CHAPTER 26 INTERFERENCE - Texas A&M 26 chapter 26 interference and diffraction interference

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Transcript of CHAPTER 26 INTERFERENCE - Texas A&M 26 chapter 26 interference and diffraction interference

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    CHAPTER 26 INTERFERENCE AND DIFFRACTION

    INTERFERENCE

    CONSTRUCTIVE

    DESTRUCTIVE

    YOUNG’S EXPERIMENT

    THIN FILMS

    NEWTON’S RINGS

    DIFFRACTION

    SINGLE SLIT

    MULTIPLE SLITS

    RESOLVING POWER

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    IN PHASE 180 0 OUT OF PHASE

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    OR THE BOOKS DISCRIPTION

    CONSTRUCTIVE INTERFERENCE

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    DESTRUCTIVE INTERFERENCE

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    When two waves arrive at a point where path

    differences from the sources differ by one

    wavelength, two wavelengths, three wavelengths,

    etc. there will be constructive interference.

    Or mathematically:

    �� − �� = �� ℎ��� � = 0,±1,±2,±3,….

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    When two waves arrive at a point where path

    differences from the sources differ by � � ��������ℎ, �� ��������ℎ, �� ��������ℎ, ���. (Or half integer number of wavelengths) there will

    be destructive interference.

    Or mathematically:

    �� − �� = �� + ��! �

    Where ℎ��� � = 0,±1,±2,±3,….

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    Many ways to produce two rays of coherent light.

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    Consider path differences

    "��ℎ #$%%������ = # &$�'

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    Constructive Interference

    # &$�' = �� ℎ��� � = 0,±1,±2,±3,….

    Destructive Interference

    # &$�' = �� + ��! �

    ℎ��� � = 0,±1,±2,±3,….

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    We can obtain the distance ( a bright line is below (or above) the center of the screen.

    Use

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    If ) ≫ #

    tan ' ≈ ()

    Therefore

    ( = ) tan '

    And for the �/0 bright line (1 ≈ ) tan '1

    For small angles

    tan ' ≈ sin '

    Therefore

    (1 ≈ ) sin '1

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    Use this with

    # &$�' = ��

    Or

    &$�'1 = ��#

    To get

    (1 ≈ )��#

    ℎ��� � = 0,±1,±2,±3,…

    This is the equation that Young used to measure

    the wave length of light.

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    As was said many ways of producing two rays of

    coherent light. We have been discussing

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    Can also get two coherent rays from thin films.

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    We will discuss the interference between rays

    coming from the two surfaces on opposite sides of

    a space filled with liquid (thin film) or air.

    First: we need to know that when light reflects

    from a surface sometimes it undergoes a phase

    shift of 180 0 .

    For example if a ray is in air 4� = 1.005and reflects at the surface with water 4� = 1.335back into the air it does not undergo a phase shift.

    But if a ray is in glass 4� = 1.335and reflects at the surface with air 4� = 1.005 back into the glass it undergoes a 180

    0 phase shift.

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    Rule

    If ray goes from low n material and reflects at

    surface of high n material no phase shift.

    If ray goes from high n material and reflects at

    surface with low n material 180 degree phase

    shift.

    6% �7 89:;91 ?@@@@@@@@@@@A �B ��# �B > �7

    No phase shift.

    But

    6% �B 89:;91 ?@@@@@@@@@@@A �7 ��# �B > �7

    180 0

    (half wavelength) phase shift

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    Now consider we have a film as shown

    Assume perpendicular.

    Film thickness where shown �.

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    No phase shift at either surface or phase shift at

    both surfaces

    Constructive interference if

    2� = �� � = 0, 1, 2, 3, …

    But if one surface has phase shift and other does

    not

    Destructive interference.

    Or

    No phase shift at either surface or phase shift at

    both surfaces

    Destructive interference if

    2� = �� + ��! � � = 0, 1, 2, 3, …

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    Example 26.3

    Find separation of interference fringes.

    One reflection (top) – no phase shift other

    (bottom) – phase shift.

    Glass

    Air

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    Equation for destructive interference

    2� = �� � = 0, 1, 2, 3, …

    Where glass touches – destructive interference.

    From figure

    � D =

    ℎ �

    So

    � = Dℎ�

    Therefore

    2 EDℎ� F = ��

    D = � E �2ℎF � = � �� 2ℎ

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    D = � G40.1�54500 D 10IJ �5240.02 D 10IK �5 L

    D = � 41.25 ��5

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    EXAMPLES OF INTERFERENCE PHENOMONO

    Newton’s Rings

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    Nonreflective Coatings

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    DIFFRACTION

    Light goes in straight line.

    Therefore would expect sharp shadow of object on

    a screen.

    But

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    Consider light going through single slit.

    This shows a ray from the top of the slit and one

    from half way down the slit (middle).

    Or looking only at the slit

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    The difference in path distance to the screen is

    "��ℎ M$%%. = �2 sin '

    If the path difference is one half wave length –

    destructive interference.

    Therefore the requirement for a dark fringe is

    � 2 sin ' = ±

    � 2

    sin ' = ± ��

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    If you divide the slit into 4 regions

    If the path difference is one half wave length –

    destructive interference.

    Therefore the requirement for a dark fringe is

    � 4 sin ' = ±

    � 2

    sin ' = ±2 ��

    Becomes a/4

    Becomes O/Q sinθ

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    Divide the slit into more sections and can show

    The requirements for dark fringes are

    sin ' = � �� 4� = ±1,±2,±3,… 5

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    If the wavelength of light is much smaller than the

    slit width (and normally it is) then

    ��� ≪ 1 4���( &����5

    Therefore

    ' ≪ 1 4���( &����5

    And

    sin ' ≈ '

    Therefore for dark fringes

    ' ≈ ��� 4� = ±1,±2,±3,… 5

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    Also for

    Use for distance to screen ) = D

    Then tan ' = ST

    And for the � �ℎ dark band

    tan '1 = (1)

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    If the screen is a distance from the slit such that

    ) ≫ (1

    And it normally is

    Then ' ≪ 1 4���( &����5

    And tan '1 ≈ '1

    Therefore

    '1 ≈ (1)

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    Then using

    ' ≈ ��� 4� = ±1,±2,±3,… 5

    '1 ≈ (1) ≈ �� � 4� = ±1,±2,±3,… 5

    Solve for (1

    (1 ≈ )��� 4� = ±1, ± 3, ± 5,… 5

    This is equation 26.10 in the book.

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    Multiple Slits

    Path difference will be # sin '

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    Constructive interference occurs for

    # sin ' = �� 4� = 0, ±1,±2,±3,… . 5

    More complicated than two slit diffraction

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    Eight Slit

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    More – 16 slits

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    DIFFRACTION GRATINGS

    Prisms refract light different amounts depending

    on the wavelength of the light.

    Note short wavelengths (blue) are refracted more

    than he long wavelengths (red).

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    Diffraction Gratings will do much the same.

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    The equation for the maxima is the same as for

    multiple slits.

    # sin ' = �� 4� = 0, ±1,±2,±3,… . 5

    Note

    sin ' = ��#

    Thus long wavelengths diffracted more than short

    wavelengths. (Opposite to prisim.)

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    X-RAY DIFFRACTION

    First experiments in 1912.

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    Crystal is uniform arrangement of atoms forming

    planes for diffraction.

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    Gives pattern

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    Consider the ato