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Typical variation of refractive index

Frequency

Refractive index Absorption Limiting value = 1

Infrared

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X-ray

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Two Superposed Waves a . cos Ď&#x2030; 1 . t

y Ď&#x2030; 1,Ď&#x2030; 2,k 1,k 2,x,t

k 1 .x

a . cos Ď&#x2030; 2 . t

k 2 .x

Take the angular frequencies to be 10 .Ď&#x20AC; and 11.5 . Ď&#x20AC; , wavevectors to be 10 . Ď&#x20AC; and 12 . Ď&#x20AC; 2 1

0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

1 2

time t=0 2 1

0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

1 2

time t=0.15

The carrier has edged slightly ahead of the envelope.

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Groupv.mcd

1

04/02/99 12:29

10 . Ď&#x20AC; 11.5 . Ď&#x20AC; Phase velocity : = 0.977 10 . Ď&#x20AC; 12 . Ď&#x20AC; 11.5 10 Group velocity: = 0.75 12 10

We expect the envelope to lag behind

2 1

0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

1 2

{/ OTs3:<2e7°Â&#x153;Â&#x17E;Â?Â&#x2122;l>=<ÂĄyÂ&#x153;yRN7'UK4HL7dI}CG0tUe:<Â&#x201A;§728Â&#x201A;§03Ue7_U8OT=<7oCG4a37-Ux U8RN0ÂŤC OT=<s.I8R<7'>4H282eOP72Â&#x17E;4H=N_ 7-=;aH7zT03Â&#x201A;ÂĽ7°4bI4Â&#x192;zÂ&#x20AC;4XIK72Â&#x201E;I8OPL7Hl The carrier wavelength is

2 .Ď&#x20AC; = 0.182 11 . Ď&#x20AC;

By t 0.4 the envelope has travelled to x 0.3 but the carrier has moved to x 0.4 , i.e. the carrier is ahead by half a wavelength.

Â&#x153;Â&#x17E;Â?ÂŤWIs Groupv.mcd

2

04/02/99 12:30

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