698 L E T T E R S TO T H E EDITOR Vol. 62

Envelope Factors Due to Laser Modulation in Time-Average, Holographic,

Vibration Analysis KARL A. STETSON

Scientific Research Staff, Ford Motor Company, Dearborn, Michigan 48121 (Received 1 December 1971)

INDEX HEADINGS: Holography; Interferometery; Modulation of beams.

This letter presents an analysis of the relationship between the laser modulation and the fringes that appear in the reconstruction of a time-average hologram. Its specific objective is to evaluate the relative gain of fringe brightness due to laser modulation against the increase of exposure time that such modulation requires. Many people have objected to the decrease of brightness of the high-order fringes obtained in the reconstructions from time-average holograms of vibrating objects. These fringes follow a zero-order Bessel function, and the irradiance of any bright fringe is inversely proportional to fringe-order number. Strobo-scopic techniques have been suggested1-3 that compensate for this by illuminating the object only when it is nearly stationary, that is, at the extreme positions of its vibration cycle. In the limit, such techniques would yield fringes following a cosine function, for which fringes of all orders would have equal brightness. Unless a pulsed laser were used that could be synchronized to the the vibration frequency of the object, it would be very difficult to approach this limit without requiring an infinitely long time to obtain an exposure suitable for developing the hologram. With a continuous gas laser, the actual system must operate between the limits of no laser modulation and modulation such as to limit the illumination to infinitely short pulses. Although elaborate modula­tion schemes have been developed to attack the fringe-brightness problem,4'5 the analyses presented have not dealt with the question of cost in exposure time vs benefit in fringe brightness of elementary modulation systems. For that reason, the discussion of this letter will center on modulation of the laser output that is predisposed to enhance the brightness of high-order fringes in the reconstruction of the time-average hologram. Such modulation will be expected to leave the laser output maximum when the

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object occupies its extreme positions; that is, it will have a period half that of the object's vibration, and it is expected to be a symmetrical function with respect to time about any instant at which the object is stationary.

If we record a time-average hologram of an object that vibrates sinusoidally, while modulating the irradiance of the laser, the normalized characteristic function of the reconstructed fringes may be defined as4

where M (Ω) is the characteristic fringe function, Ω is the fringe-locus function relating the object motion to the fringe loci, T is the exposure time, ƒi(t) is the irradiance modulation (0≤ƒ i ( t )≤1) , ω is the object's vibration frequency in radians, t is time, and

If fi(t) is periodic, with period equal to π/ω so that it executes two cycles in every cycle of the object's vibration, and if fit) is even, i.e., fi(t) =ƒ i(—t), then in the limit of large T, Eq. (1) may be rewritten in the series form

where θ=ωt and

The coefficients of the even-ordered Bessel functions that form this series are, in fact, the Fourier cosine-series coefficients of the modulation function, ƒi(/). Defining these coefficients as

we may rewrite Eq. (3) as

The characteristic function has the form of the familiar zero-order Bessel function, plus varying amounts of the even-ordered Bessel functions. If the amount of vibration is large, i.e., Ω≫0, so that the fringes are of high order, then we may use the approxi­mate form of Bessel functions of the first kind to obtain

This indicates that, whether the laser is modulated under these conditions or not, and whenever the vibration amplitude is large, the zeros of the fringe function always occur at the same values of the fringe-locus function, at least to first-order approximation. This means, in turn, that the fringes will remain fixed upon the object and continue to denote the same amounts of vibration, even though their contrast may change.

Returning to Eq. (5), we see that the value of the fringe func­tion will be unity where the object is not moving, i.e., where Ω = 0, so that the quantity in brackets in Eq. (6) represents the average field amplitudes in any high-order bright fringe, relative to what it would be in the absence of modulation. Let us define the quantity in brackets to be an envelope factor. This factor multi­plies the envelope of the Bessel function, in the region of high-order fringes, and affects their brightness without changing their over-all appearance or their location. We may see by inspection that this factor equals the sum of the Fourier coefficients of ƒi(t) divided by the zeroth coefficient.

Now let us assume that the laser is completely turned on at each instant the object occupies an extreme position of its vibra­tion cycle. This requires that ƒ (0) = 1 in addition to the conditions stated above. If we express fi(t) in terms of its cosine series, i.e.,

fi(θ) = ∑ n = 0∞ Cn cosnθ, we can see that ƒi(0) = 1 = ∑,n=0

∞ Cn. The irradiance of the bright fringes will be proportional to the square of the fringe function expressed in Eq. (6), so we may write

This indicates that the irradiance of the high-order fringes does not depend upon the shape of the modulation function, provided that it is unity when the object occupies an extreme position, is even, and always transmits the same fraction of the total laser power. For example, if the laser is modulated according to a square wave at twice the vibration frequency so that it is on for half the time and off for half, the envelope factor will be 2 and the irra­diance in the high-order fringes will be four times that of simple Jo fringes. Sinusoidal modulation at the same frequency, between complete extinction and complete transmission, would also give an envelope factor of 2, and the fringes would have the same appearance as with the square-wave modulation. Even more ironically, sinusoidal modulation at four times the vibration fre­quency, again between transmittance of zero and one, would also give an envelope factor of 2 and fringes of identical appearance.

Now let us consider the case when the object has low reflectance so that only a small fraction of the light with which it is illuminated is reflected to the hologram plate. Maximum reconstruction effi­ciency for the hologram will be obtained by making the ratio between the object and reference beams unity, and the reconstruc­tion efficiency will tend to decrease with the reciprocal of the beam ratio, as the reference beam is increased. If the object has low reflectance, then the irradiance of the hologram by light reflected from the object will vary only slightly as the reference-beam strength is increased, assuming that the ratio is controlled by a variable, lossless beam splitter. Thus the total irradiance of the hologram will vary approximately with the beam ratio. We may set the beam ratio K equal to 1/C0 to keep the exposure time of the hologram constant. Substituting this into Eq. (7) yields the result that by modulating the laser, and compensating for the decreased laser power by increasing the beam ratio, the brightness of the high-order fringes will still increase in direct proportion to the envelope factor. Similarly, if the normal brightness of high-order Jo fringes can be tolerated, we may set K = 1 /C0

2 and use the combination of modulation and beam-ratio adjustment to shorten exposure time.

Some attention should be given to the case where ƒi(t) is not an even function. If it is not, it may nonetheless be expressed as the sum of an even and an odd function. Examination of Eq. (1) will show that the odd function will contribute to Eq. (3) a series of odd-order Bessel functions, whose coefficients will be propor­tional to the Fourier sine series of the odd function, times the square root of — 1. As a result, the fringe function will be complex and of the form

The irradiance of the fringes will be proportional to the fringe function times its complex conjugate,

As before, we have sinusoidal fringes, represented by the terms in brackets, multiplied by an envelope function. If we calculate the visibility of the sinusoidal fringes in Eq. (9), we can conclude that the addition of the odd function to the laser modulation has lowered the visibility of the fringes and shifted their locations on the surface of the object.

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In summary, it has been shown that laser-irradiance modulation may increase the brightness of high-order fringes in the recon­struction of a time-average hologram more rapidly than it will increase exposure time. It was also shown that, when the modula­tion is subject to certain practical restrictions, the zeros of the high-order fringes are unaffected by the modulation, whereas the brightness of the high-order fringes is proportional to the reciprocal of the square of the average energy transmitted by the modulator.

1 P. Shajenko and C. D. Johnson, Appl. Phys. Letters 13, 44 (1968). 2 P. A. Fryer, Appl. Opt. 9, 1216 (1970). 3 P. A. Fryer, Repts. Progr. Phys. 33, 489 (1970). 4 F. M. Mottier, in Applications De VHolographie, edited by J. C. Viénot, J. Bulabois, and J. Pasteur (Univ. de Besangon, France, 1970), p. 6. 5 C. C. AleksofT, Appl. Opt. 10, 1329 (1970). 6 K. A. Stetson, J. Opt. Soc. Am. 60, 1378 (1970).

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