Path-Length-Difference Effects in FM Laser Photomixing P. J. Titterton

Sylvania Electronic Systems—Western Operation, Mountain View, California 94042. Received 22 August 1967.

In a recent article in this journal, Foreman1 discussed the path-length-difference effects observed in heterodyning with a multi-mode gas laser. He assumed the envelope of the amplitudes (of the many axial modes considered) to be a Gaussian. In his treatment the relative phases of the modes are irrelevant, since it is assumed that the phases do not change rapidly enough to af­fect the system. Moreover, in his experimental situation, each mode is only effectively heterodyned with itself.

When a multimode laser is internally phase modulated, with the frequency of modulation near a multiple of the axial mode inter­val, the modes can become coupled to produce two distinctly different types of outputs.2 ,3 Modulation at almost exactly the axial mode spacing results in a phase-locked output where the relative phases of the modes are locked together. The intensity when considered as a function of time becomes a series of regular spikes, separated by the inverse of the modulation frequency and whose width and height depend on the number of modes coupled and the oscillating line width of the laser transition. On the other hand, modulation at a frequency slightly removed from the axial mode spacing (say about 200 kHz for a He-Ne laser with an axial mode interval of about 100 MHz) results in an intensity that is almost completely free of random fluctuations. Moreover, the ratio of the mode amplitudes becomes equal to ratios of Bessel functions of the first kind Jn.

Now, will Foreman's solution apply to these very different lasers, as well as to the ordinary free-running multimode laser he explicitly treats?

Indeed his solution will apply to a phase-locked laser whose modes have phases locked together and which describe an en­velope function very nearly equal to a Gaussian curve.3

Consider now the second type, the F M laser. For it, the nor­malized optical electric field traveling in the positive z direction is, to first order,4

where Ω0 is the carrier frequency, n0 the mode number of the center mode, L the length of the cavity, Γ the modulation index, and vm the modulation frequency.

We consider the same experimental setup as Foreman, i.e., this optical beam is first passed through a beam splitter. Then one of the two resulting beams is passed through a single-sideband modulator and the beams are recombined and fall on a photo-detecting surface. For ωs, the frequency offset of the single-side­band modulator, SM the pathlength traversed by the modulated beam and SR that traversed by the unmodulated beam,

206 APPLIED OPTICS / Vol. 7, No. 1 / January 1968

where ER and EM represent the optical electric fields in the un­modulated and modulated beams, respectively.

Assuming the response of the photodetector to be describable by I = αE2, we find for the mixing term

Retaining only those terms tha t oscillate a t ωs (and neglecting the term arising from p = -2n0 - l), we find Imix(ωs) = I(δ) cos(ωst + ε) implies I(δ) = [A2(δ) + B2(δ)]½ for

and

where δ = SM - SR. Doing the sums,5 we find I(δ) = |J0

{2Γ sin(πδ/2L)}| . Recall that Foreman's result was I(δ) = (U2 + V2)½, where

and

for γp = e x p [ - (ωp - Ω0)2/0.7214 W2], where W is the full an­gular frequency width of the gain curve between the half-power points, and N is the number of modes present.

He evaluates I(δ) numerically for two, three, four, five, six, seven, and twenty-one modes. Our result is only comparable with his for odd numbers of modes, since we took the center of the F M spectrum to lie a t line center, as it must for stability.

Note that I(δ) = 1 for J0 of zero argument, and hence the sig­nal is a maximum for path length differences of δ = 0, 2L, 4 L , . . . .

Moreover, owing to the closed form of our answer, all of the extrema are immediately identifiable as solutions of the equation cos(πδ/2L)J1[2Γ sin(πδ/2L)] = 0, i.e., δ = L/2 , 3 L / 2 , . . . , as well as δ = 0, 2L, 4L, . . ., as well as the values of delta arising from 2Γ sin(πδ/2L) = xi, where χi is the ith zero of the first order Bessel function.

Since the sine is always less than or equal to one, the magnitude of Γ prescribes how many zeroes will arise. This is the only rela­tion the number of zeros (and peaks) has to the number of modes present, since the more modes that we are trying to make into a stable F M spectrum, the greater Γ must be. As a rule of thumb,

Fig. 1. The intensity I(δ) plotted as a function of the path length difference, δ = S M - SR, divided by L, the length of the laser

cavity for any L and Γ = 2.0.

Fig. 2. The intensity I(δ) for arbitrary L and Γ = 3.0.

Fig. 3. The intensity I(δ) for arbitrary L and Γ = 7.0.

Γ is usually a little less than half the number of modes oscillating Thus, Fig. 1 for Γ = 2.0 corresponds to Foreman's Fig. 5, and Fig. 2 for Γ = 3.0 corresponds to Foreman's Fig. 7. As an indication of the behavior for many modes running, Fig. 3 is plotted for Γ = 7.0.

Besides the extremely useful closed form of our result, we note the following differences between the results for the F M and free-running multimode laser.

(1) For a given number of modes, the F M result may have fewer zeros.

(2) For a given number of modes, the first zero of the F M re­sult occurs at a smaller value of δ/L than that of the free-running answer.

(3) For a given number of modes, the interior peaks are far higher for the F M than the free-running laser. For Γ = 2.0, the value at δ/L = 1 is approximately twelve times as high as that in the free-running case, while for Γ = 3.0, the peak is approximately five times as high as that in the free-running case at δ = 1.

(4) The free-running case has a maximum at δ/L = 1 for N = 5, 7, and 21, whereas the F M case has minima for Γ = 2.0 and 7.0, and a maximum for Γ = 3.0.

Owing to these differences, it is seen that there are more usable positions for a photodetector if an F M laser is used as the source in a heterodyne path difference experiment. Moreover, the closed form of the result means the usable positions are easily deter­mined.

References 1. J. W. Foreman, Jr., Appl. Opt. 6, 821 (1967). 2. S. E. Harris and R. Targ, Appl. Phys. Letters 5, 202 (1964). 3. E. O. Ammann, B. J. McMurtry, and M. K. Oshman, I E E E

J. Quantum Electron. 1, 263 (1965).

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4. S. E. Harris and O. P. McDuff, IEEE J. Quantum Electron. 1, 245 (1965).

5. E. Jahnke and F. Emde, Tables of Functions (Dover Publica­tions, New York, 1945), 4th ed., p. 144.