A uniform film of TiO₂, 1036 nm thick and having index of refraction 2.62, is spread uniformly over the surface of crown glass of refractive index 1.52. Light of wavelength 520.0 nm falls at normal incidence onto the film from air. You want to increase the thickness of this film so that the reflected light cancels. What is the minimum thickness of TiO₂ that you must add so the reflected light cancels as desired?

검증된 단계별 안내

Step 1: Understand the concept of thin-film interference. Reflected light cancels when destructive interference occurs. This happens when the optical path difference between the two reflected rays is an odd multiple of half the wavelength in the film.

Step 2: Calculate the wavelength of light inside the TiO2 film. The wavelength in the film is given by \( \lambda_{film} = \frac{\lambda_{air}}{n_{film}} \), where \( \lambda_{air} \) is the wavelength in air (520.0 nm) and \( n_{film} \) is the refractive index of TiO2 (2.62).

Step 3: Determine the condition for destructive interference. For destructive interference, the optical path difference must satisfy \( 2t = (m + \frac{1}{2}) \lambda_{film} \), where \( t \) is the thickness of the film, \( m \) is an integer (order of interference), and \( \lambda_{film} \) is the wavelength in the film.

Step 4: Calculate the additional thickness needed. The current thickness of the film is 1036 nm. To achieve destructive interference, find the minimum additional thickness \( \Delta t \) such that the total thickness \( t_{total} = t_{current} + \Delta t \) satisfies the interference condition.

Step 5: Solve for \( \Delta t \). Rearrange the interference equation to isolate \( \Delta t \), ensuring that the total thickness corresponds to the next odd multiple of half the wavelength in the film. Use \( \Delta t = \frac{(m + \frac{1}{2}) \lambda_{film}}{2} - t_{current} \) to find the minimum additional thickness.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Interference of Light

Interference occurs when two or more light waves overlap, resulting in a new wave pattern. In thin films, constructive or destructive interference can happen depending on the path difference between reflected waves. For destructive interference, the path difference must equal an odd multiple of half the wavelength, leading to cancellation of reflected light.

Refractive Index

The refractive index is a dimensionless number that describes how light propagates through a medium. It is defined as the ratio of the speed of light in a vacuum to the speed of light in the medium. In this scenario, the refractive indices of TiO2 and crown glass are crucial for determining how light reflects and refracts at the boundaries, affecting the interference pattern.

Thin Film Thickness

The thickness of a thin film plays a critical role in determining the conditions for interference. For a film to cause destructive interference, its thickness must be adjusted so that the additional path length of light reflected from the top and bottom surfaces leads to a phase shift that results in cancellation. The minimum thickness can be calculated using the wavelength of light and the refractive indices involved.