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Wave Optics
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General Wave Equation Solution
𝐸 𝑟⃗ , 𝑡 = 𝑒⃗ 𝐴 𝑟⃗ , 𝑡 𝑒 ! ” $⃗ , &
Polarization Amplitude Phase
§ Amplitude A and phase 𝜙 are functions of both Time & Space.
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Directionality of A Wave
§ Generally, light is a transverse wave (unlike sound = longitudinal)
§ Polarization of the EM field (made up of electrical and magnetic field) is defined as the direction along the electric field vector points.
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Classification of Polarization
There is always a basis 𝑥”, 𝑦” for decomposing the filed into 2 polarizations (eigen modes); equivalently (right, left) circular polarization is also a basis.
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Representations of Polarization States
§Natural lightàunpolarizedàsuperposition 𝐸! = 𝐸” with no phase relationship between the two
§Circular polarizedà𝐸! = 𝐸”, ∅! − ∅” = 𝜋⁄2 §Matrix formalism of polarization transformation
(Jones – 2 x 2, complex & Muller – 4 x 4, real)
𝐸!# 𝐸! 𝐽$$ 𝐽$% 𝐸! 𝐸 “# = 𝐉 𝐸 ” = 𝐽 % $ 𝐽 % % 𝐸 ”
𝑆⃗ = 𝐼 , 𝑄 , 𝑈 , 𝑉 & ;
𝐼=𝐸”+𝐸”, 𝑄=𝐸”−𝐸” !#!#
𝑈 = 2 R e 𝐸 ! 𝐸 #∗ , 𝑉 = − 2 I m 𝐸 ! 𝐸 #∗ 5
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Definitions: Amplitude & Absolute Phase
This is a simple way of writing the solution to the wave equation:
𝐸 𝑧,𝑡 =𝐴cos 𝑘𝑧−𝜔𝑡−𝛿
𝐴 = Amplitude (This is related to the wave’s energy)
𝛿 = Absolute phase (or “initial phase: the phase when 𝑧 = 𝑡 = 0”)
Absolute phase = 0
Absolute phase = 2𝜋⁄3
Position, 𝑧 at 𝑡 = 0
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Definition: The Phase of a Wave
The phase, 𝜙, is everything inside the cosine.
𝐸 𝑧,𝑡 =𝐴cos 𝑘𝑧−𝜔𝑡−𝛿 =𝐴cos 𝜙
Do NOT confuse “the phase” with “the absolute phase” (or “initial phase”)
The angular frequency and wave vector can be expressed as derivatives of the phase:
𝜔 = −𝜕𝜙⁄𝜕𝑡 𝑘 = 𝜕𝜙⁄𝜕𝑧
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𝑃 = 𝑥 + 𝑖𝑦 = 𝐴 cos 𝜙 + 𝑖 𝐴 sin 𝜙
Complex Numbers
Considerapoint,𝑃= 𝑥,𝑦,ona2DCartesiangrid.
Let the x-coordinate be the real part and the y-coordinate the imaginary part of a complex number.
So instead of using an ordered pair, we write:
where: 𝑖 = −1
𝑥 = 𝐴 cos ∅
𝑦 = 𝐴sin∅ 𝑿
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𝑒*+ =cos𝜙+𝑖sin𝜙
Complex Numbers – Euler’s Formula
So the point, 𝑃 = 𝐴 cos 𝜙 + 𝑖 𝐴 sin 𝜙, can be written:
𝐴 = Amplitude
𝑥 = 𝐴 cos ∅
𝑦 = 𝐴sin∅ 𝑿
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𝐸𝑧,𝑡 =Re𝐴exp𝑖𝑘𝑧−𝜔𝑡−𝛿 𝐸𝑧,𝑡=𝐴2exp𝑖𝑘𝑧−𝜔𝑡−𝛿 +𝑐.𝑐.
Waves using Complex Numbers
We have seen that the 𝐸 field of a light wave can be written: 𝐸 𝑧,𝑡 =𝐴cos 𝑘𝑧−𝜔𝑡−𝛿
Since𝑒*+ =cos𝜙+𝑖sin𝜙,𝐸 𝑧,𝑡 canalsobewritten:
where “+𝑐. 𝑐.” means “plus the complex conjugate.”
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𝐸𝑡 =Re𝐸,exp𝑖𝜔𝑡
The Fourier Transform in Optics
𝐸𝑧,𝑡 =Re𝐸,exp𝑖𝑘𝑧−𝜔𝑡
𝐸𝑧 =Re𝐸,exp𝑖𝑘𝑧
lim𝐸𝑡 =? -→/
What is wrong with this description?
lim 𝐸 𝑡 = 0 We must always expect this. -→/
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§ Monochromatic wave has only one frequency, 𝜔.
§ The light wave has many frequencies and the frequency increases in time (from red to blue).
Light Electric Field
Light Electric Field
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The Spectrum of a Light Wave
The spectrum of a light wave is defined as:
whereF 𝐸 𝑡 denotes𝐸 𝜔 ,theFouriertransformof𝐸 𝑡 .
Note that Fourier transform of 𝐸 𝑡 is usually a complex quantity: F𝐸𝑡 =𝐸𝜔=𝐸𝜔𝑒*01
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The Fourier Transform & its Inverse
Many of you have seen this in other classes:
𝐹 𝜔 = Q𝑓 𝑡 exp −𝑖𝜔𝑡 𝑑𝑡 2/
𝑓𝑡 =2𝜋 Q𝐹𝜔exp𝑖𝜔𝑡𝑑𝜔
:FourierTransform
: Inverse Fourier Transform
We often denote the Fourier transform of function 𝑓 𝑡 by F 𝑓 𝑡 , and the inverse transform of a function 𝑔 𝜔 by F:; 𝑔 𝜔
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Example: The Fourier Transform of Rectangle Function
𝐹𝜔= Gexp−𝑖𝜔𝑡𝑑𝑡=−𝑖𝜔exp−𝑖𝜔𝑡 :<⁄"
−𝜏⁄2 𝜏⁄2 𝐹𝜔
= 1 exp−𝑖𝜔𝜏⁄2−exp𝑖𝜔𝜏⁄2 −𝑖𝜔
1 exp 𝑖𝜔𝜏⁄2 −exp −𝑖𝜔𝜏⁄2
=𝜏 sin 𝜔𝜏⁄2 𝜔𝜏⁄2
2𝑖 =𝜏sinc 𝜔𝜏⁄2
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Example: The Fourier Transform of Gaussian Function
F exp −𝑎𝑡" = G exp −𝑎𝑡" exp −𝑖𝜔𝑡 𝑑𝑡 = 𝑎exp −𝜔"⁄4𝑎
𝑓 𝑡 =exp −𝑎𝑡"
𝐹 𝜔 = 𝜋𝑎exp −𝜔"⁄4𝑎 ∝exp −𝜔"⁄4𝑎
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Coherence Overview
Coherence:
Measure of the correlation between the phases measured at different (temporal and spatial) points on a wave.
Temporal coherence:
Measure of the correlation of light wave’s phase at different points along the direction of propagation.
Spatial coherence:
Measure of the correlation of a light wave’s phase at different points traverse to the direction of the propagation.
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Temporal & Spatial Coherence
Noncoherent
Emission Spatially
Wavelength Coherent
Arc Pinhole Lamp Aperture
Wavelength Filter
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Interference
§ Conditions for interference
1. Propagationinthesamedirection
2. Samepolarization
3. Smallerphasedifferencethancoherencelength.
§ The main property is the phase difference between two waves 𝛿$% = 𝜙$ − 𝜙%
§Interference of two waves
𝐼=𝐼$+𝐼%+2 𝐼$𝐼%cos𝛿$%
𝐼=2𝐼, 1+cos𝛿$% :specialcaseofequalintensities
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Coherence Function
§Coherence function: Correlation of statistical field (complex) Γ 𝑟⃗ , 𝑟⃗ , 𝜏 = 𝐸 𝑟⃗ , 𝑡 + 𝜏 X 𝐸 ∗ 𝑟⃗ , 𝑡
for identical locations 𝑟⃗ = 𝑟⃗ = 𝑟⃗ ;"
Γ𝑟⃗,𝑟⃗ =𝐼𝑟⃗ :Intensity §Normalized Γ : Degree of coherence
𝛾 𝜏=𝛾 𝑟⃗,𝑟⃗,𝜏= $% $% $ %
Γ𝑟⃗,𝑟⃗,𝜏 $ %
𝐼 𝑟⃗ 𝐼 𝑟⃗ $ %
oCoherentlimit: 𝛾!" =1 oIncoherentlimit:𝛾 =0
o Partial coherence: 0 < 𝛾!" < 1
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Diffraction
§Huygens-Fresnel principle:
Each point disturbed by the advancing wavefront can be viewed as a source of a spherical wave, new wavefront is enveloped of these secondary spherical wavefronts.
Single-slit diffraction
• One of the examples of diffraction effects is the passage of light through a hole of finite size.
• If light passes through narrow, point like hole, almost exact spherical wave forms. However if the hole has a finite size, an interference pattern forms behind the hole.
• The pattern near the slit, where exact wavefront profiles are important is called near-field, or Fresnel, diffraction.
• The pattern at the screen far away, where we can use geometrical optics rays and approximations is called far-field, or Fraunhofer diffraction.
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Circular-aperture Diffraction
§ Light of wavelength λ passes through a circular aperture of diameter D, and is then incident on a viewing screen a distance L behind the aperture, L>>D.
§ The diffraction pattern has a circular central maximum, surrounded by a series of secondary bright fringes shaped like rings.
Circular aperture
p=2 𝜃! p=1
The angle of the first minimum in the intensity is
𝜃! = 1.22𝜆 𝐷
The width of the central maximum on the screen is
𝑤 = 2𝐿tan𝜃! ≈ 2.44𝜆𝐿 𝐷
Central maximum
Light intensity
p=1 p=2 p=3
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