Second-harmonic interferometers II

Volume 36, number 6

OPTICS COMMUNICATIONS

15 March 1981

SECOND-HARMONIC INTERFEROMETERS II F.A. HOPF, A. TOMITA, G.A1-JUMAILY, M. CERVANTES and T. LIEPMANN

Optical Sciences Center, University of Arizona, Tucson, AZ 85721, USA Received 5 November 1980

Experiments on interferometers based on second harmonic generation of light are described. These make use of the distortions of phase and amplitude produced by the phase mismatch of angle-matched crystals to provide information. The interferometers are directly sensitive to small wavefront tilts and do not require additional reference wavefronts.

In a previous publication [1 ], we discussed second harmonic interferometers (SHI) using phase-matched doublers. In this letter, we discuss SHI's in which the phase mismatch of an angle-tuned doubler [2] is the key to forming interference patterns. In SHI's the information resides in the wavefront o f the incident fundamental, which induces an identical wavefront in the induced SH polarization. In phase matched operation, the radiated SH wave has the same wavefront as the fundamental. In phase-mismatched operation, both the amplitude and the phase of the SH are affected, which are the basis o f the interferometric concepts described in this work. The equation that governs the production o f second harmonic light under conditions of low efficiency reads [3] : 87r2L 2 6 G - ;kGn G d etf 6 IR

I sin( AkL/2) l ×[ ~ j (exp(iAkL/2)}.

(1)

We specialize our notation to the doubling of infrared (IR) 1.06/am light to ~G = 0.53/am, which is the case of the experiments described later. The length of the crystal is L, CIR = IC IRI exp(i¢iR) is the incident infrared wavefront (whose information resides in ¢IR), nG is the index in the green, CG is the complex amplitude o f the second harmonic defined in the same way as ~ IR, and d etf is the nonlinear coefficient. The factor Ak = k G - 2klR is given for a Type I

doubler by Ak = 2zr(n G - nlR)/~.G. (In a Type II doubler the algebra is more complicated and is not discussed explicitly.) In the case of angle or critical phase-matching, (CPM) one of the indices (usually riG) depends on the angle of tilt, 0, between the phase front normal and the optical axis, and this dependence is utilized to set Ak = 0 (at the angle 0 = 0pM ). Tilts of the wavefront cause Ak to depart from zero (phase mismatch). This results in the amplitude and phase perturbations in the green beam described by the two terms in brackets in eq. (1) which are the basis of the two interferometers described here. The first is based on the effect of the mismatch introduced by the wavefront aberration on the amplitude (first bracket) and is referred to as a Maker fringe interferometer (MF). The other, called the nonlinear induced shear interferometer (NLIS), depends upon the effect of the mismatch on the phase term (second bracket). The principle o f the MF is based on the fact that the first term in brackets in eq. (1) vanishes whenever AdcL/2 = mrr (m can be all integers but zero), giving dark fringes in the intensity of the SH. This is the familiar Maker fringe, which is discussed in standard texts [3]. Suppose we take the example of an ooe (Type T) interaction in which case n G is angledependent and nlR is angle-independent. Let us take the coordinate z to describe the direction of propagation o f the light, and let the optic axis lie in the yz plane. We then cally and z the critical and noncritical dimensions respectively. The Maker fringe then appears 487

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in the y-direction whenever 3 2 2 . 1 L nIR(no -- he)Sin 20pM (0 -- 0pM ) = m 2 XG n2n 2 O

(2)

e

Here nlR is the ordinary index at XIR , n o and n e are the ordinary and extraordinary indices at )'GThis interferometer is well suited to measuring small wavefront distortions (GX/2) that extend across the full width of a laser beam. Cases where this is useful is in the fine alignment o f collimators and the measurement of small amounts of self-focusing. The shearing interferometer, which is a logical alternative for these tasks, is significantly less sensitive than the MF. Let us now consider the effect of the term in the second bracket (i.e. phase), and let us suppose, first, that AkL/2 < rnrr everywhere in the pattern, i.e. there are no Maker fringes. In this case the phase term causes a phase aberration in the green whose local magnitude is equal to ~ times the left side of eq. (2). The NLIS can be constructed using two SHG crystals. If the first crystal is noncritically phase matched (NCPM) it generates wavefront in the green that is a faithful replica of the wavefront o f the infrared [1], which is used as a reference wave. The SH generated from the second crystal, which is critically phase-matched, has an additional phase distortion due to the phase-mismatch. By observing the interference pattern between these two SH's, one can measure the amount o f the induced phase distortion, which is related to the local wavefront tilt according to eq. (2). A somewhat more complicated version which we use in the experiments since we do not have a NCPM crystal, is to use for the first crystal a CPM crystal whose critical dimension is orthogonal to that of the second doubler. The interferometer is then sensitive to both dimensions, since the aberrations induced by the two crystals are orthogonal to each other. A potential application of the NLIS is to enhance a phase distortion whose depth is small to be measured by conventional interferometers (e.g. T w y m a n - G r e e n ) . Since the induced phase distortion depends on the local wavefront tilt, which can be changed by the lateral magnification of the wavefront, the induced phase change can exceed the original phase distortion. In the experiments, we use a 100 kW Q-switched 1.06 #m YAG laser and angle phase-matched cubicshaped crystals 1 cm on each side. Our primary anal488

15 March 1981

yzing crystal is always a CPM Type I LilO 3. When a second crystal is used it is a CPM Type II LiCHO 2 or KDP. The first experiment consists of the setup shown in fig. 1. A collimating beam expander using lenses of 23 and 260 m m focal lengths was set up, in which the second lens was an achromat. The KDP doubler was placed before the first lens which, when phase matched, produced ample green light, and an aluminum foil was placed at the focal plane of the first lens, where a pinhole was burned. This pinhole eliminated the chromatism of the first lens, and resulted in the collimator being achromatic. There was a slight residual chromatism due to the fact that the achromat was not optimized for our wavelengths. A 2 ° glass wedge, whose purpose is explained below, and a LilO 3 crystal was placed after the collimator. In order to illustrate the principles of the interferometers, we tested the beam coming from the collimator. When the KDP was tuned far from phase matching, no green light emerged from the collimator and the doubling o f the infrared by the LilO 3 doubler produced the pictures in figs. 2 a - c , where the colli-

I ×1

Ll

P

L2

W

] X2

I nterferograms

MF (No Green produced by c r y s t a l X I . )

NLIS (Crystal X I producing Green.)

Fig. 1. Schematic of experiment: X 1,2: angle matched crystals, L 1,2 : lenses, P: pinhole, W: 2° wedged glass blank. In the MF interferometer (see figs. 2a-c) the crystal X 1 is mistuned to generate no second harmonic. In the NLIS the crystal is tuned to generate second harmonic. In fig. 3 the data is taken with X 1 between L2 and the wedge W, and the object is between L2 and X l .

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OPTICS COMMUNICATIONS

15 March 1981

Fig. 2. (a)-(c) Maker fringe interferograms. Collimator giving slightly converging (a), collimated (b), and slightly diverging (c) beams. (d)-(e) Interferograms taken by NLIS, in which green light is generated before the collimator. Adjustments of the collimator for (d)-(f) correspond to those for (a)-(c) and the horizontal fringes in (d), (e) and (f) are the Fresnel fringes produced by the edges of the crystal and should be ignored. mator gave slightly converging (fig. 2a), collimated (fig. 2b), and slightly diverging (fig. 2c) light. The Maker fringes are visible in figs. 2a and 2c as horizontal black bands. Bright fringes above and below the central spot are visible to the eye but the film does not have the dynamic range to record them. Right after each picture was taken the KDP was phasematched to produce green light. This gave the interference patterns in figs. 2 d - f which correspond to the figs. 2 a - c respectively. The vertical fringes are due to the tilt between the IR and green produced by chromatism of the wedge, and the slight curvature in the fringes is an artifact of the residual chromatism of the achromat (this will be ignored in subsequent discussion). When the beam is a plane-wave the fringes are vertical as seen in fig. 2e. When the wavefronts are curved, the fringes are tilted as seen in figs. 2d and 2f. This tilt comes from the displacement of the disturbance due to the linear dependence of Ak on 0 in eq. (2). The displacement effect is similar to those o f shearing interferometer. When the "sinc" function in eq. (1) changes sign, the bright and dark fringes inter-

change which gives an apparent motion of X/2 to the fringes at the location of the Maker fringe. This is seen in figs. 2d and 2f. Depending on circumstances, this can be viewed either as a very useful way of locating the Maker fringes (e.g. in a case where the object is scratched or pitted, so that the Maker fringe contrast is poor), or as a confusing feature of the NLIS. We next show that the NLIS is capable of giving a larger fringe motion than is present in the original phase distortion. This is due to the dependence of Ak on the slope rather than on the depth of the wavefront distortion. We begin by removing X 1 (i.e. the KDP) from the setup in fig. 1 and use instead a LiCHO 2 crystal that we locate just in front of the wedge. The critical dimension of the LiCHO 2 is orthogonal to that o f the LilO3, and we use a one dimensional phase object oriented so that the LiCttO 2 acts as if it were noncritically-phase matched (i.e. it generates a wavefront at 0.53/am that is a faithful replica of the wavefront at 1.06/am). The phase-object is a groove of 0.14 mm width polished in a microscopic 489

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15 March 1981

Fig. 3. (a) Interferogram of index-matched groove taken by a Tamin interferometer. (b) NLIS interferogram of the same object. slide. It is partially index matched to have a XG/10 depth as shown by a Jamin interferometer in fig. 3a. In order to increase the slope, we demagnify the object b y a factor of two with a collimating lens pairs. The resulting NLIS interference pattern is shown in fig. 3b. One sees that there is a XG/5 fringe motion, which is twice that of the fringe m o t i o n in fig. 3a. Hence we have demonstrated that the NLIS is capable of enhancing the fringe motion, which cannot be done with conventional shearing interferometers. We have performed a number o f tests to compare the sensitivity o f the SHI's with a wedged shearing interferometer. While the overall comparison is complicated, as a rule, the shearing interferometer reaches marginal sensitivity for phase front curvatures that are easily measured b y the MF and NLIS. The NLIS is more sensitive than the MF and is much more sensitive than the shearing interferometer. Note however, that the sensitivity of NLIS depends on the difference between n o and n e. Crystals with large (n o - he) such as LilO 3 are more sensitive to tilts than those with small (n o - ne) such as KDP. In conclusion, we have tested the concept of two SHI's in which phasemismatching of the doubling process is used to measure local wavefront tilt. As with the other SHI's [1 ], they give interferograms in 490

the visible when the laser operates at near infrared such as a YAG laser and they are self-referencing. They are suited to testing the degree of collimation of a wavefront with small curvature. The NLIS is able to detect a phase front distortion which is ~<0.1 X. The MF is easy to operate, and is particularly useful for detecting small changes in the wavefront curvature of an unexpanded gaussian beam and for locating the beam waist accurately, tasks which shearing interferometers do not perform as well. We have used b o t h of these interferometers in looking for self-focusing, which will be reported in a future publication. This work is supported by the Air Force Office of Scientific Research under the grant AFOSR-76-3077. One of the authors (MC) gratefully acknowledges the financial support provided by a CONACYT followship.

References [1] F.A. Hopf, A. Tomita and G. A1-Jumaily, Optics Lett. 5 (1980) 386. [2] P.D. Maker, R.W. Terhune, M. Nisenoff and C.M. Savage, Phys. Rev. Lett. 8 (1962) 21. [3] F. Zernike and J.E. Midwinter, Applied non-linear optics (Wiley, New York, 1973).