## Abstract

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Full Article | PDF Article**Journal of the Optical Society of America**- Vol. 22,
- Issue 6,
- pp. 307-332
- (1932)
- •doi: 10.1364/JOSA.22.000307

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- Plane Waves of Light, I, J.O.S.A. 15, 137–161; 1927; and II, J.O.S.A. 16, 1–25; 1928.

[Crossref] - The term “absorbing power” usually implies that the incident beam is in free space.

- The Vectorial Photoelectric Effect in Thin Films of Alkali Metals, H. E. Ives, Phys. Rev. 38, 1209–1218; 1931; The Photoelectric Effect From Thin Films of Alkali Metal on Silver, H. E. Ives and H. B. Briggs, Phys. Rev. 38, 1477–1489; 1931.

[Crossref] - In this and many equations which follow, we replace the velocity of light, which was denoted by c in §11, by its equivalent pΛ0/2π, in order to avoid confusion with the direction cosine c that appears constantly throughout this part of our work.

- The third member of each of these equations is obtained from the second by means of the fundamental identity(63)∣a∣2-∣b∣2=∣a+b∣2Rea-ba+b,which is satisfied by any complex numbers a and b. Re x means the real part of x.There is another identity of a similar kind which we shall have occasion to use later, and which may best be listed here for reference. It is(64)Re (1/a)=∣1/a∣2Re a.

- It need hardly be mentioned that these equations are quite general and therefore apply to any medium provided suitable subscripts are attached. That is, in the lower medium we have q1=2π(N1+iK10)/Λ0, etc., the “sub 1” being affixed to the N and K0 which vary from material to material, but not to Λ0, the “ether wave length” of the light, which does not.

- When the media are both dielectrics, g/g1 reduces to ∊/∊1, and (72) becomes the well-known rule that the normal component of displacement is continuous.If the lower medium is a perfect conductor, g1 is infinite, and hence E1z=0, as it should be.In the more general case, in which both conduction and displacement currents exist, neither of these rules is valid. Instead, the general relation is found from the second of equations (1) to be that σEz+∊∂Ez/∂t must be continuous; for the tangential components of H, and therefore the z-component of curl H, are known to be continuous. As we are dealing with plane waves in which the time factor is e−ipt, this rule requires continuity of (σ−∊ip)Ez; or what amounts to the same thing, of gEz.Finally, we may remark that if the two media are of like permeability, so that μ=μ1, g/g1 is identical with q2/q12. This explains the appearance of the factor q2/q12 in the paper referred to in Footnote 3.

- To establish this, we first note that, by definition,(74)κ/κ1=(c2k1/c12k).Also, it follows from Table 2 thatc2/c12=c2+(r2q2c2/q12c12).We therefore have(75)Reκκ1=c2Rek1k+r2Re(q2c2q12c12k1k+i∊),the term i∊ having no effect on the real part of the expression as long as ∊ is a real member. But we may write the last term in the form(76)r2Re[k1k(q2c2q12c2+i∊qcq1c1μ1μ)],in which form it is easy to give ∊ such a value that the term in parenthesis will be real. To do this, call qc/q1c1=u+iv. Then the parenthesis factors into(u+iv) (u+iv+i∊μ1μ)which will obviously be real and equal to u2+v2=|qc/q1c1|2 provided we set ∊ equal to −2vμ/μ1, as we may do since this value is real. Thus (76) becomesr2∣qc/q1c1∣2Re(k1/k);and hence by (64) and (75),Reκ1κ=c2|κ1κ|2Rek1k+r2|gg1|2Rek1k.

- Certain quantities which measure intrinsic properties of the three substances necessarily have the same values for the Incident Beam and Beam 2, and also for Beams 1 and 3. The index of refraction N, the extinction coefficient K0 and the propagation constant q belong to this class. In the case of such quantities the subscript 1 is used consistently for the film, even when we are talking about the internally reflected Beam 3, and no subscript at all for air, even when dealing with Beam 2.

- The negative sign is explained as follows: Since the beam is directed downward, c, k, and therefore Sz also, are negative. But by “amount of energy” we evidently mean an essentially positive quantity. Hence we must replace the negative numbers c and k by positive numbers −c and −k.

- This is the lamellar reflecting power of the film in the presence of the base. It reduces to the bulk reflecting power of the base when the thickness of the film is made zero (this corresponds to X=1), and to the bulk reflecting power of the film material alone, provided the thickness of the film is made infinite (this corresponds to X=0). As we have said, the negative sign merely reflects the fact that the energy flows downward in the incident beam and upward in the reflected one.

- This is also true even when the base is a dielectric, if total reflection takes place at the lower boundary of the film. Our argument, though nominally carried out for a metallic base, is good for dielectric bases also, whether totally reflecting or not, since real values of k4 merely imply that η=0.

- The physical significance of this assumption is easily seen. As explained in §§4 and 6, the real part of iq1c1ζ measures the amount by which light is attenuated in its travel from the upper to the lower surface of the film. The imaginary part measures the phase change that takes place in the same vertical distance (which is less than the phase change which takes place in the same distance when measured in the direction of propagation). To say that ∊ is small, therefore, means that the film is thin enough to be nearly transparent, and also that it is thin when compared with the wave length of the light in the film (not its ether wave length). In order to meet the latter restriction ζ ought not exceed some such value as Λ/20.In the experimental study with which the present paper is mainly concerned, the film is usually a monomolecular layer of an alkali metal. For such films the conditions are probably met for ether wave lengths greater than 100A.

- That is, thin enough to justify the use of the approximations (85). (See Footnote 13.)

- If we had kept the scale factors in mind, instead of plotting everything in arbitrary units as we have done in Fig. 23, the bulk curve would be just the integral of the lamellar curves with respect to depth.

- The constants (adopted from Duncan’s data) are N1=0.0690; K10=0.815.

The Vectorial Photoelectric Effect in Thin Films of Alkali Metals, H. E. Ives, Phys. Rev. 38, 1209–1218; 1931; The Photoelectric Effect From Thin Films of Alkali Metal on Silver, H. E. Ives and H. B. Briggs, Phys. Rev. 38, 1477–1489; 1931.

[Crossref]

Plane Waves of Light, I, J.O.S.A. 15, 137–161; 1927; and II, J.O.S.A. 16, 1–25; 1928.

[Crossref]

[Crossref]

Plane Waves of Light, I, J.O.S.A. 15, 137–161; 1927; and II, J.O.S.A. 16, 1–25; 1928.

[Crossref]

[Crossref]

In this and many equations which follow, we replace the velocity of light, which was denoted by c in §11, by its equivalent pΛ0/2π, in order to avoid confusion with the direction cosine c that appears constantly throughout this part of our work.

The third member of each of these equations is obtained from the second by means of the fundamental identity(63)∣a∣2-∣b∣2=∣a+b∣2Rea-ba+b,which is satisfied by any complex numbers a and b. Re x means the real part of x.There is another identity of a similar kind which we shall have occasion to use later, and which may best be listed here for reference. It is(64)Re (1/a)=∣1/a∣2Re a.

It need hardly be mentioned that these equations are quite general and therefore apply to any medium provided suitable subscripts are attached. That is, in the lower medium we have q1=2π(N1+iK10)/Λ0, etc., the “sub 1” being affixed to the N and K0 which vary from material to material, but not to Λ0, the “ether wave length” of the light, which does not.

When the media are both dielectrics, g/g1 reduces to ∊/∊1, and (72) becomes the well-known rule that the normal component of displacement is continuous.If the lower medium is a perfect conductor, g1 is infinite, and hence E1z=0, as it should be.In the more general case, in which both conduction and displacement currents exist, neither of these rules is valid. Instead, the general relation is found from the second of equations (1) to be that σEz+∊∂Ez/∂t must be continuous; for the tangential components of H, and therefore the z-component of curl H, are known to be continuous. As we are dealing with plane waves in which the time factor is e−ipt, this rule requires continuity of (σ−∊ip)Ez; or what amounts to the same thing, of gEz.Finally, we may remark that if the two media are of like permeability, so that μ=μ1, g/g1 is identical with q2/q12. This explains the appearance of the factor q2/q12 in the paper referred to in Footnote 3.

To establish this, we first note that, by definition,(74)κ/κ1=(c2k1/c12k).Also, it follows from Table 2 thatc2/c12=c2+(r2q2c2/q12c12).We therefore have(75)Reκκ1=c2Rek1k+r2Re(q2c2q12c12k1k+i∊),the term i∊ having no effect on the real part of the expression as long as ∊ is a real member. But we may write the last term in the form(76)r2Re[k1k(q2c2q12c2+i∊qcq1c1μ1μ)],in which form it is easy to give ∊ such a value that the term in parenthesis will be real. To do this, call qc/q1c1=u+iv. Then the parenthesis factors into(u+iv) (u+iv+i∊μ1μ)which will obviously be real and equal to u2+v2=|qc/q1c1|2 provided we set ∊ equal to −2vμ/μ1, as we may do since this value is real. Thus (76) becomesr2∣qc/q1c1∣2Re(k1/k);and hence by (64) and (75),Reκ1κ=c2|κ1κ|2Rek1k+r2|gg1|2Rek1k.

Certain quantities which measure intrinsic properties of the three substances necessarily have the same values for the Incident Beam and Beam 2, and also for Beams 1 and 3. The index of refraction N, the extinction coefficient K0 and the propagation constant q belong to this class. In the case of such quantities the subscript 1 is used consistently for the film, even when we are talking about the internally reflected Beam 3, and no subscript at all for air, even when dealing with Beam 2.

The negative sign is explained as follows: Since the beam is directed downward, c, k, and therefore Sz also, are negative. But by “amount of energy” we evidently mean an essentially positive quantity. Hence we must replace the negative numbers c and k by positive numbers −c and −k.

This is the lamellar reflecting power of the film in the presence of the base. It reduces to the bulk reflecting power of the base when the thickness of the film is made zero (this corresponds to X=1), and to the bulk reflecting power of the film material alone, provided the thickness of the film is made infinite (this corresponds to X=0). As we have said, the negative sign merely reflects the fact that the energy flows downward in the incident beam and upward in the reflected one.

This is also true even when the base is a dielectric, if total reflection takes place at the lower boundary of the film. Our argument, though nominally carried out for a metallic base, is good for dielectric bases also, whether totally reflecting or not, since real values of k4 merely imply that η=0.

The physical significance of this assumption is easily seen. As explained in §§4 and 6, the real part of iq1c1ζ measures the amount by which light is attenuated in its travel from the upper to the lower surface of the film. The imaginary part measures the phase change that takes place in the same vertical distance (which is less than the phase change which takes place in the same distance when measured in the direction of propagation). To say that ∊ is small, therefore, means that the film is thin enough to be nearly transparent, and also that it is thin when compared with the wave length of the light in the film (not its ether wave length). In order to meet the latter restriction ζ ought not exceed some such value as Λ/20.In the experimental study with which the present paper is mainly concerned, the film is usually a monomolecular layer of an alkali metal. For such films the conditions are probably met for ether wave lengths greater than 100A.

That is, thin enough to justify the use of the approximations (85). (See Footnote 13.)

If we had kept the scale factors in mind, instead of plotting everything in arbitrary units as we have done in Fig. 23, the bulk curve would be just the integral of the lamellar curves with respect to depth.

The constants (adopted from Duncan’s data) are N1=0.0690; K10=0.815.

The term “absorbing power” usually implies that the incident beam is in free space.

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The electric vector in (_{0}=4893 _{0}=0.868. The numbers on the curves denote depths (in cm×10^{−6}) below the surface of the plate.

The electric vector in (a) is normal to, and in (b) parallel to, the plane of incidence.

Optical constants of silver (Minor’s values).

Total electric intensity vs. wave length.

Lamellar absorbing power of silver on silver. Computed from Minor’s constants. For a film of thickness ^{5}

Bulk absorbing power of silver. Computed from Minor’s constants.

The ^{2} sec I vs. angle of incidence.

^{2} sec I.^{2} sec I.

The ^{2} sec I vs. angle of incidence.

^{2} sec I.^{2} sec I.

**Table 3** Reflection and refraction from a film.

**Table 4** Reflection and refraction from a film.

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$$Q=\sigma {E}^{2}/4\pi .$$

$$Q=\frac{\sigma}{4\pi}{{E}_{0}}^{2}({u}^{2}\hspace{0.17em}{\text{cos}}^{2}\hspace{0.17em}pt+{\mathcal{u}}^{2}\hspace{0.17em}{\text{sin}}^{2}\hspace{0.17em}pt).$$

$$\overline{Q}=\frac{p}{2\pi}{\int}_{0}^{2\pi /p}Q\hspace{0.17em}dt=\frac{\sigma}{8\pi}{{E}_{0}}^{2}({u}^{2}+{\mathcal{u}}^{2}),$$

$$\overline{Q}=\frac{\sigma}{8\pi}(\mid {{E}_{x}}^{2}\mid +\mid {{E}_{y}}^{2}\mid +\mid {{E}_{z}}^{2}\mid ).$$

$$\sigma =2N{K}_{0}p/\mu $$

$$\overline{Q}=\frac{N{K}_{0}p}{4\pi \mu}(\mid {{E}_{x}}^{2}\mid +\mid {{E}_{y}}^{2}\mid +\mid {{E}_{z}}^{2}\mid ).$$

$${\mathit{\u0112}}^{2}={\scriptstyle \frac{1}{2}}(\mid {{E}_{x}}^{2}\mid +\mid {{E}_{y}}^{2}\mid +\mid {{E}_{z}}^{2}\mid );$$

$${A}_{I}=\sigma /4\pi =N{K}_{0}p/2\pi \mu .$$

$${A}_{B}=1-\mid {E}_{2}/E{\mid}^{2};$$

$${A}_{B\perp}=1-{\left|\frac{k-{k}_{1}}{k+{k}_{1}}\right|}^{2}={\left|\frac{2k}{k+{k}_{1}}\right|}^{2}Re\frac{{k}_{1}}{k},$$

$${A}_{B\Vert}=1-{\left|\frac{\kappa -{\kappa}_{1}}{\kappa +{\kappa}_{1}}\right|}^{2}={\left|\frac{2\kappa}{\kappa +{\kappa}_{1}}\right|}^{2}Re\frac{{\kappa}_{1}}{\kappa},$$

$$h=2\pi i\mu /{\mathrm{\Lambda}}_{0},$$

$$q=(2\pi /{\mathrm{\Lambda}}_{0})(N+i{K}_{0}).$$

$$k=(N+i{K}_{0})c/\mu $$

$$\kappa =-\mu c/(N+i{K}_{0}).$$

$${\mathfrak{E}}_{y}=(E+{E}_{2}){e}^{iqrx-ipt};$$

$$\frac{{\mathfrak{E}}_{y}}{E}=\frac{2k}{k+{k}_{1}}.$$

$$\frac{{\mathfrak{E}}_{x}}{E}=-\frac{2{\kappa}_{1}}{\kappa +{\kappa}_{1}}\sqrt{1-{r}^{2}},$$

$$\frac{{\mathfrak{E}}_{z}}{E}=-\frac{2\kappa}{\kappa +{\kappa}_{1}}r.$$

$$\frac{{E}_{1z}}{E}=-\frac{2\kappa}{\kappa +{\kappa}_{1}}\frac{g}{{g}_{1}}r.$$

$${E}_{1z}/{\mathfrak{E}}_{z}=g/{g}_{1}.$$

$${A}_{B\perp}={\left|\frac{{\mathfrak{E}}_{y}}{E}\right|}^{2}Re\frac{{k}_{1}}{k};$$

$${A}_{B\Vert}=\left({\left|\frac{{\mathfrak{E}}_{x}}{E}\right|}^{2}+{\left|\frac{g}{{g}_{1}}\right|}^{2}{\left|\frac{{\mathfrak{E}}_{z}}{E}\right|}^{2}\right)Re\frac{{k}_{1}}{k}.$$

$${A}_{L}=1-\frac{\mid {S}_{2z}\mid +\mid {S}_{4z}\mid}{\mid {S}_{z}\mid}.$$

$${A}_{L}=1+\frac{{S}_{2z}}{{S}_{z}}-\frac{{S}_{4z}}{{S}_{z}}.$$

$${S}_{z}=(p{\mathrm{\Lambda}}_{0}/16{\pi}^{2}){[EH]}_{z},$$

$${S}_{z}=(p{\mathrm{\Lambda}}_{0}/16{\pi}^{2})\mid EH\mid c.$$

$${S}_{z}=(p{\mathrm{\Lambda}}_{0}/16{\pi}^{2})\mid {E}^{2}\mid k=-(p{\mathrm{\Lambda}}_{0}/16{\pi}^{2})\mid {H}^{2}\mid \kappa ,$$

$$-p{\mathrm{\Lambda}}_{0}k/8{\pi}^{2}=-p{\mathrm{\Lambda}}_{0}Nc/8{\pi}^{2}\mu .$$

$${S}_{2z}=-(p{\mathrm{\Lambda}}_{0}/16{\pi}^{2})\mid {{E}_{2}}^{2}\mid k=(p{\mathrm{\Lambda}}_{0}/16{\pi}^{2})\mid {{H}_{2}}^{2}\mid \kappa .$$

$$\frac{{S}_{2z}}{{S}_{z}}=-{\left|\frac{(k-{k}_{1})({k}_{1}+{k}_{4}){X}^{2}+(k+{k}_{1})({k}_{1}-{k}_{4})}{(k+{k}_{1})({k}_{1}+{k}_{4}){X}^{2}+(k-{k}_{1})({k}_{1}-{k}_{4})}\right|}^{2}.$$

$$\frac{{S}_{2z}}{{S}_{z}}=-{\left|\frac{(\kappa -{\kappa}_{1})({\kappa}_{1}+{\kappa}_{4}){X}^{2}+(\kappa +{\kappa}_{1})({\kappa}_{1}-{\kappa}_{4})}{(\kappa +{\kappa}_{1})({\kappa}_{1}+{\kappa}_{4}){X}^{2}+(\kappa -{\kappa}_{1})({\kappa}_{1}-{\kappa}_{4})}\right|}^{2}.$$

$${{H}_{4}}^{\prime}{{l}_{4}}^{\prime}/{{E}_{4}}^{\prime}=-i{q}_{4}{c}_{4}/{h}_{4}=-{k}_{4}.$$

$${k}_{4}=R{e}^{i\eta},$$

$$\begin{array}{l}{S}_{4z}=(p{\mathrm{\Lambda}}_{0}/16{\pi}^{2})\mid {{E}_{4}}^{\prime}{{H}_{4}}^{\prime}{{l}_{4}}^{\prime}\mid \text{cos}\hspace{0.17em}\eta \\ =(p{\mathrm{\Lambda}}_{0}/16{\pi}^{2})\mid {{E}_{4}}^{\prime}{\mid}^{2}R\hspace{0.17em}\text{cos}\hspace{0.17em}\eta \\ =(p{\mathrm{\Lambda}}_{0}/16{\pi}^{2})\mid {{E}_{4}}^{\prime}{\mid}^{2}Re\hspace{0.17em}{k}_{4}.\end{array}$$

$$\frac{{S}_{4z}}{{S}_{z}}={\left|\frac{4k{k}_{1}X}{(k+{k}_{1})({k}_{1}+{k}_{4}){X}^{2}+(k-{k}_{1})({k}_{1}-{k}_{4})}\right|}^{2}Re\hspace{0.17em}({k}_{4}/k).$$

$$\frac{{S}_{4z}}{{S}_{z}}={\left|\frac{4\kappa {\kappa}_{1}X}{(\kappa +{\kappa}_{1})({\kappa}_{1}+{\kappa}_{4}){X}^{2}+(\kappa -{\kappa}_{1})({\kappa}_{1}-{\kappa}_{4})}\right|}^{2}Re\hspace{0.17em}({\kappa}_{4}/\kappa ).$$

$$\begin{array}{l}{A}_{L\perp}=2[\mid ({k}_{1}+{k}_{4})X+({k}_{1}-{k}_{4}){\mid}^{2}\mid k(X+1){\mid}^{2}Re\frac{{k}_{1}(X-1)}{k(X+1)}\\ +\hspace{0.17em}\mid ({k}_{1}+{k}_{4})X-({k}_{1}-{k}_{4}){\mid}^{2}\mid k(X-1){\mid}^{2}Re\frac{{k}_{1}(X+1)}{k(X-1)}]\\ \xf7\hspace{0.17em}\mid (k+{k}_{1})({k}_{1}+{k}_{4}){X}^{2}+(k-{k}_{1})({k}_{1}-{k}_{4}){\mid}^{2}.\end{array}$$

$$\u220a=i{q}_{1}{c}_{1}\zeta ={k}_{1}{h}_{1}\zeta ,$$

$$X\doteq 1,\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}X-1\doteq \u220a,\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}X+1\doteq 2;$$

$${A}_{L\perp}\doteq {\left|\frac{2k}{k+{k}_{4}}\right|}^{2}\left[Re\frac{{k}_{1}\u220a}{k}+{\left|\frac{{k}_{4}\u220a}{{k}_{1}}\right|}^{2}Re\frac{{k}_{1}}{k\u220a}\right].$$

$${A}_{L\Vert}\doteq {\left|\frac{2\kappa}{\kappa +{\kappa}_{4}}\right|}^{2}\left[Re\frac{{\kappa}_{1}\u220a}{\kappa}+{\left|\frac{{\kappa}_{4}\u220a}{{\kappa}_{1}}\right|}^{2}Re\frac{{\kappa}_{1}}{\kappa \u220a}\right].$$

$$k=Nc/\mu $$

$$\begin{array}{c}Re({k}_{1}/k\u220a)=0,\\ Re({k}_{1}\u220a/k)=-4\pi \mu {N}_{1}{K}_{10}\zeta /c{\mu}_{1}{\mathrm{\Lambda}}_{0}N.\end{array}$$

$${A}_{L\perp}\doteq \frac{\left(\frac{{N}_{1}{K}_{10}p}{2\pi {\mu}_{1}}\right)\left({\left|\frac{2k}{k+{k}_{4}}\right|}^{2}\right)\zeta}{\left(\frac{N{\mathrm{\Lambda}}_{0}p\hspace{0.17em}\text{cos}\hspace{0.17em}I}{8{\pi}^{2}\mu}\right)}.$$

$${A}_{L\Vert}\doteq {\left|\frac{2\kappa}{\kappa +{\kappa}_{4}}\right|}^{2}Re\frac{{\kappa}_{1}\u220a}{\kappa}+{\left|\frac{2{\kappa}_{4}}{\kappa +{\kappa}_{4}}\right|}^{2}Re\frac{\kappa \u220a}{{\kappa}_{1}}.$$

$$Re\frac{{\kappa}_{1}\u220a}{\kappa}={r}^{2}{\left|\frac{g}{{g}_{1}}\right|}^{2}\left(\frac{{N}_{1}{K}_{10}p}{2\pi {\mu}_{1}}\right)\left(\frac{8{\pi}^{2}\mu}{N{\mathrm{\Lambda}}_{0}p\hspace{0.17em}\text{cos}\hspace{0.17em}I}\right)\zeta ,$$

$$Re\frac{\kappa \u220a}{{\kappa}_{1}}=(1-{r}^{2})\left(\frac{{N}_{1}{K}_{10}p}{2\pi {\mu}_{1}}\right)\left(\frac{8{\pi}^{2}\mu}{N{\mathrm{\Lambda}}_{0}p\hspace{0.17em}\text{cos}\hspace{0.17em}I}\right)\zeta .$$

$${A}_{L\Vert}\doteq \frac{\left(\frac{{N}_{1}{K}_{10}p}{2\pi {\mu}_{1}}\right)\left({\left|\frac{2{\kappa}_{4}}{\kappa +{\kappa}_{4}}\right|}^{2}(1-{r}^{2})+{\left|\frac{g}{{g}_{1}}\right|}^{2}{\left|\frac{2\kappa}{\kappa +{\kappa}_{4}}\right|}^{2}{r}^{2}\right)\zeta}{\left(\frac{N{\mathrm{\Lambda}}_{0}p\hspace{0.17em}\text{cos}\hspace{0.17em}I}{8{\pi}^{2}\mu}\right)}.$$

$${A}_{L}\doteq \frac{\left(\frac{{N}_{1}{K}_{10}p}{2\pi {\mu}_{1}}\right)\left({\left|\frac{{\mathfrak{E}}_{x}}{E}\right|}^{2}+{\left|\frac{{\mathfrak{E}}_{y}}{E}\right|}^{2}+{\left|\frac{g}{{g}_{1}}\right|}^{2}{\left|\frac{{\mathfrak{E}}_{z}}{E}\right|}^{2}\right)\zeta}{\left(\frac{N{\mathrm{\Lambda}}_{0}p\hspace{0.17em}\text{cos}\hspace{0.17em}I}{8{\pi}^{2}\mu}\right)},$$

$$\mid {\mathfrak{E}}_{\perp}/E{\mid}^{2}\text{sec}\hspace{0.17em}I=\mid {\mathfrak{E}}_{y}/E{\mid}^{2}\text{sec}\hspace{0.17em}I,\mid {\mathfrak{E}}_{\Vert}/E{\mid}^{2}\text{sec}\hspace{0.17em}I=(\mid {\mathfrak{E}}_{x}/E{\mid}^{2}+\mid {\mathfrak{E}}_{z}/E{\mid}^{2})\hspace{0.17em}\text{sec}\hspace{0.17em}I$$

$$\begin{array}{l}\frac{{E}_{1}}{E}=\frac{2k({k}_{1}+{k}_{4}){X}^{2}}{(k+{k}_{1})({k}_{1}+{k}_{4}){X}^{2}+(k-{k}_{1})({k}_{1}-{k}_{4})},\\ \frac{{E}_{2}}{E}=\frac{(k-{k}_{1})({k}_{1}+{k}_{4}){X}^{2}+(k+{k}_{1})({k}_{1}-{k}_{4})}{(k+{k}_{1})({k}_{1}+{k}_{4}){X}^{2}+(k-{k}_{1})({k}_{1}-{k}_{4})},\\ \frac{{E}_{3}}{E}=\frac{2k({k}_{1}-{k}_{4})}{(k+{k}_{1})({k}_{1}+{k}_{4}){X}^{2}+(k-{k}_{1})({k}_{1}-{k}_{4})},\\ \frac{{{E}_{4}}^{\prime}}{E}=\frac{4k{k}_{1}X}{(k+{k}_{1})({k}_{1}+{k}_{4}){X}^{2}+(k-{k}_{1})({k}_{1}-{k}_{4})}.\end{array}$$

$$\begin{array}{l}\frac{{H}_{1}}{H}=\frac{2\kappa ({\kappa}_{1}+{\kappa}_{4}){X}^{2}}{(\kappa +{\kappa}_{1})({\kappa}_{1}+{\kappa}_{4}){X}^{2}+(\kappa -{\kappa}_{1})({\kappa}_{1}-{\kappa}_{4})},\\ \frac{{H}_{2}}{H}=\frac{(\kappa -{\kappa}_{1})({\kappa}_{1}+{\kappa}_{4}){X}^{2}+(\kappa +{\kappa}_{1})({\kappa}_{1}-{\kappa}_{4})}{(\kappa +{\kappa}_{1})({\kappa}_{1}+{\kappa}_{4}){X}^{2}+(\kappa -{\kappa}_{1})({\kappa}_{1}-{\kappa}_{4})},\\ \frac{{H}_{3}}{H}=\frac{2\kappa ({\kappa}_{1}-{\kappa}_{4})}{(\kappa +{\kappa}_{1})({\kappa}_{1}+{\kappa}_{4}){X}^{2}+(\kappa -{\kappa}_{1})({\kappa}_{1}-{\kappa}_{4})},\\ \frac{{{H}_{4}}^{\prime}}{H}=\frac{4\kappa {\kappa}_{1}X}{(\kappa +{\kappa}_{1})({\kappa}_{1}+{\kappa}_{4}){X}^{2}+(\kappa -{\kappa}_{1})({\kappa}_{1}-{\kappa}_{4})}.\end{array}$$

$$\mid a{\mid}^{2}-\mid b{\mid}^{2}=\mid a+b{\mid}^{2}Re\frac{a-b}{a+b},$$

$$Re\hspace{0.17em}(1/a)=\mid 1/a{\mid}^{2}Re\hspace{0.17em}a.$$

$$\kappa /{\kappa}_{1}=({c}^{2}{k}_{1}/{{c}_{1}}^{2}k).$$

$${c}^{2}/{{c}_{1}}^{2}={c}^{2}+({r}^{2}{q}^{2}{c}^{2}/{{q}_{1}}^{2}{{c}_{1}}^{2}).$$

$$Re\frac{\kappa}{{\kappa}_{1}}={c}^{2}Re\frac{{k}_{1}}{k}+{r}^{2}Re\left(\frac{{q}^{2}{c}^{2}}{{{q}_{1}}^{2}{{c}_{1}}^{2}}\frac{{k}_{1}}{k}+i\u220a\right),$$

$${r}^{2}Re\left[\frac{{k}_{1}}{k}\left(\frac{{q}^{2}{c}^{2}}{{{q}_{1}}^{2}{c}^{2}}+i\u220a\frac{qc}{{q}_{1}{c}_{1}}\frac{{\mu}_{1}}{\mu}\right)\right],$$

$$(u+iv)\hspace{0.17em}\left(u+iv+i\u220a\frac{{\mu}_{1}}{\mu}\right)$$

$${r}^{2}\mid qc/{q}_{1}{c}_{1}{\mid}^{2}Re({k}_{1}/k);$$

$$Re\frac{{\kappa}_{1}}{\kappa}={c}^{2}{\left|\frac{{\kappa}_{1}}{\kappa}\right|}^{2}Re\frac{{k}_{1}}{k}+{r}^{2}{\left|\frac{g}{{g}_{1}}\right|}^{2}Re\frac{{k}_{1}}{k}.$$

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