February 1972 L E T T E R S T O T H E E D I T O R 297

Fringes of Hologram Interferometry for Simple Nonlinear Oscillations

KARL A. STETSON* Optical Metrology Division, National Physical Laboratory,

Teddington, Middlesex, England (Received 31 July 1971)

INDEX HEADINGS : Holography; Interferometry.

In a previous article,1 a solution was presented for the character­istic-fringe function of hologram interferometry for general non­linear oscillations of an object that possessed a single degree of freedom. This result was,

where M is the characteristic-fringe function, Ω is the fringe-locus function, Co is a constant chosen to make M (0) = 1, ƒ is the time function according to which all points on the object move, G0 is a constant equal to the value of G(ƒ) at t=0, and G(ƒ) is

where g(f) is the nonlinear restoring force of the system. One of the simplest and most common cases of nonlinear

oscillators is described by Duffing's equation,2 in which the restoring force is the sum of a linear and a cubic function of displacement. This communication presents a series solution for this case that offers helpful physical insight into the relation­ship between hologram interference fringes and the nonlinearities of the oscillation.

Let us assume that the restoring force has the form,

where K and β are constants. The function G(ƒ) becomes

and we assume that ƒ is a periodic function of time, bounded by ± 1 , such that ƒ= —1 and dƒ/dt = 0 at t=0. Under these con­ditions, the constant Go becomes

298 L E T T E R S T O T H E E D I T O R Vol. 62

Substituting, we obtain from Eq. (1)

Noting that

and letting

we may factor the denominator in the integrand of Eq. (5) to obtain

If we make the transformation of variables

Eq. (6) becomes

The exponential function within the integrand may be expanded in a Fourier series, with Bessel-function coefficients. Interchanging summation and integration, and recognizing that only the even-order terms survive the integration, we obtain

where the coefficients, Cm, are defined as

[Co equals the integral of Eq. (9) evaluated at m = 0 . ] From a handbook of elliptic integrals5 we obtain the following power series for the coefficients:

It is possible to see that the Bessel-function series of Eq. (8) converges quite rapidly for small values of the nonlinearity coefficient β. For large values of the fringe-locus function Ω we may approximate the Bessel functions by

Thus, the fringe function becomes, approximately,

Because, M(0) = 1, the quantity in brackets is an envelope factor that determines the irradiance of the higher-order fringes relative to the zero-order fringe. It is significant to notice that the zeros of these fringes are approximately the same as for sinusoidal-vibra tion fringes, and that only the irradiance of the higher-order fringes is affected by the nonlinearities. If we take the leading terms of Eq. (8), and substitute for ς, the result is

Correspondingly, Eq. (12) becomes

Clearly, the irradiance of the higher-order fringes decreases or increases depending upon whether the coefficient of nonlinearity is positive or negative, respectively, that is, whether the spring is hard or soft. Because the irradiance of the fringes is proportional

to the square of the fringe function, we may estimate that when the nonlinear component is 8% of the linear component, the fringes exhibit about 2% change of irradiance.

The author is greatly indebted to J. R. Bell and to G. F. Miller of the Mathematics Division of this laboratory for providing the method of solution presented herein, and to A. E. Ennos for helpful discussions of the manuscript.

* Present address: Ceramics and Glass Department, Scientific Research Staff, Ford Motor Co., Box 2053, Dearborn, Mich. 48121.

1 K. A. Stetson, J. Opt. Soc. Am. 61, 1359 (1971). 2 J. J. Stoker, Nonlinear Vibrations in Mechanical and Electrical Systems

(Wiley-Interscience, New York and London, 1950), Ch. 4, pp. 83-90. 3 P. F. Byrd and M. D. Friedman, Handbook of Elliptic Integrals for

Engineers and Scientists (Lange, Maxwell, and Springer, London and New York, 1954), p. 292, formula No. 806.01.