 0°, which provides the STF analog of the CLC devices considered by Kopp and Genack [14]. The center-wavelength of the Bragg regime is then estimated as 1090 nm, and the full-width-at-half-maximum bandwidth as 72 nm [21]. Fig. 2 shows a narrow spectral hole (about 2 nm bandwidth) in the total co-polarized reflectance P ðnÞ RRR ¼ n RRR and, correspondingly, a peak Pin the ðnÞ total co-polarized transmittance TRR ¼ n TRR , excited by a normally incident RCP plane wave when L ¼ 27X. The plots in Fig. 2 are similar to those discussed by Hodgkinson et al. [12]. However, on extrapolating from Kopp and Genack [14], a spectral hole must also be generated by an incident LCP plane wave, provided the ratio L=X is sufficiently large. Indeed, Fig. 3 shows an ultra-narrow (about 0.02 nm bandwidth) spectral hole total co-polarPin the ðnÞ ized transmittance TLL ¼ n TLL and a corre-

F. Wang, A. Lakhtakia / Optics Communications 215 (2003) 79–92

(a)

85

(b)

P ðnÞ P ðnÞ Fig. 2. (a) Total reflectances RRR ¼ n RRR , etc., and (b) total transmittances TRR ¼ n TRR , etc., computed for the unslanted chiral STF device with the following parameters: a ¼ 0°, vs ¼ 30°, pa ¼ 2:0, pb ¼ 2:6, pc ¼ 2:1, ka ¼ kc ¼ 140 nm, kb ¼ 150 nm, Na ¼ Nb ¼ Nc ¼ 40,000, X ¼ 300 nm, h ¼ 1, /t ¼ 90°, L ¼ 27X and hinc ¼ winc ¼ 0°.

(b)

(a)

Fig. 3. Same as Fig. 2, but for L ¼ 91X.

sponding in the total co-polarized reflectance P peak ðnÞ RLL ¼ n RLL excited by an incident LCP plane wave when L ¼ 91X. Therefore, there are two types of spectral holes of opposite circular polarization states, depending on different values of L=X. Furthermore, one type of hole is found in co-handed reflectance – when the handedness of the incident circularly polarized plane wave is the same as the structural handedness of the chosen thin film – and the second in cross-handed transmittance (when the handednesses of the incident plane wave and the chiral STF do not coincide). Fig. 4 shows the plots of TRR and RLL versus the ratio L=X at the peak wavelength kp ¼ 1090:328 nm, which is the center-wavelength of the 0:02 nm spectral hole of Fig. 3. If Lco is defined as the crossover thickness at which TRR ¼ RLL , spectral holes of the second type appear only for L > Lco . The crossover thickness is seen in Fig. 4 to be

Lco  54:5X. As L increases above Lco , the spectral reflection hole wanes and the spectral transmission hole enhances to steady state.

P ðnÞ Fig. 4.P Total transmittance TRR ¼ n TRR and total reflectance ðnÞ RLL ¼ n RLL versus L=X at the peak wavelength kp ¼ 1090:328 nm. See Fig. 2 for other parameters. The crossover thickness Lco  54:5X.

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F. Wang, A. Lakhtakia / Optics Communications 215 (2003) 79–92

3.3. Slanted chiral STF device There is no distinction in the direction of propagation of Floquet harmonics of orders n ¼ 0 and n 6¼ 0, when a ¼ 0° [21]. But a distinction appears for a 6¼ 0° between Floquet harmonics of orders n ¼ 0 and n 6¼ 0, the former being classified as specular and the latter as nonspecular in the literature on diffraction gratings [38]. Regardless of the value of a in our studies, non-negligible remittances were found to be only of orders n ¼ 2 (nonspecular) and n ¼ 0 (specular). The remittances of other orders turned out to be negligible (<0.01) in the wavelength-regime focused on. Plots of only the non-negligible remittances are presented in this paper. All presented results apply for negative values of a, provided all remittances of order n are considered as the remittances of order n. Let us commence with a ¼ 15°. The centerwavelength of the Bragg regime can be estimated as the solution of the equation [21]

kBr 0

qffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi     ¼ ðX cos aÞ
c kBr þ
~d kBr ; 0 0

ð32Þ

where
~d ðk0 Þ ¼


a ðk0 Þ
b ðk0 Þ :
a ðk0 Þ cos2 v þ
a ðk0 Þ sin2 v

ð33Þ

In comparison with Figs. 2 and 3 therefore, Figs. 5 and 6 display a blue-shifted Bragg regime with Br kBr 0  1053 nm, while k0  1090 nm in Figs. 2 and 3. Fig. 5 shows the remittance spectrums for L ¼ 27X. A hole centered at kBr 0 in the spectrum of ð2Þ RRR , and a corresponding peak in the spectrum ð0Þ of TRR , are clearly evident therein. The bandwidth of the hole is still about 2 nm. Thus, a major effect of a 6¼ 0° is to produce a nonspecular spectral reflection hole. That reflection hole is absent in the remittance spectrums of Fig. 6, for which L ¼ 91X. Instead, a ð0Þ hole appears in the spectrum of TLL , which is purely specular; and it is accompanied by signifi-

(a)

(b)

(c)

(d)

Fig. 5. (a,b) Nonspecular and specular reflectances and (c,d) nonspecular and specular transmittances of order n, computed for the slanted chiral STF device with the following parameters: a ¼ 15°, vs ¼ 30°, pa ¼ 2:0, pb ¼ 2:6, pc ¼ 2:1, ka ¼ kc ¼ 140 nm, kb ¼ 150 nm, Na ¼ Nb ¼ Nc ¼ 40,000, X ¼ 300 nm, h ¼ 1, /t ¼ 90°, L ¼ 27X and hinc ¼ winc ¼ 0°. Remittances of maximum magnitudes less than 0:01 are not shown.

F. Wang, A. Lakhtakia / Optics Communications 215 (2003) 79–92

(a)

(b)

(c)

(d)

87

Fig. 6. Same as Fig. 5, but for L ¼ 91X. ð2Þ

ð0Þ

cant peaks in the spectrums of RLL ; RLL and ð2Þ TLL . The peak wavelength kp ¼ 1052:80 nm and the bandwidth is 0.15 nm. The crossover thickness of the slanted chiral STF device can be estimated from the plots of various remittances against L=X in Fig. 7. Actually, the peak wavelength turns out to be a function of both a and L. For a ¼ 0°; kp ¼ 1090:328 nm for all L. In contrast, kp varies from 1052.65 to 1053:75 nm as L changes from 25X to 100X, when a ¼ 15°. The remittances at the peak wavelengths seem to be varying somewhat irregularly with L in Fig. 7, in contrast to that in Fig. 4 for the unslanted chiral STF device. However, the values of ð0Þ ð2Þ ð0Þ þ TRR and RLL þ RLL þ TLL at 1052.80 nm do vary quite smoothly with L. From their plots therefore, the crossover thickness is determined as Lco ¼ 42:0X for a ¼ 15°. Only for L < Lco does the ð2Þ spectral reflection hole exist in RRR ; when L > Lco , ð0Þ the spectral transmission hole in TLL takes over just as for the unslanted chiral STF device. Though the chiral STF device, whether slanted or unslanted, is very weakly dissipative, significant absorption occurs only for LCP incidence in the

wavelength-regime of the spectral transmission hole (L > Lco ). In contrast, the absorbance for RCP incidence first increases to a small value (< 0:1) as L increases to Lco , and then drops to a minuscule value ð< 0:01Þ. Fig. 8 shows the absorbance spectrums for LCP incidence in both the unslanted and the slanted chiral STF devices when L ¼ 91X. Clearly, absorbance is higher for the unslanted chiral STF device than for the slanted one, in their respective spectral transmission hole regimes. When dissipation in the chiral slanted STF is enhanced – for example, by choosing smaller Na;b;c ð0Þ in (3) – the spectral hole in TLL for L > Lco fades ð2Þ away, although the spectral hole in RRR for L < Lco still exists. We concluded thus from several calculations not reported here. This conclusion is not surprising because the transmission hole is affected by the entire thickness 2L of the thin film, whereas the reflection hole is affected only by the first few thickness periods [39,40]. While the transmission hole always occurs in ð0Þ the spectrum of TLL for L > Lco , the corresponding reflectance peaks appear in different Floquet harmonics as the slant angle a changes. Fig. 9 shows

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(a)

(a)

(b)

(b)

ð2Þ

ð0Þ

Fig. 7. (a) Reflectances RLL and RLL as well as transmittance ðþ2Þ ð0Þ ð2Þ ð0Þ TLL , and (b) transmittance TRR and the sum RLL þ RLL þ ðþ2Þ TLL , versus L=X at the peak wavelength kp ¼ 1052:80 nm. The curves are obtained by least-squares fitting of fifth-order polynomials to the computed data (shown by heavy dots).

the peaks in the co-polarized reflectances of different orders for a ¼ 5°; 10°; 15° and 16:7°, respectively, excited by an incident LCP plane wave ð0Þ when L ¼ 91X. The spectral hole in TLL shown in Fig. 9 is nearly unaffected by a, though its bandwidth increases with a (with the exception of a ¼ 16:7°). But the reflectance peak first shifts ðþ2Þ from a nonspecular mode (RLL ) to the specular ð0Þ mode (RLL ), and then is shared by the specular and ð0Þ ð2Þ the other nonspecular modes (RLL and RLL ), as a increases up to 15°. A further increase of a returns the peak in Fig. 9 to the specular mode. In fact, as a increases beyond 15°, the circular Bragg phenomenon is subverted by a Rayleigh– Wood anomaly, which is the conversion of the nth-order Floquet harmonic from propagating to evanescent, or vice versa. This conversion occurs at the wavelength [21] 2X kRW : ð34Þ 0n ¼ jnj sin a

P ðnÞ ðnÞ ðnÞ ðnÞ Fig. 8. Absorbance AL ¼ 1  n ½RRL þ RLL þ TRL þ TLL for LCP incidence for (a) the unlanted chiral STF device and (b) the slanted chiral STF device (a ¼ 15°), when L ¼ 91X. See Fig. ð0Þ 2 for other parameters. For comparison, TLL and TLL , respectively, are also plotted.

At a fixed a, the nth-order Floquet harmonic is evanescent along the z axis for k0 > kRW 0n . As a increases from 0°, in the present instance, kRW 02 decreases from ‘‘infinity’’ and begins to approach the center-wavelength kBr 0 of the Bragg reBr gime predicted by (32). Both kRW 02 and k0 blue-shift RW as a increases, but k02 decreases more rapidly than kBr 0 , and the Rayleigh–Wood anomaly eventually wipes out the circular Bragg phenomenon. ð2Þ No wonder, the spectral hole in RRR is absent in Fig. 10 for a ¼ 16:7° and L ¼ 27X. But the spectral ð0Þ hole in TLL survives at about kp ¼ 1043:98 nm when L ¼ 91X; see Fig. 9(d). For a > 17:1°, the circular Bragg phenomenon vanishes completely, and in consequence, neither of the two types of spectral holes exists. Finally, the twist angle /t 6¼ 90° affects the spectral holes too. Most significantly, the spectral holes are not located roughly in the center of the

F. Wang, A. Lakhtakia / Optics Communications 215 (2003) 79–92

(a)

(b)

(c)

(d)

ðnÞ

ð0Þ

89

Fig. 9. Reflectances RLL (n ¼ 0; 2) and transmittance TLL , computed for the slanted chiral STF device with (a) a ¼ 5°, (b) a ¼ 10°, (c) a ¼ 15° and (d) a ¼ 16:7°. See Fig. 6 for other parameters. Reflectances of maximum magnitude less than 0.01 are not shown.

(a)

(b)

(c)

(d)

Fig. 10. Same as Fig. 5, but for a ¼ 16:7°.

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(a)

(b)

(c)

(d)

Fig. 11. Same as Fig. 5, but for /t ¼ 45°.

(a)

(b)

(c)

(d)

Fig. 12. Same as Fig. 6, but for /t ¼ 45°.

F. Wang, A. Lakhtakia / Optics Communications 215 (2003) 79–92

Bragg regime as for /t ¼ 90°, but are shifted towards the edges of the Bragg regime when / 6¼ 90°. This is illustrated by Figs. 11 and 12 – which show the spectral holes of the chiral slanted STF device as before, but with /t ¼ 45°. Evidently, the spectral holes, located close to 1071 nm, are blue-shifted in the Bragg regime. 4. Concluding remarks In this paper, we have theoretically analyzed an optical device made by introducing a central twist defect in a slanted chiral sculptured thin film. A slanted chiral STF is periodically nonhomogenous in two directions: normal and parallel to the substrate plane. The normal and transverse periodicities are intimately coupled by the slant angle a 6¼ 0°. A coupled wave theory was employed for the planewave response of the slanted chiral STF device of finite thickness. The twist angle /t , if equal to 90°, gives rise to two types of spectral holes roughly centered in the Bragg regime: one is in the nonspecular reflectance excited by an incident RCP plane wave, and the other in the specular transmittance excited by an incident LCP plane wave. The thickness of the device determines which of the two hole is observed. Both types of holes eventually disappear – as the Bragg regime also does – as a increases, having been subverted completely by the Rayleigh–Wood phenomenon. Corresponding to the spectral reflection hole, a peak occurs in the co-handed and specular transmittance for all a. However, a reflectance peak accompanying the spectral transmission hole turns out to be cross-handed and can be evinced in specular and/or nonspecular directions. The spectral reflection hole is less susceptible than the spectral transmission hole to the dissipative properties of the chiral STF. References [1] A. Thelen, J. Opt. Soc. Am. 56 (1966) 1533. [2] W.H. Southwell, J. Opt. Soc. Am. A 5 (1988) 1558. [3] H.A. Haus, C.V. Shank, IEEE J. Quantum Electron. 12 (1976) 532.

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[4] G.P. Agrawal, N.K. Dutta, Semiconductor Lasers, Van Nostrand Reinhold, New York, NY, USA, 1993. [5] S. Chandrasekhar, Liquid Crystals, second ed., Cambridge University Press, Cambridge, UK, 1992 (Chapter 4). [6] P.G. de Gennes, J. Prost, The Physics of Liquid Crystals, second ed., Clarendon Press, Oxford, UK, 1993 (Chapter 6). [7] A. Lakhtakia, Mater. Sci. Eng. C 19 (2002) 427. [8] A. Lakhtakia, R. Messier, Opt. Photon. News 12 (9) (2001) 26. [9] S.D. Jacobs, K.A. Cerqua, K.L. Marshall, A. Schmid, M.J. Guardalben, K.J. Skerrett, J. Opt. Soc. Am. B 5 (1988) 1962. [10] I.J. Hodgkinson, Q.h. Wu, Adv. Mater. 13 (2001) 889. [11] A. Lakhtakia, M.W. McCall, Opt. Commun. 160 (1999) 457. [12] I.J. Hodgkinson, Q.H. Wu, K.E. Thorn, A. Lakhtakia, M.W. McCall, Opt. Commun. 184 (2000) 57. [13] V.I. Kopp, A.Z. Genack, U.S. Patent 6 396 859, 2002. [14] V.I. Kopp, A.Z. Genack, Phys. Rev. Lett. 89 (2002) 033901. [15] A. Lakhtakia, W.S. Weiglhofer, Proc. R. Soc. Lond. A 453 (1997) 93, erratum: 454 (1998) 3275. [16] V.C. Venugopal, A. Lakhtakia, Proc. R. Soc. Lond. A 456 (2000) 125. [17] A. Lakhtakia, V.C. Venugopal, M.W. McCall, Opt. Commun. 177 (2000) 57. [18] K. Rokushima, J. Yamakita, J. Opt. Soc. Am. A 4 (1987) 27. [19] K.C. Chang, V. Shah, T. Tamir, J. Opt. Soc. Am. 70 (1980) 804. [20] R. Petit (Ed.), Electromagnetic Theory of Gratings, Springer, Heidelberg, Germany, 1980. [21] F. Wang, A. Lakhtakia, R. Messier, Eur. Phys. J. AP 20 (2002) 91. [22] K. Robbie, M.J. Brett, A. Lakhtakia, J. Vac. Sci. Technol. A 12 (1995) 2991. [23] F. Liu, M.T. Umlor, L. Shen, J. Weston, W. Eads, J.A. Barnard, G.J. Mankey, J. Appl. Phys. 85 (1999) 5486. [24] I.J. Hodgkinson, Q.H. Wu, B. Knight, A. Lakhtakia, K. Robbie, Appl. Opt. 39 (2000) 642. [25] R. Messier, V.C. Venugopal, P.D. Sunal, J. Vac. Sci. Technol. B 18 (2000) 1538. [26] M. Suzuki, Y. Taga, Jpn. J. Appl. Phys. Pt. 2 40 (2001) L358. [27] R. Messier, A. Lakhtakia, V.C. Venugopal, P.D. Sunal, Vac. Technol. Coating 2 (10) (2001) 40. [28] K. Rokushima, J. Yamakita, J. Opt. Soc. Am. 73 (1983) 901. [29] J.M. Jarem, P.P. Banerjee, Computational Methods for Electromagnetic and Optical Systems, Marcel Dekker, New York, NY, USA, 2000 (Chapter 3). [30] I.J. Hodgkinson, Q.H. Wu, Appl. Opt. 38 (1999) 3621. [31] V.C. Venugopal, A. Lakhtakia, in: O.N. Singh, A. Lakhtakia (Eds.), Electromagnetic Fields in Unconventional Materials and Structures, Wiley, New York, NY, USA, 2000 (Chapter 5).

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[32] C. Kittel, Introduction to Solid State Physics, fourth ed., Wiley Eastern, New Delhi, India, 1974 (Chapter 13). [33] A. Lakhtakia, Eur. Phys. J. AP 8 (1999) 129. [34] I.J. Hodgkinson, Q.h. Wu, Birefingent Thin Films and Polarizing Elements, World Scientific, Singapore, 1997 (Chapter 16). [35] T.K. Gaylord, M.G. Moharam, Proc. IEEE 73 (1985) 894. [36] A. Lakhtakia, V.K. Varadan, V.V. Varadan, J. Opt. Soc. Am. A 6 (1989) 1675, erratum: 7 (1990) 951.

[37] J.D. Jackson, Classical Electrodynamics, third ed., Wiley, New York, NY, USA, 1999 (Section 7.3). [38] D. Maystre (Ed.), Selected Papers on Diffraction Gratings, SPIE Optical Engineering Press, Bellingham, WA, USA, 1993. [39] J.B. Geddes III, A. Lakhtakia, Eur. Phys. J. AP 14 (2001) 97, erratum: 16 (2001) 247. [40] I.J. Hodgkinson, Q.h. Wu, L. De Silva, Proc. SPIE 4806 (2002) 118.