diffraction by a circular aperture

diffraction by a circular aperture

sin(θ_R) = (j_(1, 1) λ)/(2 π a) | θ_R | Rayleigh criterion angle λ | wavelength a | aperture radius (valid in far-field limit (Fraunhofer diffraction))

wavelength | 500 nm (nanometers) aperture radius | 0.01 mm (millimeters)

Rayleigh criterion angle | 1.747° (degrees) = 30.5 mrad (milliradians) = 0.0305 radians = 1 degree 44 arc minutes 50.35 arc seconds

Calculate the Rayleigh criterion angle using the following information: known variables | | λ | wavelength | 500 nm a | aperture radius | 0.01 mm Convert known variables into appropriate units using the following: 1 nm = 1×10^-9 m: 1 mm = 0.001 m: known variables | | λ | wavelength | 1/2000000 m a | aperture radius | 1×10^-5 m The relevant equation that relates Rayleigh criterion angle (θ_R), wavelength (λ), and aperture radius (a) is: sin(θ_R) = (j_(1, 1) λ)/(2 π a) Take the inverse sine of both sides: θ_R = sin^(-1)((j_(1, 1) λ)/(2 π a)) Substitute known variables and constants into the equation: known variables | | λ | wavelength | 1/2000000 m a | aperture radius | 1×10^-5 m constants | | π | pi | 3.14159 j_(1, 1) | first zero of the Bessel function \!\(\*SubscriptBox[ StyleBox["J", FontSlant->"Italic"], "1"]\)(\*StyleBox["x", FontSlant->"Italic"]) | 3.83174 | : θ_R = sin^(-1)((5×10^-7 m×3.83174)/(2×1×10^-5 m×3.14159)) Cancel any units in sin^(-1)((5×10^-7 m×3.83174)/(2×1×10^-5 m×3.14159)) and add the unit rad for angles: θ_R = sin^(-1)((5×10^-7×3.83174)/(2×1×10^-5×3.14159)) rad Evaluate sin^(-1)((5×10^-7×3.83174)/(2×1×10^-5×3.14159)): θ_R = 0.030496 rad Convert 0.030496 rad into degrees using the following: 1 rad = 57.296°: Answer: | | θ_R = 1.747°

Diffraction pattern

(diffraction pattern has circular symmetry and is a function of diffraction angle only)

order of zero | diffraction angle 1 | 1.747° 2 | 3.2° 3 | 4.644° 4 | 6.086° 5 | 7.531° 6 | 8.98° 7 | 10.43° 8 | 11.9° 9 | 13.36° 10 | 14.84° (39 zeros of I_θ for 0° < θ < 90°)