Lecture 12

Optical parametric oscillators and amplifiers. Singly- and doubly- resonant oscillators.

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Optical parametric process

Only one strong input is present

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1 Optical parametric processes

Amplifier: Weak signal at � is amplified, and � is generated or weak signal at � is amplified, and � is generated

Generator: Both � and � (signal and idler) are generated from quantum noise

Oscilaltor: Both � and � (signal and idler) are generated from 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

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

∗ ∆ NL coupling coefficient =−�� ��� g = ∗ ∆ =−�� � � � (5.7) =−��� � �∆

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) � to be ∗ real: �=� ; we also assume � is constant – undepleted pump approximation, ∆�=0 (no phase mismatch), and no absorption.

define � = �� and get: � =−���∗ �� =−���∗ *

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for the normalized filed 2 2 2 2 intensity |�| |�| =� |�| |�| �

2 Optical parametric amplification

2 2 � � � � à − �� = 0 and − �� = 0 ��2 ��2

The general solutions ± to this equation are: � or ���h(��) & ���h(��)

� = ����h(��)+�sinh(��) Look for solutions in the form: � = ����h(��)+�sinh(��)

� � For initial conditions: � = � � = � � � � � Solution: ∗ � = ����h(��) − ��sinh(��) ∗ (12.1) � = ����h(��) − ��sinh(��)

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Optical parametric amplifier (OPA)

Now if we have input at only one wave:

Initial conditions: � = � � � � = 0 � Solution: � � � = � ���h(��) weak input (12.2) (‘seed’) ∗ � = −�� sinh(��)

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

The corresponding photon-flux densities are: In terms of intensities:

2 2 Φ1 = Φ10���h (��) �1 = �10 ���h (��) (12.4) 2 (12.3) � Φ = Φ ���ℎ (��) 2 2 10 �2 = �10 ���ℎ (��) �

These expressions are symmetric under the interchange of � and � Also, check for Manley- Rowe 6 6

3 Optical parametric amplification

for ��>2 ���h2(��)= (�+�)2 ≈ �/4 (12.5)

– so that the gain increases exponenially with ��. This behavior of monotonic growth of both waves is different from that of sum-frequency generation, where x oscillatory behavior occurs. �� 2� Recalling that � = � � à � = 2 ��� The gain � coefficient � ��� ����� � � = �� = � = � ( �) ��� � ��� ��� � (12.6) 8� � = ( ) � ���� �

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Optical parametric process

∗ ∆ =−���� Now, allow for the phase mismatch: ∆� ≠ 0 à (12.7) =−���∗ �∆

∆/ ∆/ make a substitution: � = �� , � = �� 2 � � Then (12.7) is reduced (see Stegeman) to the equation − � � = 0 (12.8) ��2

where Δ� � = � − ( ) 2

For initial conditions: � = � The solution is: � = 0 ∆� � = � [���h(��) + � ���h(��)]�∆/ 2� (12.9) � = −��∗ sinh(��) �∆/

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4 Optical parametric process

In terms of intensities:

∆� � = � |���h(��) + � ���h(��)|2 � 10 2� ∆� =� [�����(��) + ������(��)] �� �� (12.10)

� � � � � �� = ��� ���� (� �) �� �

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Optical parametric process

What happens if � − ( ) < 0 ?

periodic solution instead of steadily growing one

cosh → cos and sinh → sin � − ( ) < 0 � − ( ) > 0

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5 Optical parametric process

Test (12.10) for � ≪ and � ≪ � – Low efficiency DFG process to produce �

Δ� � ��� 2�� � the ‘usual’ � → � � = �� = � = ( ) � � � � � � � � gain 2 increment

Δ� sinh(��) → sin( z) 2

() From (12.10) 2 2 2 2 2 �2 = �10 sinh (� �) → �10 sin ( �)= �10 � =

� � 2� � � sin2( �) 2� 2 � � = � � �2 2 = � � �2 sin�2( �) 10 10 30 � �� � � 2 �� � 2 2 �

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

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Optical parametric process Sensitivity to phase mismatch at high gain

Δ� � − ( ) > 0 � = � − ( ) 2 �� = 30; gain = 3 1025 When the gain increment � �� = 10; becomes large, the interaction gain = 108 becomes less sensitive to phase

mismatch � gain �� = 3; gain = 100

�� = 0.9; normalized gain = 2 gain = ���h2(�′L) Δ� � 2 −2� −� 0 � 2�

� − ( ) < 0 -oscilaltory

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6 Optical parametric process

Four types of the optical parametric devices

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Optical parametric amplifier (OPA)

strong pump ω3 out ω2

Pout(ω1) = G x Pin(ω1) cosh2 is a fast growing function input signal at ω1 out ω1 nonlinear crystal

Parametric gain coefficient � = cosh2(��)

where 2�� � - on-axis gain � = ( ) � increment �� �

2) � – pump intensity (power density) (W/cm d=deff – nonlinear coefficient (pm/V) fractional gain For (γL) <<1, cosh2(γL) ≈ 1+(γL)2

For (γL) >>1, cosh2(γL) ≈ ¼ exp(2γL)

Pump Pump intensity Parametric gain

CW laser, few W ~ 5 kW/cm2 few % Nanosecond pulses ~ 1 mJ/pulse ~ 1 MW/cm2 ~ 10 Intense ps or fs pulses ~ 1 mJ/pulse ~ 1 GW/cm2 ~ 1010

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7 Optical parametric generator (OPG)

strong pump ω3 From (12.3) and (12.4) and it follows that if there ? is no input : � = � = 0, the output is zero

nonlinear crystal

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

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

2 2 � = �cosh �� + (1 + �)���ℎ ��

2 2 � = �cosh �� + (1 + �)���ℎ �� 2� � � � = ( ) � where � and � are the inputs at � and �; and (as usual) �� �

Thus, for � = � = 0, the output is non-zero and is seeded by quantum noise

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Optical parametric generator (OPG)

strong pump ω3 out ω2 High pumping intensity needed, typically > 1 GW/cm2 6 quantum noise – to achieve high (> 10 ) gain out ω1 nonlinear crystal

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)

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8 Optical parametric generator and quantum noise

The quantum noise is associated with the vacuum state, which is equivalent to one photon per mode How to make sense of one photon per mode?

cavity length L NLO crystal

gain bandwidth �� � Cavity length L à mode spacing Δ� = � �

Total longitudianl modes (assume one transverse mode) � = �/��=� L/c

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

������� = �ℏ� /� =Δ� L/c ℏ�/(L/c)= ℏ� x �� since L drops out, this is a universal formula

� Example: Δ�=1THz, � = 1 µ� (ℏ� = 1.87 ×10 �) à ������� = ��� �� ~ �� �

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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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9 Optical parametric process

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Optical parametric oscillator (OPO) Singly -resonant OPO (SRO) (d2/n3) Let us find the oscillation threshold for a SRO. The OPO cavity is resonant for ��. Assume the fractional intensity roundtrip loss at R1 R2 resonating signal wave � to be ���� ≪ 1. For example it can be NLO crystal associated with non-unity reflection of the incoupling/outcoupling � OPO mirrors (���� = 1 − � � ) or loss in the NLO crystal. 1 2 � � Then the threshold condition can be simply written as: Gain resonant wave 2 (1 − ����,) ×cosh (��)=1; fractional power gain � = �� For small OPO gain (��)2 ≪ 1 per pass g = cosh2(��) ≈ 1 + (��)2 Simply put: the fractional gain per pass must equal Assuming that the loss is small, ≪ 1, and ∆�=0, we can write to the fractional energy loss per pass. (1 − ����)(1 + (��)2) = 1

� � or approximately (��) ≈ ������ (12.11) the threshold intensity is 2�� � �� ���� 1 proportional to 2 � ≈ ~ ���� (12.12) replacing � from (12.6) ( ) � � ≈ ���� or 2 2 3 the loss at the � � � 2��� (d /n ) resonating wave

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

Now the OPO cavity is resonant for both �� and ��. R1 R2 Assume the fractional intensity roundtrip loss NLO crystal at � is ���� so that the field loss is ���� = ���� � at � is ���� so that the field loss is ���� = ���� � �

Then the threshold condition can be found in this way: resonant waves As before in this Lecture, we write the coupled equations for the two resonating waves

in a DRO a fractional gain ... to get field increase ∗ needed to achieve the after passing the crystal =−��� by adding the ∗ threshold is very small, so we Δ� =−����� − ���� ∗ losses we get can assume that on a single Δ� =−���� field increase ∗ ∗ =−��� pass Δ� ≪ � and integrate aftear a roundtrip Δ� =−���� − ���� � these equations ∗ Δ� =−����

The threshold condition is when Δ� = Δ� = 0 ∗ 0 =−����� − ���� ∗ 0= −���� − ����� *

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

For nontrivial solution the determinant ���� + ��� should = 0, so we get −��� + ����

� � � � � � (12.13) �� = ������������ = � ������������

2 Comparing (12.11) and (12.13) one can see that the DRO threshold � ~ �� – can be dramatically lower than that for SRO

e.g. for the intensity fractional loss at � ����=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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11 Optical parametric process

So why there is such a big difference in thresholds?

DRO: both waves are coherent and resonate

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

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Optical parametric process

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12 Optical parametric process

Calc. threshold for 3 OPO types

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Optical parametric process

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16 Literature

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