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

In summary, the conversation is about a question regarding Macleod's expression for phase shift upon reflection. The conversation includes equations (1) to (13), which involve definitions of tilted optical admittance, reflection coefficient, and characteristic matrix. The person eventually arrives at Macleod's expression for phase-shift upon reflection, which can be simplified to a more appropriate expression. The person is seeking feedback on their understanding of this topic.
  • #1
DivGradCurl
372
0
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!

[tex]\rm E_i + E_r = E_t \qquad (1)[/tex]
[tex]\rm H_i - H_r = H_t \qquad (2)[/tex]

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

[tex]\rm \eta_0 E_i - \eta_0 E_r = \eta_1 E_t \qquad (3)[/tex]

Substitute (1) into (3):

[tex]\rm \eta_0 E_i - \eta_0 E_r = \eta_1 \left( E_i + E_r \right) \qquad (4)[/tex]

Rearrange (4):

[tex]\rm E_i \left( \eta_0 - \eta_1 \right) = E_r \left( \eta_0 + \eta_1 \right) \qquad (5)[/tex]

Apply the definition of reflection coefficient [tex]\rho[/tex]:

[tex] \rho \equiv \rm \frac{E_r}{E_i} = \frac{ \eta_0 - \eta_1}{ \eta_0 + \eta_1} \qquad (6)[/tex]

Replace [tex]\eta_1[/tex] by the input optical admittance [tex]Y[/tex] in (6):

[tex] \rho = \frac{ \eta_0 - Y}{ \eta_0 + Y} \qquad (7)[/tex]

By definition, [tex]Y \equiv C/B[/tex]. Then

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

From the general form of the characteristic matrix

[tex]\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)
[/tex]

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

(problems previewing all of this latex code in one post - I'll break it up - please wait)
 
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  • #2
OK. Part II...

Therefore

[tex]\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) [/tex]

so

[tex]\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)[/tex]

By inspection of (11), it follows

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

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

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

Note: [tex]\eta_m[/tex] is the tilted optical admittance of the "substrate" or outgoing medium, whereas [tex]\eta_0[/tex] 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!
 
Last edited:
  • #3


Hi there! I would like to commend you for your thorough understanding and application of Macleod's expression for phase shift upon reflection. Your derivation is correct and well-explained.

To answer your question, the phase shift upon reflection is an important phenomenon in thin film optics. It is caused by the difference in optical admittance between the incident and reflected waves, which is determined by the characteristic matrix of the thin film.

In simple terms, the phase shift occurs because the thin film acts as a waveguide for the reflected wave, causing it to travel a longer path than the incident wave. This results in a phase difference between the two waves, which can be calculated using the reflection coefficient (ρ) as shown in your derivation.

I hope this helps to clarify your understanding of phase shift upon reflection in thin films. Keep up the good work in your studies!
 

1. What is the Macleod's phase-shift upon reflection?

Macleod's phase-shift upon reflection refers to a phenomenon in which a thin film, such as a layer of metal or dielectric material, causes a phase change in the reflected light. This phase shift can be controlled by adjusting the thickness and refractive index of the thin film.

2. How does Macleod's phase-shift upon reflection affect the reflected light?

The phase shift caused by Macleod's phase-shift upon reflection can change the intensity and polarization of the reflected light. This can be useful in applications such as anti-reflective coatings, where the goal is to minimize the amount of light reflected from a surface.

3. Can Macleod's phase-shift upon reflection be observed in all types of thin films?

Yes, Macleod's phase-shift upon reflection can occur in any type of thin film, as long as it has a non-zero refractive index. This includes films made of metals, dielectric materials, and even biological materials.

4. How is Macleod's phase-shift upon reflection calculated?

The phase shift caused by Macleod's phase-shift upon reflection can be calculated using the Fresnel equations, which describe the reflection and transmission of light at an interface between two materials. The phase shift is determined by the thickness and refractive index of the thin film, as well as the incident angle and wavelength of the light.

5. What are the practical applications of Macleod's phase-shift upon reflection?

Macleod's phase-shift upon reflection has many practical applications in optics and photonics. It can be used to design anti-reflective coatings, optical filters, and other devices that manipulate the phase of light. It is also important in the field of thin film interference, which is used in many optical instruments and technologies.

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