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Perturbation theory for frequency doubling and tripling of electric field amplitude and phase ripples

J. M. Auerbach D. Eimerl J. T. Hunt D. Milam

J. B. Trenholme P. W. Milonni

S olid-S tacz

This paper was prepared for submittal to the 1st Annual International Conference on

asers for Application to Inertial Confinement Fusion (ICF) Monterey, CA

May 30-June 2,1995

June 27,1995

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Perturbation theory for frequency doubling and tripling of electric field amplitude and phase ripples

J.M. Auerbach, D. Eimerl, J.T. Hunt, D. Milam, J.B. Trenholme

Lawrence Livermore National Laboratory Livermore, CA 94550

P.W. Milonni

Los Alarnos National Laboratory Los Alamos, NM 87545

ABSTRACT

A perturbation theory has been developed to calculate the transfer of electric field amplitude and phase ripples from the first harmonic to either the second harmonic or the third harmonic. The theory is restricted to steady-state conversion processes. In the case of small phase gradients, the real and imaginary parts of the output harmonic ripple are related to the real and imaginary parts of the input perturbation by a 2x2 matrix. To confirm the validity of the perturbation theory, we have performed an initial set of experiments on the Optical Sciences Laser to investigate the transfer of a weak ripple from the first harmonic to the second harmonic.

2. RIPPLE PERTURBATION THEORY

The theory describes the conversion of a ripple on the electric field input to the converter to the ripple on the output harmonic. The main part of the input electric field is assumed to be a plane wave; hence, the coefficients of the output harmonic ripple depend on the plane-wave solution of the frequency conversion equations. The full ripple perturbation theory accounts for the following ripple properties:

Amplitude of ripple Phase of ripple Direction of ripple (k-vector)

For current converter designs for Beamlet and NIF,1-2 ripple direction affects the conversion only when ripple angles are a few ten’s of rnicroradians. For phase ripples of magnitudes of a few hundredths of a wave, these angles occur only when the transverse scale lengths of the ripples are no more than a few tenths of a millimeter. Most of the modulation in Beamlet beams and simulated NIF beams has scale lengths much larger than this. Hence these beams can be analyzed with a simpler version of the ripple transfer theory which ignores ripple direction. This is called the “print-through’, model. In this model, the amplitude and phase distributions keep their initial spatial distribution but are scaled by parameters determined from the plane-wave solution for the converter. This part of the theory will be the subject of this paper. The theory will be described in terms of tripling. The converter is considered to consist of 2 units, a doubler and a tripler.

.

‘

The initial lo electric field is taken to be of the form:

p is the perturbation and is a complex quantity.

The output 3 0 field can be expressed as the product of a field conversion efficiency and a phase factor:

q is in general a function of intensity and beam direction. In the case where ripple direction is neglected, is only a function of intensity so it can be written to first order in the perturbation quantities as:

q1/2 can then be written to first order in p as:

?11/2 = qo1'2 [1 -k (A - I)] PR

where

A=-- 1 &770 r7 31

The phase of the 30 field, ~3~ at the tripler output is given by:

1 930 = 2910, tn + qlo,rnid + z(dK24 + M3S)

where

q ~ ~ ~ . ~ ~ = phase of l w wave at doubler input qio.rnjd = phase of l w wave at tripler input

AI&, 4 = phase mismatch, thickness, of doubler respectively M3, 4 = phase mismatch, thickness, of tripler respectively

(4)

The plane-wave part of the phase is a constant. The problem is now to relate the components of the perturbation to the phase. To first order in p , the phase of the 3 0 field depends on the perturbation quantities as follows:

qs0 = vp, + 3p, + constant

The factor of 3 arises from the fact that in tripling the initial lo phase is increased by a factor of 3. The factor of v is due to the fact that the residual la field coming out of the doubler has a phase3 which depends on the lo intensity in the doubler. v is given by the following expression:

where q 2 = doubler conversion efficiency and

AIS2 is' the plane-wave phase mismatch factor and is non-zero when the doubler is detuned. Thus to first order in p, the 3 0 electric field is given by:

= E , q ~ [ l + ( A + i v ) p , +3ip,]

Eo includes the plane-wave phase changes.

An interesting mapping of the ripple transfer occurs when one considers perturbations of the form:

p = Rei9

R is a constant and 0 varies between 0 and 27c. Figure 1 shows the mapping between the lo values of p (circle) and the corresponding values of the 30 perturbation for the case of the NIF baseline converter

Fig. 1. The mapping of the p and p' perturbations in the complex plane. This is for a converter consisting of an 1 1-mm KDP doubler detuned at 232 p a d and a 9-mm KD*P tripler. The plane wave intensity operating

point is Io = 3.25 GW/cm2. Doubling detuning causes the small clockwise rotation of the ellipse.

and for a plane-wave operating point of 4 GW/cm2. The mapping shows that the circle has been mapped onto an ellipse with an eccentricity of about 1:3. The parameter A has a value close to 1 at the 4 GW/cm* operating point. The clockwise rotation of the ellipse is caused by the non-zero value of v. If the doubler detuning is zero, the ellipse is oriented vertically. From Equation 1 1, one can see that the real and imaginary parts of the 3 0 perturbation, p', are related to the real and imaginary parts of the 10 perturbation by a 2x2 matrix.

The ripple transfer matrix indicates that on the flat part of the piane-wave frequency tripling curve (A approximately equals l), a phase ripple amplifies much more than an amplitude ripple. These larger phase ripples in turn affect the focusability of the 3 0 beam.

3. RIPPLE TRANSFER EXPERIMENTS

We have performed a first set of experiments on the Optical Sciences Laser to verify the ripple transfer theory. The experiments were performed for frequency doubling. A 13-mm KD*P crystal served as the doubler. The input lo field consisted of a strong main beam and a weak wave tilted at 760 p a d to the main wave and having an intensity down by a factor of 0.0014 relative to the main wave. The form of the input electric field is thus of the form:

where L = 1.3 1 mm and a =0.0374.

For this type of input field, the ripple theory utilizes an input field of the form:

E,; = E, (1 + aeia + pe-im)

and the output field is of the form:

= E~,(I+ areia +p/e-'")

For frequency doubling, the perturbation coefficients are related by a 2x2 matrix:

where

1 2

A =-(A + 2)

and

1 2

B = -(A - 2 )

A has been defined above (Equation 6). For the 13 mm KD*P doubler, A will be large and positive for low ktensities whereas B will be large and negative for operating points with large back conversion (high intensities and large detuning angles). For this latter situation, the ripple theory predicts for just an initial weak wave at 760 prad that there will be a “conjugate” weak wave in the green. That is to say, a wave at negative half the angle of the input weak wave. In the experiment, we observed a strong conjugate wave when the lo intensity was about 5 GW/cm2 and the detuning angle was 500 prad. At low intensities (approximately 1 GW/crn2), the conjugate wave was not observed and the doubled wave at an angle of 380 pad was strong. We have made a comparison between the measured and calculated amplitudes of the wave at 380 prad for the case of low intensities. The ratio of the intensities of the wave at 380 p a d to the main wave were calculated using the ripple theory expression for A and compared to the ratio of the intensity of the measured wave at 380 p a d and the main wave. The results are shown in Table I. The first entry in the table is the mean intensity of the input lo pulse and the second entry is the doubler detuning angle (prad). There is good agreement within the errors of the measurement. A more detailed analysis calculating A at each time value of the pulse and then averaging those values will yield better agreement.

.

~~ 1 .81 I 57 1 2.62 1 3.0 2-05

I 1.13 1 57 1 3.24 I 3.6 kO.5

I 1.34 I IOU I 3.14 1 2.9-tO.4

i 4.48 1 150 1 3.07 I 2.5 & 0.4

Fig. 2. Comparison of calculated and measured intensity perturbation values at an angle of 380 prad as a function of mean lo intensity I(GW/crn2) and internal doubler detuning angle, G D (pad).

4. ACKNOWLEDGMENT

We acknowledge the efforts of Walter Sell in carrying out the experiments on the Optical Sciences Laser. This work was performed under the auspices of the US Department of Energy by the Lawrence Livermore National Laboratory under Contract No. W-7405-ENG-48.

5. REFERENCES

1. B.M. Van Wonterghem, C.E. Barker, W.C. Behrendt, J.R. Murray, I.C. Smith, J.H. Campbell, D.F. Browning, D.R. Speck, “System Description and Initial Performance Results for Beamlet,” ICF quarterly Report, 5( l), 1- 17, Lawrence Livermore National Laboratory, Livermore, CA, UCRL-LR- 10582 1-95- 1 ( 1994)

2. C.E. Barker, D. Milam, R. Boyd, “High Fluence Third Harmonic Generation,” ICF quarterly Report, 3(2), 55-62, Lawrence Livermore National Laboratory, Livermore, CA, UCRL-LR- 10582 1-93-2 (1993)

3. J.A. Armstrong, N. Bloembergen, J. Ducuing, and P.S. Pershan, “Interactions between Light Waves in a Nonlinear Dielectric,” Physical Review, 127, 1918-1939 (1962)

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