25 Optics of Metals

25 Optics of Metals 25.1 INTRODUCTION We have been concerned with the propagation of light in nonconducting media We now turn our attention to describing the interaction of light with conducting materials, namely, metals and semiconductors Metals and semiconductors, absorbing media, are crystalline aggregates consisting of small crystals of random orientation Unlike true crystals they not have repetitive structures throughout their entire forms The phenomenon of conductivity is associated with the appearance of heat; it is very often called Joule heat It is a thermodynamically irreversible process in which electromagnetic energy is transformed to heat As a result, the optical field within a conductor is attenuated The very high conductivity exhibited by metals and semiconductors causes them to be practically opaque The phenomenon of conduction and strong absorption corresponds to high reflectivity so that metallic surfaces act as excellent mirrors In fact, up to the latter part of the nineteenth century most large reflecting astronomical telescope mirrors were metallic Eventually, metal mirrors were replaced with parabolic glass surfaces overcoated with silver, a material with a very high reflectivity Unfortunately, silver oxidizes in a relatively short time with oxygen and sulfur compounds in the atmosphere and turns black Consequently, silver-coated mirrors must be recoated nearly every other year or so, a difficult, timeconsuming, expensive process This problem was finally solved by Strong in the 1930s with his method of evaporating aluminum on to the surface of optical glass In the following sections we shall not deal with the theory of metals Rather, we shall concentrate on the phenomenological description of the interaction of polarized light with metallic surfaces Therefore, in Section 25.2 we develop Maxwell’s equations for conducting media We discover that for conducting media the refractive index becomes complex and has the form n ¼ n(1Ài) where n is the real refractive index and  is the extinction coefficient Furthermore, Fresnel’s equations for reflection and transmission continue to be valid for conducting (absorbing) media However, because of the rapid attenuation of the optical field within an absorbing medium, Fresnel’s equations for transmission are inapplicable Using the complex refractive index, we develop Fresnel’s equations for reflection at normal incidence Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved and describe them in terms of a quantity called the reflectivity It is possible to develop Fresnel’s reflection equations for non-normal incidence However, the forms are very complicated and so approximate forms are derived for the s and p polarizations It is rather remarkable that the phenomenon of conductivity may be taken into account simply by introducing a complex index of refraction A complete understanding of the significance of n and  can only be understood on the basis of the dispersion theory of metals However, experience does show that large values of reflectivity correspond to large values of  In Sections 25.3 and 25.4 we discuss the measurement of the optical constants n and  A number of methods have been developed over the past 100 years, nearly all of which are null-intensity methods That is, n and  are obtained from the null condition on the reflected intensity The best-known null method is the principle angle of incidence/principle azimuthal angle method (Section 25.3) In this method a beam of light is incident on the sample and the incidence angle is varied until an incidence angle is reached where a phase shift of /2 occurs The incidence angle where this takes place is known as the principle angle of incidence An additional phase shift of /2 is now introduced into the reflected light with a quarter-wave retarder The condition of the principal angle of incidence and the quarter-wave shift and the introduction of the quarter-wave retarder, as we shall see, creates linearly polarized light Analyzing the phase-shifted reflected light with a polarizer that is rotated around its azimuthal angle leads to a null intensity (at the principal azimuthal angle) from which n and  can be determined Classical null methods were developed long before the advent of quantitative detectors, digital voltmeters, and digital computers Nulling methods are very valuable, but they have a serious drawback: the method requires a mechanical arm that must be rotated along with the azimuthal rotation of a Babinet–Soleil compensator and analyzer until a null intensity is found In addition, a mechanical arm that yields scientifically useful readings is quite expensive It is possible to overcome these drawbacks by reconsidering Fresnel’s equations for reflection at an incidence angle of 45 It is well known that Fresnel’s equations for reflection simplify at normal incidence and at the Brewster angle for nonabsorbing (dielectric) materials Less well known is that Fresnel’s equations also simplify at an incidence angle of 45 All of these simplifications were discussed in Chapter assuming dielectric media The simplifications at the incidence angle of 45 hold even for absorbing media Therefore, in Section 25.4 we describe the measurement of an optically absorbing surface at an incidence angle of 45 This method, called digital refractometry, overcomes the nulling problems and leads to equations to determine n and  that can be solved on a digital computer by iteration 25.2 MAXWELL’S EQUATIONS FOR ABSORBING MEDIA We now solve Maxwell’s equations for a homogeneous isotropic medium described by a dielectric constant ", a permeability , and a conductivity  Using material equations (also called the constitutive relations): D ¼ "E B ¼ H j ¼ E Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved ð25-1aÞ ð25-1bÞ ð25-1cÞ Maxwell’s equations become, in MKSA units, @E ¼ E @t @H =ÂEþ ¼0 @t  =ÁE¼ " =ÁH¼0 =ÂHÀ" ð25-2aÞ ð25-2bÞ ð25-2cÞ ð25-2dÞ These equations describe the propagation of the optical field within and at the boundary of a conducting medium To find the equation for the propagation of the field E we eliminate H between (25-2a) and (25-2b) We take the curl of (25-2b) and substitute (25-2a) into the resulting equation to obtain =  ð=  EÞ þ ð"Þ @2 E @E ¼0 þ  @t @t ð25-3Þ Expanding the =  ð=ÂÞ operator, we find that (25-3) becomes =2 E ¼ " @2 E @E þ  @t @t2 ð25-4Þ Equation (25-4) is the familiar wave equation modified by an additional term From our knowledge of differential equations the additional term described by @E=@t corresponds to damping or attenuation of a wave Thus, (25-4) can be considered the damped or attenuated wave equation We proceed now with the solution of (25-4) If the field is strictly monochromatic and of angular frequency ! so that E  Eðr, tÞ ¼ EðrÞ expði!tÞ, then substituting this form into (25-4) yields =2 EðrÞ ¼ ðÀ"!2 ÞEðrÞ þ ði!ÞEðrÞ which can be written as h   i =2 EðrÞ ¼ ðÀ!2 Þ " À i EðrÞ ! ð25-5Þ ð25-6Þ In this form, (25-6) is identical to the wave equation except that now the dielectric constant is complex; thus,  "¼"Ài ð25-7Þ ! where " is the real dielectric constant The correspondence with nonconducting media is readily seen if " is defined in terms of a complex refractive index n (we set  ¼ since we are not dealing with magnetic materials): " ¼ n2 ð25-8Þ We now express n in terms of the refractive index and the absorption of the medium To find the form of n which describes both the refractive and absorbing behavior of a propagating field, we first consider the intensity I(z) of the field after it has Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved propagated a distance z We know that the intensity is attenuated after a distance z has been traveled, so the intensity can be described by IðzÞ ¼ I0 expðÀzÞ ð25-9Þ where  is the attenuation or absorption coefficient We wish to relate  to , the extinction coefficient or attenuation index We first note that n is a dimensionless quantity, whereas from (25-9)  has the dimensions of inverse length We can express z as a dimensionless parameter by assuming that after a distance equal to a wavelength  the intensity has been reduced to IðÞ ¼ I0 expðÀ4Þ Equating the arguments of the exponents in (25-9) and (25-10), we have   4 ¼  ¼ 2k  ð25-10Þ ð25-11Þ where k ¼ 2/ is the wavenumber Equation (25-9) can then be written as   ! 4 IðzÞ ¼ I0 exp À z ð25-12Þ  From this result we can write the corresponding field E(z) as   ! 2 EðzÞ ¼ E0 exp À z  ð25-13Þ or EðzÞ ¼ E0 expðÀkzÞ ð25-14Þ Thus, the field propagating in the z direction can be described by EðzÞ ¼ E0 expðÀkzÞ exp½ið!t À kzފ The argument of (25-15) can be written as     ! k k i! t À zþi z ! ! ! k ¼ i! t À f1 À igz ! But k ¼ !/v ¼ !n/c, so (25-16b) becomes h i n i! t À f1 À igz c h  n i ¼ i! t À z c ð25-15Þ ð25-16aÞ ð25-16bÞ ð25-17aÞ ð25-17bÞ where n ¼ nð1 À iÞ Thus, the propagating field (25-15) can be written in the form: h   n  i EðzÞ ¼ E0 exp i! t À z c Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved ð25-18Þ ð25-19Þ Equation (25-19) shows that conducting (i.e., absorbing) media lead to the same solutions as nonconducting media except that the real refractive index n is replaced by a complex refractive index n Equation (25-18) relates the complex refractive index to the real refractive index and the absorption behavior of the medium and will be used throughout the text From (25-7), (25-8), and (25-18) we can relate n and  to  We have  " ¼ n2 ¼ n2 ð1 À iÞ2 ¼ " À i ð25-20Þ ! which leads immediately to n2 ð1À2 Þ ¼" ð25-21aÞ   ¼ n2  ¼ 2! 4 ð25-21bÞ where  ¼ !=2 We solve these equations to obtain "rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi #   2 n2 ¼ þ" "2 þ 4 "rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi #   2 2 À" n  ¼ " þ 4 ð25-22aÞ ð25-22bÞ Equation (25-22) is important because it enables us to relate the (measured) values of n and  to the constants " and  of a metal or semiconductor Because metals are opaque, it is not possible to measure these constants optically Since the wave equation for conducting media is identical to the wave equation for dielectrics, except for the appearance of a complex refractive index, we would expect the boundary conditions and all of its consequences to remain unchanged This is indeed the case Thus, Snell’s law of refraction becomes sin i ¼ n sin r ð25-23Þ where the refractive index is now complex Similarly, Fresnel’s law of reflection and refraction continue to be valid Since optical measurements cannot be made with Fresnel’s refraction equations, only Fresnel’s reflection equations are of practical interest We recall these equations are given by Rs ¼ À Rp ¼ sinði À r Þ E sinði þ r Þ s tanði À r Þ E tanði þ r Þ p ð25-24aÞ ð25-24bÞ In (25-24) i is the angle of incidence and r is the angle of refraction, and Rs, Rp, Es, and Ep have their usual meanings We now derive the equations for the reflected intensity, using (25-24) We consider (25-24a) first We expand the trigonometric sum and difference terms, Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved substitute sin r ¼ n sin r into the result, and find that ! Rs cos i À n cos r ¼ Es cos i þ n cos r ð25-25Þ We first use (25-25) to obtain the reflectivity, that is, the normalized intensity at normal incidence The reflectivity for the s polarization, (25-25) is defined to be  2 R  Rs   s  ð25-26Þ Es At normal incidence i ¼ 0, so from Snell’s law, (25-23), r ¼ and (25-25) reduces to ! Rs 1Àn ¼ ð25-27Þ 1þn Es Replacing n with the explicit form given by (25-18) yields ! Rs ð1 À nÞ þ in ¼ ð1 þ nÞ À in Es ð25-28Þ From the definition of the reflectivity (25-26) we then see that (25-28) yields " # ðn À 1Þ2 þ ðnÞ2 Rs ¼ ð25-29Þ ðn þ 1Þ2 þ ðnÞ2 We observe that for nonabsorbing media ( ¼ 0), (25-29) reduces to the well-known results for dielectrics We also note that for this condition and for n ¼ the reflectivity is zero, as we would expect In Fig 25-1 a plot of (25-29) as a function of  Figure 25-1 Plot of the reflectivity as a function of  The refractive indices are n ¼ 1.0, 1.5, and 2.0, respectively Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved is shown We see that for absorbing media with increasing  the reflectivity approaches unity Thus, highly reflecting absorbing media (e.g., metals) are characterized by high values of  In a similar manner we can find the reflectivity for normal incidence for the p polarization, (25-24b) Equation (25-24b) can be written as Rp sinði À r Þ cosði þ r Þ ¼ Ep sinði þ r Þ cosði À r Þ ð25-30Þ At normal incidence the cosine factor in (25-30) is unity, and we are left with the same equation for the s polarization, (25-24a) Hence, Rp ¼ Rs ð25-31Þ and for normal incidence the reflectivity is the same for the s and p polarizations We now derive the reflectivity equations for non-normal incidence We again begin with (25-24a) or, more conveniently, its expanded form, (25-25) ! Rs cos i À n cos r ¼ ð25-25Þ Es cos i þ n cos r Equation (25-25) is, of course, exact and can be used to obtain an exact expression for the reflectivity Rs However, the result is quite complicated Therefore, we derive an approximate equation, much quoted in the literature, for Rs which is sufficiently close to the exact result We replace the factor cos r by ð1 À sin2 r Þ1=2 and use sin i ¼ n sin r Then, (25-25) becomes qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3 2 Rs 6cos i À n À sin i qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi5 ¼4 ð25-32Þ Es cos i þ n2 À sin2 i Equation (25-32) can be approximated by noting that n2 ) sin2i, so (25-32) can be written as Rs cos i À n ¼ Es cos i þ n ð25-33Þ We now substitute (25-18) into (25-33) and group the terms into real and imaginary parts: ! Rs ðcos i À nÞ þ in ¼ ð25-34Þ ðcos i þ nÞ À in Ep The reflectivity Rs is then " # ðn À cos i Þ2 þ ðnÞ2 Rs ¼ ðn þ cos i Þ2 þ ðnÞ2 ð25-35Þ We now develop a similar, approximate, equation for Rp We first write (25-24b) as Rs sinði À r Þ cosði þ r Þ ¼ Ep sinði þ r Þ cosði À r Þ Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved ð25-30Þ The first factor is identical to (25-24a), so it can be replaced by its expanded form (25-25): ! sinði À r Þ cos i À n cos r ¼ ð25-36Þ sinði þ r Þ cos i þ n cos r The second factor in (25-30) is now expanded, and again we use cos r ¼ ð1 À sin2 r Þ1=2 and sin i ¼ n sin r: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 cosði þ r Þ ðcos i Þ n À sin i À sin i qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ð25-37Þ cosði À r Þ ðcos i Þ n2 À sin2 i þ sin2 i Because n2 ) sin2i, (25-37) can approximated as cosði þ r Þ n cos i À sin2 i ffi cosði À r Þ n cos i þ sin2 i ð25-38Þ We now multiply (25-36) by (25-38) to obtain !   Rp cos i À n n cos i À sin2 i ¼ cos i þ n n cos i þ sin2 i Ep ð25-39Þ Carrying out the multiplication in (25-39), we find that there is a sin2 i cos i term This term is always much smaller than the remaining terms and can be dropped The remaining terms then lead to Rp Àn cos i þ ð25-40aÞ ¼ n cos i þ Ep or Rp cos i À 1=n ¼À cos i þ 1=n Ep ð25-40bÞ Replacing n by n(1 À i), grouping terms into real and imaginary parts, and ignoring the negative sign because it will vanish when we determine the reflectivity, gives Rp ðn À 1= cos i Þ À in ð25-41Þ ¼ Ep ðn þ 1= cos i Þ þ in Multiplying (25-41) by its complex conjugate, we obtain the reflectivity Rp : Rp ¼ ðn À 1= cos i Þ2 þ ðnÞ2 ðn þ 1= cos i Þ2 þ ðnÞ2 For convenience we write the equation for Rs , (25-35), here also: " # ðn À cos i Þ2 þ ðnÞ2 Rs ¼ ðn þ cos i Þ2 þ ðnÞ2 ð25-42Þ ð25-35Þ In Figs 25-2 through 25-5 plots are shown for the reflectivity as a function of the incidence angle i of gold (Au), silver (Ag), copper (Cu), and platinum (Pt), using (25-35) and (25-39) The values for n and  are taken from Wood’s classic text Physical Optics Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved Figure 25-2 Reflectance of gold (Au) as a function of incidence angle The refractive index and the extinction coefficient are 0.36 and 7.70 respectively The normal reflectance value is 0.849 Figure 25-3 Reflectance of silver (Ag) as a function of incidence angle The refractive index and the extinction coefficient are 0.18 and 20.2, respectively The normal reflectance value is 0.951 Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved Figure 25-4 Reflectance of copper (Cu) as a function of incidence angle The refractive index and the extinction coefficient are 0.64 and 4.08, respectively The normal reflectance value is 0.731 Figure 25-5 Reflectance of platinum (Pt) as a function of incidence angle The refractive index and the extinction coefficient are 2.06 and 2.06, respectively The normal reflectance value is 0.699 Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved sin B C cos2 sin cos C B cos C M¼ B ð25-58Þ 2B sin2 0C @ sin sin cos A 0 0 Multiplying (25-57) by (25-58), we obtain the intensity of the beam emerging from the analyzing polarizer: Ið Þ ¼ ð1 þ P2 Þ À ð1 À P2 Þ cos þ 2P sin ð25-59Þ or Ið Þ ¼ A À B cos þ C sin [...]... analyzing polarizer: IðÞ ¼ ð1 þ P2 Þ À ð1 À P2 Þ cos 2 þ 2P sin 2 25- 59Þ or IðÞ ¼ A À B cos 2 þ C sin 2 25- 60aÞ where A ¼ 1 þ P2 25- 60bÞ 2 B¼1ÀP 25- 60cÞ C ¼ 2P 25- 60dÞ Equation (25- 60a) is now written as B C IðÞ ¼ A 1 À cos 2 þ sin 2 A A ! 25- 61aÞ We set B A C sin ¼ A cos ¼ 25- 61bÞ 25- 61cÞ so that (25- 61a) can now be written as IðÞ ¼ A½1 À cosð À 2... values of the reflection coefficients rp and rs Then, we can write rp ¼ Rp ¼ p expðip Þ Ep 25- 43aÞ rs ¼ Rs ¼ s expðis Þ Es 25- 43bÞ Equation (25- 43) can be transformed to the Stokes parameters The Stokes parameters for the incident beam are S0 ¼ cos i ðEs Esà þ Ep EpÃ Þ 25- 44aÞ S1 ¼ cos i ðEs Esà À Ep EpÃ Þ 25- 44bÞ S2 ¼ cos i ðEs Epà þ Ep EsÃ Þ 25- 44cÞ S3 ¼ i cos i ðEs Epà À Ep EsÃ Þ 25- 44dÞ... unity for many metals Born and Wolf have shown that this is a natural consequence of the simple classical theory of the electron and the dispersion theory The theory provides a theoretical basis for the behavior of n and  Further details on the nature of metals and, in particular, the refractive index and the extinction coefficient (n and ) as it appears in the dispersion theory of metals can be found... Rights Reserved 25- 51Þ Substituting (25- 51) into (25- 50), we find the Stokes vector of the reflected light to be 1 0 0 01 S0 1 þ P2 Á B S0 C 2 I B À 2 C p 0BÀ 1 À P C B 1C 25- 52Þ ¼ C B B 0C @ S2 A 2 @ 2P cos Á A S03 À2P sin Á The ellipticity angle  is     1 À1 S03 1 À1 À2P sin Á  ¼ sin ¼ sin 2 2 S00 1 þ P2 Similarly, the orientation angle   1 À2P cos Á ¼ tanÀ1 2 1 À P2 25- 53aÞ is 25- 53bÞ We see... ellipse corresponding to (25- 52) is in its standard, nonrotated, form For Á ¼ /2 the Stokes vector, (25- 52), becomes 0 1 0 01 S0 1 þ P2 ÁC B À B 0C B S1 C 2p I0 B À 1 À P2 C B C B C¼ 25- 54Þ C B S0 C 2 B 0 @ A @ 2A S03 À2P and  and of the polarization ellipse corresponding to (25- 54) are,     1 À1 S03 1 À1 À2P  ¼ sin ¼ sin 2 2 S00 1 þ P2  0 1 À1 S2 ¼ tan ¼0 2 S01 25- 55aÞ 25- 55bÞ We must now transform... Stokes parameters for the reflected beam are defined as S00 ¼ cos i ðRs RÃs þ Rp RÃp Þ 25- 45aÞ S01 ¼ cos i ðRs RÃs À Rp RÃp Þ 25- 45bÞ S02 ¼ cos i ðRs RÃp þ Rp RÃs Þ 25- 45cÞ S03 ¼ i cos i ðRs RÃp À Rp RÃs Þ 25- 45dÞ Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved Substituting (25- 43) into (25- 45) and using (25- 44) yields 0 2 10 1 0 01 s þ 2p 2s À 2p 0 0 S0 S0 CB S C 2 2 2 2 B S01 C 1... and by Mott and Jones 25. 3 PRINCIPAL ANGLE OF INCIDENCE MEASUREMENT OF REFRACTIVE INDEX AND EXTINCTION COEFFICIENT OF OPTICALLY ABSORBING MATERIALS In the previous section we saw that optically absorbing materials are characterized by a real refractive index n and an extinction coefficient  Because these constants describe the behavior and performance of optical materials such as metals and semiconductors,... can then be determined In Fig 25- 6 we show the measurement configuration To derive the equations for n and , we begin with Fresnel’s reflection equations for absorbing media: Rs ¼ À Rp ¼ sinði À r Þ E sinði þ r Þ s tanði À r Þ E tanði þ r Þ p Copyright © 2003 by Marcel Dekker, Inc All Rights Reserved 25- 24aÞ 25- 24bÞ Figure 25- 6 Measurement of the principal angle of incidence and the principal... Reserved 25- 57Þ which is the Stokes vector for linearly polarized light The Mueller matrix for a linear polarizer at an angle is 0 1 1 cos 2 sin 2 0 B C cos2 2 sin 2 cos 2 0 C 1 B cos 2 C M¼ B 25- 58Þ 2B sin2 2 0C @ sin 2 sin 2 cos 2 A 0 0 0 0 Multiplying (25- 57) by (25- 58), we obtain the intensity of the beam emerging from the analyzing polarizer: IðÞ ¼ ð1 þ P2 Þ À ð1 À P2 Þ cos 2 þ 2P sin 2 25- 59Þ... of obtaining this condition We first write (25- 48b) as s ¼ Pp 25- 49Þ Substituting (25- 49) into (25- 46), we obtain the Stokes vector of the reflected light to be À Á 0 01 0 10 1 1 þ P2 À 1 À P2 S0 0 0 S0 Á À Á B 0C B C C 2B À B S1 C p B À 1 À P2 CB S1 C 1 þ P2 0 0 B C¼ B CB C B S0 C CB C 2B @ 2A @ 0 0 2P cos Á 2P sin Á A@ S2 A S3 S03 0 0 À2P sin Á 2P cos Á 25- 50Þ For incident þ45 linearly polarized