Double Boundary Problem Solved Using Fresnel Coefficients

Consider a slab of material sandwiched between two other materials as depicted in Fig. 6.1.

Because there are multiple reflections inside the middle layer, we have dropped the subscripts i, r, and t used in chapter 3 and instead use the symbols
and to indicate forward and backward traveling waves, respectively. Let n1 stand for the refractive index of the middle layer. In preparation for our treatment of many-layer systems, we use n₀ and n₂to represent the indices of the other two regions. For simplicity, we assume that indices are real. As with the single-boundary problem, we are interested in finding the transmitted fields E^(s)2
and E2
^(p) in terms of the incident fields E0
^(s) and E0
^(p). Similarly, we can also find the reflected fields E0^(s) and E0^(p) in terms the incident fields E0^(s)
and E0^(p)
.

Both forward and backward-traveling plane waves exist in the middle material. Our intuition rightly tells us that in this region there are many reflections, bouncing both forward and backwards between the two surfaces. It might therefore seem that there should be an infinite number of fields represented, each corresponding to a different bounce. Fortunately, the forward-traveling plane waves arising from the many bounces in the middle layer all travel in the same direction. Similarly, the backwards-traveling plane waves arising from the many bounces travel in a single direction. Hence, these many fields join neatly into a net forward-moving and a net backwards-moving plane wave field.

As of yet, we do not know the amplitudes and phases of the two resulting plane waves in the middle layer, but we can denote them by E1
^(s) and E1^(s)or by E1
^(p) and E1^(p), separated into their s or p-components, as usual. Similarly, E0^(s) and E0^(p) as well as E2
^(s) and E2
^(p) are understood to include all fields which “leak” through the surfaces on each of the repeated bounces. All of these are included in the overall reflection and transmission of the fields.

Thus, we need not concern ourselves with the infinite number of plane wave fields arising

6.2 Double Boundary Problem Solved Using Fresnel Coefficients 133 from the many bounces; we need only consider the five plane waves depicted in Fig. 6.1.

The fields at the boundaries are connected via the Fresnel coefficients (3.18)–(3.21), which are direct consequences of Maxwell’s equations. At the first surface we define

r_s^0
1 ≡ sin θ1cos θ0− sin θ₀cos θ1

sin θ₁cos θ₀+ sin θ₀cos θ₁ t⁰_s
¹ ≡ 2 sin θ1cos θ0

sin θ₁cos θ₀+ sin θ₀cos θ₁ r_p^0
1 ≡ cos θ1sin θ1− cos θ₀sin θ0

cos θ₁sin θ₁+ cos θ₀sin θ₀ t^0
1_p ≡ 2 cos θ₀sin θ₁

cos θ₁sin θ₁+ cos θ₀sin θ₀

(6.1)

The notation 0
1 indicates the first surface from the perspective of starting on the incident side and propagating towards the middle layer. The coefficients (6.1) are written as though the problem involves only a single interface. They do not take into account any “feedback”

from the second surface.

Similarly, the single-boundary Fresnel coefficients for light approaching the first interface from within the middle layer are

r_s^1
0 = −r^0
1_s

t^1
0_s ≡ 2 sin θ₀cos θ₁ sin θ₀cos θ₁+ sin θ₁cos θ₀ r_p^1
0 = −r⁰_p
¹

t^1
0_p ≡ 2 cos θ1sin θ0

cos θ₀sin θ₀+ cos θ₁sin θ₁

(6.2)

The notation 1
0 indicates connections at the first interface, but from the perspective of beginning inside the middle layer. Finally, the single-boundary coefficients for light approaching the second interface are

r_s^1
2 ≡ sin θ₂cos θ₁− sin θ₁cos θ₂ sin θ2cos θ1+ sin θ1cos θ2

t^1
2_s ≡ 2 sin θ2cos θ1

sin θ2cos θ1+ sin θ1cos θ2

r_p^1
2 ≡ cos θ2sin θ2− cos θ₁sin θ1

cos θ2sin θ2+ cos θ1sin θ1

t^1
2_p ≡ 2 cos θ1sin θ2

cos θ₂sin θ₂+ cos θ₁sin θ₁

(6.3)

The notation 1
2 indicates connections made at the second interface from the perspective of beginning in the middle layer.

Our task is to connect the five plane waves depicted in Fig. 6.1 using the various Fresnel coefficients (6.1)–(6.3). For simplicity, we will consider s-polarized light, but the analysis can be extended to p-polarized light simply by changing the subscripts in the derivation.

We begin at the second interface, which looks like a single-boundary problem (i.e. only one plane wave on the transmitted side). The field E1
^(s) represents the forward-traveling field of

the middle region evaluated at the origin (y, z) = (0, 0), which we arbitrarily define to be located at the first interface. At the second interface, the forward traveling wave is given by E1
^(s)e^ik1
·r, where r = ˆzd and k1
= k₁(ˆy sin θ₁+ ˆz cos θ₁). The transmitted field in the third medium is related to the forward-traveling field of the middle region via

E₂
^(s)= t¹_s
²E₁
^(s)e^{ik1d cos θ1} (6.4) where we have adjusted the phase of the field in (6.4) by k1
· r = k₁d cos θ₁.

Keep in mind that (6.4) represents the connection made at the point (y, z) = (0, d) on the second interface. In the case of the transmitted field, we let E2
^(s) stand for the transmitted field at the point (y, z) = (0, d); its phase is built into its definition. The factor t^1
2_s is the single-boundary Fresnel transmission coefficient at the interface (6.3), and we have used it in a manner consistent with our previous analysis in chapter 3.

We have written (6.4) for s-polarized light. The equation looks the same for p-polarized light; just replace the subscript s with p. Through the remainder of this section and the next, we will continue to economize by writing the equations only for s-polarized light with the understanding that they apply equally well to p-polarized light.

The backward-traveling plane wave in the middle region arises from the reflection of the forward-traveling plane wave in that same region. In this case, the connection using the appropriate Fresnel coefficient gives

E₁^(s)e^{−ik1d cos θ1} = r_s^1
2E₁
^(s)e^{ik1d cos θ1} (6.5) Here again we have chosen to let E0^(s) represent a plane wave field referenced to the ori-gin (y, z) = (0, 0). Therefore, the factor e^{−ik1d cos θ1} is needed at (y, z) = (0, d) (i.e.

r = ˆzd) since the k-vector for the reverse-traveling field in the middle region is k1 = k₁(ˆy sin θ₁− ˆz cos θ₁).

We next connect the two plane waves in the middle region with the incident plane wave.

In this case we must simultaneously connect E1
^(s) with both E0
^(s) and E1^(s) since they each give a contribution:

E₁
^(s) = t^0
1_s E₀
^(s)+ r_s^1
0E₁^(s) (6.6) Since all fields in (6.6) are evaluated at the origin (y, z) = (0, 0), there is no need for any phase factors like in (6.4) or (6.5). The relation (6.6) shows that the forward traveling wave in the middle region arises from both a transmission of the incident wave and a reflection of the backwards-traveling wave in the middle region. (We could also write an expression involving the overall reflected field E0^(s), but we refrain.) In summary, we have used the single-boundary Fresnel coefficients to construct the necessary connections in the double-boundary problem.

We next solve (6.4)–(6.6) to find the final transmitted field in terms of the incident field. We do this by eliminating E1
^(s) and E1^(s) from the expressions. Equation (6.4) can be inverted as follows:

E₁^(s)

= E2
^(s)

t^1
2_s e^{ik1d cos θ1} (6.7)

When this is substituted into (6.5), we obtain

E₁^(s)= r_s^1
2e^{ik1d cos θ1}

t^1
2_s E₂
^(s) (6.8)

6.2 Double Boundary Problem Solved Using Fresnel Coefficients 135 Substitution of (6.7) and (6.8) into (6.6) yields

E2^(s)

t^1
2_s e^{ik1d cos θ1} = t^0
1_s E₀
^(s)+ r_s^1
0r_s^1
2e^{ik1d cos θ1}

t^1
2_s E₂
^(s) (6.9) This can be simplified to

E2
^(s)

E0
^(s)

= t⁰_s
¹t¹_s
²

e^{−ik1d cos θ1} − r^1
0_s r^1
2_s e^{ik1d cos θ1} (6.10)

where the factor

k1d cos θ1 = 2πn1d cos θ1

λvac

(6.11)

represents the phase acquired by either plane wave in traversing the middle region (see (2.24) and (2.26)).

Actually, we are mainly interested in the fraction of the power that emerges through the final surface. As in (3.29), the fraction of power transmitted is given by

T_s^tot= n₂cos θ₂ n₀cos θ₀    

E2
^(s)

E0
^(s)

2

(θ₂ real). (6.12)

Of course the relationship

T_s^tot+ R^tot_s = 1 (6.13)

still applies, but it is convenient for us to compute T_s^tot directly through (6.12) instead of indirectly from R^tot_s .

When the transmitted angle θ₂ is real, we may write the fraction of the transmitted power as

T_s^tot= n2cos θ2

n0cos θ0

|t^0
1_s |²|t^1
2_s |²

|e^{−ik1d cos θ1}− r_s^1
0r_s^1
2e^{ik1d cos θ1}|² (θ2 real) (6.14) in accordance with (6.10) and (6.12). As was mentioned, (6.14) applies equally well to p-polarized light (just change the subscripts). Equation (6.14) is valid also even if the angle θ₁ is complex. Thus, it can be applied to the case of evanescent waves “tunneling” through a gap where θ₀ is beyond the critical angle for total internal reflection from the middle layer.

This will be studied further in section 6.4. Note that even if θ1 is complex, the angle θ2 is still real if the critical angle in the absence of the middle layer is not exceeded.