Diffraction and Resolution - Union Collegeminerva.union.edu/.../slides/diffraction_resolution.pdf · 2004-04-01 · Resolution of Small Apertures Two point sources far from the aperture

Diffraction and Resolution

� Fraunhofer Diffraction Pattern of a Single Slit

� Resolution of Single-Slit and Circular Apertures

� The Diffraction Grating

� Homework

Fraunhofer Diffraction - Qualitative

� Each part of the slit acts as a source of waves.

� Light from one part of the slit can interfere with light fromanother part.

� The resulting intensity on the screen depends on the angle�.

Fraunhofer Diffraction - Quantitative

� If we divide the slit into two parts, waves from the upper halfinterfere destructively with waves from the lower half when�

������ � � � �� ����� � � �

� If we divide the slit into four parts, we get

����� � � ��

Fraunhofer Diffraction - Quantitative (Continued)

� If we divide the slit into six parts, we get

����� � � ��

� Therefore, the general condition for destructive interference is

���� ���������� � � �� � � ��� � � �!� � �#"$"#"&%

Single-Slit Diffraction Example

A slit of width � is illuminated by red light of wavelength � = 650nm. For what value of � will the first minima appear at

�= 15 ' ?

Single-Slit Diffraction Example

A slit of width � is illuminated by red light of wavelength � = 650nm. For what value of � will the first minima appear at

�= 15 ' ?

���� � � � �

� � ������ � � �(% �*)�+�,.- � %

����� � + ' � + �/� - � � " +10 �

Resolution of Small Apertures

� Two point sources far from the aperture each produce a diffrac-tion pattern.

� If the angle subtended by the sources at the aperture is largeenough, the diffraction patterns are distinguishable as shownin Fig. (a).

� If the angle is small, the diffraction patterns can overlap so thatthe sources are not well resolved as shown in Fig. (b).

Condition of Resolution

� The limiting condition of resolution is know as Rayleigh’s cri-terion.

� When the central maximum of the diffraction pattern of onesource falls on the first minimum of the diffraction pattern ofanother source, the sources are said to be resolved.

� We found that the first minimum in a single-slit diffractionpattern occurs when

����� � � �

� Therefore, the limiting angle of resolution (in radians) for aslit of width a is �$24365 � �where we have applied the small angle approximation ����� �87 �

Resolution for a Circular ApertureThe limiting angle of resolution for a circular aperture of diameter9

is � 24365 �/" �/� �9

Well resolved

Just resolved

Not resolved

The Diffraction Grating

� For light incident normally on the plane of the grating withspacing d, the condition for maxima is: ����� ��;<� 3>=�?A@ � � � � , �#��� � �$"#"$">%

Diffraction Grating Interference Pattern

Homework Set 3 - Due Mon. Apr. 5

� Read Sections 27.6 - 27.8

� Answer Questions 27.14 & 27.22� Do Problems 27.19, 27.26, 27.29 & 27.32