Problems

Problem 9.1 Phase Contrast Microscopy.

(a) A phase contrast microscope is used to observe a transparent biological cell that introduces a phase shift of $\varphi = 0.2$ radians. For the idealized phase plate and small-phase model in this chapter, calculate the image intensity relative to a unit background intensity.

(b) Explain why a standard bright-field microscope cannot image a pure phase object. What modification does the phase contrast microscope introduce to solve this problem?

Problem 9.2 Confocal Microscopy.

(a) A confocal microscope uses a 50 $\mu$m diameter pinhole in an image plane with magnification $M=60$. For detection wavelength $\lambda=550$ nm and objective numerical aperture $\mathrm{NA}=1.4$, estimate the image-plane Airy-disk diameter $1.22M\lambda/\mathrm{NA}$ and express the pinhole diameter in Airy units. Explain why this number alone does not fix a unique lateral resolution improvement.

(b) Describe how the pinhole in a confocal microscope improves both lateral and axial resolution. What happens to the signal intensity as the pinhole size decreases?

Problem 9.3 Fluorescence Microscopy.

(a) A fluorescent molecule (GFP) has an excitation wavelength of 488 nm and an emission wavelength of 509 nm. Calculate the Stokes shift in wavelength and the energy lost per photon, $hc/\lambda_{\mathrm{exc}}-hc/\lambda_{\mathrm{em}}$.

(b) Explain the role of the dichroic mirror in a fluorescence microscope. Why is it essential for separating excitation and emission light?

Problem 9.4 Scanning Near-Field Optical Microscopy (SNOM).

(a) A SNOM probe has an aperture diameter of 50 nm and operates at $\lambda=500$ nm. Compare this aperture scale with the Rayleigh distance $0.61\lambda/\mathrm{NA}$ for a conventional microscope of $\mathrm{NA}=1.4$. Why can the actual SNOM resolution not be inferred from aperture diameter alone?

(b) Explain the difference between collection mode and excitation mode SNOM. What are the advantages and limitations of each configuration?

Problem 9.5 Resolution Limits.

(a) An optical microscope has a numerical aperture of 1.3 (oil immersion) and operates at $\lambda=550$ nm. Calculate the Rayleigh resolution limit.

(b) A confocal microscope is claimed to have 1.4× better lateral resolution than a conventional microscope. Under the simplified model in which the illumination and detection intensity point-spread functions are identical Gaussians of width $\sigma$ and the pinhole is very small, justify this factor by multiplying the two profiles. State one reason a real instrument might not reach it.