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Modern Optics LabLab 8: Diffraction by Periodic and Non-Periodic Structures

Transmission grating spectrometer:• Measuring and calculating the angular dispersion. • Understanding resolving power.

Reflection grating spectrometer:• Using a machinist scale to measure the laser wavelength.

The far-field (Fraunhofer) diffraction pattern of randomly placed identical particles:

• Measuring the particle size from the diffraction pattern.

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Transmission Grating – Normal Incidence

m

m

a

maAB sin DifferencePath

AB

.....3,2,1,0 ; sin : MaximaIntensity mma m

What determines whether m=positive is above the dashed line or below the dashed line?


Transmission Grating – Oblique Incidence

m

m

a

im aaCDAB sinsin DifferencePath

A

B

.....3,2,1,0 ; sinsin : MaximaIntensity mma im

Our convention: ccw angles are positive; cw angles are negative.

In the example above: both i and m are positive.

i

CD

i


Angular Dispersion

Different colors (wavelengths) of light have their maxima at slightly different transmitted angles given a particular transmission grating and incident angle. (White light gets “broken up” into rainbow colors at the maxima for m 0. The m = 0 maximum remains white.)

Angular dispersion quantifies the change in the (transmitted) angle at which the maxima occur per unit wavelength change:

d

d DispersionAngular m


Calculating the Angular Dispersion

m

m

cosd

d DispersionAngular :be shouldresult final The

a

m


sinsin

. offunction a as ofThink

rule.chain theuse and respect to with sidesboth ateDifferenti

:Hints

m

md

da

d

dim


Resolving Power

m

m

cosd

d DispersionAngular :be shouldresult final The

a

m

slits dilluminate ofnumber N where, Nm :power Resolving

imim Naa

sinsinsinsin m Using

Under which conditions can you resolve the sodium doublet?Hint: You are given and. You can make a (numerical) statement involving N, a, m and i.


White Light Source

Lens (138mm)Screen

Diffraction Grating

VIII.A Measurement of Angular Dispersion with White Light Source

138mm

Lens (48mm)

48mm

Look at “green” light to get some average wavelength. Measure from red to blue (400nm). Calculate the angular dispersion.


VIII.A Measurements with the Helium-Neon Laser

Laser

View from topScreen

m=0

m=1

m=-1

m=2

m=-2

What happens as the grating is rotated?How do the maxima move? Do they?

Laser

Screen

?


VIII.A Wavelength Measurements using i=0 and i=30

View from top

Know what you measure when doing the 30 incident angle measurement! An example:

Laser

Screen

i

m=0

m=+1

m=-1

i

m=-1

m=+1

m=+1

...,2,1,0 ; sinsin mma im

ccw.) isit - example in this positive is :(note :1mFor 111 mmim

cw.) isit - example in this negative is :(note :1mFor 111 mmim

m=-1


...,2,1,0 ; sinsin :Remember mma im

Once you have figure out what m=-1 and m=-1 are, you can calculate the wavelength as follows:

sinsin :1mFor 1 ima

sinsin :1mFor 1 ima

(Subtract the first equation from the second on each side)

sinsinsinsin 11 imim aa

11 sinsin2 mm

a


Reflection Grating Normal Incidence

Laser

m

m

ma sin

maAB sin DifferencePath

.....3,2,1,0 ; sin : MaximaIntensity mma m

A

B

Convention: ccw angles are positive; cw angles are negative.

In this example:m is negative m negative


Reflection Grating Normal Incidence

Laserm=0

m=1

m=-1

m=2

m=-2


Reflection Grating Oblique Incidence

m

A

C

i

Laser

B

D

im aaCDAB sinsin DifferencePath


Convention: ccw angles are positive; cw angles are negative.

In this example:i is positivem is negative.



Laser

m=0

i

m

imim aa 0sinsin :0mfor DifferencePath



Laser

m=0

m=-1

m=1

m=-2

m=2


Let’s rotate the picture to see what we do in VIII.B

m=0

m=1

m=2

Laser

Machinist Scale

d L

Screen

xo

xo

xm

2

20

2

m

2

:) x; (small incidence grazingFor

L

xx

m

d

L

m


Evaluation of Wavelength

md

Lxx

L

xx

m

d

L

mm

220

22

20

2

m

2

2

:) x; (small incidence grazingFor

2mx

m

Get wavelength from slope


Machinist Scale: The “grid” spacing d depends on where the laser hits the scale!

Example:

d


Another example:

d

An example of how not to do it:

d=?


VIII.C Measuring the Size and Shape of Randomly Distributed Small Particles

Look at single slit pattern first. What effect does moving the slit left or right have on the far-field diffraction pattern?


Imagine randomly placed slits of the same width:

Just as we found with the double slit pattern: We would see an intensity variation in the far field diffraction pattern due to each slit (a single slit pattern).

What about the diffraction pattern due to the interaction between the slits (like the double slit pattern for a double slit, or the diffraction grating pattern with regularly placed slits)?


VIII.C Measuring the Size and Shape of Randomly Distributed Small Particles

For identical particles ,the diffraction pattern from each individual object will look the same in the far field.How about the pattern due to the interaction between the objects?What if the objects were regularly spaced in a pattern?

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