Lecture 14

Second-harmonic generation inside a resonator cavity.

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Second harmonic generation

Second harmonic generation is instrinsically a weak process. In the first observations of SHG by Franken et al., 2� photons (in the UV) were hardly detectable. They were made with pulsed ruby lasers and a nonlinear crystal (crystalline quartz) in which the dispersion in velocity between the fundamentalight and the harmonic light limited the interaction length to about 14 microns.

NL crystal

2� �

P. A. Franken, A.-E. Hill, C. W. Peters, and G. Weinreich, “Generation of optical harmonics, Phys. Rev. Lett., vol. 7, pp. 118-119. August 1961.

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1 Second harmonic generation

How to improve SHG efficiency at a given pump laser power ?

• Tighter focusing? • Longer crystals? 5cm? 10 cm? 2� � • Shorter pulses? • Finding crystals with higher nonlinearity deff ? • Waveguides?

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Second harmonic generation

Put many crystals in a sequence?

2� �

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2 Second harmonic generation

Put many crystals in a sequence with periodic refocusing

2� �

5 5

Second harmonic generation

Put a retroreflector for a double pass?

�

2�

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3 Second harmonic generation with a feedback

NL crystal

2� �

J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric”, Phys. Rev., vol. 127, pp. 1918-1939, September 1962.

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Resonant Optical Second Harmonic Generation

Nobel prize in Physics 2018

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4 Resonant Optical Second Harmonic Generation

.... Experimental and theoretical results are described on the enhancement of optical second harmonic generation (SHG) and mixing in KDP by the use of optical resonance. Both resonance of the harmonic and of the fundamental are considered.

... Large enhancements are possible for resonators with low loss. Using a plano-concave harmonic resonator containing 1.23-cm KDP, the authors achieved a loss < 4 percent per pass. This resulted in an enhancement of ~ 500 times the harmonic power internal to the resonator and ~ 10 times external to the resonator.

...When resonating, the fundamental enhancements of ~ 5 were observed.

... The experimental results are in substantial agreement with the theory.

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Resonant Optical Second Harmonic Generation

Used a low power (mW’s) continuous-wave (CW) HeNe laser operating at λ=1.15 µm

Idea #1 ... optical feedback or resonance can be used effectively to extend the interaction length and enhance the amount of harmonic conversion. ... the output harmonic beam is reflected back and refocused by an optical resonator in such a phase that it can continue to interact with the fundamental power. In t'his way the effective length of interaction can be increased greatly...

Idea #2 .... resonating the incident fundamental beam. This is simply a means of storing the incident energy in a high Q cavity to increase the flux of fundamental power passing through the crystal, thereby increasing the generation of second harmonic power...

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5 Resonant Optical Second Harmonic Generation

Idea #1

second harmonic fundamental

=−� � 2 Recall (6.9) from Lecture 6

� For single pass and low conversion limit, the SH field increases by Δ� = � = −� � 2 � 2 � Thus the SH power generated in a single pass is Δ� = � ~ |� |2 = ( )|� |4� 2

� Now imagine we have a SH input field � ; the SH field increases by the same amount: Δ� = −� � 2 � 2 What is an extra SH power generated in Δ� ~ Δ|� |2 = 2� Δ� ≫ |Δ� |2 a single pass with non-zero input? if the input SH field is already

large enough, � > Δ�

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Resonant properties of passive optical cavities (after Siegman, Lasers)

refl

Ring circ

inc trans

M1 M2

M3

inc circ

Planar trans refl

M1 M2

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6 Resonanting Second Harmonic

r & t for 2ω SHG inc ω

inc ω circ 2ω trans 2ω

M1 M2 circ 2ω circ 2ω

M3

Ideal scenario:

M1 - transmits 100% of the pump (ω) M1 & M3 - reflect 100% of the SH (2ω) M2 - highly reflects 2ω with field reflection coeff. r and transm. coeff. t

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Resonating Second Harmonic

r2 & t2 at 2ω r1 =1 at 2ω SHG inc ω

inc ω circ 2ω trans 2ω

M1 M2 circ 2ω circ 2ω

Δ�~ Δ� r3 =1 at 2ω

M3 � + � =1 in E-field Write balance equation for a ��, + (1 − �)�, = Δ� in intensity roundtrip (self-consistency equation) � + � = 1 (or power) 2� field all other roundtrip for the E- field at 2ω : added due losses in the reflection cavity roundtrip loss to SHG

Δ� à �, = (14.1) circulating SH E-field 1 − � + �

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7 Resonating Second Harmonic

2 |Δ�| �, circulating SH power � = = (14.2) �=�, , 2 2 1 − � + � 1 − � + �

( ) can be on the order of few % à intracavity SHG enhancement > 1000

outcoupled � � = � (14.3) ideal mirror: (transmitted) , 2 , 1 − � + � SH power � + � =1 in E-field

in intensity � + � = 1 SHG output � � (power) ��ℎ. ������ = , = = enhancement 2 �, 1 − � + � factor 1 − � 2(1 − �) = ≈ (14.4) 2 2 1 − � + � 1 − � + �

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Resonating Second Harmonic

When other losses are zero: , () ��ℎ. ������ = = = → ∞ as �→ 1 �=0, ��ℎ. ������ can be any: ,

��ℎ. ������ is maximized when outcoupling loss is equal to other losses: 1 − � = � �, 2 1 − � 2� 1 1 ��ℎ. ������ = ≈ = = = (14.5) 2 2 so-called �, 1 − � + � 2� 2� � ‘impedance matching’ roundtrip passive loss in the cavity, e.g. loss in the crystal or diel. mirrors

Example: �=1% gives you 100 times in SHG eficiency

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8 Resonating Fundamental

Idea #2 refl ω r2=1 at ω t2=1 at 2ω high r1 at ω SHG circ ω circ ω

inc ω 2ω trans 2ω

M1 M2 circ ω circ ω

r3=1 at ω M3

Ideal scenario:

M1 - high and optimized reflection r1 (in E-field) at ω M2 - reflects ω (~100%) and transmits 2ω (~100%) M3 - reflects ω (~100%)

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Resonating Fundamental

lossless incoupling mirror

� + � =1 in E-field Write self-consistency equation ��, + (1 − �)�, = ��, in intensity for the E- field at ω: � + � = 1 (or power) all other roundtrip 2� field reflection losses in the added due roundtrip loss cavity to the pump �=�,

��, �, = à 1 − � + � (14.6) circulating E-field at ω

circulating power enhancement at ω � |� |2 � 1 − � 2(1 − � ) , = , = = ≈ 2 2 2 2 (14.7) �, |�,| 1 − � + � 1 − � + � 1 − � + �

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9 Resonating Fundamental

Enhancement for circulating � 2 1 − � 2� 1 1 power at ω is maximized when , ≈ → = = (14.8) 2 2 outcoupling loss is equal to �, 1 − � + � 2� 2� � other losses: 1 − � = � roundtrip power passive loss in the cavity, e.g. loss ‘impedance matching’ - can get ‘crazy’ intracavity intnsities in the crystal or diel. mirrors

�, 1 ��� ��ℎ. ������ = | | = �, �

However, optimum conversion requires impedance matching* of the resonator – the input coupler transmission should be equal to the sum of all other losses, including the power-dependent conversion losses (which will increase with intracavity power). 1 Thus most likely ��� ��ℎ. ������ ~ � ∗ � = � = 1 − � ≈ 2 1 − � = 2� = � 19 19

Resonating Fundamental

infrared input power: 700 mW

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10 Resonating Fundamental

infrared input power: ~ 100 mW

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SHG inside a laser cavity

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11 Resonant Optical Second Harmonic Generation

what is the catch?

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Resonating Fundamental

Optimum conversion requires impedance matching of the resonator, i.e., zero reflected pump power. This is achieved when the input coupler transmission equals the sum of all other losses, including the power-dependent conversion losses.

The laser frequency needs to be actively frequency stabilized to the fundamental- wave (or SH-wave) cavity resonance.

FSR ~ 1 GHz

cavity length needs to be stable to better than 1-nm ~ 1.5 MHz accuracy for cavity loss ~1%

cavity resonances

optical frequency

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