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Fiber- and microresonator-based parametric generation.
Phase conjugation, optical limiting, all-optical switching.
Lecture 20
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Four-wave mixing, one pump wave.Continued.
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Four-wave mixing (FWM), one pump wave
๐" ๐๐ ๐%One strong wave ๐๐ creates two new waves ๐" & ๐%
in a 4-wave process ๐๐๐ = ๐๐ + ๐๐
๐๐ = ๐๐๐ โ ๐๐ โ ๐๐ + nonlinear SPM and XPM terms
k
ฯ
๐%
๐"
๐/
ฯ
๐%
๐"๐/
Dispersion of a fiber or a waveguide
k
dispersive phase matching
โzero group dispersionโ
โnormal dispersionโ
k
ฯ
๐%
๐"๐/
โanomalous dispersionโ
๐/ ๐/๐/
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Phase matching terms due to SPM and XPM
Example: fused silica: ๐1 = 3.2ร10819 ๐1/๐
๐พ>? =@ABCDEFF
= 1.3 x 10-3 (Wm)-1 3.3e-20 *1.2e15 /3e8 / 1e-10=ฮป=1.56 ยตm10 ยตm mode diam.
For chalcogenide(As2Se3) nano-wires: ๐พ>? โ93 (Wm)-1
Thus, for strong pump field ๐๐ with power ๐, SPM term gives
โ๐ = ๐9ฮ๐๐ฟ =๐๐๐1๐ผ๐ฟ =
๐๐๐1
๐๐ดPQQ
๐ฟ = ๐พ>?๐๐ฟ
๐พ>? =๐๐1๐๐ดPQQ
=2๐๐1๐๐ดPQQ
In guided wave optics, the nonlinearity is expressed with the factor ๐พ [ in (Wm)-1 ]
โ๐ = ๐พ>?๐๐ฟ ร โ๐ = ๐พ>?๐
For weak fields ๐๐ and ๐๐ , XPM terms giveโ๐ = ๐1(๐ผ/ + 2๐ผ1)Recall formula (17.10a)
probe beam pump beam, co-polarized
๐1 =3๐ "
4๐91๐9๐- self phase modulation ceff.
For strong ๐๐ field, SPM term gives
โ๐ = ๐9ฮ๐๐ฟ =๐๐ ๐1๐ผ๐ฟ =
๐๐ ๐1
๐๐ดPQQ
๐ฟ = ๐พ>?๐๐ฟ
One also needs to take into accound nonlinear phase related to SPM and XPM
โ๐ = 2๐พ>?๐ for both ๐" and ๐%
2(๐/ + ๐พ>?๐) = ๐" + 2๐พ>?๐ + ๐% + 2๐พ>?๐Hence phase matching condition becomes
2๐/ = ๐" +๐% +2๐พ>?๐
(20.1)
=fiber nonlinear parameter
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Four-wave mixing (FWM), one pump wave
g = "Z [
\CB]BAB[B^
@]@A@[@^โ "Z [ BA
\C@A
๐พ = | ๐ด/|1๐
From previous lecture, see (19.18)signal and idler intensities grow as:
๐ผ3 = ๐ผ30 ๐๐๐ h2(๐พ๐ง)
๐ผ4 =๐%๐"
๐ผ30 ๐ ๐๐โ2(๐พ๐ง)
where
gain increment
Let us express the gain increment ๐พ through power ๐ at ๐๐
๐3 = ๐30 ๐๐๐ h2(๐พ๐ง)
๐4 =๐%๐"
๐30 ๐ ๐๐โ2(๐พ๐ง)ร
๐พ = | ๐ด/|1๐ = | ๐ด/|13๐ " ๐1
8๐๐1 =2๐ผ๐๐๐9
3๐ " ๐1
8๐๐1 =3๐ " ๐4๐9๐1๐1
๐๐ดPQQ
=๐1๐๐๐ดPQQ
๐ = ๐พ>?๐recall: ๐ผ = BCij
1๐ด 1
๐1 =3๐ "
4๐91๐9๐
๐พ>?
( in m-1 )
Thus, phase matching condition is power dependent (unlike ๐ 1 processes!)
2๐/ = ๐" +๐% +2๐พ>?๐ ๐3 = ๐30๐๐๐ h2(๐พ>?๐๐ฟ)
(20.2)
(20.3)
=fiber nonlinear parameter
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Degenerate four-wave mixing in a fiber
Generated radiation in the visible and mid-IR regions: 1064 nm ร 673 nm + 2539 nm
Let us estimate the fiber NL parameter and 4-wave gain in this experiment
Gain = ๐๐๐ h2(๐พ>?๐๐ฟ) = ๐๐๐ h2(2eโ3โ1e5โ1.4)=๐๐๐ h2(280) ร exp(560)
๐พ>?= 1q@ArDEFF
=2 x10-3 (Wm)-1
NL contribution to the k-vector sum ร ฮ๐st = 2๐พ>?๐ = 2โ2eโ3โ1e5=400 m-1
(Linear mismatch ฮ๐ in the biulk SiO2 would be larger : 8500 m-1)
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Four-wave mixing on a photonic chip
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Four-wave mixing on a photonic chip
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Kerr-nonlinearity induced optical parametric oscillation in a toroidal SiO2 microcavityD โ 50 ยตm
Signal /idler spectrum
Idler emission versus pump power. The differential conversion efficiency from pump to idler was 17% (and correspondingly 34% for pump to signal and idler).
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Optical phase conjugation
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Optical phase conjugation
It is possible, using nonlinear optical processes, to exactly reverse the propagation direction and phase variation of a beam of light.
The reversed beam is called a conjugate beam, and thus the technique is known as optical phase conjugation.
It is also called time reversal, wavefrontreversal.
Comparison of a phase-conjugate mirror with a conventional mirror.
With the phase-conjugate mirror the image is restored when passing back through an aberrating element
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Optical phase conjugation
PHASE-CONJUGATE MIRROR is able to compensate for distortions imposed on an image of a cat.
In both photographs the image was distorted by transmitting it through a piece of frosted glass. Reflection of the image back through the same piece of glass by an ordinary mirror yielded an unrecognizable image (top).
Reflection of the image back through the frosted glass by a phase-conjugate mirror, on the other hand, corrected the distorted image (bottom).
It did so because a phase-conjugate mirror produces a beam that propagates back through the distorting glass in a "time-reversed" sense: its trajectory retraces that of the original beam and thereby undoes the distortions.
phase-conjugate mirror
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Optical phase conjugation
Let us regard a degenerate four-wave mixing process is โ all four interacting waves have the same frequency.
In this process, a lossless nonlinear medium with a third-order nonlinear susceptibility ฯ(3) is illuminated by two strong counterpropagating pump waves E1 and E2 and by a signal wave E3.
a new wave E4 is created that is the phase conjugate of E3.
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Optical phase conjugationTwo strong counterpropagating pump waves E1 and E2 at ฯ
In a 4-wave process ๐ = ๐ โ ๐ + ๐
โฐ(๐ก) = /1 ( ๐ธ/๐
yB]z + ๐ธ1๐yBAz + ๐ธ"๐yB[z + ๐ธ%๐yB^z + ๐. ๐.)
๐ซst(๐ก) =/\๐9๐
" ( ๐ธ/๐yB]z + ๐ธ1๐yBAz + ๐ธ"๐yB[z + ๐ธ%๐yB^z + ๐. ๐.)3
total field
total NL polariz.
๐ซst ๐ก |๐ =/\๐9๐
" ( 6๐ธ/๐ธ1๐ธ"โ๐yBz 8y(๐๐|๐๐8๐๐)๐ + ๐. ๐.)="%๐9๐
" ( ๐ธ/๐ธ1๐ธ"โ๐yBz|y๐๐๐ + ๐. ๐. )
Signal wave E3 at ฯ
The nonlinear polarization at ฯ produced within the medium by all three input waveswill be:
๐๐ ๐๐๐ธ/ = ๐ธ/๐yBz8y๐๐๐
๐ธ1 = ๐ธ1๐yBz8y๐๐๐
๐๐+๐๐=0
since ๐๐+๐๐=0
We see that this contribution to the nonlinear polarization has a spatial dependence that allows it to act as a phase-matched source term for a conjugate wave E4 having wavevector โk3, and thus we see that the wavevectors of the signal and conjugate waves are related by k3 = โk4.
๐"
๐๐ ๐๐
๐"๐%
๐ธ" = ๐ธ1๐yBz8y๐๐๐
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Optical phase conjugation
๐% =32 ๐9๐
" ( ๐ธ/๐ธ1๐ธ"โ)ร Fourier components for ๐st
Now plug into SVEA equations - get coupled equations:
๐๐ธ%๐๐ง =
๐๐2๐๐๐9
32 ๐9๐
" ( ๐ธ/๐ธ1๐ธ"โ) = ๐๐พ๐ธ"โ
Since the process is phase matched, the reflected conjugate field E4 will be amplified in โz direction (๐ธ"โ serving as a seed), and its power can even exceed the incoming power E3, that is the reflectivity of a phase conjugate mirror can be >100%
๐ธ" = ๐ธ1๐yBz8y๐๐๐
๐ธ% = ๐ธ1๐yBz|y๐๐๐๐" =
32 ๐9๐
" ( ๐ธ/๐ธ1๐ธ%โ)
๐๐ธ"๐๐ง =โ
๐๐2๐๐๐9
32 ๐9๐
" ( ๐ธ/๐ธ1๐ธ%โ) = โ๐๐พ๐ธ%โ
๐ธ% grows in the opposite direction: -z
0 L
๐ธ"๐ธ%
๐พ =34๐๐ "
๐๐ ๐ธ/๐ธ1 ๐ธ"9
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Optical phase conjugation, time reversed wavefront
๐ธ% ~ ๐ธ"โ๐yBz|y๐๐๐ + ๐. ๐.
= ๐ธ"(๐)๐8yBz8y๐๐๐ + ๐. ๐.
Compare to ๐ธ"~ ๐ธ"(๐)๐yBz8y๐๐๐ + ๐. ๐.
- time is reversed: t โ -t
The generated wavefront exactly replicates the incident wavefront but propagates in the opposite direction. For this reason, optical phase conjugation is sometimes referred to as the generation of a time reversed wavefront
๐
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Optical phase conjugation, one more interpretation
The object beam E3 is a spherical wave and the other two are reference (strong) beams. The beam E4, an output beam, is the desired phase conjugate of the object beam.
The interaction of the object beam with one of the reference beams E1 (blue) produces an interference pattern in the medium. The second reference beam E2 (red) is reflected by the interference pattern. Since the second reference beam comes from a direction opposite to that of the first reference beam, the reflected beam E4 is the phase conjugate of the object beam. In reality all the processes occur simultaneously.
E3E4
E1 E2
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Optical phase conjugation
First in, first out (ordinary mirror) Last in, first out (conjugate mirror)
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Optical power limiting
In a material with a strong nonlinear effect (e.g. two-photon absorption), the absorption of light increases with intensity such that beyond a certain input intensity the output intensity approaches a constant value. Such a material can be used to limit the amount of optical power entering a system. This can be used to protect expensive or sensitive equipment such as sensors, can be used in protective goggles, or can be used to control noise in laser beams.
ExperimentTheory
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All-optical switching
All-optical switch in the form of a MachโZehnder interferometer containing a nonlinear element.
nonlinear element
Nonlinear phase shift ฯNL needs to be ฯ radiansto switch from one state to another
~ intensity
On-a-chip version
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