## Abstract

The delayed self-heterodyne interferometric technique, first proposed in the context of semiconductor lasers, has been commonly used for over 20 years in the determination of the optical linewidth of lasers. We examine this technique in the light of recent work on fiber lasers, and point out further constraints in the applicability of these measurements. An approximate but simple and intuitive expression is provided for assessing the self-heterodyne technique when applied to fiber lasers.

©2006 Optical Society of America

## 1. Background

The delayed self-heterodyne interferometer (DSHI) has been widely used for the measurement of laser linewidth since it was introduced in 1980 [1]. Originally used in the context of diode lasers, it has been very effective in characterizing the linewidth behavior of these lasers. Its experimental simplicity has led to its adoption for the measurement of linewidth of other laser types, in particular to fiber lasers [2–5], leading to the reporting in some instances of very narrow linewidth fiber lasers. However, in spite of its apparent simplicity, the data that the DSHI yields can be difficult to interpret correctly. It is the purpose of this work to conduct a closer examination on the applicability of this measurement technique, and point out new constraints on the delay parameter when applied to fiber lasers.

## 2. Analysis

The basic operation behind the delayed self-heterodyne interferometric technique is simple: the laser light is split into two paths, one of which is delayed and the light frequency-shifted with respect to the other before both beams are recombined, and the resulting beat response is measured. The laser linewidth is usually inferred from the width of this beat spectrum. At first glance, it seems reasonable to assume that the measurement will be accurate so long as the delay path is greater than the coherence length of the laser light, as in that case the two combining beams would be effectively uncorrelated. The validity and the resolution Δν_{res} of the DSHI is thus often cited as

where *τ*_{coherence}
is the laser coherence time and *L*_{d}
(*τ*_{d}
) is the fiber delay length (time). The validity of the above relations has been variously analyzed, e.g. see [6]; however, the analyses are generally based on a white frequency noise spectrum as the dominant contribution to the laser linewidth. For diode lasers, this is a valid assumption, as the optical linewidth stems primarily from spontaneous emission-induced refractive index changes in the semiconductor [7]. However, in the case of fiber lasers, the spontaneous emission contribution is extremely small (in the region of Hz [8]), and the linewidth behavior is instead overwhelmingly dominated by other (colored) noise sources.

To understand the consequences of the latter noise behavior on the measurement, we start with the beat spectrum *S(f)* measured by the DSHI [9]

where

ℑ denotes the Fourier transform and *S*_{F}*(f)* the frequency (FM) noise spectrum. We first note that, in the case where *S*_{F}*(f)* represents white (frequency-independent) noise, i.e. *S*_{F}*(f)* = *S*_{F0}
, the DSHI beat spectrum can be simplified to [10]

where *f*
_{o} is the laser center frequency. It is easy to see from the exponent in Eq. (5) that the Lorentzian laser lineshape (with the linewidth given simply by *S*_{F0}
= *l*/*τ*_{coherence}
) is faithfully reproduced for *τ*_{d}
>> *l*/*S*_{F0}
, providing a sound basis for relation (1).

In the case of fiber lasers, however, there is considerable work indicating that the dominant contribution to the frequency noise and linewidth is not white in nature, but has a low frequency (kHz) spectrum [8, 11], arising in no small part from pump-noise induced temperature fluctuations. Equation (5) is thus not applicable, and it is necessary to examine more closely the adequacy of (1) and (2) in the current context.

For a noise spectrum with a low frequency cut-off *f*
_{c}, we can to a reasonable approximation restrict the integration in Eq. (4) to the frequency interval [0,*f*
_{c}]. Expanding the term sin^{2}(*πντ*)/*ν*
^{2} in powers of τ, we have

Next, we define a time *τ*_{g}
by

and write Eq. (6) as

From Eq. (6) it can be seen that the higher order terms are much smaller than one for *τ*<*τ*_{g}
if ${f}_{c}^{2}$
${\tau}_{g}^{2}$<<1, i.e., if

where

can be viewed as an average FM noise level in the relevant frequency range.

On the other hand, for very large values of *τ*, the term sin^{2}(*πντ*) in Eq. (4) is rapidly fluctuating and averages to ½. We thus find

where the last inequality again holds if Eq. (9) is fulfilled.

Generally, the function *I*(τ) will increase from *I*(*τ*_{g}
)=1 to much larger values with increasing *τ*. However, the exact behavior will depend on the form of *S*_{F}*(f)* and *I*(*τ*) may show some oscillations in the intermediate regime. An example of this behavior is found if the noise spectrum is modeled simply as a low pass function with a single cut-off frequency, i.e.

$$\phantom{\rule{2.6em}{0ex}}=\mathit{0}\phantom{\rule{3.em}{0ex}}\mathrm{for}\phantom{\rule{.2em}{0ex}}f>{f}_{c}$$

*I*(*τ*) is readily computed for arbitrary values of *τ*_{d}
, as shown in Fig. 1. In all cases, *I*(*τ*) is parabolic for *τf*_{c}
<1/*π* and approaches a constant for large *τ*. The oscillations for intermediate times never exceed about 20% of the asymptotic value.

Summarizing the above discussion, we may therefore rewrite Eq. (3) to a good approximation as

for *all* values of *τ* with only exponentially small corrections. The laser spectrum is thus given by

which corresponds to a Gaussian lineshape.

Finally, from Eq. (14), we can obtain the FWHM of the Gaussian frequency spectrum

Relation (9) underlines the fact that the Gaussian lineshape is associated with high noise levels over a narrow frequency range (e.g. temperature fluctuations). The Gaussian approximation in turn fails for broadband, low-level noise (e.g. spontaneous emission noise), in agreement with a Lorentzian lineshape for the latter noise behavior.

## 3. Results and discussion

Equation (15) yields an approximate but intuitive expression that affords very useful insight towards clarifying the validity of the DSHI technique when used to measure the linewidth of fiber lasers. Effectively, we see that the DSHI technique ‘samples’ the fiber laser frequency noise spectrum via the area under the term sin^{2} (*πfτ*_{d}
). It should be evident, therefore, that for the measurement to be reliable, the delay time *τ*_{d}
has to be long enough to generate a sufficient number of cycles of sin^{2} (*πfτ*
_{d}) within the frequency noise spectrum. This is illustrated more clearly in Fig. 2.

As an instructive example, we consider again the case where the laser frequency noise spectrum is modeled as a low pass function with a single cut-off frequency, as in Eq. (12). Equation (15) is then readily integrated to yield the analytic linewidth result

where *x* = 2*πf*_{c}*τ*_{d}
. This linewidth behavior with delay time is quite distinctive: it initially increases linearly with *x* (*τ*_{d}
), and for large *x* > *π*, the linewidth tends towards the final value *(4log(2)f*_{c}*S*_{F0}
)^{1/2}. The corresponding requirement for *τ*_{d}
is therefore simply *τ*_{g}
> *(2f*_{c}*)*^{-1}
which, for a low frequency cut-off of 1 kHz, yields a considerable delay time (fiber length) of 0.5 ms (100 km).

Fiber delay lines of 100km or more are not commonly used in DSHI due to propagation losses. However, a recirculating 11 km delay line incorporating an EDFA for loss compensation was experimentally demonstrated over a decade ago, and applied to characterize a fiber laser [12, 13]. To our knowledge, however, a quantitative explanation for the measured linewidths with loop distance has yet to be adequately provided. It is therefore interesting to compare our current analysis with the data reported there. Figure 3 shows the fit of Eq. (16) with the data in [12]; there is very good general agreement. We note in this case that it takes a fiber delay length of 200km for the linewidth to approach its final value; from Eq. (16), we can also infer that the fiber laser tested had a frequency noise spectrum with a low frequency cut-off *f*_{c}
~ 0.5kHz., and a laser FM noise level *S*_{F0}
~ 11.5 kHz.

Our analysis clearly indicates that condition (1) is not sufficient for the DSHI technique to be accurate when applied to the linewidth determination of fiber lasers, as their noise behavior is not frequency-independent. Furthermore, our findings indicate the DSHI measured linewidth may not necessarily be valid even if the fiber laser linewidth is broad due to a high FM noise level *S*_{F0}
, if the condition *τ*_{d}
> *(2f*_{c}*)*^{-1}
is not satisfied as, unlike a spontaneous emission (or white noise) induced linewidth which is governed by the parameter *S*_{F0}
, the fiber laser linewidth is determined by the product (*f*_{c}*S*_{F0}
)^{1/2}. Therefore, the observation of a linewidth larger than the ‘instrument resolution’ commonly specified by relation (2) does not in itself validate the accuracy of the measurement, as the delay time may still not be long enough to have fully sampled the frequency noise spectrum, and the actual linewidth may be broader yet.

Finally, we note that in many applications, the noise spectra at very low frequencies is not always of concern, as very slow fluctuations can be tracked or compensated easily. Equation (15) is still useful, however, as one can simply replace the lower limit of the integral with a low (non-zero) frequency appropriate to the application, and arrive at the relevant linewidth.

## 4. Conclusions

We have analyzed the delayed self-heterodyne interferometric technique for measuring the linewidth of fiber lasers. Unlike diode lasers, fiber lasers are not dominated by white noise; instead, their noise characteristic more closely resembles a low pass filter function. The differences between the spectral behavior of the dominant noise sources for fiber lasers and diode lasers call into question the conventional criteria governing the resolution of this measurement technique. We have presented an approximate but simple and intuitive expression for the delayed self-heterodyne measured linewidth and its dependence on delay time, from which additional criteria is inferred for improving the usefulness and accuracy of this approach when used in assessing fiber lasers.

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