The film thickness of (Lamda / 4) means the beam of light wave (in this case beam "b") will go through an extra optical path of 2 x (Lamda/4) = (Lamda/2) up on reflection from the second surface before it goes through optical interference with the first beam or "a". Therefore, the optical interference involves "a" and "b", which are out of phase by (Lamda/2), or destructive interference. This is the reason for being anti-reflection coating for the reflected beam. However, for two parallel beams when both are trespassing through the thin film of any thickness, they will still be i phase, so they won't cancel each other for the transmitted beam.

Bahram Roughani Thank you so much for your answer. I probably understand your explanation, but I wish I can ask you about more detail.

Below is updated figure, if there is not destructive interference between the transmission beam "c"&"d" ? Why does them have same phase?

You're correct that an anti-reflective coating works by causing destructive interference of reflected light. This process increases the transmission by decreasing the amount of light that is reflected. However, it seems there is a misunderstanding of how this process affects the transmitted light.

When light encounters a boundary between two different materials, some is reflected, and some is transmitted. In the case of an anti-reflective coating, the coating is designed so that the reflected light from the top surface of the coating and the reflected light from the boundary between the coating and the surface beneath it interfere destructively, effectively cancelling each other out. This decreases the amount of reflected light.

This destructive interference of reflected light does not decrease the amount of transmitted light. Rather, it increases it. This is because the total amount of light must be conserved. If less light is being reflected, then more must be transmitted. Therefore, an anti-reflective coating increases the transmission of light through the coated surface by decreasing the reflection.

To address the part about transmission interference: the key here is phase shift upon reflection versus transmission. For normal incidence, there is a phase shift of 180 degrees upon reflection from a medium of higher refractive index, but no phase shift for transmission. Therefore, in the case of an anti-reflective coating, the two reflected waves from the front and back surfaces of the film undergo destructive interference, while the transmitted waves do not. They constructively interfere to give higher transmission.

Hence, it's not quite accurate to say that the transmitted waves destructively interfere in the same way that the reflected waves do, due to the phase shift differences in reflection and transmission. As a result, the transmission of light increases, as less light is reflected away due to the destructive interference occurring with reflection.

The best example of this is eyeglasses with an anti-reflective (AR) coating.

Without an AR coating, light can reflect off the surfaces of the eyeglasses, which can be distracting and reduce the overall clarity of vision. This reflected light can come from any light source, including computer screens, overhead lighting, or sunlight.

When an AR coating is applied to the lenses of the eyeglasses, the coating is engineered to be a quarter of the wavelength of the light to be eliminated. This means that when light hits the lens, it's reflected off both the outer surface of the AR coating and the boundary between the AR coating and the lens surface beneath it. Because of the carefully engineered thickness of the AR coating, these two reflected light waves interfere destructively and cancel each other out, resulting in very little reflected light.

This destructive interference for reflection does not reduce the transmitted light - it increases it. The transmitted light waves do not destructively interfere because of the phase shift differences in reflection and transmission I mentioned in the previous response. So, more light can pass through the lens to the eye, which improves vision clarity and reduces distractions from reflected light.

The outcome is that you get clearer vision with less glare, as more of the light can transmit through the lens, rather than being reflected away.

Another example can be solar panels. Solar panels also use anti-reflective coatings to ensure that as much light as possible is absorbed by the panel and converted into electricity, rather than being reflected away. By minimizing the reflection, the transmission of light into the panel increases, boosting its efficiency.

In both these examples, the principle is the same - the anti-reflective coating reduces the amount of reflected light through destructive interference, which in turn increases the transmission of light.