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Wave-Particle Duality
“Classical”: Also known as the wave theory of light. Proponents – Young & Maxwell. Concept of light interacting with light.
“Quantum”: Also known as the particle theory of light. Proponents – Newton & Einstein. Albert Einstein – Photoelectric Effect (“Photons”: Energy Packets). “Quantum” (From Planck) Energy is hc/λ or E = hυ. Concept of light interacting with materials.
Speed of light can be determined empirically (by experiment) And by solving Maxwell’s equations. Both yield the same answer.
James Clerk Maxwell – Existence of TEM waves (c = 3 x 10 8 m/s). Maxwell combined the laws of Faraday, Gauss, and Ampere
EM Spectrum
(Wavelength image from Universe by Freedman and Kaufmann.)
Chapter 1 – Wave Nature of Light
The wave description reconciles phenomena such as interference and diffraction of light.
Light is an electromagnetic (EM) wave with time varying electric (E) and magnetic (B) fields.
E ⊥ B
E and B are ⊥ to direction of propagation.
As shown on the next figure.
Ex
z
Direction of Propagation
B (^) y
z
x
y
k
An electromagnetic wave is a travelling wave which has time varying electric and magnetic fields which are perpendicular to each other and the direction of propagation, z. © 1999 S.O. Kasap, Optoelectronics (Prentice Hall)
Fig 1.1 page 2
The simplest example of this propagating wave is a sinusoidal traveling wave.
E (^) x = E 0 cost(ωt – kz + φ 0 )
Where z is the direction of propagation E (^) x is the electric field at position z and time t k = 2π/λ = defn. – the propagation constant λ = the wavelength of the light wave ω = 2πν = defn. – the angular frequency ν = the frequency in Hertz E 0 = the max amplitude of the wave φ 0 = an arbitrary phase constant with reference to a specified time point of the origin.
ωt → ω rads/sec t sec so ωt → rads
kz → k rads/meter z meter so kz → rads
Note the units of the variables.
φ 0 → rads
so (ωt – kz + φ 0 ) = the phase of the wave in rads at a particular position z and time t.
Thus E (^) x = E 0 cos(ωt – kz + φ 0 ) describes a monochromatic (single color) plane wave of infinite extent traveling in the positive z direction.
In a plane ⊥ to the direction of propagation (along z) the phase is a constant = a wavefront.
Wavefront
z
Ex = Eo sin( ω t–kz )
Ex
z
Propagation
E
B
k
E and B have constant phase in this xy plane; a wavefront E
A plane EM wave travelling along z , has the same E (^) x (or By ) at any point in a givenThe xy xy planes are of infinite extent in the plane. All electric field vectors in a given x and y directions. xy plane are therefore in phase. © 1999 S.O. Kasap, Optoelectronics (Prentice Hall)
By Faraday’s law – a time varying electric field results in a time varying magnetic field.
Faraday’s Law
t
B
E
∇× =
But in photonics practice the time varying magnetic field is small and usually neglected.
By Euler’s Identity
Euler’s Identity
Thus cos θ = Re[ejθ]
θ
e cos j sin
j
±
So the sinusoidal wave can be expressed as an equivalent exponential.
E x(z,t) = Re[ E 0 ej (^ ω t − kz^ +φ^0 )^ ] =Re[ E 0 ej φ^0 ej (ω t − kz )]
The text then defines E (^) c = E 0 Re[ejφo^ ] which combines the amplitude and constant phase info.
Vector k – defn.- The wave vector is the mathematical indication of the direction of propagation.
Wave Vector
Thus in polar coordinates the EM wave is:
Note that k • r = k (^) x x + k (^) y y + k (^) z z
E(r,t) = E 0 cos(ωt – k • r + φ 0 ) (^) y
z
k
Direction of propagation
r
O
θ
r ′^ E ( r , t )
A travelling plane EM wave along a direction k © 1999 S.O. Kasap, Optoelectronics (Prentice Hall)
Phase Velocity
The velocity with which the wave propagates through space is:
Phase velocity = v = change in distance/change in time
v( ) ( 0 ) a constant
d velocity t kz dt
= ω − + φ =
This ν is the symbol for frequency. Be careful with v and ν. (differences in program type fonts do not help but use the context of the symbol)
v for velocity ν for frequency
dt k
dz dt
kdz dt
= ω dt − = 0 ⇒ =^ ω
Wave Equation
All EM waves must obey Maxwell’s EM wave equations. for an isotropic and linear dielectric medium εr = a constant for all directions
Wave equation 2
2 2 0 0
2 2
2 2
2
t
E
u
z
E
y
E
x
E
r ∂
εε
Where u 0 = vacuum permeability (magnetic) = 4π x 10 -7^ H/m
ε 0 = vacuum permittivity (electric) = 8.8542 x 10 -12^ F/m
Initial conditions and/or boundary conditions are required to solve the wave equation.
LLR System
There is enough dispersion of the beam that it is about 7 kilometers in diameter when it reaches the Moon and 20 kilometers in diameter when it returns to Earth. Because of this very weak signal, observations are made for several hours at a time. By averaging the signal for this period, the distance to the Moon can be measured to an accuracy of about 3 centimeters (the average distance from the Earth to the Moon is about 385,000 kilometers).
The 2.7 meter telescope system at the McDonald Observatory (University of Texas operates the observatory in the Davis Mountains) used a Korad ruby laser system, routinely produced lunar laser ranging normal point data with an accuracy in the range 10-15 cm. Wavelength of the ruby laser is about 694 nm.
LLR System
McDonald Observatory - University of Texas operates the observatory in the Davis Mountains 450 west of Austin.
The 2.7 meter telescope system at the McDonald Observatory used a Korad ruby laser system. The observatory routinely produced lunar laser ranging normal point data with an accuracy in the range 10- cm. Wavelength of the ruby laser is about 694 nm.
Korad is the company of Theodore Harold Maiman who is credited with the development of the first successful laser.
LLR Calculations
Beam Divergence = 2θ =
π w 0
rads m
(^9) m (^) 327. 27109 ( 2. 7 )
2 =^4 (^694 ×^10 − )= × − π
θ
The following gives the spot size radius for a beam: ω(z) = z(2θ) where z is the distance and 2θ is the total divergence.
ω( z )= ( 384400 × 103 m^ )( 327. 27 × 10 −^9 rad )= 125. 8 m
m rads^ For the smaller telescope.
(^9) m (^) 1. 16106 ( 0. 762 )
2 =^4 (^694 ×^10 − )= × − π
θ
For the larger telescope.
For the 2.7m telescope.
ω ( z )= ( 384400 × 103 m^ )( 1. 16 × 10 −^6 rad )= 445. 9 m For the 0.76m telescope.
Section 1.2 Refractive Index
An EM wave travels thru a dielectric medium and polarizes the molecules of that medium.
This polarization mechanism retards the propagation of the EM wave.
The speed of the light wave limits the time for interaction of the atom/molecule with the light wave. Only the electrons and the photons have sufficient time to interact. Ionic polarization is just too slow. The measure of the ease with which the medium becomes polarized is called the relative permittivity εr.
Polarization is the property of electromagnetic waves, such as light, that describes the direction of their transverse electric field. More generally, the polarization of a transverse wave describes the direction of oscillation in the plane perpendicular to the direction of travel.
Index of refraction
In free space εr = 1 Phase velocity in this vacuum is: = c = 3 x 10 8 m/s
When light travels thru a medium it slows down.
Ratio of index of refraction
For a non-magnetic medium with associated εr the phase velocity is:
00
( )^1 u
vvelocity ε r ε
=
00
( )^1 u
vvacuum ε
=
n vel
c medium
vacuum (^) =
r
r
medium
vacuum
u
u vel
c n ε
εε
ε = = =
00
00 1
Frequency in a medium
Experiments show that it is the wavelength that changes when the light (EM wave) enters a medium with εr ≠ 1.
The frequency is the same as in a vacuum. λ
νλ ν
c c = → =
Thus n
vacuum medium
λ λ =
And (^) vacuum medium vacuum vacuum
medium n nk
n
k = = = = λ
π λ
π λ
2 π 2 2
Isotropic and Anisotropic
Isotropic material – same material structure in all directions. Thus n is also the same in all directions.
Examples include most gases, glasses and liquids.
Anisotropic material – atomic/molecular arrangements and interatomic bonding vary by lattice direction. The relative permittivity εr can vary with respect to the crystal direction (optical axis).
Examples include crystals such as quartz, calcite, and tourmaline which have crystallographically distinct axes.
xyz rxy z n ,, ,, = ε
Magnetic Medium
u 0 → Permeability of free space with units Henrys/meter.
u (^) r → Is the relative permeability of a medium that is magnetic.
Then for a magnetic medium:
nmagnetic = ur ε r
This situation in not encountered often in electro-optics.
Section 1.3 Group Velocity
Even though we talk about monochromatic waves there is always a slight range of wavelengths (and thus a spread of frequencies).
ω ± Δω k ± Δk
These closely separated wavelengths interfere and thus form a wave packet with a mean frequency ω that is amplitude modulated at a rate of Δω
Fig 1.6 on page 9
E max E max δ k
Wave packet
Two slightly different wavelength waves travelling in the same direction result in a wave packet that has an amplitude variation which travels at the group velocity. © 1999 S.O. Kasap, Optoelectronics (Prentice Hall)
Group Velocity
This modulated amplitude moves with wavevector Δk and has a velocity that is called the group velocity that is given by:
Group Velocity
dk
d
v g
The group velocity therefore defines the speed with with energy or information is propagated.
Note that the phase variation in the E field still move at phase velocity v (^) vel = c/n
Group Velocity in a vacuum
What is the group velocity in a vacuum?
vacuum = = c =^ kc
Thus
Recall c = υλ or υ = c/λ
kc c
dk
d
dk
d
v g = = ( )=
In a vacuum, the phase velocity = group velocity = c There is no dependence on ω or k in a vacuum.
z
Propagation direction
E
B
k
Area A
v Δ t
A plane EM wave travelling along k crosses an area A at right angles to the direction of propagation. In time Δ t , the energy in the cylindrical volume Av Δ t (shown dashed) flows through A.
© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)
Poynting Vector
Poynting Vector: ( ) 0 ( )
2
0
(^2) E B n
c S vel r E B ⎟ r × ⎠
⎞ ⎜ ⎝
⎛ = εε × = εε
For an isotropic medium, energy flow is in the direction of the wave propagation.
Irradiance
Magnitude of |S| at any instant = the irradiance at that instant.
The average irradiance is found by averaging S over one period (for a periodic signal).
2 0 0
2 0 0 2
1
2
1 I = Saverage = v ε ε rE = c ε nE
All practical measurements yield the average irradiance since all detectors have a much slower response rate than the frequency of the light wave.
Example 1.3.1 pg 13
Given: Red HeNe Laser I = 1 mW cm-
Units check: (^2) =
1
cm
mW 2
10
m
W − 2 2 = ( 10 )
1
m
mW
× −
−
4 2
3
10
1 10
m
W
What is the magnitude of the electric field in air? n (^) air = 1
m
V
m
F x s
m x
m
W
c n
I E 87 ( 310 )( 8. 8510 )( 1 )
210 2
8 12
2
0
0 =
⋅ = = ε −
Example 1.3.1 pg 13
The corresponding magnetic field is
Telsa
c
E
B 0 =^0 = 0. 29 × 10 −^6
What is the magnitude of the electric field in glass? n (^) air = 1.
m
V
m
F x s
m x
m
W
c n
I E 72
( 310 )( 8. 8510 )( 1. 45 )
210 2
8 12
2
0
0 =
⋅ = = ε −
Section 1.5 Snell’s Law
Consider a traveling plane EM wave in a medium n = n 1 propagating towards a medium n = n 2. There will be an incident wave, a reflected wave and a transmitted wave.
z^ n^2
y O θ i
n 1
A (^) i
λ λ
θ i θ r
Incident Light Bi Ar
Br
θ t θ t
λ t Refracted Light
Reflected Light
k t
A (^) t Bt
A B ′ B
A ′
A ′′
θ r k i k r
A light wave travelling in a medium with a greater refractive index (reflection and refraction at the boundary. n 1 > n 2 ) suffers © 1999 S.O. Kasap, Optoelectronics (Prentice Hall)
Snell’s Law
The geometrical arguments yield: 1
2
sin
sin
n
n
transmitted
incident =
θ
θ
This is Snell’s Law.
θtransmitted is angle of the refracted wave with respect to the normal to the plane of incidence.
θincident is angle of the incident wave with respect to the normal to the plane of incidence.
θincident = θreflected by the fundamental laws of geometric optics.
Or equivalently: n 1 sin θincident = n 2 sin θtransmitted
Refraction Angle
When n 1 > n 2 then the transmitted wave is “bent away” from the normal to the plane of incidence.
1
2
sin
sin
n
n
transmitted
incident =
θ
θ
When n 1 < n 2 then the transmitted wave is “bent toward” from the normal to the plane of incidence.
These results are from the phase matching conditions at the boundary interface resulting from the wave traveling at different velocities in medium one versus medium two.
Total Internal Reflection
When n 1 > n 2 then the wave is traveling from a more dense medium to a less dense medium (optically).
n 2 θ i
n (^) 1 > n 2 θ i
Incident light
θ t
Transmitted (refracted) light
Reflected light
k t
θ c θ i > θ c TIR
θ c
Evanescent wave
k i k (^) r
(a) (b) (c)
Light wave travelling in a more dense medium strikes a less dense medium. Depending on the incidence angle with respect to θ c , which is determined by the ratio of the refractive indices, the wave may be transmitted (refracted) or reflected. (a) θ i < θ c (b) θ i = θ c (c) θ i > θ c and total internal reflection (TIR).
© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)
Critical Angle
When the refraction angle θt reaches 90°, the incidence angle is called the critical angle θ c.
1
sin^2
sin 90
sin
sin
sin
n
n
c
i
transmitted
incident (^) = θ = θ = θ
θ D
At this angle the incident wave suffers Total Internal Reflection. There will two waves resulting from the wave impinging on this interface. One will be the TIR wave returning into medium 1 and an evanescent wave in medium 2. The evanescent quickly dies out exponentially in both distance and travel time into medium w.
k (^) i
n 2
n 1 > n (^) 2
θ t =90° (^) Evanescent wave
Reflected wave
Incident wave
θ i θ r
E (^) r ,//
E (^) r ,⊥ E (^) i ,⊥
E (^) i ,//
E (^) t ,⊥
(b) θ i > θ c then the incident wave suffers total internal reflection. However, there is an evanescent wave at the surface of the medium.
z
y
x into paper θ i θ r
Incident wave
θ t
Transmitted wave
E (^) i, // E (^) i ,⊥ E (^) r ,//
Et ,//
E (^) t, ⊥
E (^) r ,⊥
Reflected wave
k (^) t
k (^) r
Light wave travelling in a more dense medium strikes a less dense medium. The plane of incidence is the plane of the paper and is perpendicular to the flat interface between the two media. The electric field is normal to the direction of propagation. It can be resolved into perpendicular (⊥) and parallel (//) components
(a) θ i < θ c then some of the wave is transmitted into the less dense medium. Some of the wave is reflected.
E (^) i ,⊥
© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)
Section 1.6 Fresnel’s Equations
The discussion results in the formulation of reflection and transmission coefficients for both the perpendicular and parallel electric fields.
We can resolve the field Ei of the incident wave into two components. One in the plane of incidence E//. The other perpendicular to the place of incidence E⊥.
Transverse electric field (TE) waves are perpendicular to the direction of propagation (z direction in the text.)
Transverse magnetic field (TM) waves have their magnetic field components perpendicular to the direction of propagation (z direction in the text.)
Reflectance R
Reflectance is a measure of the intensity of the reflected light with respect to the incident light.
R⊥ = |r⊥|^2 R// = |r// |^2
At normal incidence R = R⊥ = R// =
2
1 2
1 2
n n
n n
For example, calculate the reflectance for an air-glass interface at normal incidence. n (^) air = 1 n (^) glass = 1.
( 0. 2 ) 0. 044 %
5
5
51
(^512) ⎟^2 = (^2) = = ⎠
⎜ ⎞ ⎝
⎟=⎛ ⎠
⎜ ⎞ ⎝
⎛
R = −
Transmittance T
Transmittance is a measure of the intensity of the transmitted wave with respect to the incident light.
T⊥ = (n 2 /n 1 ) |t⊥|^2 T// = (n 2 /n 1 ) |t (^) //|^2
At normal incidence T = T⊥ = T// = ( )
2 1 2
n n
nn
All the light energy must be either reflected or transmitted into the next medium. Thus R + T = 1
Example Air – Quartz interface
55. 78 D
(at 400 nm) tan tan^1 1
n
n θ p
Find the Brewster’s angle at two wavelengths for light propagating from air onto quartz. n (^) air = 1 n (^) quartz = 1.4702 at 400 nm n (^) quartz = 1.4624 at 500 nm
55. 64 D
(at 500 nm) tan tan^1 1
n
n θ p
Δθp = 0.14°
Example Glass Prism
Light in a glass prism is incident to the air interface. If θc = 45°, then find the index of refraction n 2 for the prism.
2
1 moredense
lessdense sin n
n θ c = =
sin
2 (^ )=^1 = =
c
n
n glass
What if another prism is submerged in water with n = 1.33 but the critical angle remains as before θc = 45°?
sin
2 (^ )=^1 = =
c
n n glass
Section 1.7 Multiple Interference and Optical Resonators
Optical resonator – defn. – stores energy or filters light only at certain frequencies (or equivalently wavelength).
The Fabry-Perot optical cavity is such an optical resonator. It is a parallel mirror cavity (mirrors are spaced apart with some separation and medium between them).
The cavity results in a series of allowed stationary or standing EM waves (cavity modes) given by:
m = L
Where m = 1, 2, 3,…
A
B L
M 1 M (^2) m = 1
m = 2
m = 8
Relative intensity
υ
δυ m
υ m - 1 υ m υ m + 1 (a) (b) (c)
R ~ 0.
1 υ f R ~ 0.
Schematic illustration of the Fabry-Perot optical cavity and its properties. (a) Reflected waves interfere. (b) Only standing EM waves, modes, of certain wavelengths are allowed in the cavity. (c) Intensity vs. frequency for various modes. R is mirror reflectance and lower R means higher loss from the cavity. © 1999 S.O. Kasap, Optoelectronics (Prentice Hall)
Resonant Frequency
λ
ν
c Recall: =
L
c m 2
Resonant Frequencies: νm
L
c
m
c
L
m
m
L
m L
Thus:
FundamentalModem 1 :ν 1 2 ⎟= ν f ⎠
L
c
This fundamental mode υf is also called the “free spectral range” of the cavity.
Etalon
Etalon is French for standard. The Fabry-Perot optical cavity serves to store energy only a certain frequencies (i.e. as compared to a standard).
The intensity of light inside the cavity is the result of a large number of waves interfering both constructively and destructively.
whereA magnitudeandr Rreflectance 1
2 cavity =^ − r (^2) e − j 2 kL = =
A
E
2 2 2 0
(^20) whereI ( 1 ) 4 sin( )
A
R R kL
I
I (^) cavity Ecavity = − +
Maximum Intensity in the Cavity
The maximum intensity inside the cavity is when sin 2 (kL) = zero which is the same as sin 2 (mπ) where m = 1, 2, 3, …
The Finesse of the cavity is a measure of the sharpness of the mode peaks. The mode peaks are the width of frequencies allowed around each mode. Higher finesse = greater frequency selectivity of the cavity.
m π R
I
I =
= wheneverkL ( 1 )^2 m
0 max
R
R F −
= 1
Finesse
π
Spectral Width - FWHM
The spectral width δυm is the full width at half maximum of an individual mode intensity. FWHM is the familiar half power measure.
The Finesse of the cavity is a measure of the sharpness of the mode peaks. The mode peaks are the width of frequencies allowed around each mode. Higher finesse = greater frequency selectivity of the cavity.
F
f m
ν δν =
R
R F −
= 1
Finesse
π
Figure 1.17 on page 31
L λ λ m - 1 λ m Fabry-Perot etalon
Partially reflecting plates
Input light Output light
Transmitted light
Transmitted light through a Fabry-Perot optical cavity.
© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)
Example 1.7.1 on page 31
Consider a Fabry-Perot optical cavity of air (n =1) of length L = 100um. Mirror reflectance R = 0.9.
Calculate the cavity mode nearest to 900 nm.
- 22 90010
2 2 ( 10010 ) 9
6
×
= = × −
− m
m L m λ
Due to boundary conditions, only integer modes exist. Fractions of a modes do not exist in the cavity. m nm m
L m (^) 222 900.^90 λ=^2 =^2 (^100 ×^10 −^6 ) =
Calculate the separation modes. v^ m vf cL m (^6) m^12 Hz
8
- 510 2 ( 10010 )
310 2
= × ×
Δ = = = × −
L
Input light Output light
Calculate the cavity’s finesse. (^29). 8
9
9 1
= −
= −
=π π R
F^ R
Fraunhofer Diffraction
In Fraunhofer diffraction the EM wave are approximately pane waves.
The intensity at the far field observation is the sum of all the waves arriving from all point sources in the aperture. Incident plane wave
Newwavefront
A secondarywave source
(a) (b)
Another newwavefront (diffracted) θ z
(a) Huygens-Fresnel principles states that each point in the aperture becomes a source ofsecondary waves (spherical waves). The spherical wavefronts are separated by λ. The new wavefront is the envelope of the all these spherical wavefronts. (b) Another possiblewavefront occurs at an angle θ to the z -direction which is a diffracted wave. © 1999 S.O. Kasap, Optoelectronics (Prentice Hall)
Fraunhofer Diffraction
Thus for a single slit diffraction I(θ) = I(0) sinc^2 (β) where β = ½ ka sin θ
θ
A y sin θ
y
Y
θ θ = 0
δ y
δ y z
Incidentlight wave Screen
θ
R = Large
θ
b c
Light intensity
a
y
y z
(a) (^) (b) (a) The aperture is divided intoamplitude ∝ δ y. (b) The intensity distribution in the received light at the screen far away N number of point sources each occupying δ y with from the aperture: the diffraction pattern
Incidentlight wave
© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)
Zero Intensity points are located at m = +/-1, +/-2, … is the order.
sin whereaisthewidthoftheslit. a
θ = m^ λ
Fraunhofer Diffraction
For a circular aperture, the observer sees a central white spot called the Airy disk surrounded by dimmer rings called Airy rings.
sin 1. 220 whereDisthediameterofthecircularaperture. D
θ =^ λ
The above equation is for the central white spot. For the center of higher order rings, the solution to the Bessel functions gives us:
sin = 2. 233 form= 2 D
θ^ λ
sin = 3. 238 form= 3 D
θ^ λ
Example 1.10.1 on page 41
Resolving power of imaging systems.
Δθ
S 1
S 2
S 1
S 2 A 1
A 2
I
y
Screen
s Δθ
L
Resolution of imaging systems is limited by diffraction effects. As points S 1 and S 2 get closer, eventually the Airy disks overlap so much that the resolution is lost. © 1999 S.O. Kasap, Optoelectronics (Prentice Hall)
Example 1.10.
Limit of the resolving power of imaging systems is the Rayleigh criterion when the principal maximum of one diffraction pattern coincides with the minimum of the other.
sin( (^) min) 1. 220 D
Δ θ =^ λ Angular limit of resolution
For the human eye D is about 2 mm with n = 1.22 (water)
Given that the light source is “green” at 550 nm. The two objects to be resolved are 30 cm from the eye.
mm
nm n D 2
1
- 33
thus sin( ) 1. 220 1 1. 220550 min ⎟⎠ ⎜ ⎞ ⎝
⎟ = ⎛ ⎠
⎜ ⎞ ⎝
Δθ = ⎛λ
giving 0. 0145 D Δθmin =
Example 1.10.
The minimum physical separation is s = 2 L tan(Δθ/2)
s L θ^ mm ) 0. 076 mm 76 μ m 2
) 2 ( 300 )tan(^0.^0145 2
= 2 sin(Δmin^ = = =
D
giving Δθmin = 0. 0145 D
Diffraction Grating
A diffraction grating in its simplest form is an optical device that has a periodic series of slits in an opaque screen.
d z
y Incident light wave
Diffraction grating
One possiblediffracted beam
θ
a
Intensity
y
m = 0
m = 1
m = -
m = 2
m = -
Zero-order
First-order
First-order Second-order
Second-order
Single slit diffraction envelope
d sin θ
(a) (b ) (a) A diffraction grating with N slits in an opaque scree. (b) The diffracted light pattern. There are distinct beams in certain directions (schematic) © 1999 S.O. Kasap, Optoelectronics (Prentice Hall)
Diffraction Grating
The incident beam of light is diffracted in certain well-defined directions depending on both the wavelength of the light and the grating’s properties.
Waves will interfere constructively when the optical path difference d sin θ is a whole multiple of wavelength λ.
d = separation of the slits
Grating equation: d sin θ = m λ where m = +/-1, +/-2, …
This equation is valid for light normal to the grating.
Transmission and Reflection Grating
Transmission grating – incident and diffracted beams are on opposite sides of the grating.
Reflection grating – incident and diffracted beams are on the same side of the grating.
Incident light wave (^) m = 0
m = -
m = 1 Zero-order First-orde
First-order
(a) Transmission grating (^) (b) Reflection grating
Incident light wave
Zero-order
First-order
First-order
(a) Ruled periodic parallel scratches on a glass serve as a transmission grating. (b) A reflection grating. An incident light beam results in various "diffracted" beams. The zero-order diffracted beam is the normal reflected beam with an angle of reflection equal to the angle of incidence. © 1999 S.O. Kasap, Optoelectronics (Prentice Hall)
End of Chapter One
