Finding an angular resolution be for light with a wavelength

Question

A space telescope had a diffraction limited angular resolution of 0.08" for light (542 nm). What would the angular resolution be for light with a wavelength of 2.8 μm? Report your answer to three decimal places and include the word arcsec as your unit.

Answer 1

The diffraction limited angular resolution, θ, of a telescope is given by:

 

θ = 1.22λ/d

 

where λ is the wavelength in cm and d is the telescope's diameter in cm.  Let

• θ₁ = the resolution for 542 nm = 0.08 arcsec
• θ₂ = the resolution for 2.8 μm (2800 nm):

 

θ₂/θ₁ = (1.22·2800/d)/(1.22·542/d)

θ₂/(0.08 arcsec) = 2800/542

 

Solve for θ₂.  The answer will have units of arcseconds (").  The longer wavelength (2800 nm) will have  a larger angle of resolution because it diffracts more as it enters the telescope. 

 

Strictly speaking, the formula for angular resolution will give an answer in radians, but since we are taking a ratio (θ₂/θ₁), the conversion factors to arcseconds will cancel out in the same way that the 1.22 and the d canceled out.