Techniques for measuring 1-μm diam Gaussian beams: comment Rita Csomor

MIKI Instrumentation, Budapest, P.O. Box 183, Buda­pest, Hungary. Received 4 February 1985. 0003~6935/85/152295-04$02.00/0. © 1985 Optical Society of America. Cohen et al. have described three different techniques for

measuring the diameter of Gaussian beams in the 1-μm range.1 They give two approximate expressions for the beam diameters in case of the Ronchi ruling technique. They suppose that the beam is scanned across a ruling of reduced contrast (RE = reflectivity of the bars; RS = reflectivity of the spacings), both the bars and spacings having the same width W. Their expressions are valid only if 1.3 < d/W < 2.7, where d is the beam diameter at the exp(—2) intensity points. Broockman et al.2 have extended these approximate results for asymmetric ruling.

The present paper gives the intensity of light ID reflected from the Ronchi ruling as a function of the position of the beam center (ξ). The calculation was carried out for a non­symmetrical grating having the grating period 2W, but the width of the bars is W(1 — ε) and that of the spaces is W(1 + ε) (Fig. 1).

As the intensity distribution of the Gaussian beam in the x direction is

(where IT = total power propagating in the beam), and the reflectivity of the grating is

we can detect the reflected intensity3

Fig. 1. Relative position of a Gaussian beam and a nonideal ruling. R(x) = reflectivity of the ruling; I(x) = intensity distribution of the

beam.

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This is the convolution of R(x) and I(x) and can be calculated by means of the convolution theorem:

where F denotes the Fourier transform and F-1 is the inverse Fourier transform of the function in the braces. According to the known definition of the Fourier transformation,

The key to the calculations is given if we find R (f). We have to consider that

Fig. 2. Nonideal ruling as the superposition of two rulings having maximum contrast.

Now we apply the following identity, Similarly,

to Eq. (7) for x = 2fW:

The Fourier transform of the Gaussian distribution is

and since Finally the reflected intensity is

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we can omit the even members in both sums:

where Kgr = RS/RE. As a simple verification of Eq. (23) let us compose our ruling

as the superposition of two rulings having ηgr = 1 presented in Figs. 2(b) and (c). If we denote the intensity extrema in case of T7gr = 1 and ε = 0 by I0

max and I0min, we can read from Fig.

According to Fig. 1, we can measure the maximum reflected intensity in case of ξ = 0 (the beam is at the center of the bar) and the minimum intensity in case of ξ = W. (The beam is at the center of a space.)

We can define the modulation function

According to Eq. (16), we can write

and substituting these into Eq. (17) we have Eq. (23). At this point we can compare our results with those of

Broockman et al.2 They have made an approximate calcu­lation only of/max and Imin, for the general case of ε ≠ 0, RS ≠ 0, and RE ≠ 1, taking into account that part of the ruling (Fig. 2, in Ref. 2) which covers the central portion of the beam containing almost its whole power. It is a very good approx­imation (within 1%) for d/W < 3.3333.

It is rather interesting that from the approximate calcula­tion for K0 and KL [Eqs. (15) and (16) in Ref. 2], the exact formulas (24) and (25) can be deduced. Neither Eq. (11) for KL in Ref. 2 nor the corresponding Eqs. (20) and (24) in this paper can be inverted and solved for d/W via analytic math­ematical methods. Their expression contains the first three members of the direct integration of Eq. (3), represented by the known tabulated error function with the unknown d/W as the argument. The mentioned convolution integral was carried out completely, but the result is an infinite sum of transcendent functions also containing d/W in the arguments. Fortunately, K0 = K0(d/W) can be very well approximated by the linear expression1'2

where in the above mentioned domain 1.3 < d/W < 2.7. Figure 3 shows d/W as a function of η0 and K0.

So we obtain for the modulation

where ηgτ is the modulation factor of the grating:

In the case of a symmetrical grating (ε = 0)

where η0(d/W) = η0(ε = 0;d/W). At the same time η0 is the modulation of a symmetrical grating with ideal reflectivity (RE = 1, RS = 0, Y ηgr = 1). As we can see, in the case of a symmetrical but nonideal grating the modulation can be factorized to the modulation of an ideal grating and to the own modulation of the grating, that is, to ηgτ.

Using the intensity ratio KL = Imin/Imax = (1 - ηsym)/ (1 + ηsym) for a nonideal grating of reduced contrast, and K0 = (1 — η0)/(l + η0) for an ideal grating, we obtain from Eq. (23) the following relations:

Fig. 3. d/W ratio as a function of modulation η0 and intensity ratio K0 in case of a symmetrical ruling having ideal reflectivity.

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The final results (16), (20), (21), (24), and (25) offer a con­venient computing process executable with a programmable calculator for the most general cases of nonsymmetrical gratings having reduced contrast without the laborious searching of the tabulated error function values.

These results have other applications where the scanning beam distribution can be approximated by a Gaussian func­tion, e.g., for the determination of the electron beam diameter in a TV camera from resolution data. If in the case of a bar code reader or a similar device we compare the measured ID(Ξ function and its theoretical value, we can find the distortions of the optical signal due to the uneven light deflection or un-sufficiency of the detector plus amplifier unit.

References 1. D. K. Cohen, B. Little, and F. S. Luecke, "Techniques for Mea­

suring 1-μm diam Gaussian Beams," Appl. Opt. 23, 637 (1984). 2. E C. Broockman, L. D. Dickson, and R. S. Fortenberry, "Gener­

alization of the Ronchi Ruling Method for Measuring Gaussian Beam Diameter," Opt. Eng. 22, 643 (1983).

3. Gy. Akos and R. Csomor, "Fókuszált lézernyaláb átméröjének közvetett meghatározása" ("An Indirect Method for Measuring the Diameter of a Focused Laser Beam,") MIKI Közleményei 23,. 49 (1984).

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