l12 optical parametric oscillators and amplifiers · 2019. 12. 12. · 2 3 optical parametric...

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Optical parametric oscillators and amplifiers.

Singly- and doubly- resonant oscillators.

Lecture 12

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Optical parametric process

Only one strong input is present

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Optical parametric processes

Amplifier: Weak signal at 𝜔" is amplified, and 𝜔# is generatedor weak signal at 𝜔# is amplified, and 𝜔" is generated

Generator:Both 𝜔" and 𝜔# (signal and idler) are generatedfrom quantum noise

Oscilaltor:Both 𝜔" and 𝜔# (signal and idler) are generatedfrom quantum noise and resonate (at least one of them). Tune wavelength via crystal angle, temperature or poling period.

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Optical parametric amplification

g = &'

()(*(+,),*,+

&-)&.

=−𝑖𝑔 𝐴3𝐴#∗ 𝑒67∆9.

&-*&.

=−𝑖𝑔 𝐴3𝐴"∗𝑒67∆9.

&-+&.

=−𝑖𝑔𝐴"𝐴#𝑒7∆9.

Again, recall coupled-wave equations for 3 waves – Lecture #5 (5.7)

NL coupling coefficient

Now, assume that A3 field (‘pump’) is much stronger than A1 and A2 fields (‘signal’ and ‘idler’). We can also assume (by the proper choice of time origin) 𝐴3 to be real: 𝐴3=𝐴3∗ ; we also assume 𝐴3 is constant – undepleted pump approximation, ∆𝑘=0 (no phase mismatch), and no absorption.

for the normalized filed intensity |𝐴"|2 |𝐴"|2=𝑔#|𝐴3|2|𝐴#|2𝐿#

(5.7)

&-)&.

=−𝑖𝛾𝐴#∗

&-*&.

=−𝑖𝛾𝐴"∗

and get:define 𝛾 = 𝑔𝐴3

*

𝑑𝑑𝑧

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Optical parametric amplification 𝑑2𝐴"𝑑𝑧2 − 𝛾

#𝐴" = 0

𝑒±C. or 𝑐𝑜𝑠h(𝛾𝑧) & 𝑐𝑜𝑠h(𝛾𝑧)

𝐴" = 𝑎"𝑐𝑜𝑠h(𝛾𝑧)+𝑏"sinh(𝛾𝑧)

à

𝜔"

𝜔3 𝜔3For initial conditions: 𝐴" = 𝐴"P𝐴# = 𝐴#P

𝜔"𝜔#

𝑑2𝐴#𝑑𝑧2 − 𝛾

#𝐴# = 0and

𝐴# = 𝑎#𝑐𝑜𝑠h(𝛾𝑧)+𝑏#sinh(𝛾𝑧)

𝜔#

Solution:𝐴" = 𝐴"P𝑐𝑜𝑠h(𝛾𝑧)− 𝑖𝐴#P∗ sinh(𝛾𝑧)

𝐴# = 𝐴#P𝑐𝑜𝑠h(𝛾𝑧)− 𝑖𝐴"P∗ sinh(𝛾𝑧)

Look for solutions in the form:

The general solutions to this equation are:

(12.1)

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Optical parametric amplifier (OPA)

𝜔"

𝜔3 𝜔3Initial conditions: 𝐴" = 𝐴"P𝐴# = 0

𝜔"𝜔#Solution:

𝐴" = 𝐴"P 𝑐𝑜𝑠h(𝛾𝑧)

𝐴# = −𝑖𝐴"P∗ sinh(𝛾𝑧)

weak input(‘seed’)

The OPA uses three-wave mixing in a nonlinear crystal to provide optical gain of the ’seed’ at 𝜔".Simultneously, the wave at 𝜔# grows from zero and is amplified.

The corresponding photon-flux densities are:

Φ1 = Φ10𝑐𝑜𝑠h2(𝛾𝑧)

Φ2 = Φ10𝑠𝑖𝑛ℎ2(𝛾𝑧)

In terms of intensities:

𝐼1 = 𝐼10 𝑐𝑜𝑠h2(𝛾𝑧)

𝐼2 =𝜔#𝜔"

𝐼10 𝑠𝑖𝑛ℎ2(𝛾𝑧)

(12.2)

(12.3) (12.4)

Now if we have input at only one wave:

These expressions are symmetric under the interchange of 𝜔" and 𝜔#Also, check for Manley- Rowe

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Optical parametric amplification

𝑐𝑜𝑠h2(𝛾𝑧)= "V(𝑒C.+𝑒6C.)2 ≈ 𝑒#C./4

𝛾𝑧>2

– so that the gain increases exponenially with 𝛾𝑧. This behavior of monotonic growth of both waves is different from that of sum-frequency generation, where oscillatory behavior occurs.

The gain coefficient

𝐼( =𝑐𝜀P2 𝜔 𝐴( #Recalling that à 𝐴( # =

2𝐼(𝑐𝜀P𝜔

𝜸 = 𝑔𝐴3 =𝑑𝑐

𝜔"𝜔#𝜔3𝑛"𝑛#𝑛3

𝐴3 =𝟐𝝎𝟏𝝎𝟐𝜺𝟎𝒄𝟑

(𝒅𝟐

𝒏𝟑) 𝑰𝝎𝟑

=8𝜋#

𝜆"𝜆#𝜀P𝑐(𝑑#

𝑛3) 𝐼(+

(12.5)

(12.6)

for

x

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Now, allow for the phase mismatch: ∆𝑘 ≠ 0&-)&.

=−𝑖𝛾𝐴#∗ 𝑒67∆9.

&-*&.

=−𝑖𝛾𝐴"∗𝑒67∆9.

The solution is:

𝐴" = 𝐴"P [𝑐𝑜𝑠h(𝛾m𝑧) + 𝑖∆𝑘2𝛾m 𝑠𝑖𝑛h(𝛾

m𝑧)]𝑒67∆9./#

𝐴# = −𝑖𝐴"P∗CCn sinh(𝛾

m𝑧) 𝑒67∆9./#

(12.9)

à (12.7)

Then (12.7) is reduced (see Stegeman) to the equation 𝑑2𝑎"𝑑𝑧2 − 𝛾

m#𝑎" = 0

where 𝛾m# = 𝛾# − (

Δ𝑘2 )

#

(12.8)

For initial conditions: 𝐴" = 𝐴"P𝐴# = 0

𝐴" = 𝑎"𝑒67∆9./#, 𝐴# = 𝑎#𝑒67∆9./#make a substitution:

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In terms of intensities:

𝑰𝟏 = 𝐼10 |𝑐𝑜𝑠h(𝛾m𝑧) + 𝑖∆𝑘2𝛾m 𝑠𝑖𝑛h(𝛾

m𝑧)|2

=𝑰𝟏𝟎 [𝒄𝒐𝒔𝒉𝟐(𝜸m𝒛)+ ∆𝒌𝟐𝜸n

𝟐𝒔𝒊𝒏𝒉𝟐(𝜸m𝒛)]

𝑰𝟐 =𝝎𝟐𝝎𝟏

𝑰𝟏𝟎𝜸𝜸m

𝟐 𝒔𝒊𝒏𝒉𝟐(𝜸m𝒛)

(12.10)

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What happens if 𝛾# − (w9#)# < 0 ?

cosh → cosand

sinh → sin

periodic solution instead of steadily growing one

𝛾# − (w9#)# < 0 𝛾# − (w9# )

# > 0

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𝐼2 =(*

()𝐼10

CCn

2 sinh2(𝛾m𝐿) → (*

()𝐼10

C}~*

2 sin2(w9# 𝐿)=(*

()𝐼10 𝛾2

��#(}~* �)}~* �

#=

Test (12.10) for 𝛾 ≪ w9# – Low efficiency DFG process to produce 𝐴#and 𝐴# ≪ 𝐴"P

𝛾m → 𝑖Δ𝑘2

𝛾 = 𝑔𝐴3 =𝑑𝑐

𝜔"𝜔#𝜔3𝑛"𝑛#𝑛3

𝐴3 =2𝜔"𝜔#𝜀P𝑐3

(𝑑#

𝑛3) 𝐼(+the ‘usual’ gain increment

From (12.10)

sinh(𝛾m𝑧) → sin(Δ𝑘2z)

=𝜔#𝜔"

𝐼102𝜔"𝜔#𝜀P𝑐3

𝑑#

𝑛3 𝐼(+𝐿2sin2(Δ𝑘2 𝐿)Δ𝑘2 𝐿 2

= 𝐼10𝐼302𝜔#2

𝜀P𝑐3𝑑#

𝑛3 𝐿2 sin𝑐2(Δ𝑘2 𝐿)

- get the ‘usual’ formula for low efficiency DFG process with phase mismatch

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Optical parametric processSensitivity to phase mismatch at high gain

When the gain increment 𝛾becomes large, the interaction becomes less sensitive to phase mismatch Δ𝑘

𝛾𝐿 = 3;gain = 100

𝛾𝐿 = 10;gain = 108

𝛾𝐿 = 30;gain = 3 � 1025

gain = 𝑐𝑜𝑠h2(𝛾′L)𝛾𝐿 = 0.9;gain = 2

norm

aliz

edga

in

Δ𝑘2𝐿

𝜋 2𝜋−2𝜋 −𝜋 0

𝛾# − (w9#)# < 0

𝛾# − (w9# )# > 0

-oscilaltory

𝛾m# = 𝛾# − (Δ𝑘2)#

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Four types of the optical parametric devices

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Optical parametric amplifier (OPA)

nonlinear crystal

strong pump ω3

out ω1

out ω2

where - on-axis gain increment

𝐼(+ – pump intensity (power density) (W/cm2)

d=deff – nonlinear coefficient (pm/V)

For (γL) <<1, cosh2(γL) ≈ 1+(γL)2

For (γL) >>1, cosh2(γL) ≈ ¼ exp(2γL)

Parametric gain coefficient 𝐺 = cosh2(𝛾𝐿)

Pout(ω1) = G x Pin(ω1)

Pump Pump intensity Parametric gain

CW laser, few W

Nanosecond pulses ~ 1 mJ/pulse

Intense ps or fs pulses ~ 1 mJ/pulse

~ 5 kW/cm2

~ 1 MW/cm2

~ 1 GW/cm2

few %

~ 10

~ 1010

input signal at ω1

𝛾 =2𝜔"𝜔#𝜀P𝑐3

(𝑑#

𝑛3) 𝐼(+

cosh2 is a fast growing function

fractional gain

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Optical parametric generator (OPG)

nonlinear crystal

strong pump ω3

where 𝑛"P and 𝑛#P are the inputs at 𝜔" and 𝜔"; 𝛾 =2𝜔"𝜔#𝜀P𝑐3

(𝑑#

𝑛3) 𝐼(+

From (12.3) and (12.4) and it follows that if there is no input : 𝐴"P = 𝐴#P = 0, the output is zero

However if no field is injected into the NLO medium, the input is the quantum noise associated withthe vacuum state, which is equivalent to one photon per mode.

?

𝑛" = 𝑛"Pcosh2 𝛾𝐿 + (1 + 𝑛#P)𝑠𝑖𝑛ℎ2 𝛾𝐿

𝑛# = 𝑛#Pcosh2 𝛾𝐿 + (1 + 𝑛"P)𝑠𝑖𝑛ℎ2 𝛾𝐿

Quantum theory tells (see Yariv “Quant. Electron.” book) that in fact that the amount of the output photons (per mode), 𝑛" and 𝑛# at 𝜔" and 𝜔", is given by

and (as usual)

𝐴"P = 𝐴#P = 0, the output is non-zero and is seeded by quantum noiseThus, for

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Optical parametric generator (OPG)

The principle of operation of a travelling-wave 'superfluorescent' optical parametric generator (OPG) is based on a single-pass high-gain (106–1010) amplification of quantum noise in a nonlinear crystal pumped by intense short laser pulses.

The main characteristics of the travelling-wave OPGs are:

• Simplicity (no cavity needed). One pump laser.• Ability to produce high peak power outputs (>1 MW) in the form of a single pulse.• Broad tunability, restricted only by the phase-matching and crystal transparency.• No build-up time. This allows generating synchronized, independently tunable pulses for time-

resolved spectroscopy.• The output builds from quantim noise ( ~nW) to reach the peak power comparable to that of the pump

(~ MW)

nonlinear crystal

strong pump ω3

out ω1

out ω2

quantum noise

High pumping intensity needed, typically > 1 GW/cm2

– to achieve high (> 106) gain

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Optical parametric generator and quantum noiseThe quantum noise is associated with the vacuum state, which is equivalent to one photon per modeHow to make sense of one photon per mode?

NLO crystal

gain bandwidth 𝜟𝝂

cavity length L

Cavity length Là mode spacing Δ𝜈𝑚 =𝑐𝐿

Total longitudianl modes (assume one transverse mode) 𝑁 = Δ𝜈/Δ𝜈𝑚=Δ𝜈 L/c

Total quantum noise ‘power’ entering the crystal = (total number of noise photons) x ℏ𝜔 / (period of circulation) =

𝑷𝑸𝒏𝒐𝒊𝒔𝒆 = 𝑁ℏ𝜔 /𝑇 =Δ𝜈 L/c ℏ𝜔/(L/c)= ℏ𝝎 x 𝜟𝝂 since L drops out, this is a universal formula

Example: Δ𝜈=1THz, 𝜆 = 1 µ𝑚 (ℏ𝜔 = 1.87 ×106"� 𝐽) à 𝑷𝑸𝒏𝒐𝒊𝒔𝒆 = 𝟐𝟎𝟎 𝒏𝑾~ 𝟏𝟎6𝟕𝑾

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Spontaneous parametric down conversion as a source of entangled photons Typically, low-power lasers (e.g. CW lasers) are used in this case

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Optical parametric oscillator (OPO)Singly -resonant OPO (SRO)

g = &'

()(*(+,),*,+

𝛾 = 𝑔𝐴3For small OPO gain (𝛾𝐿)2 ≪ 1

Let us find the oscillation threshold for a SRO. The OPO cavity is resonant for 𝝎𝟐. Assume the fractional intensity roundtrip loss at resonating signal wave 𝜔# to be 𝑙𝑜𝑠𝑠(#¡ ≪ 1. For example it can be associated with non-unity reflection of the incoupling/outcoupling OPO mirrors (𝑙𝑜𝑠𝑠(#¡ = 1 − 𝑅1𝑅2) or loss in the NLO crystal.

Then the threshold condition can be simply written as:

𝜔3

𝜔"𝜔#

(1 − 𝑙𝑜𝑠𝑠¡,(#) ×cosh2(𝛾𝐿)=1;

R1 R2

cosh2(𝛾𝐿) ≈ 1 + (𝛾𝐿)2

Assuming that the loss is small, ≪ 1, and ∆𝑘=0, we can write

(1 − 𝐿𝑜𝑠𝑠(#¡ )(1 + (𝛾𝐿)2) = 1

or approximately

resonant wave

NLO crystal

(12.11)(𝜸𝑳)𝟐 ≈ 𝒍𝒐𝒔𝒔𝝎𝟐𝑰

2𝜔"𝜔#𝜀P𝑐3

(𝑑#

𝑛3) 𝐼(+𝐿2 ≈ 𝑙𝑜𝑠𝑠(#¡replacing 𝜸 from (12.6)

the threshold intensity is proportional to the loss at the resonating wave

Gain

Simply put: the fractional gain per pass must equal to the fractional energy loss per pass.

fractional power gain

per pass

𝐼(+¥¦§ ≈

𝜀P𝑐3𝑙𝑜𝑠𝑠(#¡

2𝜔"𝜔#𝐿21

(d2/n3)~ 𝑙𝑜𝑠𝑠(#¡ (12.12)or

(d2/n3)

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Optical parametric oscillator (OPO)Doubly -resonant OPO (DRO)

Now the OPO cavity is resonant for both 𝝎𝟏 and 𝝎𝟐.

𝜔3

𝜔"𝜔#

R1 R2

resonant waves

NLO crystal

Then the threshold condition can be found in this way:

&-)&.

=−𝑖𝛾𝐴#∗

&-*&.

=−𝑖𝛾𝐴"∗

As before in this Lecture, we write the coupled equations for the two resonating waves

Assume the fractional intensity roundtrip loss at 𝜔" is 𝑙𝑜𝑠𝑠("¡ so that the field loss is 𝑙𝑜𝑠𝑠("¨ ="

#𝑙𝑜𝑠𝑠("¡

at 𝜔# is 𝑙𝑜𝑠𝑠(#¡ so that the field loss is 𝑙𝑜𝑠𝑠(#¨ ="#𝑙𝑜𝑠𝑠(#¡

in a DRO a fractional gain needed to achieve the threshold is very small, so we can assume that on a single pass Δ𝐴 ≪ 𝐴 and integrate these equations

Δ𝐴" =−𝑖𝛾𝐿𝐴#∗

Δ𝐴# =−𝑖𝛾𝐿𝐴"∗

by adding the losses we get field increase aftear a roundtrip

Δ𝐴" =−𝑙𝑜𝑠𝑠("¨ 𝐴" − 𝑖𝛾𝐿𝐴#∗

Δ𝐴# =−𝑖𝛾𝐿𝐴"∗ − 𝑙𝑜𝑠𝑠(#¨ 𝐴#

The threshold condition is when Δ𝐴" = Δ𝐴# = 00 =−𝑙𝑜𝑠𝑠("¨ 𝐴" − 𝑖𝛾𝐿𝐴#∗

0= −𝑖𝛾𝐿𝐴"∗ − 𝑙𝑜𝑠𝑠(#¨ 𝐴#

... to get field increase after passing the crystal

*

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Optical parametric oscillator (OPO)Doubly -resonant OPO (DRO)

For nontrivial solution the determinant 𝑙𝑜𝑠𝑠("¨ + 𝑖𝛾𝐿

−𝑖𝛾𝐿 + 𝑙𝑜𝑠𝑠(#¨should = 0, so we get

𝜸𝑳 𝟐 = 𝒍𝒐𝒔𝒔𝝎𝟏𝑬 𝒍𝒐𝒔𝒔𝝎𝟐𝑬 = 𝟏𝟒 𝒍𝒐𝒔𝒔𝝎𝟏

𝑰 𝒍𝒐𝒔𝒔𝝎𝟐𝑰 (12.13)

Comparing (12.11) and (12.13) one can see that the DRO threshold 𝐼(+¥¦§ ~ 𝛾𝐿 2

– can be dramatically lower than that for SRO

e.g. for the intensity fractional loss at 𝜔" 𝑙𝑜𝑠𝑠("¡ =4%, the DRO threshold is 100 times smaller than for SRO

However in DRO both signal and idler waves have to belong to the cavity modes simultaneously(an overconstrained system). DRO is very sensitive (interferometrically) to the tiny cavity length fluctuations and hence is less preferrd in practical applications.

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So why there is such a big difference in thresholds?

DRO: both waves are coherent and resonate

SRO: the idler wave (𝜔") does not resonate and every roundtrip has to start from scratch

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Calc. threshold for 3 OPO types

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Summary

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Literature

• Wikipedia ....

• .....

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Home work

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l12 optical parametric oscillators and amplifiers · 2019. 12. 12. · 2 3 optical parametric...

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