Electromagnetic waves

Transcription

1 Electromagnetic waves Electromagnetic waves are generated by electric charges in non-uniform motion. In next chapter we shall deal with phenomenon of electromagnetic radiation. Here we shall discuss some basic properties of electromagnetic waves without touching the problem of their generation. Maxwell has observed that the sourceless electromagnetic field equations possess a wave solution. This theoretical prediction was confirmed in experiments realized by H. Hertz in November 886. The most important Hertz s results were published in 888 (Ann. Phys. 34 6, Ann. Phys , Ann. Phys. 36 ) Electromagnetic waves in non-dispersive dielectric media 7.. A strength field approach The sourceless Maxwell s equations in non-conducting continuous media have the following µ H µ F F F = (7..) 4854 where (F µ )= E E E 3 E B 3 B E B 3 B E 3 B B C A, (Hµ )= D D D 3 D H 3 H D H 3 H D 3 H H C A. 367

2 7. ELECTROMAGNETIC WAVES In order to get solution one has to include the constitutive relations which provide some relations between pairs (E, B) and (D, H). In the simplest case of isotropic linear media those relations have the form D = "E, H = µ B (7..) 4858 where " = const and µ = const. Equivalently, one can cast equations (7..) in the D + r H = (7..3) r D = B + r E = (7..5) r B = (7..6) 4859 Acting with r on equations (7..3) and (7..5) we obtain "µ@ r E+ r r B {z } {z B r(r B) r r B+ r r E {z } {z } "µ@ E r(r E) r E = (7..7) =. (7..8) 486 If one defines the d Alembert operator "µ t r (7..9) 486 then the result can be cast in the form E =, and B =. (7..) Equations (7..) are just wave equations, therefore the Maxwell s theory supports the ex- istence of electromagnetic waves. The characteristic velocity which appears in the d Alembet operator depends on properties of continuous media and it reads v := c p "µ (7..) Note that for " ==µ the continuous medium is reduced to an empty space and then v = c. The fact that a speed of electromagnetic waves in dielectric media is lower than the speed of light in vacuum does not violate the Einstein s postulate because the effect originates in effective description of dielectric media. On quantum level photons are absorbed end emitted all the time so the velocity v cannot be interpreted as a velocity of a single photon. 368

3 7. Electromagnetic waves in non-dispersive dielectric media A potential approach It follows that electromagnetic potentials A µ! (', A) also satisfy the wave equation. Indeed, substituting expressions E A r', B = r A (7..) 4873 into first pair of Maxwell s equations (7..3) and (7..4) we obtain r )A + ' + r A) = (r A) r ' = (7..4) 4874 Then imposing the Lorenz condition ' + r A = (7..5) 4875 one gets that the four-potential satisfies the d Alembert equation A µ =. (7..6) The equation (7..6) lead to conclusion that the free electromagnetic field is represented by transverse waves. In order to see it, we we split the solution of the equation (7..6) into two components A µ! (', A) =(', ) +(, A). (7..7) {z } {z } A µ A µ From linearity they are also solutions of the equation (7..6). The four-potential A µ 4879 expressions leads to 488 E = r', B =, (7..8) 488 whereas A µ gives E A, B = r A. (7..9) It follows from (7..8) that r E =and from (7..9) and Lotenz condition (7..5) for A µ that r E =. These conditions imply that the potential A µ represent longitudinal degrees of freedom whereas A µ represent transverse degrees of freedom. Note that such distinguishing on longitudinal and transverse degrees of freedom in the case of time dependent fields Note that for electric field E = ne (n x,t) one gets r E =(ê i n)@ i E (n x,t)= (ê i n)n s E (s, t) =(n n)@ s E (s, t) =, where s := n x. Taking E = n E (n x,t) with n n =one gets r E =. 369

4 7. ELECTROMAGNETIC WAVES is meaningless for the fields and it can be introduced only for electromagnetic potentials. The longitudinal degrees of freedom can be eliminated by properly chosen gauge transformation. In order to see it we consider the gauge transformation ' = A = A + r. (7..) E E z } { z } { E = [ r']+ A] E z } { z } { = [ ) r@ (A + r r ]. (7..) E 4889 The appropriate choice of (t, x), namely ' = (7..) allows to eliminate E. The condition =is necessary in order to potentials (', A ) satisfy the Lorenz condition. Note, that the choice ' is compatible with the fact that ' =. The Lorenz condition (7..5) imposed on the potentials (', A ) reduces to the form ' =, r A = (7..3) for given by (7..). It restricts a wave solutions to transverse waves. It is important to notice that the gauge fixing (7..) exist only for some free and time-dependent fields. When electromagnetic field is not free then the condition (7..3) is not a gauge condition anymore. Instead it is a kind of restriction which allows to separate out the transversal (radiation) part of the electromagnetic field. The longitudinal part represents static fields and does not satisfy (7..3). Notice that such decomposition is not invariant under Lorentz transformations Plane waves Phase velocity A special group of electromagnetic waves is given by waves characterized by constant phase surfaces. Such surfaces are solutions of the equation (t, x) =const. (7..4) The form of the function determines geometry of the surface of constant phase. For instance for a plane wave that propagates in a dielectric homogeneous medium the surface of constant phase is given by equation n x vt = const (7..5) 37

5 7. Electromagnetic waves in non-dispersive dielectric media where n is a constant unit vector in direction of propagation of the wave. For spherical wave the vector n is replaced by a radial spherical versor ˆr and for cylindrical wave by a radial versor in cylindrical coordinates ˆ. The phase velocity v p is defined as a velocity of translocation of the phase surface. It can be obtained from d =which leads to the t dt+ r dx = 49 Dividing by r one can cast this equation in the form v p = r r dx dt t r. (7..6) It follows that the phase velocity is just a projection of dx 49 the point x belonging to the surface of constant phase and r dt on r r dx, where dt is the velocity of is a vector normal to this surface. 493 r 494 The phase velocity of a plane wave with the surface = const given by (7..5) reads v p = v = c p "µ. (7..7) Solution of wave equation in + dimensions Equations E = and B = can be represented by a single equation = where ={E,E,E 3,B,B,B 3 }. For the case of plane waves the surface of constant phase depends in fact on a single coordinate. One can always choose one of the axes of the Cartesian reference frame in direction of propagation of the wave. In such a case the function depends only on two variables (t, x). Defining two light-cone coordinates x ± := x ± vt (7..8) one gets and therefore t + @ = v(@ ) x + @ + (7..3) 4@ The (x +,x )= (7..3) 493 has a general solution = + (x + )+ (x ), then (t, x) = + (x + vt)+ (x vt). (7..3) 37

6 7. ELECTROMAGNETIC WAVES This solution describes superposition of two waves that propagate in positive (x vt) and negative +(x + vt) direction of axis x. Let us observe that also for spherical waves one can also get an explicit form of the solution. The wave equation in this case reads t Substituting (t, r) = r (t, r), one gets r r = (7..33) r apple r =. (7..34) where r (t, r) =r@ r (t, r) (t, r). It follows that (t, r) satisfies =, with x ± := r ± vt then the solution reads (t, r) = + (r + vt) r + (r vt). (7..35) r The solution is a superposition of ingoing + and outgoing spherical wave Spectral decomposition We shall consider here a plane wave which is superposition of many plane waves which differ by the angular wave frequency!. The frequency of a single component enters to the solution through one of the factors cos!t, sin!t, exp ( i!t). The problem of polarization of such waves will be discussed in further part. In this section we shall study the case where all vectors A (or E) assigned to different frequencies oscillate in one common direction. Mathematically such superposition of waves can be represented by a Fourier transform. Any function f(x) of class L i.e. and which satisfies a condition Z f(x) dx < (7..36) f(x) = [f(x ) + f(x + )] (7..37) at the discontinuity points can be represented by a Fourier integral f(x) = Z The expansion coefficients are given by the Fourier transform of f(x) F (k) F[f(x)](k) := dk F (k)e ikx. (7..38) Z dx f(x)e ikx. (7..39) 37

7 7. Electromagnetic waves in non-dispersive dielectric media If the function f(x) depends on all spacetime coordinates x then f(x) = Z ( ) 3 d 3 xf(k)e ik x, (7..4) R Z 3 F (k) = d 3 kf(x)e ik x. (7..4) R 3 In study of electromagnetic waves it is convenient to work with complex functions instead of the real ones. However, since the electromagnetic field is a real-valued field, then one has to interpret its complex version as an auxiliary field whose physical content is encoded in either its real or imaginary part. The electromagnetic plane wave can be represented by a Fourier decomposition on a plane monochromatic waves A(t, x) = Z ( ) 3 d 3 k a(t, k)e ik x (7..4) R 3 where Z a(t, k) = d 3 x A(t, x)e ik x. (7..43) R 3 In our approach A is a complex-valued vector function. An assumption that the wave is a superposition of many waves with different frequencies leads to the following form of the coef- ficients a(t, k) =a(k)e i!(k)t, (7..44) where k := k is a wave number and k is called wave vector. The electromagnetic potential takes the form A(t, x) = Z ( ) 3 d 3 k a(k)e i(k x!(k)t). (7..45) R 3 Let us observe that each monochromatic component must be a solution of wave equation "µ t r e i(k x!(k)t) = ) "µ! c k =. (7..46) The last equality is called the dispersion relation and it can be written in terms of wave number ck = p "µ!(k). The characteristic expression p "µ is called the refraction coefficient and it is usually denoted by n := p "µ. Note, the Fourier coefficients must obey relation a (t, k) =a(t, k) for a real-valued electromagnetic potential. A factor e i!(k)t can be introduced only for complex-valued fields. For real-valued fields it must be replaced by either cos(!(k)t) or cos(!(k)t). 373

8 7. ELECTROMAGNETIC WAVES If the x axis is parallel to the vector k then a(k) =a(k )( ) (k ) (k 3 ) 4956 and consequently the Fourier integral can be written in the form A(t, x) = Z dk a(k )e i(k x!( k )t). (7..47) 4957 Suppose that the coefficients a(k ) vanish outside the interval k k <, where " is a small 4958 number. If k > then the integral contains only contributions from k >. In such a case k := k k. Figure 7.: The amplitude coefficient a(k ) Expanding!(k) in Taylor series in a neighborhood of k d!!(k) =!(k )+(k k ) +... (7..48) dk k=k and defining! :=!(k ) and one gets v g := d! (7..49) dk k=k A(t, x) = Z k + k dk a(k)e i(kx! t (k k )v gt) Z k + = e i(k x! t) dk a(k)e i(k k )(x v gt) k {z } A (t,x) (7..5) (7..5) For k < analysis is very similar with k = k. 374

9 7. Electromagnetic waves in non-dispersive dielectric media The expression e i(k x! t) represent a dominant frequency oscillation term. The amplitude term A (t, x) assumes constant values on the planes x v g t = const. This expression give a profile (envelope) of a wave packet. The velocity with which the envelope moves is given by (7..49). For this reason it is termed a group velocity. When the frequency is a linear function of the wave number! = vk, then the phase factor is = kx vkt and one can conclude that the group velocity is equal to the phase velocity v g = d! dk = v, v p t r =! k = v. (7..5) 4969 Let us observe that relation! = k v p leads to v g = d(k v p) dk =v p + k dv p dk =v p dv p d (7..53) 497 where := k is the length of the wave Monochromatic wave in homogeneous dielectrics Let us consider a monochromatic electromagnetic wave with the angular frequency! that propagates in a homogeneous dielectric medium characterized by a constant permeabilities " and µ. Such wave is described by auxiliary complex fields E = E e i(k x!t), B = B e i(k x!t) (7..54) where E and B are some constant complex amplitude vectors. The fields (7..54) are solu- tions of the wave equation under condition n! c = k. (7..55) It is enough to consider k as a real vector. In order to get information about mutual orientation of E, B and k one has to plug such solutions to Maxwell s equations and analyze the resulting algebraic equations. Considering that re i(k x!t) = ik e i(k x!t) one gets r E = i k E, r E = i k t E = i!e (7..56) 498 and similarly for B. Then r E = ) k E = (7..57) r B = ) k B = (7..58) 375

10 7. ELECTROMAGNETIC WAVES r B n te = ) k B + n! c E = (7..59) r E + tb = ) k E c B = (7..6) Two first equations (7..57) and (7..58) imply that vectors E and B lies in the plane orthog- onal to the wave vector k. Taking into account that the wave number k = n! 4983 c is given by the 4984 dispersion relation (7..5) one gets from (7..59) and (7..6) E = n ˆk B, B = n ˆk E, ˆk k k. (7..6) 4985 A scalar product of amplitudes read E B = [ˆk E B (ˆk E )(ˆk B )] = E B (7..6) 4986 and therefore E B =. (7..63) Then one can conclude that electric and magnetic field vectors are mutually perpendicular. The amplitudes of the fields are proportional, what follows from the expression E E = n [ˆk B ] [ˆk B ]= n [ˆk (B B ) (ˆk B )(ˆk B )] 4989 which gives E = n B. (7..64) Moreover, since both vector amplitudes (7..6) are related by multiplication by a real vector n ˆk then it follows that the electric and magnetic field have the same phase. The square of E is a complex number so it can be represented in the form E = E e i'. The phase of this number was chosen as i. The square of the complex magnetic field vector reads (B ) = n (ˆk E ) = n E e i' ) B = n E e i' what clearly shows that complex electric and magnetic field have the same phase. Some of the above statements do not hold in conducting media where the vector k must be replaced by a complex vector. 376

11 7. Electromagnetic waves in non-dispersive dielectric media Polarization of electromagnetic waves In order to study polarization of electromagnetic wave we choose a given point in space and observe behaviour of the electric field E at this point. The magnetic field can be discarded in this analysis because it not an independent field. Its orientation is determined by by an expression like (7..6) Totally polarized electromagnetic wave The monochromatic wave is totally polarized because its amplitude vector E ia a constant vector i.e. its orientation in a space remains unchanged in time. In this section we shall consider such a wave. Let E = E e i(k x!t) be an complex-valued electric field vector describing the electromagnetic plane wave with a single frequency. The amplitude E is a complex constant vector and a physical electric field is given by its real part Re(E). At the fixed point the field E is a time dependent function. It follows from Gauss law that E k =. Since E is a complex function then its square E E is a complex number. Following the previous section we shall parametrize this number as E E = E e i'. (7..65) 5 The ampliud vector E can be parametrized by two real vectors e and e as follows E =(e + ie )e i'. (7..66) The square of (7..66) reads E E =(e e +ie e )e i' and it is equivalent to (7..65) for mutually perpendicular real vectors e and e. Then we have e e =, e k =, e k =. (7..67) The lengths of the vectors e a := e a, a =, can be expressed in terms of the amplitudes and phase shift of the components of electric field in a given reference frame. Without loss of generality we can choose two Cartesian versors ˆx and ŷ as vectors being parallel to the plane defined by e and e. The orientation of this versors is determined by a measure aparathus associated with the laboratory. A third versor ẑ is determined by ẑ = ˆx ŷ. The electric field amplitude decomposes as follows E = Ae i ˆx + Be i ŷ (7..68) 377

12 7. ELECTROMAGNETIC WAVES 5 where A, B,, are real-valued quantities. Then E E = A + B (7..69) h E E = AB e i( ) e i( )i ẑ =iab sin( )ẑ. (7..7) 5 where :=. On the other side E E = (e ie ) (e + ie )=e + e (7..7) E E = (e ie ) (e + ie )=i e e =ie e ê ê. (7..7) The vector ê ê is an unit vector. One can always choose both versors ê, ê in a way that ê ê = ẑ. Comparing both decompositions one gets two equations e + e = A + B, e e = AB sin (7..73) 55 which can be cast in the form (e ± e ) = A + B ± AB sin (7..74) 56 The sum and difference of square roots of (7..74) gives e = hp A + B +AB sin + p A + B AB sin i (7..75) 57 e = hp A + B +AB sin p A + B AB sin i (7..76) The expression (7..75) and (7..76) gives respectively the lengths of the major and minor semi-axes of the polarization ellipse. The orientation of the ellipse is given by the angle # between vectors ˆx and ê ˆx ê = cos #, ˆx ê = sin # (7..77) ŷ ê =sin#, ŷ ê = cos #. (7..78) 53 The angle # can be determined from the identity which can be checked immediately Re[(E ê )(E ê )] (7..79) [e i' (e + ie ) ê ][e i' (e ie ) ê ]= ie e. 378

13 7. Electromagnetic waves in non-dispersive dielectric media Figure Plugging (7..68) to (7..79) one gets Then from (7..8) we get Re[(Ae i ˆx ê + Be i ŷ ê )(Ae i ˆx ê + Be i ŷ ê )] = Re[(A cos # + Be i sin #)( A sin # + Be i cos #)] = Re[(B A )sin# cos # + AB(e i cos # e i sin #)] = (A B )sin(#)+ab cos cos(#) (7..8) tan(#) = AB A cos. (7..8) B Let us consider some particular cases.. The linear polarization When = {, }, cos = ± then e = p A + B, e =, tan(#) =± AB A B The complex electric field vector reads E =(e + ie )e i, :=!t k x + ' (7..8) 538 what leads to the physical electric field Re[E] = p A + B cos ê. (7..83) 379

14 7. ELECTROMAGNETIC WAVES At a given space point the electric field oscillates with the frequency! and orientation of the vector Re[E] is fixed in direction of the vector ê. For B =the field oscillate in direction ê = ˆx and for A =it oscillates in direction ê = ŷ. For A = B the vector of the electric field form angle # = /4 with ˆx for =and # = /4 for =.. The circular polarization When = ± and A = B one gets e = A, e = ±B ±A, tan(#) =undetermined. (7..84) 545 The electric field vector reads E = A(ê ± iê )(cos i sin ) (7..85) 546 and then Re[E] =A[cos ê ± sin ê ]. (7..86) where :=!t k x + '. At a given space point the electric field vector rotates with the frequency omega. The rotation is characterized by a positive helicity (anti-clockwise) for = / and by a negative helicity (clockwise) for = /. The amplitude of the vector Re[E] remains constant. 3. The elliptical polarization If non of the listed above cases is present then the electromagnetic wave has elliptic polarization. The electric field vector rotates and oscillates simultaneously. Let us observe that the wave with elliptic polarization Re[E] =Re[(e + ie )(cos i sin )] = e ê cos + e ê sin (7..87) can be interpreted as a superposition of two linearly polarized waves whose planes of polarization are mutually perpendicular. Each linear polarization can be decomposed on a combination of two circular polarizations. Let us define the circular polarization vectors ê ± := ê cos ± ê sin, (7..88) 558 then plugging ê cos = (ê + + ê ), ê sin = (ê + ê ) (7..89) 559 to the formula (7..87) one gets Re[E] =e ê cos + e ê sin = e + e ê + + e e ê. (7..9) 38

15 7. Electromagnetic waves in non-dispersive dielectric media It follows that the elliptically polarized electromagnetic wave can be decomposed either on two linearly polarized waves whose directions of polarization are orthogonal or on two circularly polarized waves with opposite helicities Partially polarized electromagnetic wave In many realistic situations the electromagnetic wave is not perfectly monochromatic. Its frequencies belong to a narrow interval! around some frequency!. A single monochromatic wave is polarized, however, superposition of such waves with different polarizations needs a special treatment. At fixed space point the electric field of such a wave is of the form E = E (t)e i!t (7..9) where the amplitude E (t) is slow-varying function of time. Since the vector of amplitude describes the polarization of the wave then the change of its orientation means that the polarization of the wave changes slowly. In experimental study of polarization one can measure the intensity of light beam that pass through a polarizing filters in dependence on orientation of the filter. The intensity of light is a quadratic function of electric field. For this reson we shall consider the quadratic functions of the electric field components E i (t). Since the auxiliary electric field is a complex field we have the following possibilities E i E j = E i E j e i!t, E i E j = E i E j ei!t, E i E j = E i E j However, whats matter are not actual values of this quantities but rather theirs time average values defined by the expression hf(t)i := T Z T dt f(t). (7..9) The characteristic time scale of variation of amplitudes and the functions e ±i!t are different. Then time averaging over intervals T such that the phase factor oscillates many times whereas the amplitude is essentially unchanged leads to conclusion that terms which depends on dom- inant frequency! gives zero in averaging process. On the other hand, the average values E i E j D E = do not vanish. It follows that properties of partially polarized electro- E i E j magnetic wave are completely characterized by the tensor D J ij := EE i j E. (7..93) Since the vector E is perpendicular to the wave vector k then J ij has only four components. We choose the Cartesian coordinates x, x in the plane perpendicular to the wave vector so 38

16 7. ELECTROMAGNETIC WAVES i, j = {, } in (7..93). The trace of this tensor reads TrĴ = X J ii = E + E = E (7..94) i= and it represent an intensity of the wave (density of the energy flux). This quantity is not related to polarization properties of the wave and therefore one should replaced a tensor (7..93) by another one which contains relative intensities instead of absolute ones. We define the polar- ization tensor ij := J ij TrĴ = D E i E j h E i It follows from definition of the polarization tensor that:. Tr ˆ =, + =,. ˆ =ˆ,, R, =. E (7..95) There are two limit cases: total polarization of the electromagnetic wave and the absence of polarization. When the electromagnetic wave is totally polarized the amplitude vector E = const what means that the time averaging drops out ij = Ei E j E. (7..96) 597 The determinant of the polarization tensor vanishes in such a case det ˆ = E 4 (E E )(E E ) (E E )(E E ) =. (7..97) When the electromagnetic wave is not polarized (natural light beam) the average intensity of the electromagnetic wave is the same in all directions what gives E E = E E = E. (7..98) 5 On the other side the components E (t) and E (t) are not correlated in absence of polarization 5 what leads to vanishing of expressions E E == E E. (7..99) 5 Substituting this results to (7..95) we get the following form of the polarization tensor ij = ij. (7..) 38

17 7. Electromagnetic waves in non-dispersive dielectric media which leads to det ˆ = It follows that the determinant of the polarization tensor vanishes for 54 totally polarized electromagnetic wave and it equals to /4 for absence of polarization. One can 55 define the grade of polarization P [, ] det ˆ = 4 ( P ) (7..) where P =corresponds to absence of polarization and P =represent a maximal grade of polarization. It is convenient to decompose the polarization tensor on its symmetric S ij and anti-symmetric A ij parts defined as follows S ij := ( ij + ji )= ( ij + ij) R (7..) A ij := ( ij ji )= ( ij ij) i " ija I (7..3) 5 where A R, then ij = S ij i " ija (7..4) Polarization tensor for totally polarized wave. Let us study the meaning of components of the polarization tensor in the case of totally polarized electromagnetic wave E = E e i(k x!t) =(E ˆx + E ŷ)eik x e i!t where E = Aei and E = Bei. At given x the expression e ik x is a fixed complex number. It has no influence on the tensor ij because it is an overall phase factor and therefore it does not contribute to =. Plugging this components to (7..96) one gets " # A AB e i ij = A + B AB e i B " # " # A AB cos i AB = A + B AB cos B A + B sin {z } {z } A S ij The parameter A vanishes for = {, }. We have seen that in this case the electromagnetic wave is linearly polarized. When it happens the polarization tensor is symmetric " # A ±AB ij = A + B ±AB B (7..5) 383

18 7. ELECTROMAGNETIC WAVES 56 and it takes the following form for particular cases of linear polarization " # " # " ± # ij = ij = ij = ± (7..6) corresponding respectively to polarizations along the axis x, y and the diagonal one. The circular polarization is obtained for = ± and A = B what gives " # " # ij = i (±). (7..7) In general, the coefficient A has interpretation of degree of circular polarization. Its extremal values A = and positive helicity. and A = +correspond to circularly polarized waves with respectively negative Stokes parameters Going back to our observation that the polarization tensor is a Hermitian matrix we conclude that it can be decomposed on a set of Pauli matrices a and the identity matrix ˆ = [I + a a] (7..8) 55 where the Pauli matrices have the form " # " =, = i i #, 3 = " # and a are termed the Stokes parameters. The determinant of the polarization tensor is related to the polarization degree P (7..). Explicitly " # 4 ( P )= 4 det + 3 i = + i 3 4 [ ( + + 3)] 58 then P = q (7..9) It means that all states with given grade of polarization form a spherical surfaces in the space of Stokes parameters. The states of maximal polarization form a sphere with the radius P =and the state in which polarization is absent correspond to the origin = = 3 =. In order to understand the meaning of the Stokes parameters we consider states P =and express a in terms of parameters A, B and. Since Tr( a b )= ab, Tr( a )= 384

19 7. Electromagnetic waves in non-dispersive dielectric media 534 then a =Tr( a ˆ ). (7..) Plugging to this expression the polarization tensor " A AB (cos i sin ) ˆ = A + B AB (cos + i sin ) B one gets # = AB A + B cos, = AB A + B sin 3 = A B A + B. (7..) The parameter = A, so it describes the grade of circular polarization. For B =we have = =and 3 =+what correspond to polarization in direction of x-axis. Similarly, for A =we have = =and 3 = one gets polarization in direction of y-axis. Then we conclude that 3 describes polarization in directions x and y. Finally in the case A = B and =the wave is polarized in direction of a line which form the angle # = /4 with the axis x. This situation correspond to values of the Stokes parameters = 3 =and =+. If =is replaced by = the parameter 3 takes the value 3 = (polarization in direction of a line which form the angle # = /4 with the axis x). We have shown that for a totally polarized electromagnetic wave P =it holds e + e = A + B, e e = AB sin, tan(#) = AB A B cos. One can express the Stokes parameters in terms of lengths of parameters that characterize the ellipse of polarization = tan(#), = e e 3 e +. (7..) e 548 The parameter and p + 3 are invariant under Lorentz transformations Decomposition of partially polarized wave on polarized and unpolarized components Let us split the tensor J ij = E i E j on component J (n) ij that represent the unpolarized electromagnetic wave and J (p) ij corresponding to the polarized electromagnetic wave component. We have shown that for unpolarized component (n) ij := J (n) ij J (n) = ij ) J (n) ij = J (n) ij (7..3) 385

20 7. ELECTROMAGNETIC WAVES Figure 7.3: The space of Stokes parameters and the meaning of points at the surface of the sphere with the radius P = where J (n) Tr(Ĵ (n) ). For a polarized component the time averaging is redundant J (p) E i(p) E j(p), so the expression J ij J (n) ij = J (p) ij reads ij = J ij J (n) ij = E i(p) E j(p). (7..4) The determinant of the rhs of this expression vanishes according to (7..97). It leads to the equation apple det J ij J (n) ij = (7..5) 386

21 7. Electromagnetic waves in non-dispersive dielectric media 558 where J ij = J ij and J Tr(Ĵ). It constitute an equation which allows to determine the intensity of non polarized component J (n) 559 det " J J (n) J J J J (n) = J [ ] {z } = 4 det ˆ = 4 ( P ) # =(J J (n) )(J J (n) ) J + 4 (J (n) ) J (n) J [ + ] {z } Tr ˆ = h (J (n) ) JJ (n) +( P )J i = (7..6) 56 Since J (n) <J, then the physical solution reads J (n) =( P )J (7..7) The intensity of the polarized component equals to J (p) = PJ as a consequence of decomposition J (p) ij = J ij J (n) ij. One can easily establish connection with the Stokes parameters. The only difference is that the intensity of a polarized component is a fraction of a total intensity i.e. e + e = A + B = PJ, then 3 = tan(#), = e e PJ. (7..8) Decomposition of partially polarized wave on two incoherent elliptically polarized waves From Hermiticity of ˆ it follows that eigenvalues a, a =, of this tensor are real-valued. The eigenvectors n (a) of the polarization tensor are given by two complex versors n (a) n (a) = satisfying equations ij n (a) j = a n (a) i. (7..9) 57 Multiplying this equation by n (a) i a = ij n (a) = J i = J D En i (a) i E > n (a) j and summing over i we get D E EE i j n (a) i n (a) j = D J (E i n (a) i )(E j n(a) j ) E then both eigenvalues are real-valued and positive. The eigenvalues by P. Indeed, the equation a can be parametrized det[ˆ I] =, Tr ˆ {z} + det ˆ = (7..) {z } 4 ( P ) 387

22 7. ELECTROMAGNETIC WAVES has two solutions, = 573 ( ± P ). 574 One can show that eigenvectors are mutually orthogonal. Multiplying the equation with a =by n () and the complex conjugated equation with a =by n () ( ij n () j = n () i /n () i ij n () j = n () i /n () i and subtracting the resultant equations one gets ( ij ji) n () j n () {z } i =( )n () i n () i. (7..) Since 6= it follows that n () n () =. It means that the complex eigenvectors form the orthonormal set n (a) n (b) = ab. (7..) The matrix whose columns are formed by the eigenvectors is an unitary matrix " # h i n U := n () n () U () :=. n () It follows from this definition and the fact that n (a) are eigenvectors of ˆ that " # " # U ˆ U = The last equality reads, ˆ = U ij = n () i n () j + n () i n () U (7..3) j. (7..4) A complex vector amplitude can always be chosen in the way that one of two mutually perpen- dicular components is real whereas the other one is imaginary. Let e and e be two perpendic- ular real vectors. We consider the following decomposition n () := e ê + ie ê (7..5) 585 then the normalization condition n () n () =leads to equation e + e =. The second vector n () is orthogonal to the first one, than substituting n () = ê + ê one gets e + i e =. The solution of the last equation which leads to normalized the second vector reads = ie and = e n () := ie ê + e ê. (7..6) The vectors (7..5) and (7..6) describes two identical ellipses (the same ratio of semiaxes). The major semi-axes of these ellipses form an angle /. The polarization components associated which each ellipse are mutually incoherent. It follows from the fact that in decomposition (7..4) there are no cross terms. 388

23 7. Electromagnetic waves in non-dispersive dielectric media Figure 7.4: The polarization ellipses Energy and momentum flux of electromagnetic waves The energy density u and the momentum flux density S of the electromagnetic field is given by expressions u = [E D + H B] (7..7) S = c [E H] (7..8) where all fields are real-valued. We assume that the continuum medium is linear, homogeneous and isotropic. It follows that D = "E and B = µh. Considering the electromagnetic field as a real part of the complex auxiliary field E C, B C Re E = (E + E ) Re B = (B + B ) (7..9) the expressions (7..7) and (7..8) are substituted by u = apple "(Re E) + 8 µ (Re B) S = c [Re E Re B]. (7..3) 4 µ When fields oscillate quickly with a frequency! the instantaneous values of u and S are less relevant than theirs time average values hui and hsi. Taking a monochromatic wave in the form E = E e i(k x!t) B = B e i(k x!t) (7..3) 389

24 7. ELECTROMAGNETIC WAVES 54 one gets that expressions (Re E) = 4 E +he E i + E = E E = E E (Re B) = 4 B +hb B i + B = B B = B B does not contain oscillating terms because E e i!t =, E e i!t =. Similarly the following expression hre E Re Bi = 4 [he B i + he Bi + he Bi + he B i] {z } {z } = Re [E B ]= Re [E B ] (7..3) 57 also does not contain terms behaving as e ±i!t. Then hui = 6 [E D + H B ] (7..33) 58 hsi = c 8 Re [E H ]. (7..34) 59 We have already shown that Maxwell s equations lead to following algebraic equations E = n ˆk B, B = n ˆk E, E = n B (7..35) where ˆk := k k. Then average value of the energy density reads hui = apple " E + 6 µ B = h "µ i 6 µ n + B = 8 µ B. Similarly one gets average value of the Poynting vector hsi = c apple 8 Re n (ˆk B) µ B The last two expressions are proportional = c 8 µ n Re( B )ˆk = c 8 µ n B ˆk. hsi ˆk hui = c n = v (7..36) 53 and the proportionality coefficient is equal to a velocity of propagation of electromagnetic wave. 39

25 7. Electromagnetic waves in non-dispersive dielectric media Reflection and refraction of light at the surface of interface In current section we shall deal with description of electromagnetic wave on the border of two different homogeneous dielectrics. Such dielectrics are characterized by electric permittivities ", " and magnetic permeabilities µ and µ. The refractive indices are given by n := p " µ and n := p " µ. The surface of contact of two dielectric media is termed surface of interface. Especially simple solution is obtained for an idealized situation i.e when the surface of interface is approximated by an infinite plane. Let ˆn be an unit vector, normal to the surface of interface that point out into medium. As there are not free charges and free currents the fields have to satisfy sourceless Maxwell s equations r D =, r H td = (7..37) r B =, r E + tb = (7..38) 53 and the boundary conditions ˆn (D D )=, ˆn (H H )= (7..39) ˆn (B B )=, ˆn (E E )= (7..4) which requires continuity of normal components D n and B n and continuity of tangent components H t and E t. We choose a z axis as being perpendicular to the plane and oriented in direction of ˆn, i.e. ẑ = ˆn. The incoming electromagnetic waves, characterized by the wave vector k, propagates in medium. We shall denote by k the wave vector of reflected electromagnetic wave (in medium ) and by k the wave vector of transmitted electromagnetic wave (in medium ). For general case of oblique incidence the vectors k and ˆn are not parallel therefore they define plane of incidence. The angles between wave vectors k, k, k and the vector ˆn are denoted by, and. Without loss of generality we choose versor ˆx parallel to the incidence plane (and perpendicular to ẑ). The last versor ŷ is defined as ŷ = ẑ ˆx Relations between the angles of incidence, reflection and refraction The incident wave, reflected wave and transmitted wave are given respectively by E = E e i(k r!t), H = n µ ˆk E, (7..4) E = E e i(k r!t), H = n µ ˆk E, (7..4) E = E e i(k r!t), H = n µ ˆk E. (7..43) 39

26 7. ELECTROMAGNETIC WAVES 536 These fields have to satisfy some continuity conditions at the surface of interface z = ˆn [E + E ] = ˆn E ˆn [H + H ] = ˆn H. 537 The common factor e i!t cancels out on both sides of the equations giving ˆn [E e ik r + E e ik r ] = ˆn E e ik r ˆn [H e ik r + H e ik r ] = ˆn H e ik r (7..44) (7..45) The solution of these conditions cannot depend on coordinates x and y on the surface z =. One can satisfy this condition if it holds e ik r = e ik r = e ik r, where r = x ˆx + y ŷ at the surface of interface. Alternatively k r = k r = k r (7..46) where by assumption there is no component k y i.e. k ŷ =. It follows from (7..46) and this assumption that k ŷ ==k ŷ (7..47) what means that vectors k and k belongs to the plane of incidence. The condition (7..46) reduces to the following one k ˆx = k ˆx = k ˆx (7..48) 545 where k a ˆx = k a cos ( a + /) = k a sin a, a =,,, then k sin = k sin = k sin. (7..49) 546 The dispersion relation can be written in the form! c = k n = k n = k n (7..5) which implies that k = k and k = n n k. The first equality of (7..49) leads to equality of the incidence and the reflection angles sin =sin ) =. (7..5) 549 The second equality in known as the Snell s law sin sin = n n. (7..5) 39

27 7. Electromagnetic waves in non-dispersive dielectric media The Snell law determines the angle of refraction in dependence on the angle of incidence. Note that from sin = n n sin it follows that there are some limitations for the solutions. When n >n then there exist solution for any. However, for n <n there exist a critical value c = arcsin n n and for > c there is no solution. The relation between the incidence angle and the refraction angle is shown in Fig.7.5. Figure 7.5: The refraction angle = arcsin( n n sin ) in dependence on the angle of incidence for n <n, n = n and n >n Conditions for amplitudes The continuity conditions for parallel components of the fields E and H read ˆn [E + E ]=ˆn E (7..53) 557 ˆn [H + H ]=ˆn H. (7..54) Since B = n ˆk E 558 and consequently E = relations ˆk B n we get for E and H = µ B the following 559 H = Z ˆk E, E = Z ˆk H (7..55) q where Z µ n = µ 56 " is called impedance of the medium. The amplitudes of the fields satisfy 56 E = ZH. It can be seen taking a square of any of equations (7..55). 56 The equations (7..53) and (7..54) determine amplitudes of the electric field. The 563 generic monochromatic wave is a superposition of waves described by E perpendicular to the 564 plane of incidence and by E parallel to this plane. 393

28 7. ELECTROMAGNETIC WAVES Fresnel s formulas for electric field vector perpendicular to the pane of incidence 567 We shall consider E as being perpendicular to the plane of incidence. In such a case it is 568 convenient to express H by E according to (7..55). The continuity conditions read ˆn [E + E ] = ˆn E (7..56) ˆn [ˆk E + ˆk E ] = Z Z ˆn [ˆk E ]. (7..57) Figure 7.6: The electric field vector perpendicular to the plane of incidence Since the electric field vectors are perpendicular to the plane of incidence we can choose them in the form E a = E aŷ, a =,,. It follows that the first condition (7..56) gives E + E = E. (7..58) Making use of orthogonality of vectors ˆn E a =and of the fact that ˆn ˆk = the second condition (7..57) in the form ˆn ˆk we get Z (ˆn ˆk )E + Z (ˆn ˆk )E = Z (ˆn ˆk )E (7..59) 573 Multiplying (7..58) by Z (ˆn ˆk ) and adding to (7..59) we get E = Z (ˆn ˆk ) Z (ˆn ˆk )+Z (ˆn ˆk ) E (7..6) 394

29 7. Electromagnetic waves in non-dispersive dielectric media and then E = Z (ˆn ˆk ) Z (ˆn ˆk ) Z (ˆn ˆk )+Z (ˆn ˆk ) E (7..6) One can express these formulas in terms of the angles and. Note that Z Z = µ µ n n = µ µ sin sin, (7..6) where the last equality is true for all angles of incidence if only n <n. Otherwise, it holds only for being smaller then the critical angle. When the solution exist, all three wave vectors are real-valued and have the following expansion on the Cartesian versors ˆk = sin ˆx + cos ẑ ˆk = sin ˆx cos ẑ ˆk = sin ˆx + cos ẑ. 579 Then the formulas take the form E E = Z cos Z cos Z cos + Z cos, E E = Z cos Z cos + Z cos (7..63) 58 or alternatively E E = µ tan µ tan µ tan + µ tan µ =µ = sin( ) sin( + ), (7..64) 58 E E = µ tan µ tan + µ tan µ =µ = cos sin sin( + ). (7..65) These formulas were derived in 88 by Fresnel who analyzed oscillations of light in ether. The result was obtained before a final formulation of Maxwell s equations Each ratio of electric field amplitudes E /E and E /E can be studied either for n <n or n > n. In Fig.7.7 we show these ratios in dependence on the angle of incidence for µ = µ. 587 For n <n the ratio E /E is always negative. It means that electric field vector changes 588 its phase by when wave reflects on the surface of interface. The amplitude of electric field 589 E is always less than E. Both ratios are related by E /E E /E =according to the continuity condition of the parallel (to the surface of interface) electric field component. For a normal incidence the amplitudes take values E E = µ n µ n µ n + µ n, E E = µ n µ n + µ n. (7..66) 395

30 7. ELECTROMAGNETIC WAVES (a) (b) Figure 7.7: The ratios of amplitudes of electric field perpendicular to the plane of incidence for µ = µ. Note that E E E E = For n > n the reflected wave and incident wave have the same phase. However, the refracted wave exist only for < c. This phenomenon is called total internal reflection and we shall comment about it in section Fresnel formulas for electric field vector parallel to the pane of incidence 596 We shall assume H perpendicular to the plane of incidence, namely we choose H = H ŷ. In 597 this case it is more convenient to express the electric field as E = Zˆk H then the continuity 598 conditions read ˆn [ˆk H + ˆk H ] = Z Z ˆn [ˆk H ]. (7..67) ˆn [H + H ] = ˆn H. (7..68) All fields H a are parallel to the vector ŷ. Taking into account that ˆn ˆk = substituting the amplitude ZH = E one gets from (7..67) ˆn ˆk and (ˆn ˆk )E +(ˆn ˆk )E =(ˆn ˆk )E. (7..69) 53 The second condition (7..68) became Z E + Z E = Z E (7..7) 396

31 7. Electromagnetic waves in non-dispersive dielectric media Figure 7.8: The electric field vector perpendicular to the plane of incidence The solution of the last two equations reads E = Z (ˆn ˆk ) Z (ˆn ˆk )+Z (ˆn ˆk ) E (7..7) E = Z (ˆn ˆk ) Z (ˆn ˆk ) Z (ˆn ˆk )+Z (ˆn ˆk ) E (7..7) For real-valued refraction angles (n >n ) and for < in the case n <n the above formulas take the form E E = Z cos Z cos Z cos + Z cos, E E = Z cos Z cos + Z cos. (7..73) 535 Plugging Z Z = µ µ sin sin to the first formula in (7..73) one gets E E = µ sin( ) µ sin( ) µ =µ tan( ) = µ sin( )+µ sin( ) tan( + ), (7..74) where sin( ) sin( ) sin sin( )+sin( ) = The second formula became h i i sinh + h i cos + cos h i = tan( ) tan( + ) E E = 4µ cos sin µ =µ cos sin = µ sin( )+µ sin( ) sin( + ) cos( ). (7..75) 397

32 7. ELECTROMAGNETIC WAVES The ratio of the electric field components corresponding to the incident, reflected and refracted wave are shown in figure Fig.7.9, where µ = µ. For n <n the ratio E /E is positive for < B and negative for > B. The angle B is defined by condition B + = /. According to Snell s law n sin B = n sin = n sin B = n cos B ) tan B = n n. (7..76) The angle B is called the Brewster angle. For the incident angle equal to the Brewster angle (a) (b) Figure 7.9: The amplitudes of electric field parallel to the plane of incidence for µ = µ. There is no reflected wave for the Brewster angle B the reflected electromagnetic wave vanishes. A general electromagnetic wave is a superposi- tion of the wave having both components of the electric field vector - perpendicular and parallel to the plane of incidence. When the wave vector of the incident wave form the Brewster angle with the optic axis then the reflected wave has only a component perpendicular to the plane of incidence. It means that the wave is polarized. This method is used to obtain polarized elec- tromagnetic wave. The Brewster effect is associated with the fact that electromagnetic radiation cannot be emitted in direction of oscillation of electric charges. The incident electromagnetic 53 wave forces electrons of the material to oscillate in direction of the vector of electric field E. 53 For the Brewster angle the refracted wave is emitted in direction perpendicular to direction of propagation of the incident wave. It means that electrons oscillate along the line parallel to direction in which the reflected wave would propagate. However, since there is not radiation emitted in such direction consequently there is no reflected wave. 398

33 7. Electromagnetic waves in non-dispersive dielectric media Total reflection It is convenient to write the formulas that involve and in a way that they are true also for n >n and > c. According to our previous results, all tangent components (to the surface of interface) of the wave vector k a of the incident a =, reflected a =and refracted a = wave must be equal i.e. 533 k sin {z } = k sin = k {z } sin {z } k t =k x k t =k x k t =k x where k ax = k a ˆx = k a (ˆk a ˆx) =k a cos ( a + /). The normal components of the incident 533 and reflected wave are opposite k n = k n. Together with the expression kn = k kt = 533 k one gets k t k t = k sin (7..77) " # kn = k k sin = k n sin. (7..78) n These formulas are valid for any values of the angle and the value of the ratio n /n. The angle can be derived from these formulas substituting k t = k sin and k n = k cos. It gives n sin = n sin (7..79) cos = n n sin. (7..8) Expression (7..79) is just a Snell law. The point is that the angle may or not to have a geometric meaning. In both cases it is determinable from the formula (7..8). An important characteristic of total reflection is given by ratios of amplitudes of electric field vectors. For incident and refracted waves they read 534 E = E e i[k r!t] = E e i[(k xx+k z z)!t] = E e i[k ( x sin +z cos )!t] E = E e i[k r!t] = E e i[(k xx+k z z)!t] = E e i[( k x sin +k z z)!t] (7..8) (7..8) where r = xˆx + zẑ, k x k t = k t and k z k n. For n >n and > c the rhs of (7..8) takes negative values, then cos <. It follows that has no geometric meaning. Let us define " n " sin s := kn = k sin # = k # (7..83) n sin c 399

34 7. ELECTROMAGNETIC WAVES (a) (b) Figure 7.: A total internal reflection where n sin c = n. It follows from this expression that kz = ±is. Plugging this expression to E one gets E = E e sz i[ k x sin!t] where there term e+sz is not allowed because it reads to unlimited grows of amplitude for z! (what is not consistent with conservation of the energy). The inverse of the expression s n s = k sin (7..85) n is denoted by and it is called penetration depth := r = s n n (7..84) e (7..86) sin where we made use of relation between wave number and wave length k = /. It follows from this expression and the formula (7..84) that refracted electromagnetic waves penetrate the second medium only on depth being of the order of maginitude of the wavelength. The phase of the refracted wave is = k x sin!t so the phase velocity of the refracted wave c c/n sin c vp = = = = v. (7..87) = r k sin n sin (n /n ) sin sin where v = c/n is a phase velocity of the plane wave in the second medium. The phase velocity vp is smaller than v because c <. The wave described by (7..84) is not a plane 4

35 7. Electromagnetic waves in non-dispersive dielectric media wave because its amplitude and phase is not constant on the plane perpendicular to the vector of propagation in the second medium. The phase of this wave clearly depends on properties of the medium ( c ) and the angle of incidence. Substituting cos = s n into Fresnel formulas (7..63) and (7..73) one gets where E E E E? k n sin = is k (7..88) = Z cos iz (s/k ) Z cos + iz (s/k ) = ei'? (7..89) = Z cos iz (s/k ) Z cos + iz (s/k ) = ei' k (7..9) tan '? = Z (s/k ) Z cos, tan ' k = Z (s/k ) Z cos, Z Z µ µ n n. (7..9) The parametrization comes from +i Z (s/k ) Z cos +itan '? = = cos '? + i sin '? = ei'? i Z (s/k ) i tan '? cos '? i sin '? e i'? = ei'? Z cos and similarly for ' k. The ratio of the amplitudes has absolute value e i'? =and e i' k = so the magnitude of reflected wave is equal to magnitude of incident wave. The only change is in a phase. It follows that the energy flux of the incident wave is totally reflected. There is no transfer of the energy to the second medium. For this reason the phenomenon is called total reflection. The existence of penetration depth and the reflection of the whole energy do not lead to contradiction if the refracted wave goes back to the first medium. This fact was confirmed experimentally. Moreover, the small distance on which the refracted wave propagate parallel to the surface of interface in the second medium before going back to the first medium was also observed experimentally. Total reflection is responsible for formation of mirages Energy fluxes of reflected and refracted wave The energy flux is represented by time average of the Poynting vector hsi = c 8 µ Re [E B ]= 8 Z E ˆk = 8 Z E ˆk (7..9) 4

36 7. ELECTROMAGNETIC WAVES 5374 which can be decomposed on directions parallel and perpendicular to the surface of interface hsi =(hsi ˆn) ˆn +(hsi ˆt) ˆt (7..93) {z } {z } hsi n hsi t 5375 The reflection coefficient is given as the ratio R := hs i ˆn hs i ˆn = E E (7..94) where ˆk ˆn = ˆk ˆn because =. The reflection coefficient R represent a total flux of energy reflected on the surface of interface. This flux can be split on contribution coming from the wave having the electric field vector perpendicular and parallel to the plane of incidence. It leads to definition of coefficients R? and R k apple apple Z cos Z cos Z cos Z cos R? =, R Z cos + Z cos k = Z cos + Z cos. (7..95) In the case of normal incidence =and for µ = µ the reflection coefficient has the form n n R n =. (7..96) n + n The transmission coefficient is defined as follows T := hs i ˆn hs i ˆn = Z cos Z cos E E (7..97) The perpendicular and parallel transmission coefficients read T? = 4Z Z cos cos (Z cos + Z cos ) T k = 4Z Z cos cos (Z cos + Z cos ) (7..98) For normal incidence and for µ = µ one gets T n = 4n n (n + n ). (7..99) Let us observe that R? + T? =and similarly R k + T k =. The energy conservation requires that it must hold for total coefficients R + T =. (7..) Electromagnetic waves in dispersive dielectric me- dia (Napisac ten rozdzial...) 4

37 7.3 Electromagnetic waves in conducting media Electromagnetic waves in conducting media The constitutive relations in conducting media must include Ohm s law which is relation be- tween an electric strength field E and a current density J. We shall assume that D = "E, B = µh, J = E, = (7.3.) where is electric conductivity and stands for density of free electric charges. We shall consider the homogeneous and non-dispersive medium, " = const, µ = const. The Maxwell s equations take the form r E =, r B "µ te 4 c µe = (7.3.) r B =, r E + tb =. (7.3.3) Relaxation time Let us consider a plane electromagnetic wave in non-perfect conductor. Without loss of generality we can choose a Cartesian versor ê 3 in direction of propagation of the wave. Since the electric field enters to the Maxwell s equation by a current term one has to check what are restrictions on transversality of electromagnetic waves in such a case. For this reason we do not assume that E and B are perpendicular to the wave vector. The behaviour of these fields is determined by Maxwell s equation. Let us observe that r E = ê i (@ i E)=ê 3 (@ 3 E) r E = ê i (@ i E)=ê 3 (@ 3 E) 54 so the Maxwell s equations take the form ê 3 (@ 3 E)=, ê 3 (@ 3 B)= (7.3.4) 543 ê 3 (@ 3 B) "µ te = 4 c µe (7.3.5) ê 3 (@ 3 E)+ tb =. (7.3.6) 544 Taking a scalar product of (7.3.5) with ê 3 one gets ê t E = 4 " ê 3 E (7.3.7) 43

38 7. ELECTROMAGNETIC WAVES Multiplying (7.3.7) by dt and summing with first equation of (7.3.4) multiplied by dx 3 we get ê t Edt 3 Edx 3 = 4 {z } " de ê 3 Edt 547 what gives apple de ê 3 dt + 4 " E = This is in fact equation for longitudinal component of the electric field and can be cast in the form de k dt + E k =, := " 4 (7.3.8) E k := ê 3 E and is a constant with dimension of time and it is called relaxation time. Re- peating these steps for two remainig Maxwell s equations we get db k dt =. Equation (7.3.8) has solution E k (t, x 3 )=E k (,x 3 )e t It follows that in non-perfect conductors there can exist a time dependent longitudinal component of the electric field which decreases with time as e t. The Maxwell s equations allow only for static magnetic longitudinal component B k = const. It leads to conclusion that electromagnetic waves in conductors are transversal. Any time dependent longitudinal electric component vanishes exponentially with time. Note, that a longitudinal component is absent in perfect conductors = ( = ). On the other hand in empty space =( = ) therefore the longitudinal component is static. However, a static electric field cannot be a part of a wave solution so electromagnetic waves in empty space are transversal Dispersion relation The appropriate combination of Maxwell s equations leads to the second order equations for electric and magnetic field. The steps are exactly the same as for non-conductors. Taking rotational of Ampere s-maxwell equation and using other equations one can get equation for magnetic field and similarly acting with rotational on Faraday s law one gets equation for electric field. The resulting equations have the form LE =, LB = (7.3.9) 44

39 7.3 Electromagnetic waves in conducting media where L stands for a linear differential operator L := "µ t +4 µ t r. (7.3.) The first order differential operator appearing in 4 c t plays role of a dissipative term (similarly to diffusion equation). For! the operator L became a d Alembert operator. The fields E = E e i(apple r!t), B = B e i(apple r!t) (7.3.) 543 are solution of (7.3.9) if Le i(apple r!t) =. The last equation gives which gives a dispersion relation apple "µ c! i 4 µ c! + apple e i(apple r!t) = apple = "µ! c apple +i 4 "!. (7.3.) Note that rhs of (7.3.) is a complex number. It means that apple is a complex-valued vector, i.e. apple = apple C. Let us parametrize the complex number apple as apple = k + is, k, s R Plugging this expression to (7.3.) one gets k s = "µ! c, ks = µ! c. (7.3.3) Then plugging s (k) from the second equation into the first one and multiplying by k (s ) we obtain In both cases k 4 "µ! c k s 4 + "µ! c s "µ! " = + c µ! =, (7.3.4) c µ! =. (7.3.5) c 4 "! # (7.3.6) 5439 then k = "µ! c s "! 3 +5 s = "µ! c s "!

40 7. ELECTROMAGNETIC WAVES p 544 where term with must be discarded in both cases because it lads to imaginary solutions 544 for k and s. Finally we obtain k = n p! c 4 s + 4 "! 3 +5 (7.3.7) 544 s = n p! c 4 s + 4 "! 3 5. (7.3.8) Relation between amplitudes Having the form of vector apple one can obtain an algebraic relation between E and B. Let us consider an electromagnetic wave which propagate along the axis x 3. Then taking the wave vector in the form apple =(k + is)ê 3 one gets E = E e sx3 e i(kx3!t) (7.3.9) B = B e sx3 e i(kx3!t). (7.3.) 5447 The Faraday s law ê 3 E + tb =, 3 E = i(k + is)e t B = i!b gives 5448 where B = c! (k + is) ê 3 E. (7.3.) apple = apple e i, apple = p s k + s, = arctan. (7.3.) k 5449 Plugging (7.3.7) and (7.3.8) to these formulas we obtain apple = n! c " + # 4 4 "! (7.3.3) 545 and = 4 arctan. (7.3.4) "! 545 The second expression can be proved as follows. Let us denote tan = s applep u k = 4 p, u + (7.3.5) u + "! 46

41 7.3 Electromagnetic waves in conducting media 545 then tan( ) = sin( ) sin cos = cos( ) = s k s k cos sin = tan tan h pu i h pu i p p u+ u+ = p p u = u+ = [( p u) ] / = p u = 4 "!. p u+ p u+ p u+ =( p u + ) applep u p u Finally we obtain that complex electric and magnetic field amplitudes satisfy B = n " + # 4 4 "! e i ê 3 E. (7.3.6) The factor e i is responsible for time lag which manifests as a mutual phase shift of fields B and E. Since ê 3 E ==ê 3 B then B = n " + # 4 4 E. (7.3.7) "! Limit cases Let us analyze again the equation "µ t E +4 µ te r E = The second order time derivative term c t E has its origin in displacement current whereas the first order time derivative term 4 c t E comes from the conduction current. Each of these terms contribute to dispersion relation through "µ t E! "µ c!, 4 µ te! i 4 µ c!. 546 The ratio of absolute values of these two terms reads 4 µ c! "µ c! = 4 "! = tan( ). (7.3.8) 546 Case 4 "!. 47

42 7. ELECTROMAGNETIC WAVES In this case the effect of conduction current is small comparing with the effect of displacement current. Expanding k in powers of 4 "! one gets k = n p! c 4 = n! c and similarly " + 4 s + 4 "! 4 "! # = n! c = n p! c " + " + "! # 4 "! +...+# +... (7.3.9) s = n p! c 4 s + 4 "! 3 5 " = p n! + 4 c "! +...+( )# = r "µ! 4 p c "! +...= c r µ +... (7.3.3) " q where Z := µ 5466 ". Notice that the imaginary part s = Im(apple) does not depend on the frequency 5467!. For the phase shift is so the fields E and B have approximately the same phase Case 4 "!. In this limit the conduction current dominates. Such situation takes place in metals where /" 8. The conduction current dominates for frequencies less than 7 Hz (microwaves, radio-frequency, light, some range of X-ray). One gets k s c p µ! 547 then apple = p k + s p 4 µ! (7.3.3) c s = arctan arctan() = k 4. (7.3.3) 5473 The fields have relative phase shift = /4 and their amplitudes satisfy 5474 where n takes values of the order of unity whereas r 4 B n "! E (7.3.33) q 4 "! then B E. 48

43 7.4 Waveguides Distribution of electric current in conductors The displacement t D term is irrelevant for good conductors. Then equation r E 4 µ t E = 5477 is equivalent to the following one r J 4 µ t J = (7.3.34) where J = E. The last equation is known as diffusion equation. Since electric field possesses time dependence given by term e i!t then the current J must contain this dependence as well i.e. J = J (r)e i!t. Plugging this form to last equation we obtain r J + J =, 4 µ! := i c (7.3.35) 548 or considering that p i = p ( + i) = +i, = c p µ! s in the limit 4 "! q In SI units = µ!. Let us consider a conductor in a half-space x. The electromagnetic field propagates along the axis x x. For linearly polarized wave one can choose for convenience the axis x 3 in direction of the electric field vector, then J (r) =J 3(x)ê 3. The equation apple d d(x) + J 3 (x) = (7.3.36) 5486 has a solution J 3 (x) =J 3 ()e i x = J 3 ()e x e i x. (7.3.37) We have to reject the second one solution e i x because it leads to non-physical behaviour in conductor x length. The current density vanishes exponentially in the conductor. The characteristic is called skin depth. For instance, a skin depth for an electromagnetic wave with the frequency =4 9 Hz which enters to silver material ( =.58 8 m) has value = p µ = 6 m. In copper ( =.68 8 m) for UV light = 5 Hz skin depth is = 9 m. 49

44 7. ELECTROMAGNETIC WAVES Waveguides Waveguides are formed by conducting tubes filled with some dielectric material which in general posses some magnetic properties. The perpendicular cross section of the waveguide is not necessary a symmetric figure. The electric and magnetic field exist only inside the waveguide E E, B B, D D, H H 5498 and E =, B =, D =, H =. The field intensities and inductions are related by constitutive relations D = "E and B = µh. Figure 7.: A waveguide cross section. (a) (b) Figure 7.: Waveguides (a) and coaxial cable (b) 4