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Total reflection of waves propagating from a rare isotropic medium to a dense anisotropic medium
Optics Communications 233 (2004) 271–275 www.elsevier.com/locate/optcom
Total reflection of waves propagating from a rare isotropic medium to a dense anisotropic medium Yi-Jun Jen *, Yan-Ru Cheng Department of Electro-Optical Engineering, National Taipei University of Technology, No. 1, Sec. 3, Chung-Hsiao E. Road, Taipei 106, Taiwan Received 6 October 2003; received in revised form 26 January 2004; accepted 27 January 2004
Abstract The distribution diagram and the boundary conditions of wave vectors are used here to study the propagation of light between anisotropic media. Reflectance and transmittance are calculated according to the non-symmetric internal reflection phenomenon. Total reflection of light propagating from a rare medium to a dense medium may occur. Castillo and BallinasÕ challenge of the correctness of Lin and Wu is adequate only in a particular case, and this study resolves both analyses by considering a more general case. Ó 2004 Elsevier B.V. All rights reserved. PACS: 78.20.Ci; 78.20.Fm Keywords: Anisotropic medium; Total reflection
1. Introduction Recently, some interesting phenomena for waves propagating in anisotropic media are proposed. When two of the principal axes are on the plane of incidence, it is possible to have total reflection from a rare isotropic medium to an anisotropic medium [1,2]. It is also possible that the reflection angle and transmission angle are larger than 90° [3]. The wave in anisotropic medium may
*
Corresponding author. Tel.: +88634803113; fax: +88634 803113. E-mail address: [email protected] (Y.-J. Jen).
propagate backwards: the wave vector and the ray vector point at different media. However, a controversy regarding the total reflection between an isotropic medium and an anisotropic medium must be resolved. Lin and Wu [1] have claimed that total reflection of light that propagates from a rare isotropic medium to a dense medium can occur. However, the results of Castillo and Ballinas [2] do not agree with those of Lin and Wu. This paper uses a distribution diagram and the boundary conditions of wave vectors to indicate the wave vector and refraction index in an anisotropic medium. When the principle axes are rotated by an angle / from the surface coordinates, the incident medium is rarer or denser
0030-4018/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2004.01.062
272
Y.-J. Jen, Y.-R. Cheng / Optics Communications 233 (2004) 271–275
z Incident medium
z
y
z' φ
y'
x, x '
y
Anisotropic medium
0
0
0
Fig. 1. Principal axes of the anisotropic medium (x , y , z ) and the surface coordinates (x, y, z) of the system.
than the transmitted medium depending on the incident angle.
2. Theory The incident wave is assumed to be a monochromatic, linearly polarized, uniform electromagnetic plane wave of infinite extent with an optical frequency of x, and a time dependence of expðixtÞ. Fig. 1 depicts an isotropy–anisotropy system. The relationship between the principal axes (x0 ; y 0 ; z0 ) and the surface coordinates (x; y; z) is x ¼ x0 ; y ¼ y 0 cos / z0 sin /; z ¼ y 0 sin / þ z0 cos /:
ð1Þ
the wavelength in vacuum. The critical condition occurs when b ¼ bmax ¼ k0 gz and b ¼bmax ¼ k0 gz at points A and D, respectively. By drawing a tangent line through a specific wave vector point in Fig. 2, the direction of the corresponding ray vector is vertical to the tangent line. Fig. 2 shows that the ray vectors reach the boundary at points A and D. The abnormal total reflection usually occurs at the interface of two anisotropic media. Fig. 3 shows the incident and transmitted wave-vector ellipsoids of the system (anisotropic medium 1/ anisotropic medium 2). The wave vectors in both media are decided by the invariant incident parallel-interface wave vector component b. It can be visualized that wave vectors and ray vectors
The refraction indices in the three principal axes are nx0 , ny 0 and nz0 , which can be converted to tensor factors [3], kz '
g2y ¼ n2y 0 cos2 / þ n2z0 sin2 /; g2z ¼ n2y 0 sin2 / þ n2z0 cos2 /;
kz
Qr
ð2Þ
tˆ+ k y'
gyz ¼ ðn2y 0 n2z0 Þ sin / cos /: f
For TM waves, the elliptical surface of the wave vector distribution [3] in an anisotropic medium can be plotted as the variation in components kzþ and kz with the parallel interface component of wave vector b. According to laws of refraction and reflection, the parallel interface components of the incident, refracted and reflected waves have to be equal. Once the incident parallel wave vector b is decided, the direction of the refracted wave vector in anisotropic medium can be derived from the wave vector distribution as shown in Fig. 2. The wave vector k0 is defined as 2p=k0 and k0 represents
bA
k+
bB
ky b
bE
bC
bD
k– Qi
tˆ– Fig. 2. Elliptical surface of wave vector distribution in anisotropic medium.
Y.-J. Jen, Y.-R. Cheng / Optics Communications 233 (2004) 271–275
and k0 is the wave vector in vacuum. The angle of the incident ray is negative between b ¼ 0 and b ¼ bE in Fig. 2, where bE satisfies HðbE Þ ¼ 0. The intrinsic impedances (gi and gr ) of the two possible waves in the anisotropic medium for a given b are also different: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k02 g21z b2 cos Hi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi gi ¼ pffiffiffiffiffiffiffiffiffiffiffi 2ffi ; e0 =l0 g1yz k0 g1y g1z 1 g1y g1z ð5Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 g b k cos Hr 0 1z rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi gr ¼ pffiffiffiffiffiffiffiffiffiffiffi 2ffi : e0 =l0 g1yz k0 g1y g1z 1 g1y g1z
k1+z β
βc
k1−z Anisotropic medium 1
Anisotropic medium 2 β
k2−z
Fig. 3. The possible wave vectors in two adjacent anisotropic media.
(^tþ and ^t ) of the incident and reflected waves are non-symmetrical about the normal of the interface. In the calculation of the reflection coefficient, the boundary conditions for the electric and magnetic fields are given by EðiÞ cos Hi þ EðrÞ cos Hr ¼ EðtÞ cos Ht ; H
ðiÞ
þH
ðrÞ
ðtÞ
¼H ;
273
The reflectance R can be derived by considering the non-symmetric reflection power ratio. The illuminated area on the interface is A. The crosssectional areas of the incident, reflected and refracted beams are A cos Hi , A cos Hr and A cos Ht , respectively. The reflectance R and transmittance are represented as 2 Er g A cos Hr R ¼ i Ei gr A cos Hi 2 32 ðg22z ðb=k0 Þ2 Þ1=2 ðg21z ðb=k0 Þ2 Þ1=2 2 1=2 2 1=2 g1z g1y ð1ðg1yz =g1y g1z Þ Þ 7 6 g g ð1ðg =g g2z Þ Þ ¼ 4 2z 2y 2 2yz 2y2 1=2 5; ðg2z ðb=k0 Þ Þ ðg21z ðb=k0 Þ2 Þ1=2 þ g g ð1ðg =g g Þ2 Þ1=2 g g ð1ðg =g g Þ2 Þ1=2 2z 2y
2yz
2y 2z
1z 1y
1yz
1y 1z
ð3Þ
where the subscripts t, r and i indicate transmission, reflection and incidence, respectively. After the directions of the ray vectors in Fig. 2 are calculated, the angles Hi , Hr and Ht represent angles of incidence, reflection and refraction for ray vectors (^tþ , ^t and ^t0 , respectively) 0 1 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !2 u u g1yz g1yz C b B g1y Hi ¼tan1 @ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi t1 2 A; g1z g g g1z 2 1y 1z k02 g21z b 0 1 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !2 u u g1yz g1yz C b B g1y Hr ¼tan1 @ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi t1 þ 2 A; g1z g g g1z 2 2 2 1y 1z k g b 0 1z
ð4Þ
where gjy , gjz and gjyz (j ¼ 1 or 2) are the transformed indexes of refraction in surface coordinates
ð6Þ T ¼
4g
ðg22z ðb=k0 Þ2 Þ1=2
2z g2y ð1ðg2yz =g2y g2z Þ
ðg21z ðb=k0 Þ2 Þ1=2 2 1=2
Þ
ðg22z ðb=k0 Þ2 Þ1=2 g2z g2y ð1ðg2yz =g2y g2z Þ2 Þ1=2
g1z g1y ð1ðg1yz =g1y g1z Þ2 Þ1=2
þg
ðg21z ðb=k0 Þ2 Þ1=2
1z g1y ð1ðg1yz =g1y g1z Þ
2 :
2 1=2
Þ
ð7Þ The incident index of refraction n1 and the refracted index of refraction n2 of refraction depend on b qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 b2 þ ðkjz Þ nj ðbÞ ¼ ; j ¼ 1 or 2: ð8Þ k0 For isotropic media, the indexes gjy and gjz are the same as the refractive index nj0 , and index gjyz becomes zero. Whether the anisotropic medium is denser than the incident isotropic medium with refractive index depends on the difference between
274
Y.-J. Jen, Y.-R. Cheng / Optics Communications 233 (2004) 271–275
the magnitude of the incident wave vector qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 b2 þ ðk1z Þ , and that of the refractive wave vecqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 tor b þ ðk2z Þ . The transmitted anisotropic meqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 dium is denser when b2 þ ðk2z Þ > b2 þ ðk1z Þ. At critical points, the wave component b satisfies b ¼ k0 g2z and the relation for the denser condition becomes, sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 Þ b20 þ ðk1z g42z þ g22yz > : ð9Þ 2 k0 g2z Total reflection requires that the parallel wave vector b can exceed critical values. b > k0 g2z for b is positive: b < k0 g2z for b is negative:
nent b in their example cannot reach the critical point.
3. Example The same anisotropy example is considered here as was considered by Lin and Wu, with principal indices ny 0 ¼ 1:20, nz0 ¼ 1:80 and / ¼ 0:70 at a wavelength of k0 ¼ 632:8 nm. The incident angle for abnormal total reflection must exceed 70.42°. The incident refractive index n0 is in the range 1:67 > n0 > 1:57= sin h0 . Choosing the index n0 ¼ 1:59 yields a critical angle of 83.23° and the lower limit for n0 becomes 1.58, which value is reasonable. Figs. 4 and 5 show the wave vectors in the
ð10Þ
Eqs. (9) and (10) give the range of incident refractive index sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 b2 þ ðk1z Þ g42z þ g22yz > > g2z : ð11Þ 2 k0 g2z For the isotropic medium/anisotropic medium system considered by previous papers, the requirement for the system to have abnormal total reflection is sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g42z þ g22yz g2z : ð12Þ > n10 > 2 sin h10 g2z However, the inequality in Lin and WuÕs analysis does not consider the incident angle h10 . The incident angle h10 must exceed sin1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g42z =ðg42z þ g22yz Þ to yield abnormal total reflection. When n10 satisfies Eq. (12), light that propagates from a rare medium to a dense medium is totally reflected. Clearly, the index range vanishes when g2yz ¼ 0 (tilt angle / ¼ 0). Therefore, it is impossible to have abnormal total reflection in a rare-dense system under the assumption / ¼ 0 made by Castillo and Ballinas. The prerequisite for total reflection in a rare-dense system is that the optical axes are tilted an angle to the surface coordinates. The incident refractive index range in Lin and Wu is incorrect: the wave vector compo-
k (rad/µm) 19.0 18.0 k 2z +β2
17.0
n 0k 0
16.0
15.0
75
80
85
90
θ 0 (deg)
Fig. 4. Wave vectors in the isotropic medium and the anisotropic medium for light propagating toward the dense side (b > 0) in the range 0 < h0 < 90°.
k (rad/µm) kz +β2 2
16.5
16.0
n0 k0
15.5
15.0 14.5
75
80
85
90
θ 0 (deg)
Fig. 5. Wave vectors in the isotropic medium and the anisotropic medium for light propagating toward the rare side (b < 0) in the range 0 < h0 < 90°.
Y.-J. Jen, Y.-R. Cheng / Optics Communications 233 (2004) 271–275
isotropic medium and the anisotropic medium for light propagating toward the dense side (b > 0) and the rare side (b < 0) of the anisotropic medium, respectively, in the range 0 < h0 < 90°. At the critical points on both sides, the index of anisotropy is 1.67, greater than the incident index, 1.59.
4. Conclusion In this paper, the reflectance is calculated by considering non-symmetric reflection phenomenon. The incident and reflected vectors (including wave vectors and ray vectors) are not symmetric about the normal of the interface. Two studies [1,2] about the abnormal total reflection have to be resolved and corrected. It is possible to have total reflection for light propagating from a rare me-
275
dium to dense medium. The requirement to have the abnormal total reflection is derived here correctly. Acknowledgements The authors would like to thank the National Science Council of the Republic of China for financially supporting this research under Contract No. NSC. 91-2622-E-027-024-CC3.
References [1] C.L. Lin, J.J. Wu, Opt. Lett. 23 (1998) 22. [2] G.F. Torres del Castillo, C.J. Perez Ballinas, Opt. Lett. 26 (2001) 1251. [3] Y.J. Jen, C.C. Lee, Opt. Lett. 26 (2001) 190.
Total reflection of waves propagating from a rare isotropic medium to a dense anisotropic medium Yi-Jun Jen *, Yan-Ru Cheng Department of Electro-Optical Engineering, National Taipei University of Technology, No. 1, Sec. 3, Chung-Hsiao E. Road, Taipei 106, Taiwan Received 6 October 2003; received in revised form 26 January 2004; accepted 27 January 2004
Abstract The distribution diagram and the boundary conditions of wave vectors are used here to study the propagation of light between anisotropic media. Reflectance and transmittance are calculated according to the non-symmetric internal reflection phenomenon. Total reflection of light propagating from a rare medium to a dense medium may occur. Castillo and BallinasÕ challenge of the correctness of Lin and Wu is adequate only in a particular case, and this study resolves both analyses by considering a more general case. Ó 2004 Elsevier B.V. All rights reserved. PACS: 78.20.Ci; 78.20.Fm Keywords: Anisotropic medium; Total reflection
1. Introduction Recently, some interesting phenomena for waves propagating in anisotropic media are proposed. When two of the principal axes are on the plane of incidence, it is possible to have total reflection from a rare isotropic medium to an anisotropic medium [1,2]. It is also possible that the reflection angle and transmission angle are larger than 90° [3]. The wave in anisotropic medium may
*
Corresponding author. Tel.: +88634803113; fax: +88634 803113. E-mail address: [email protected] (Y.-J. Jen).
propagate backwards: the wave vector and the ray vector point at different media. However, a controversy regarding the total reflection between an isotropic medium and an anisotropic medium must be resolved. Lin and Wu [1] have claimed that total reflection of light that propagates from a rare isotropic medium to a dense medium can occur. However, the results of Castillo and Ballinas [2] do not agree with those of Lin and Wu. This paper uses a distribution diagram and the boundary conditions of wave vectors to indicate the wave vector and refraction index in an anisotropic medium. When the principle axes are rotated by an angle / from the surface coordinates, the incident medium is rarer or denser
0030-4018/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2004.01.062
272
Y.-J. Jen, Y.-R. Cheng / Optics Communications 233 (2004) 271–275
z Incident medium
z
y
z' φ
y'
x, x '
y
Anisotropic medium
0
0
0
Fig. 1. Principal axes of the anisotropic medium (x , y , z ) and the surface coordinates (x, y, z) of the system.
than the transmitted medium depending on the incident angle.
2. Theory The incident wave is assumed to be a monochromatic, linearly polarized, uniform electromagnetic plane wave of infinite extent with an optical frequency of x, and a time dependence of expðixtÞ. Fig. 1 depicts an isotropy–anisotropy system. The relationship between the principal axes (x0 ; y 0 ; z0 ) and the surface coordinates (x; y; z) is x ¼ x0 ; y ¼ y 0 cos / z0 sin /; z ¼ y 0 sin / þ z0 cos /:
ð1Þ
the wavelength in vacuum. The critical condition occurs when b ¼ bmax ¼ k0 gz and b ¼bmax ¼ k0 gz at points A and D, respectively. By drawing a tangent line through a specific wave vector point in Fig. 2, the direction of the corresponding ray vector is vertical to the tangent line. Fig. 2 shows that the ray vectors reach the boundary at points A and D. The abnormal total reflection usually occurs at the interface of two anisotropic media. Fig. 3 shows the incident and transmitted wave-vector ellipsoids of the system (anisotropic medium 1/ anisotropic medium 2). The wave vectors in both media are decided by the invariant incident parallel-interface wave vector component b. It can be visualized that wave vectors and ray vectors
The refraction indices in the three principal axes are nx0 , ny 0 and nz0 , which can be converted to tensor factors [3], kz '
g2y ¼ n2y 0 cos2 / þ n2z0 sin2 /; g2z ¼ n2y 0 sin2 / þ n2z0 cos2 /;
kz
Qr
ð2Þ
tˆ+ k y'
gyz ¼ ðn2y 0 n2z0 Þ sin / cos /: f
For TM waves, the elliptical surface of the wave vector distribution [3] in an anisotropic medium can be plotted as the variation in components kzþ and kz with the parallel interface component of wave vector b. According to laws of refraction and reflection, the parallel interface components of the incident, refracted and reflected waves have to be equal. Once the incident parallel wave vector b is decided, the direction of the refracted wave vector in anisotropic medium can be derived from the wave vector distribution as shown in Fig. 2. The wave vector k0 is defined as 2p=k0 and k0 represents
bA
k+
bB
ky b
bE
bC
bD
k– Qi
tˆ– Fig. 2. Elliptical surface of wave vector distribution in anisotropic medium.
Y.-J. Jen, Y.-R. Cheng / Optics Communications 233 (2004) 271–275
and k0 is the wave vector in vacuum. The angle of the incident ray is negative between b ¼ 0 and b ¼ bE in Fig. 2, where bE satisfies HðbE Þ ¼ 0. The intrinsic impedances (gi and gr ) of the two possible waves in the anisotropic medium for a given b are also different: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k02 g21z b2 cos Hi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi gi ¼ pffiffiffiffiffiffiffiffiffiffiffi 2ffi ; e0 =l0 g1yz k0 g1y g1z 1 g1y g1z ð5Þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 g b k cos Hr 0 1z rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi gr ¼ pffiffiffiffiffiffiffiffiffiffiffi 2ffi : e0 =l0 g1yz k0 g1y g1z 1 g1y g1z
k1+z β
βc
k1−z Anisotropic medium 1
Anisotropic medium 2 β
k2−z
Fig. 3. The possible wave vectors in two adjacent anisotropic media.
(^tþ and ^t ) of the incident and reflected waves are non-symmetrical about the normal of the interface. In the calculation of the reflection coefficient, the boundary conditions for the electric and magnetic fields are given by EðiÞ cos Hi þ EðrÞ cos Hr ¼ EðtÞ cos Ht ; H
ðiÞ
þH
ðrÞ
ðtÞ
¼H ;
273
The reflectance R can be derived by considering the non-symmetric reflection power ratio. The illuminated area on the interface is A. The crosssectional areas of the incident, reflected and refracted beams are A cos Hi , A cos Hr and A cos Ht , respectively. The reflectance R and transmittance are represented as 2 Er g A cos Hr R ¼ i Ei gr A cos Hi 2 32 ðg22z ðb=k0 Þ2 Þ1=2 ðg21z ðb=k0 Þ2 Þ1=2 2 1=2 2 1=2 g1z g1y ð1ðg1yz =g1y g1z Þ Þ 7 6 g g ð1ðg =g g2z Þ Þ ¼ 4 2z 2y 2 2yz 2y2 1=2 5; ðg2z ðb=k0 Þ Þ ðg21z ðb=k0 Þ2 Þ1=2 þ g g ð1ðg =g g Þ2 Þ1=2 g g ð1ðg =g g Þ2 Þ1=2 2z 2y
2yz
2y 2z
1z 1y
1yz
1y 1z
ð3Þ
where the subscripts t, r and i indicate transmission, reflection and incidence, respectively. After the directions of the ray vectors in Fig. 2 are calculated, the angles Hi , Hr and Ht represent angles of incidence, reflection and refraction for ray vectors (^tþ , ^t and ^t0 , respectively) 0 1 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !2 u u g1yz g1yz C b B g1y Hi ¼tan1 @ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi t1 2 A; g1z g g g1z 2 1y 1z k02 g21z b 0 1 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !2 u u g1yz g1yz C b B g1y Hr ¼tan1 @ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi t1 þ 2 A; g1z g g g1z 2 2 2 1y 1z k g b 0 1z
ð4Þ
where gjy , gjz and gjyz (j ¼ 1 or 2) are the transformed indexes of refraction in surface coordinates
ð6Þ T ¼
4g
ðg22z ðb=k0 Þ2 Þ1=2
2z g2y ð1ðg2yz =g2y g2z Þ
ðg21z ðb=k0 Þ2 Þ1=2 2 1=2
Þ
ðg22z ðb=k0 Þ2 Þ1=2 g2z g2y ð1ðg2yz =g2y g2z Þ2 Þ1=2
g1z g1y ð1ðg1yz =g1y g1z Þ2 Þ1=2
þg
ðg21z ðb=k0 Þ2 Þ1=2
1z g1y ð1ðg1yz =g1y g1z Þ
2 :
2 1=2
Þ
ð7Þ The incident index of refraction n1 and the refracted index of refraction n2 of refraction depend on b qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 b2 þ ðkjz Þ nj ðbÞ ¼ ; j ¼ 1 or 2: ð8Þ k0 For isotropic media, the indexes gjy and gjz are the same as the refractive index nj0 , and index gjyz becomes zero. Whether the anisotropic medium is denser than the incident isotropic medium with refractive index depends on the difference between
274
Y.-J. Jen, Y.-R. Cheng / Optics Communications 233 (2004) 271–275
the magnitude of the incident wave vector qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 b2 þ ðk1z Þ , and that of the refractive wave vecqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 tor b þ ðk2z Þ . The transmitted anisotropic meqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 dium is denser when b2 þ ðk2z Þ > b2 þ ðk1z Þ. At critical points, the wave component b satisfies b ¼ k0 g2z and the relation for the denser condition becomes, sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 Þ b20 þ ðk1z g42z þ g22yz > : ð9Þ 2 k0 g2z Total reflection requires that the parallel wave vector b can exceed critical values. b > k0 g2z for b is positive: b < k0 g2z for b is negative:
nent b in their example cannot reach the critical point.
3. Example The same anisotropy example is considered here as was considered by Lin and Wu, with principal indices ny 0 ¼ 1:20, nz0 ¼ 1:80 and / ¼ 0:70 at a wavelength of k0 ¼ 632:8 nm. The incident angle for abnormal total reflection must exceed 70.42°. The incident refractive index n0 is in the range 1:67 > n0 > 1:57= sin h0 . Choosing the index n0 ¼ 1:59 yields a critical angle of 83.23° and the lower limit for n0 becomes 1.58, which value is reasonable. Figs. 4 and 5 show the wave vectors in the
ð10Þ
Eqs. (9) and (10) give the range of incident refractive index sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 b2 þ ðk1z Þ g42z þ g22yz > > g2z : ð11Þ 2 k0 g2z For the isotropic medium/anisotropic medium system considered by previous papers, the requirement for the system to have abnormal total reflection is sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g42z þ g22yz g2z : ð12Þ > n10 > 2 sin h10 g2z However, the inequality in Lin and WuÕs analysis does not consider the incident angle h10 . The incident angle h10 must exceed sin1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi g42z =ðg42z þ g22yz Þ to yield abnormal total reflection. When n10 satisfies Eq. (12), light that propagates from a rare medium to a dense medium is totally reflected. Clearly, the index range vanishes when g2yz ¼ 0 (tilt angle / ¼ 0). Therefore, it is impossible to have abnormal total reflection in a rare-dense system under the assumption / ¼ 0 made by Castillo and Ballinas. The prerequisite for total reflection in a rare-dense system is that the optical axes are tilted an angle to the surface coordinates. The incident refractive index range in Lin and Wu is incorrect: the wave vector compo-
k (rad/µm) 19.0 18.0 k 2z +β2
17.0
n 0k 0
16.0
15.0
75
80
85
90
θ 0 (deg)
Fig. 4. Wave vectors in the isotropic medium and the anisotropic medium for light propagating toward the dense side (b > 0) in the range 0 < h0 < 90°.
k (rad/µm) kz +β2 2
16.5
16.0
n0 k0
15.5
15.0 14.5
75
80
85
90
θ 0 (deg)
Fig. 5. Wave vectors in the isotropic medium and the anisotropic medium for light propagating toward the rare side (b < 0) in the range 0 < h0 < 90°.
Y.-J. Jen, Y.-R. Cheng / Optics Communications 233 (2004) 271–275
isotropic medium and the anisotropic medium for light propagating toward the dense side (b > 0) and the rare side (b < 0) of the anisotropic medium, respectively, in the range 0 < h0 < 90°. At the critical points on both sides, the index of anisotropy is 1.67, greater than the incident index, 1.59.
4. Conclusion In this paper, the reflectance is calculated by considering non-symmetric reflection phenomenon. The incident and reflected vectors (including wave vectors and ray vectors) are not symmetric about the normal of the interface. Two studies [1,2] about the abnormal total reflection have to be resolved and corrected. It is possible to have total reflection for light propagating from a rare me-
275
dium to dense medium. The requirement to have the abnormal total reflection is derived here correctly. Acknowledgements The authors would like to thank the National Science Council of the Republic of China for financially supporting this research under Contract No. NSC. 91-2622-E-027-024-CC3.
References [1] C.L. Lin, J.J. Wu, Opt. Lett. 23 (1998) 22. [2] G.F. Torres del Castillo, C.J. Perez Ballinas, Opt. Lett. 26 (2001) 1251. [3] Y.J. Jen, C.C. Lee, Opt. Lett. 26 (2001) 190.