Concept explainers
(a)
Interpretation:
A spread sheet should be created to determine the Bragg wavelength for the given information.
Concept introduction:
Couple wave theory will be used to determine the behavior of holographic filters. For a holographic optical element, the Bragg wavelength
Where the grating material’s refractive index is n, grating period is d and the angle of incidence of the radiation beam is given by
Explanation of Solution
Given:
A holographic grating
The grating space is d = 17.1 µm
The refractive index n is =1.53
At angles of incidence from 0-90 degrees.
The spreadsheet for the conversion of
Angle of incidence (θ) in degree | Conversion of θ from degree to radian | Cos θ | Bragg wavelength | |||
0 | 0 | 1 | 52.326 | |||
10 | 0.17453293 | 0.984807753 | 51.53105048 | |||
20 | 0.34906585 | 0.939692621 | 49.17035608 | |||
30 | 0.52359878 | 0.866025404 | 45.31564528 | |||
40 | 0.6981317 | 0.766044443 | 40.08404153 | |||
50 | 0.87266463 | 0.64278761 | 33.63450446 | |||
60 | 1.04719755 | 0.5 | 26.163 | |||
70 | 1.22173048 | 0.342020143 | 17.89654602 | |||
80 | 1.3962634 | 0.173648178 | 9.086314545 | |||
90 | 1.57079633 | 6.12574E-17 | 3.20536E-15 | |||
CellD3=RADIANS(B7) | ||||||
CellF3=COS(D7) | ||||||
CellH3=2*1.53*17.1*F3 |
(b)
Interpretation:
The angle at which the Bragg wavelength is 462 nm should to calculated.
Concept introduction:
Couple wave theory will be used to determine the behavior of holographic filters. For a holographic optical element, the Bragg wavelength
Where the grating material’s refractive index is n, grating period is d and the angle of incidence of the radiation beam is given by
Answer to Problem 7.24QAP
The value of angle of incidence is 89.5°, when the Bragg wavelength
Explanation of Solution
The value of Bragg wavelength is 462 nm i.e. 0.462 µm.
Substituting the value for Bragg wavelength in the as 0.462 µm, the grating space d as 17.1 µm, and the refractive index n as 1.53 in the equation,
Rearranging the equation:
(c)
Interpretation:
The reason why the Bragg wavelength is also said to be “playback wavelength” at times should be determined.
Concept introduction:
Couple wave theory will be used to determine the behavior of holographic filters. For a holographic optical element, the Bragg wavelength
Where the grating material’s refractive index is n, grating period is d and the angle of incidence of the radiation beam is given by
Explanation of Solution
A hologram is constructed and reconstructed. Bragg angle is that angle at which a hologram is reconstructed.
The wavelength obtained from the Bragg angle is called the Bragg Wavelength. This wavelength is the playback wavelength.
(d)
Interpretation:
The historical significance of the Bragg wavelength should be determined.
Concept introduction:
Couple wave theory will be used to determine the behavior of holographic filters. For a holographic optical element, the Bragg wavelength
Where the grating material’s refractive index is n, grating period is d and the angle of incidence of the radiation beam is given by
Answer to Problem 7.24QAP
The Bragg wavelength signifies that the X-rays are waves and thus they have historic importance in changing the concept of X-rays.
Explanation of Solution
The thought that X-rays are waves are the main reason behind the historical significance of the Bragg wavelength. In earlier days it was considered that X-rays are composed up of particles. But then the thought was changed and it was considered and experimented that the X-rays consist of waves.
(e)
Interpretation:
The angular tuning range of the given filer should be determined.
Concept introduction:
Couple wave theory will be used to determine the behavior of holographic filters. For a holographic optical element, the Bragg wavelength
Where the grating material’s refractive index is n, grating period is d and the angle of incidence of the radiation beam is given by
Answer to Problem 7.24QAP
The angular tuning range of the filer is 0° to 40.8°.
Explanation of Solution
Given:
The Bragg wavelength λb is 1550 nm. The grating space d is 535 nm.
The filter is tunable over the wavelength range 2nd to
The refractive index is 1.53.
Thus, the wavelength range is from 1240 nm to 1640 nm.
Substituting in the equation,
Again, substituting the value of λb as 1640 nm:
(f)
Interpretation:
The Bragg wavelength and the wavelength range of the given filter should be determined.
Concept introduction:
Couple wave theory will be used to determine the behavior of holographic filters. For a holographic optical element, the Bragg wavelength
Where the grating material’s refractive index is n, grating period is d and the angle of incidence of the radiation beam is given by
Answer to Problem 7.24QAP
The wavelength range is 490 nm to 647 nm.
Explanation of Solution
Given:
Grating space is 211.5 nm.
Substituting in the equation
Thus, the Bragg wavelength is
The filter is tunable over the wavelength range 2 nd to
The refractive index is 1.53.
The grating space is 211.5 nm.
Thus, the wavelength range is from 490 nm to 647 nm.
(g)
Interpretation:
The applications of tunable holographic filters should be determined.
Concept introduction:
A hologram is an image which appears as three dimensional and it can be seen through bare eye. The technology of making and using of holograms is said to be holography.
Explanation of Solution
In single mode fiber multiplexers, tunable holographic filters are used. The filter is a volume-reflection hologram in dichromated gelatin and showcases high reflection efficiencies with very narrow spectrum of bandwidth. They have been used for position tuned wavelength demultiplexers or channel selection switches in multichannel optical fiber systems. By using an insertion loss between input and output single mode fibers of less than 3 decibel with bandwidth nearly 1% of the wavelength band a device can also be constructed.
(h)
Interpretation:
The refractive index modulation in a given film should be determined.
Concept introduction:
Couple wave theory will be used to determine the behavior of holographic filters. For a holographic optical element, the Bragg wavelength
Where the grating material’s refractive index is n, grating period is d and the angle of incidence of the radiation beam is given by
Answer to Problem 7.24QAP
The refractive index modulation is
Explanation of Solution
Given:
Wavelength = 633 nm
Thickness = 38 µm
The refractive index modulation for a given wavelength and thickness is given by the formula:
Substituting wavelength as 633 nm and the thickness as 38 µm in the above equation:
(i)
Interpretation:
The expression for ideal grating efficiency and its value should be determined.
Concept introduction:
Couple wave theory will be used to determine the behavior of holographic filters. For a holographic optical element, the Bragg wavelength
Where the grating material’s refractive index is n, grating period is d and the angle of incidence of the radiation beam is given by
Explanation of Solution
For the real grating, the refractive index modulation is described by:
Where De is the grating efficiency.
The grating efficiency is expressed in terms of % or fraction. For real grating, when efficiency is expressed in terms of fraction its value will be less than 1.
For ideal grating, the value will be 1, which is 100%.
Substituting De as 1 we get:
Now, substituting the value of p we would get:
(j)
Interpretation:
The efficiency for a given grating should be determined.
Concept introduction:
For the real grating, the refractive index modulation is described by:
Where De is the grating efficiency.
Answer to Problem 7.24QAP
The grating efficiency is 0.79 or 79 %.
Explanation of Solution
Given:
Thickness = 7.5
Refractive index modulation is 0.030
Wavelength = 633 nm
For the real grating, the refractive index modulation is described by:
Where De is the grating efficiency.
Substituting thickness as 7.5 µm, wavelength as 633 nm, and refractive index modulation as 0.030:
Taking sin on both sides:
(k)
Interpretation:
A recipe for holographic films using gelatin, the fabrication of the films and its chemical process should be determined.
Introduction:
Holographic image or patterns are micro embossed over thin plastic films. These are especially used for packaging purpose. Holographic films for fabricating filters are available commercially but these can also be prepared in laboratory using common chemicals like Gelatin.
Explanation of Solution
The recipe for dichromate gelatin for holographic films is:
The gelatin is dissolved in water at 60°C. The potassium dichromate is added to it, then triphenyl dye is added and the ingredients are mixed properly by which they dissolve completely.
In order to fabricate the film, the above solution is then pipetted out on a leveled glass plate, which is then dried for 24 hours.
Gelatin is a
The dried fabricated film is now exposed to an argon laser beam and the diffraction pattern is then measured.
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Chapter 7 Solutions
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