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International Journal of Infrared and Millimeter Waves, VoL 6, No. 3, t985 BREWSTER ANGLES FOR MAGNETIC MEDIA C. Lee Giles* and Walter J. Wild** Optical Sciences Center Universityof Arizona Tucson, A r i z o n a 85721 Received November 22, 1984 Brewster angles (angles for which there is no reflection of an incident electromagnetic plane wave) will exist for either incident polarization at a planar interface separating two media provided that either or both of these media are magnetic. Except at certain angles (normal and grazing incidence) and specific material properties, a Brewster angle cannot exist simultaneously for both incident polarizations, but always exists for one polarization or the other for real values of permittivity and permeability. The existence conditions for the Brewster angle are readily formulated in terms of relative media permittivity and permeability, or relative media refractive index and electromagnetic impedance. When the ratio of permeabilities and permittivities is not unity, the Brewster angle is no longer orthogonal to the transmitted beam angle. From the viewpoint of the dipole oscillator model, the contributions of the electric and magnetic dipoles are such that they do not independently dominate the Brewster angle existence; rather the ratio of media permittivities and/or permeabilities determine when a Brewster angle may exist. *Present address: Naval Research Laboratory, Optical Sciences Division, Washington, D.C. 20375. **Also at IIT Research Institute, 10 W. 35th St., Chicago, Illinois 60616 187 0195-9271/85/0300-0187504,50/0 zg 1985 Plenum Publishing CorporalJon 188 Giles and Wild INTRODUCTION Recently, interesting and unusual properties of the r e f l e c t i o n a n d scattering of electromagnetic plane waves in magnetic materials have been reported. It has been shown that for a planar interface and the special case of equal refractive indices for both magnetic media, the amplitude reflection coefficient is constant, and independent of the angle of incidence and incident polarization I . For the scattering of a plane wave from a magnetic sphere, the scattered wave can be zero for various scattering angles for certain specific ratios of permittivity and permeability. 2 This paper describes another interesting property of nonabsorbing magnetic materials, the existence of a Brewster angle (incident angle of total transmission or zero reflection) for either incident polarization of an electromagnetic plane wave. This variation of the Brewster angle with polarization is not well-known. 3 This is because most studies of the Fresnel reflection equations begin with the assumption of unity permeability which is true for the visible and infrared regions of the spectrum. In the submillimeter to radio-frequency parts of the spectrum one cannot necessarily ignor the permeability; indeed, the permeability may be complex.4,5, 6 For the case where dielectrics are considered, most treatments which incorporate the permeability usually go no further than a simple statement of derived equations (such as for the Brewster angles) and there is no detailed discussion of the influence of the permeability or the physical interpretation. 7,8 We expect that some of the results discussed here can be verified experimentally with materials such as the Yttrium Iron Garnets. 6 We describe the material conditions that determine which incident polarization has a Brewster angle and show that the Brewster angle cannot exist simultaneously for both incident polarizations, except at normal and grazing incidence. We also show that whether or not the Brewster angle exists for either polarization is determined by the relative strengths of the ratios of permittivity and permeability or ratios of refractive index and bulk impedance. Further, in general the angle between the Brewster angle and the transmitted wave deviates from 90 ~ whenever the ratio of permittivities and permeabilities is different from unity. ANALYSIS Brewster Angles for Magnetic Media t89 Consider the reflection and transmission of an incident electromagnetic plane wave at a planar interface separating two media, either or both of which m a y be magnetic. The constitutive relation between the electric field E and its displacement field D is linear, isotropic, homogeneous and time-independent, i.e. D = EE where e is the medium permittivity. By magnetic, we mean that the constitutive relation between the magnetic field H and its induction field B is linear, isotropic, homogeneous and time-independent, i.e. B = pH where p is the medium permeability, we assume that for either or both media the permeability p is not equal to Po, the permeability of free space and that both e and p are real and positive. Note that materials which closely approximate such properties exist at specific frequencies in the far infrared, millimeter, and microwave spectral regions. I El ,F1 E2F 2 \ B ~ \\, ! O2 PLANEOF INCIDENCE %L Fig. I. Incident polarizations, media permittivities and permeabilities, angles of incidence and refraction, and Brewster angle are defined. 190 Giles and Wild The reflection coefficients for either p o l a r i z a t i o n are easily derived. I0,11 We let r• and rll denote the electric field reflection coefficients for the polarizations respectively p e r p e n d i c u l a r and parallel to the plane of incidence. See figure I for further clarification. Define el, UI and e2, U 2 as the permittivity, p e r m e a b i l i t y pairs for the incident and reflecting media. Further analysis is simplified if we define dimensionless ratios for permittivity and p e r m e a b i l i t y such that e = e2/~ I and U = U2/UI, i.e. e is the ratio of p e r m i t t i v i t y of the reflecting medium to the p e r m i t t i v i t y of the incident medium and similarly for U. Now e and U are independent of the choice of units and can be interpreted as the relative strengths of the electric and magnetic dipole contributions of each media. 12 The reflection coefficients have the form8: r• rll (~) cos @I - cos 82 ({U/e) cos 81 + cos = (I) @2 cos 81 - ( ~ ) cos 82 cos @I + (U~7~) cos 82 = (2) where 81 and 82 are the angles of incidence and refraction. The value of the reflection angle 81 for which r• or rll is zero is defined as the Brewster angle for that p o l a r i z a t i o n (all polarizations thus denoted are for the electric field) and will be defined as 81 or 811. Using Snell's law, sin 81 = e ~ sin @2, and setting equations (1) and (2) equal to zero, we find the following expressions for the Brewster angle: polarization El : tan 2 81 = U (e-p) l-ep polarization Ell tan2 : 811 = E (pqe) I-EU (3) (4) Another form expresses the Brewster angles in terms of the relative refractive index m and relative bulk electromagnetic impedance z, where m ~ ~ and z ~ ~ / ~ . ~quations (3) and (4) may now be written as Brewster Angles for Magnetic Media 191 1-Z 2 polarization E1 : (5) tan 2 81 = m 2 I -m 2 polarization Ell : tan2 81t = m2 z2-I -z2 1-m 2 (6) From these equations, certain special cases for the Brewster angle are i m m e d i a t e l y obvious and reassuring. Equations (3) and (4) yield tan2 8jl = g 81 does not exist, le # I, H = 11 (7) tan 2 8• = P 811 does not exist. As is well-known, the Brewster angle for d i e l e c t r i c s (Pl = P2 = P = 1)exists only for the p a r a l l e l p o l a r i z a t i o n . However, the a n a l o g o u s m a g n e t i c case (g = I) has a Brewster angle only for the p e r p e n d i c u l a r p o l a r i z a t i o n . Inspection of equations (5) and (6) s h o w that for z = I (impedance matching), Brewster angles occur for both p o l a r i z a t i o n s at normal incidence. For m = I (refractive index matching), Brewster angles occur for both p o l a r i z a t i o n s at grazing incidence. I However, the existence of the latter case is d i f f i c u l t to show except in the limit of one r e f r a c t i v e index a p p r o a c h i n g the other. General c o n d i t i o n s for the existence of the Brewster angle for either or b o t h p o l a r i z a t i o n s can be readily d e r i v e d from equations (3) and (4) or equations (5) and (6) by requiring the right side of these equations to be positive and real. From equations (3) and (4), we have the following existence conditions: 81 exists if H ~ e > I/p or p ~ e ~ I/p, BII exists if e > ~ > 1/e or e ~ ~ r I/e; (8) and from equations (5) and (6): 192 Giles and Wild 8• exists if [: i> ] ~ or [: ~ I] 8,} exists if [zm I> ] ~ or [: <111> (9) These inequalities are graphically portrayed as decision regions in figures 2 and 3. Equations (8) and (9) show that for real values of m and z or e and ~, a Brewster angle always exists for one polarization. Note that many Z)o M. )IELECTRIC) L=I I 0 i E Fig. 2. Decision regions are illustrated for Brewster angles for different polarizations as determined by the relative permittivity e and relative permeability ~. The regions denoted by 81 define values of e and ~ where the Brewster angle exists for the electric field polarization perpendicular to the plane of incidence. Similar regions denoted by 8{{ are for the parallel polarization. The solid lines indicate where a Brewster angle exists for both polarizations. The dielectric case is denoted by the dotted line. Brewster Angles for Magnetic Media 193 values of (el,e 2) and (~I,~2) could represent a single p o i n t (c,~) in figure 2. A similar situation exists for (m,z) in figure 3. In each figure, the dotted line represents the d i e l e c t r i c (~I = ~2 = U = I) case. A c t u a l l y U = I does not mean that both m e d i a have to be d i e l e c t r i c s . In fact, both m e d i a may be m a g n e t i c but their p e r m e a b i l i t i e s m u s t be equal. These m a t e r i a l s are then m a g n e t i c a l l y equivalent but e l e c t r i c a l l y dissimilar. A similar situation occurs for e = I. In figure 3, the p o i n t m = z = I (both media electromagnetically identical) has a B r e w s t e r angle ~ 0 _ 0 o I m Fig. 3. Decision regions are i l l u s t r a t e d for Brewster angles for d i f f e r e n t p o l a r i z a t i o n s as d e t e r m i n e d by the relative r e f r a c t i v e index m and relative impedance z. The regions denoted by 81 define values of m and z where the Brewster angle exists for the electric field p o l a r i z a t i o n p e r p e n d i c u l a r to the plane of incidence. Similar regions d e n o t e d by 8H are for the p a r a l l e l p o l a r i z a t i o n . The solid lines indicate where a B r e w s t e r angle exists for both p o l a r i z a t i o n s . The d i e l e c t r i c case is denoted by the dotted line. 194 Giles and Wild for all angles. A similar p o i n t exists at e = ~ = I in figure 2. For both p o l a r i z a t i o n s the critical angle 8c is defined by sin 0 c = m = e / ~ and exists if m < I or e~ ~ I. From figures 2 and 3 we see that a c r i t i c a l angle can coexist w i t h a Brewster angle p r o v i d i n g the above c o n d i t i o n s are satisfied. From the d e f i n i t i o n of the c r i t i c a l and Brewster angles, we have the relations: tan 81 = {(1-~/e) tan @c tan 811 = {(1-e/~) tan 0c . Since the e x i s t a n c e c o n d i t i o n s for 81 and 8ff i m p l y that respectively ~/e < I and e/~ < I, the Brewster angle is always smaller than the critical angle. DISCUSSION With figures 2 and 3, we can i n t e r p r e t the m a t e r i a l conditions which d e t e r m i n e the existence of a Brewster angle for a particular polarization. Figure 3 implies that if both the r e f r a c t i v e index and impedance of the r e f l e c t i n g m e d i u m are j o i n t l y greater than or less than the refractive index and i m p e d a n c e of the i n c i d e n t medium, the Brewster angle exists only for the p e r p e n d i c u l a r p o l a r i z a t i o n . If, on the other hand, the relative refractive index and the relative i m p e d a n c e are not j o i n t l y greater than u n i t y or less than unity, the Brewster angle exists only for the parallel polarization. For the special cases of z = I and m = I, the Brewster angle is r e s p e c t i v e l y at 8 = 0~ and 8 = 90 ~ for both p o l a r i z a t i o n s . If we can d e s c r i b e the m a t e r i a l p r o p e r t i e s of e and ~ to be due primarily to the electric and m a g n e t i c dipole moments, 12 figure 2 offers some unusual i n t e r p r e t a t i o n s of the existence conditions. The relative m a g n i t u d e s of the e l e c t r i c or m a g n e t i c dipoles do not independently dominate the Brewster angle existence, c o n t r a r y to what one m i g h t expect. The s i t u a t i o n is more complicated. For example, one c o n d i t i o n for the e x i s t e n c e of 81 is that the relative m a g n e t i c dipole s t r e n g t h of the r e f l e c t i n g m e d i u m to the i n c i d e n t m e d i u m is greater than the similar relative electric dipole strength, plus the a d d i t i o n a l c o n d i t i o n that the p r o d u c t of the electric and magnetic dipole strengths of the reflecting medium is greater than the similar Brewster Angles for Magnetic Media 195 p r o d u c t for the i n c i d e n t medium. Replace greater by lesser in the p r e c e d i n g a r g u m e n t and we have the other e x i s t e n c e c o n d i t i o n for ~• C o n t r a s t this to a similar e x i s t e n c e condition for Ell where the relative magnetic dipole s t r e n g t h of the incident medium is also greater than the relative e l e c t r i c dipole strengths but the d i p o l e strength p r o d u c t of the r e f l e c t i n g m e d i u m is less than that of the incident medium. In summary, the m a t e r i a l p r o p e r t i e s that d e t e r m i n e the e x i s t e n c e of a Brewster angle for an i n c i d e n t E 1 (TE) : Eli ( T ~ ) : ~ - i\',\ i I l!o ~z~~ \ / / I o ;o ~ iI / ~176 #. 90 ~ 6o ~ / - " ",,~ / / " / ./o{,, 60 %# ",, ~ (DIELECTRIC) ~ 12oL..._ ', "~-..,8oo ~ .. 120 ~ Fig. 4. Curves of constant ~ ~ 8 + 82 are illustrated as a function of e and p. When either e = I or p = I, we have the w e l l - k n o w n c o n d i t i o n e = 90 ~ . If e = p (impedance m a t c h i n g ) , ~ = 8 = 0 ~ (Brewster's angle at normal incidence); and if e = I/p (refractive index m a t c h i n g ) , e = 180 ~ (Brewster's angle at g r a z i n g incidence). Giles and Wild 196 p o l a r i z a t i o n are more complex that might be expected 9 But simple i n e q u a l i t i e s b e t w e e n the relative p e r m i t t i v i t y and the relative p e r m e a b i l i t y or between the relative refractive index and the relative impedance provide d e f i n i t i v e e x i s t e n c e conditions. It is also i n t e r e s t i n g to consider how the p e r m e a b i l i t y changes the o r t h o g o n a l i t y c o n d i t i o n between the Brewster angle and t r a n s m i t t e d beam angle, i.e., how does ~+82 b e h a v e (see figure I) for general e and ~? R e f e r r i n g to e q u a t i o n s (3) and (4), we easily see that polarization El : cos(~+@ 2) = {(~/e) I-l~ (I0) I-1-I polarization Eli .- cos(8+@ 2) = ~(e/~) 1-e Figure 4 i l l u s t r a t e s a family of curves of e E 8+82 for v a r i a b l e e,~. We see that e = 90 ~ only when e = I or ~ = I. These values of e will vary from 0 ~ to 180 ~ , d e p e n d i n g on e and ~. For the case where e = ~ we have the Brewster angle at normal incidence. 4 For the case where e = I/~ (discussed at length in reference I), the Brewster angle is at extreme grazing incidence; this may be a singular situation if e1~ I < c2~ 2 (or e < I/~) since a c r i t i c a l angle also exists--the Brewster angle always precedes the c r i t i c a l angle but if e = I/~ each "coexists" at 81 = 8 = 90 ~ All curves pass through the p o i n t ~ = ~ = I; this is not a p a t h o l o g i c a l situation, rather, no e l e c t r o m a g n e t i c i n t e r f a c e exists for these m a t e r i a l parameters. Realizing that m a g n e t i c dipoles will radiate the same as electric dipoles, the c o n t r i b u t i o n of each (parameterized by ~ and e) will lead to the results shown in figure 4. We should also point out that for a n o n l i n e a r optical material w h e r e b y e and/or ~ can be e l e c t r i c a l l y (or thermally) m o d u lated, a form of m i l l i m e t e r wave optical "switch" can be devised. This, however, can only be realized if suitable m a t e r i a l s can be located or fabricated. We believe that this work will i n t e r e s t m i l l i m e t e r wave researchers concerned with the d e v e l o p m e n t and c h a r a c t e r ization of m a g n e t i c materials. Further a p p l i c a t i o n s of this and p r e v i o u s studies include interesting scattering properties of m a g n e t i c a e r o s o l s for m i l l i m e t e r and FIR lasers, and the a p p l i c a t i o n to Brewster angle m e a s u r e m e n t s BrewsterAnglesforMagneticMedia ~97 for the determination of the optical properties of magnetic media (similar to that for dielectrics in the visible). 2,8 ACKNOWLEDGEMENT We would like to acknowledge A. Lohmann. useful suggestions made by REFERENCES I. C.L. Giles and W. J. Wild, "Fresnel reflection and transmission at a planar boundary from media of equal refractive indices," Appl. Phys. Lett. 40 (3), 210 (1982). 2. M. Kerker, D-S. Wang and C. L. Giles, "Electromagnetic scattering by magnetic spheres," J. Opt. Soc. Am. 73, 765 (1983). 3. R. F. Harrington, Time-Harmonic Electromagnetic Fields (McGraw-Hill, New York, 1961) p. 59. 4. G.T. Ruck, D.E. Barrick, W.D. Stuart, and C.K. Kirchbaum, Radar Cross Section Handbook, (Plemum, New York, 1970), p. 624. 5. A. Sommerfeld, Optics (Academic Press, New York, 1949) p. 18. 6. B. Lax and K.J. Button, Microwave Ferrites and Ferrimagnetics, (New York, McGraw-Hill, 1962) Chap. 7. 7. W . T . Doyle, "Graphical Approach to Fresnal's Equation for Reflection and Refraction of Light," Am. J. Phys., 48(8), 643 (1980). 8. M. Zahn, Electromagnetic Field Theory: A Problem Solving Approach, (Wiley) p. 540, 2nd Ed. (1979). 9. R. Chiarelli, S. Faetti, and L. Frounaoui, "Determination of the Molecular Orientation at the Free Surface of Liquid Crystals from Brewster Angle Measurements," Opt. Comm., 46 (I), 9 (1983). 10. E. Hecht and A. Zajac, Optics (Addison-Wesley, Reading, 1979) p. 73. 11. R. K. Wagsness, Electromagnetic Fields 2nd Ed. (Wiley, New York, 1975) pp. 464, 469. 12. M. Born and E. Wolf, Principles of Optics 5th Ed. (Pergamon Press, New York, 1975) p. 84.