Light is incident on a glass-air interface from the glass side
(n=1.5), and researchers want to use the evanescent field on the air side (n=1.0) to excite the
molecules adhered to the surface. It is desired that the evanescent field extend a distance δ = 2 µm
into the air side when using light of free-space wavelength of 1 µm.
a. How close to critical angle must the incident beam be? (Give the difference ∆θ = θ1 - θc)
b. Considering that a beam with finite width contains rays with a spread of angles due to diffraction,
how wide must the beam be so that the angular spread is just equal to the difference in angle
found in part a?
To answer these questions, we need to understand some basic principles of optics and the concept of total internal reflection.
Total internal reflection occurs when light travels from a medium with a higher refractive index to a medium with a lower refractive index and the angle of incidence is greater than the critical angle. In this case, light is incident from a glass-air interface, where the refractive indices are given as n1 = 1.5 for glass and n2 = 1.0 for air.
a. To determine how close to the critical angle the incident beam must be, we first need to find the critical angle itself. The critical angle (θc) is given by:
θc = arcsin(n2/n1)
Substituting the values, we have:
θc = arcsin(1.0/1.5) ≈ arcsin(0.6667) ≈ 41.81 degrees
To find the difference ∆θ = θ1 - θc, we need the angle of incidence (θ1) of the incident beam. However, this information is not provided in the question. You would need to know the angle of incidence to calculate the required θ1.
b. When a beam of light has finite width, there is a range of incident angles due to diffraction. To achieve an angular spread equal to the difference in angle found in part a (∆θ), we need to consider the beam's width.
Let's assume that the beam has a Gaussian intensity distribution, which means the intensity is highest at the center and decreases gradually towards the edges.
To calculate the required beam width, we can use a rough approximation by assuming that the angular spread (∆θ) is approximately twice the standard deviation (∆σ) of the Gaussian beam.
∆θ ≈ 2∆σ
The standard deviation (∆σ) is related to the beam width (∆w) by the equation:
∆w ≈ λ/(π∆σ)
where λ is the wavelength of light.
Given that ∆θ = θ1 - θc, you would need the value of θ1 to proceed with calculating the required beam width.
Please note that without additional information, specifically the angle of incidence (θ1) of the incident beam in part a, we cannot provide a specific answer for either question.