# [thin films] (Macleod's) phase-shift upon reflection

Folks,

I'm reading the 3rd edition of Thin Film Optical Filters by Macleod, and I've derived his expression for phase shift upon reflection (page 45). Finally, it lead me to a question. Any help is highly appreciated!

$$\rm E_i + E_r = E_t \qquad (1)$$
$$\rm H_i - H_r = H_t \qquad (2)$$

Rewrite (2) using the definition of tilted optical admittance:

$$\rm \eta_0 E_i - \eta_0 E_r = \eta_1 E_t \qquad (3)$$

Substitute (1) into (3):

$$\rm \eta_0 E_i - \eta_0 E_r = \eta_1 \left( E_i + E_r \right) \qquad (4)$$

Rearrange (4):

$$\rm E_i \left( \eta_0 - \eta_1 \right) = E_r \left( \eta_0 + \eta_1 \right) \qquad (5)$$

Apply the definition of reflection coefficient $$\rho$$:

$$\rho \equiv \rm \frac{E_r}{E_i} = \frac{ \eta_0 - \eta_1}{ \eta_0 + \eta_1} \qquad (6)$$

Replace $$\eta_1$$ by the input optical admittance $$Y$$ in (6):

$$\rho = \frac{ \eta_0 - Y}{ \eta_0 + Y} \qquad (7)$$

By definition, $$Y \equiv C/B$$. Then

$$\rho = \frac{ \eta_0 - C/B}{ \eta_0 + C/B} = \frac{ \eta_0 B - C}{ \eta_0 B + C} \qquad (8)$$

From the general form of the characteristic matrix

$$\begin{bmatrix} \displaystyle B \\ \displaystyle C \end{bmatrix} = \begin{bmatrix} \displaystyle \alpha & \displaystyle i\beta \\ \displaystyle i\gamma & \displaystyle \delta \end{bmatrix} \begin{bmatrix} \displaystyle 1 \\ \displaystyle \eta_m \end{bmatrix} = \begin{bmatrix} \displaystyle \alpha + i\beta \eta _m \\ \displaystyle \delta \eta_m + i \gamma \end{bmatrix} \qquad (9)$$

let $$B = \alpha + i\beta \eta _m$$ and $$C = \delta \eta_m + i \gamma$$ in (8).

(problems previewing all of this latex code in one post - I'll break it up - please wait)

OK. Part II...

Therefore

$$\rho = \frac{\eta_0 \left( \eta_m ^2 \beta^2 + \alpha ^2 \right) - \eta _m ^2 \delta ^2 - \gamma ^2 + i \, 2 \eta_0 \left( \eta_m ^2 \beta \delta - \alpha \gamma \right)}{\eta_0 \left( \eta_m ^2 \beta^2 + \alpha ^2 \right) + 2 \eta_0 \eta_m \left( \alpha \delta + \beta \gamma \right) + \eta _m ^2 \delta ^2 + \gamma ^2} \qquad (10)$$

so

$$\varphi = \arctan \left| \frac{2 \eta_0 \left( \eta_m ^2 \beta \delta - \alpha \gamma \right)}{\eta_0 \left( \eta_m ^2 \beta^2 + \alpha ^2 \right) - \left( \eta _m ^2 \delta ^2 + \gamma ^2 \right)} \right| \qquad (11)$$

By inspection of (11), it follows

$$\varphi = \arctan \left| \frac{\eta_0 \Im \left\{ B C ^{\ast} - CB^{\ast} \right\} }{\eta_0 ^2 BB^{\ast} - CC^{\ast}} \right| \qquad (12)$$

Finally, if $$\eta_0 = \eta_m$$, I arrive at Macleod's expression for phase-shift upon reflection

$$\varphi = \arctan \left| \frac{\eta_m \Im \left\{ B C ^{\ast} - CB^{\ast} \right\} }{\eta_m ^2 BB^{\ast} - CC^{\ast}} \right| \qquad (13)$$

Note: $$\eta_m$$ is the tilted optical admittance of the "substrate" or outgoing medium, whereas $$\eta_0$$ corresponds to the incident medium. They can be made the same, but are not required to always be the same. That's the point. With that in mind, I should say expression (12) is a more appropriate description. Sounds simple, but I can be wrong. Please let me know what you think. Thanks!

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