Problem Set 2 Fall 2010

Due 2/10/2010, 12:00 PM Question 1: Optics

Figure 1 shows a simplified model of myopia (nearsighted vision). The image focal plane lies a small distance in front the retina. Point sources in the object plane do not come to a sharp focus on the retina, so the owner of this eyeball sees a blurred image of height HI.

Figure 1: Myopia

Now consider the case shown in Figure 2. A (diverging) corrective lens has been added in the front focal plane of the eye. This lens causes the image to come into sharp focus exactly on the plane of the retina.

Figure 2: Corrected Vision

With the corrective lens in place, is the height of the image on the retina

(a) bigger; (b) smaller; or (c) the same size

as the uncorrected image?

Draw a ray diagram to justify your answer. A ruler will help keep your work tidy. (Hint: will any of the rays stay the same?)

Question 2: Underwater Vision

Image of \{\}Anableps Anableps\{\} from http://www.ryanphotographic.com

The four-eyed fish (Anableps anableps, shown at right) is a surface dweller with the incredible ability to see simultaneously above and below water. Although the fish really has only two eyes, each Anableps eyeball consists of two essentially independent optical systems – one tuned for above water vision and another for below. The above- and below-water corneas, lenses, and retinas each have optical properties adapted to their purpose.

Optical models of the above- and below-water systems are shown below. Assume that the index of refraction (n) for air is 1 and water is 1.33. The fluid filling the eyes is optically similar to water. The lens material has an index of refraction around 1.5 and the cornea's index is about 1.34.

Which of the two systems do you think the fish uses for underwater vision? Why does this system work better below water? Draw a few rays and/or include a couple of bullet points to explain your answer.

HINT: consider how the fish would view a distant object.

\ System A: highly curved cornea flattened lens| System B: flattened cornea highly curved lens|

Question 3: Anti-reflective Coating

Consider an anti-reflective coating such as the kind discussed in lecture and shown in the diagram below. The coating is designed so that there is no reflection whatsoever of 600nm light that is normally incident upon the coating.

The coating is one-quarter wave thick. Assume for simplicity that n0, nl, and ns are numerically close to 1 and that the fundamental physical constants (e.g., n0, nl, ns) of the system are constant across all wavelengths. Under these assumptions, the quarter-wave coating (nl layer) has a thickness of about 600/(4 nl) ~ 600 / 4 = 150 nanometers.

In this design, 600nm light waves that reflect off the nl-ns interface travel exactly one half of a wavelength (300nm) farther than light reflected off the n0-nl interface. As such, reflected, normally-incident light waves with a wavelength of 600nm interfere destructively with incident light and cancel each other out.

What other wavelengths of light will be perfectly transmitted (i.e. not be reflected at all) by the coating?

Question 4: Microscope Resolution

Consider two infinitesimally small point sources separated by a small distance Δr in the object plane of a microscope, as shown in the figure. The objective lens has diameter D. Assume that the tube lens has infinite extent. Assume that all the lenses are in air with an index of refraction of 1.

Each point source produces an Airy disk in the image plane.

According to Lord Rayleigh’s definition, it is possible just barely to resolve two points if the peak of one Airy disk lies at the first diffraction minimum of the other. As discussed in lecture, the first minimum of the Airy disk lies at 1.22 λ / D, where λ is the wavelength of the light and D is the diameter of the aperture. Numerical Aperture (NA) is defined as n sin(θ), where n is the index of refraction and θ is the maximum angle of light collection.

   * In terms of NA, λ, and D, derive an approximate expression for the minimum separation of the two point sources Δr that can be resolved by the Rayleigh criterion. Show your work. (Hint: you may find the small angle approximation useful, but perhaps not completely justified.)
   * What is the theoretical resolution of a 40X, NA = 0.65 objective focused on a sample labeled with a red fluorescent dye (peak emission at 625 nm)?
   * What is the best possible resolution for an air objective?
   * Why are high NA, long working distance objectives expensive?