Lecture Notes on Wave Optics (03/12/14)
2.71/2.710 Introduction to Optics βNick Fang
1
Outline:
A. Superposition of waves, Interference
B. Interferometry
- Amplitude-splitting (e.g. Michelson interferometry)
- Wavefront-splitting (e.g. Youngβs Double Slits)
C. More Interferometry: Fabry Perot, etc
A. Superposition of Waves, Interference The nature of linear wave equation guarantees that waves can be superimposed: we may combine an array of waves by algebra, as far as each of them are proper solution of the wave equation. To facilitate this process, we make use of the following complex number to represent the wave field.
- Waves in complex numbers
For example, the electric field of a monochromatic light field can be expressed as:
πΈ(π§, π‘) = π΄πππ (ππ§ β ππ‘ + π) (1)
Since exp(ππ₯) = cos(π₯) + π sin(π₯) (2)
πΈ(π§, π‘) = π π{π΄ππ₯π[π(ππ§ β ππ‘ + π)]} (3) Or
πΈ(π§, π‘) =1
2{π΄ππ₯π[π(ππ§ β ππ‘ + π)]} + π. π. (πππππππ₯ πππππ’πππ‘π) (4)
- Complex numbers simplify optics! Interference with two beams
e.g. 2 plane waves propagating in +z direction
πΈ1π₯ = πΈ1π₯(0)ππ₯π[π(π1π§ β π1π‘ + π1)] (5)
πΈ2π₯ = πΈ2π₯(0)ππ₯π[π(π2π§ β π2π‘ + π2)] (6) We can define phase of each waves:
πΏ1(π§, π‘) = π1π§ β π1π‘ + π1 (7) πΏ2(π§, π‘) = π2π§ β π2π‘ + π2 (8)
For a point P located at z=z0 the combined field is:
πΈπ₯ = πΈ1π₯ + πΈ2π₯ = πΈ1π₯(0) exp[ππΏ1(π§0, π‘)] + πΈ2π₯(0) exp[ππΏ2(π§0, π‘)] (9)
Lecture Notes on Wave Optics (03/12/14)
2.71/2.710 Introduction to Optics βNick Fang
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B. Interferometry We often use optical interferometers to facilitate the study of interference. A few common setups are discussed here. Based on the operation principle to split the beams, we may find the so-called amplitude-splitting or wavefront-splitting devices.
- Michelson Interferometry
The Michelson Interferometer is named after Albert Michelson, who used it with Edward Morley in 1887, in an attempt to measure the existence of the "ether". In Michelson- Morleyβs famous experiment, the delay time βπ is achieved simply by moving a mirror along the optical axis. Moving a mirror backward by a distance L yields a delay of:
βπ =2πΏ
π (10)
(e.g. 300 Β΅m of mirror displacement yields a delay of 2 Γ 10β12s=2ps).
A Michelson Interferometer as shown in left schematic, split a beam of incident light into two arms using a thin glass window. Both beams travel to mirrors that are precisely aligned to reflect them. Before recombining them at the beam splitter, the two beams traveled with different optical path length L1 and L2.
πΌ = 2πΌ0 + 2πΌ0 β¨cos (2πL1βL2
π)β© (11)
The variation of intensity as a function of the
path length L1 gives us a measure of wavelength of light! Recent effort is to apply such technology in measurement of gravity waves.
Observation:
- Michelson Interferometer measures (auto)-correlation in time. To see this effect, we suppose the input light beam is not monochromatic. Thus
πΌ =π
2νβ¨πΈπ₯ β πΈπ₯
ββ© =π
2νβ¨(πΈ1π₯ + πΈ1π₯(π‘ β π)) β (πΈ1π₯
β + πΈ1π₯β (π‘ β π))β© (12)
Beam-splitter
Input
beam
Delay
Mirror
Mirror
L1
L2 Output
beamI0I0
Translation stage
Input
beam E(t)
E(tβt)
Mirror
Output
beam
Lecture Notes on Wave Optics (03/12/14)
2.71/2.710 Introduction to Optics βNick Fang
3
πΌ = πΌ1(π‘) + πΌ1(π‘ β π) + 2β¨πΈ1π₯ β πΈ1π₯β (π‘ β π)β© (13)
Such auto-correlation function tells the similarity of the field over given period of time. The Fourier transform of the auto-correlated signal yields the Power Spectrum:
β« πΈ1π₯ β πΈ1π₯β (π‘ β π)ππ‘
β
0
πΉπβ πΈ1π₯(π) β πΈ1π₯
β (π) = |πΈ1π₯(π)|2 (14)
This is how Fourier Transform Spectroscopy (often abbreviated as FTIR) are constructed nowadays. (See Pedrotti 21-2 for more discussion)
- In the above analysis we assumed the input beams are ideal plane waves so we only focused on intensity variation of a single spot as the arm length changes. In reality, a set of nested rings are often observed on the receiving screen as the pattern on right. This is due to the variation of phase as a function of momentum difference (π1 β π2)π§.
- Youngβs Double Slit Interferometry To analyze Youngβs experiment, we assume the screen X with two narrow slit is illuminated with a monochromatic plane wave. After the slits, two cylindrical waves are excited. This creates fringes at the observation plane Xβ , after travelling a distance π§ = π. For a position xβ on the screen,
the phase difference of the two waves becomes:
Interferogram
This interferogram is
very narrow, so the
spectrum
is very broad.4000 3200 2400 1600 800
incoming
plane wave
(on-axis)
opaque
screen
z =l
x=x0
x=βx0
observation
point xβ
Z
XβX
Β© Source unknown. All rights reserved.This content is excluded from our Creative
Commons license. For more information,see http://ocw.mit.edu/fairuse.
Β© Source unknown. All rights reserved.This content is excluded from our Creative
Commons license. For more information,see http://ocw.mit.edu/fairuse.
Lecture Notes on Wave Optics (03/12/14)
2.71/2.710 Introduction to Optics βNick Fang
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πΏ1 β πΏ2 = π(π1 β π2) + (π1 β π2) (15)
where π1 = β(π₯β² β π₯0)2 + π2, π2 = β(π₯β² + π₯0)2 + π2. When the slits are arranged symmetrically and the incoming plane wave is at normal incidence, then π1 β π2 =0 In a typical experiment the screen is placed far away from the slits (π β« π₯0) so we can further take approximation:
π1 = πβ1 +(π₯β²βπ₯0)
π2
2
β π (1 +(π₯β²βπ₯0)
2π2
2
) (16)
π2 = πβ1 +(π₯β²+π₯0)
π2
2
β π (1 +(π₯β²+π₯0)
2π2
2
) (17)
(π1 β π2) = β2π₯β²π₯0
π (18)
In order for the constructive interference to occur, cos(πΏ1 β πΏ2) = 1, then
π(π1 β π2) = βπ2π₯0π₯
β²
π= 2ππ (19)
π = 0,Β±1, Β±2,β¦. are called order number.
This gives rise to the famous condition:
2π₯0π₯β² = πππ (20)
Therefore Youngβs experiments directly measured the wavelength of light (in 18 century!) Note: Generally fringes will form by two or more beams crossing at an angle. To quantify that we can modify our phase term for the crossing beams:
πΏ1(π₯, π§, π‘) = π1π₯π₯ + π1π§π§ β π1π‘ + π1 (21) or in terms of incident angle ΞΈ:
πΏ1(π₯, π§, π‘) = π1π₯π πππ1 + π1π§πππ π1 β π1π‘ + π1 (22) πΏ1 β πΏ2 = (π1π πππ1 β π2π πππ2)π₯ + (π1πππ π1 β π2πππ π2)π§ + (π1 β π2) (23)
Fringes can vary both on x and z directions! - Note: Youngβs double slit experiment measures the correlation in space.
Assuming the input light beam is not a plane wave (i.e. inhomogeneous), at xβ=0 we measure the interference:
πΌ =π
2νβ¨πΈπ₯ β πΈπ₯
ββ© =π
2νβ¨(πΈ1π₯(π₯ β π₯0) + πΈ1π₯(π₯+π₯0)) β (πΈ1π₯
β (π₯ β π₯0) + πΈ1π₯β (π₯ + π₯0))β© (24)
Lecture Notes on Wave Optics (03/12/14)
2.71/2.710 Introduction to Optics βNick Fang
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πΌ = πΌ1(π₯ β π₯0) + πΌ1(π₯ + π₯0) + 2β¨πΈ1π₯(π₯ β π₯0) β πΈ1π₯β (π₯ + π₯0)β© (25)
Such correlation function tells the similarity of the field over a given spatial period. This effect is often used to measure the coherence of a remote star under the telescope, although the radiation is thought to be randomly distributed. A daily life example is the spatial coherence of ripples in the pool (βSpatial coherence from Ducksβ by Emil Wolf et al, Physics today 2010).
- Comparison between Michelson and Youngβs double slits:
Both can be regarded as interference of two spherical waves, but observed in different directions.
- Other wavefront splitting interferometry similar to Youngβs Double slits: o Lloydβs Mirror o Fresnelβs biprism o Fresnelβs mirror o Billetβs split lens
C. More examples of Interferometry
- Thin Film interference
Lecture Notes on Wave Optics (03/12/14)
2.71/2.710 Introduction to Optics βNick Fang
6
- Fabry-Perot Interferometry Consider two parallel reflective
surfaces separated by distance d,
the first one has a amplitude
transmission and reflection
coefficient t and r, and the
second has amplitude
transmission and reflection
coefficient tβ and rβ. Multiple
reflections between the two
surfaces results in two series of
reflected and transmitted terms. Due to the round trip travel path, there is a phase
difference between successive transmitted terms:
πΏ0 = (2ππ§π) (26)
The transmitted series is
πΈπ‘ = exp (ππΏ0/2)(π‘β²π‘πΈ0 + π‘
β²(ππβ²)π‘πΈ0 exp(ππΏ0) + π‘β²(ππβ²)2π‘πΈ0 exp(2ππΏ0) + β― )
(27) πΈπ‘ = π‘
β²π‘πΈ0exp (ππΏ0/2)[1 + (ππβ²) exp(ππΏ0) + (ππ
β²)2 exp(2ππΏ0) + β― ]
(28)
πΈπ‘ = π‘β²π‘πΈ0exp (ππΏ0/2)β (ππβ²)π exp(πππΏ0)
βπ=0 (29)
πΈπ‘ =π‘β²π‘exp (ππΏ0/2)πΈ0
1βππβ² exp(ππΏ0) (30)
The transmitted irradiance is given by:
πΌπ‘ =πβ²ππΌ0
|1βππβ² exp(ππΏ0)|2 (31)
The denominator of the last result can be expressed as:
|1 β ππβ² exp(ππΏ0)|2 = (1 β ππβ² exp(ππΏ0))(1 β π
βπβ²β exp(βππΏ0))
= 1 β (π πβ²exp(ππΏ0) + πβπβ²β exp(βππΏ0)) + π π β²
= 1 β 2βπ π β²cos (πΏ) + π π β²
= (1 β βπ π β²)2+ 2βπ π β²(1 β cos(πΏ))
= (1 β βπ π β²)2+ 4βπ π β²sin2 (
πΏ
2) (32)
Where πΏ = (2ππ§π) + ππ + ππβ² (33)
We define the coefficient of finesse F:
r, t rβ, tβ
E0
tE0
rE0
rβtE0 tβtE0
trβtE0eid
rrβtE0
rβrrβtE0
trβ(rrβ)tE0e2id
tβrrβtE0eid
tβ(rrβ)2tE0e2id
(rrβ)2tE0
rβ(rrβ)2tE0
Lecture Notes on Wave Optics (03/12/14)
2.71/2.710 Introduction to Optics βNick Fang
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β± =4βπ π β²
(1ββπ π β²)2 (34)
To express the general form of It:
πΌπ‘ =πβ²ππΌ0
(1ββπ π β²)2 [
1
1+β±sin2 (πΏ
2)] (35)
Applications: Fabry-Perot cavities are often designed to distinguish closely spaced spectral lines of a gas medium. Higher values of Finesse F give a sharper transmission pass band and greater spectral resolution. To find the half-width of the pass band, we solve:
1
1+β±sin2 (πΏ
2)=1
2 (36)
giving
π ππ (πΏ1/2
2) =
1
ββ± (37)
Figure 8.9 from Pedrotti: Transmittance of Fabry-Perot Cavity.
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