Problem 56 - University Physics with Modern Physics

Question

Reflective Coatings and Herring. Herring and related fish have a brilliant silvery appearance that camouflages them while they are swimming in a sunlit ocean. The silveriness is due to platelets attached to the surfaces of these fish. Each platelet is made up of several alternating layers of crystalline guanine \((n=1.80)\) and of cytoplasm \((n=1.333,\) the same as water), with a guanine layer on the outside in contact with the surrounding water (Fig. \(\mathrm{P} 35.56\) ). In one typical platelet, the guanine layers are 74 nm thick and the cytoplasm layers are 100 \(\mathrm{nm}\) thick. (a) For light striking the platelet surface at normal incidence, for which vacuum wavelengths of visible light will all of the reflections \(R_{1}\) , \(R_{2}, R_{3}, R_{4},\) and \(R_{5},\) shown in Fig. P35.56, be approximately in phase? If white light is shone on this platelet, what color will be most strongly reflected (see Fig. 32.4\() ?\) The surface of a herring has very many platelets side by side with layers of different thickness, so that all visible wavelengths are reflected. (b) Explain why such a "stack" of layers is more reflective than a single layer of guanine with cytoplasm underneath. (A stack of five guanine layers separated by cytoplasm layers reflects more than 80\(\%\) of incident light at the wavelength for which it is "tuned.") (c) The color that is most strongly reflected from a platelet depends on the angle at which it is viewed. Explain why this should be so. (You can see these changes in color by examining a herring from different angles. Most of the platelets on these fish are oriented in the same way, so that they are vertical when the fish is swimming.)

Step-by-Step Solution

Answer

Visible light with wavelengths around 532 nm (green) is strongly reflected. Multiple layers increase reflectivity due to enhanced constructive interference. Viewing angle changes the reflected color by altering the interference conditions.

Step 1: Understanding Reflection and Phase

When light reflects off thin film layers like those in the fish's platelets, constructive interference occurs when the optical path difference is a multiple of the wavelength. This condition is given by \( 2nt = m\lambda \), where \( n \) is the refractive index, \( t \) is the thickness of the layer, \( m \) is an integer, and \( \lambda \) is the wavelength. We use this to find the wavelength for maximum reflection.

Step 2: Calculating Optical Path Difference for Guanine

The guanine layer (refractive index \( n = 1.80 \)) leads to a path difference of \( 2 \times 1.80 \times 74 \, \text{nm} = 266.4 \, \text{nm} \). We want this path difference to be an integer multiple of the wavelength.

Step 3: Applying the Condition for Constructive Interference

For the condition \( 2nt = m\lambda \), plug in the values: \( 266.4 = m \cdot \lambda \), simplifying to \( \lambda = \frac{266.4}{m} \). Within the visible spectrum (400 nm to 700 nm), find suitable wavelengths by choosing values of \( m \).

Step 4: Determining Wavelengths Within Visible Spectrum

Choose \( m \) such that the calculated \( \lambda \) falls into the visible range. For example, if \( m = 1\), \( \lambda = 266.4 \, \text{nm}\) (not visible), try \( m = 2\): \( \lambda = 133.2 \, \text{nm}\) (not visible), go on until \( \lambda \) is visible, like for \( m = 5\), \( \lambda = 532.8 \, \text{nm} \), corresponding to green light.

Step 5: Explaining Higher Reflectivity With Multilayer Stack

Stacking multiple layers increases reflectivity through repeated constructive interference, enhancing the reflected light's intensity, unlike a single reflective layer.

Step 6: Angle Dependence Explanation

The path difference, and thereby the reflected color, changes with viewing angle due to changes in the effective thickness of the layers and the angle of incidence, altering the conditions for constructive interference, as seen in thin-film interference phenomena.

Key Concepts

Constructive Interference

Constructive interference is a key concept in understanding how certain colors are enhanced when light reflects off thin-film structures, like those found in the platelets of fish skin. When light waves reflect off different layers in a structure, they can combine and strengthen each other if their peaks align. This happens when the optical path difference between reflections is an integer multiple of the light's wavelength. In formula terms, this can be expressed as \( 2nt = m\lambda \), where \(n\) is the refractive index, \(t\) is the thickness of the layer, \(m\) is an integer, and \(\lambda\) is the wavelength of light. These aligned waves enhance specific colors, depending on how the structure of the layers interacts with the light. Understanding this concept helps to explain why some colors are more strongly reflected than others.

Refractive Index

The refractive index, denoted as \(n\), is a measure of how much light bends or slows down as it passes through a material. Different materials have different refractive indices, which affects the light's behavior as it interacts with those materials. For instance, in the case of the herring's platelets, guanine has a refractive index of 1.80, while cytoplasm has a refractive index of 1.333. This difference in refractive index between layers of guanine and cytoplasm causes variations in how light is bent and reflected.
Together, these differences in bending lead to different optical path lengths, which are central to the effect of constructive interference. By tuning these refractive indices, the platelets can strongly reflect specific wavelengths, contributing to their silvery appearance and the fish's camouflage.

Optical Path Difference

Optical path difference is a crucial factor in determining how light waves interact when reflecting off layered materials. It refers to the extra distance light travels due to the change in speed when passing through different materials, rather than just traveling through space.
Mathematically, the optical path difference can be calculated using the formula \(2nt\), where \(n\) is the refractive index of a layer, and \(t\) is the material's thickness. In the case of a guanine layer in fish platelets, this means calculating twice the product of its refractive index (\(1.80\)) and its thickness (74 nm), resulting in an optical path difference of 266.4 nm.
The optical path difference determines conditions for constructive interference and, hence, which color from the visible spectrum is enhanced when viewed under normal lighting conditions.

Visible Spectrum

The visible spectrum consists of light wavelengths that are perceptible to the human eye, typically ranging from about 400 nm to 700 nm. This range includes colors like violet, blue, green, yellow, orange, and red. Different wavelengths correspond to different colors, with shorter wavelengths towards the violet end and longer wavelengths towards the red end.
When light interacts with thin film structures, certain wavelengths within the visible spectrum can be enhanced through constructive interference. For fish platelets, this means that the arrangement and properties of guanine and cytoplasm layers manipulate light to strengthen particular wavelengths, resulting in the enhancement of specific colors such as the silvery hue seen in herring.

Reflectivity

Reflectivity refers to a material's ability to reflect light, which is significantly influenced by its structure and composition. In multilayered systems like fish platelets, reflectivity can be greatly enhanced by layering materials with different refractive indices in such a way that maximizes constructive interference.
When multiple layers are stacked, each layer adds to the overall reflection, creating a cascade effect that boosts the intensity of the reflection drastically compared to if there were only a single layer. For instance, multiple guanine layers with cytoplasm spacers reflect more than 80% of incident light at wavelengths they are tuned for. This high reflectivity is critical in creating the bright, appealing silvery appearance seen in specific marine animals, helping them to blend into their environment effectively.