C4-1
Zero Path Difference Interferometry and Coherence Properties of Semiconductor Laser Sources
The coherence properties of diode lasers are important in coherent communication and interferometric sensor applications as well as to the practical problem of balancing two lengths to micrometer level accuracy. In this project we will examine a very powerful interferometric method for precise length measurements. You will obtain a practical understanding of the coherence properties of diode lasers and their effect on measurement parameters in coherent setups.
Using a diode laser in the LED, multimode, and single-mode regimes, the principal coherence characteristics will be examined with a Michelson interferometer. Utilizing laser diode coherence properties, measurement accuracies on the on the order of tens of micrometers should be obtainable.
3.1Coherence Properties
For our purposes in this section, we will define coherence in the practical terms of being able to combine two beams of light and have them interfere constructively or destructively. This implies that there exists a phase relationship between the two beams. This phase relationship is most readily detected in the optical regime through the use of an interfering beam interferometer as done in Project 1. The Michelson interferometer setup pictured in Figure 1.6 will also be used in this project to examine coherence properties of diode lasers. Consider the two beams in the Michelson interferometer as they propagate through two different paths and are recombined by the beamsplitter and, subsequently, detected in the photodetector. The path length of one arm is assumed to be larger than the other arm by a length of L so that the light is delayed a time L/c) in one of the arms relative to the other. The electric field amplitudes at the beamsplitter are given by
E(t) A[E1(t)+E2(t)](3.1)
with the total power PE2 so that the power may be written as
P(t) {E1(t)2E 2 (t)2Re[E1(t)E*2(t-)]}(3.2)
The last term represents the interferometer interference term of interest and the average transmitted power P(t) involves the autocorrelation of the laser field [3.1], E1(t)E2*(t-). This autocorrelation function is related directly to the complex degree of coherence of the two interfering beams
(3.3)
With this function, we arrive at the desired form for the interfering fields
<P>=<P1>+<P2>+2 <P1<P2cos (3.4)
where is the phase difference between the two beams [3.1]. The coherence timec is defined to be related to the degree of coherence by the following expression
c 2d(3.5)
When L = 0, i.e., = 0, then = 1 whereas for , 0. In the case of purely monochromatic light with frequency and infinitely narrow linewidth 1 for all since the beams will always interfere. As the linewidth for a laser broadens with a Lorentzian spectral density,
e-(3.6)
where c-1. For different spectral densities, the constant relating c and varies between 1 and 2. We will use the value of for the following examples since it is a good approximation for broadband, “incoherent” sources. For =1 MHz, c = 318 nanoseconds whereas for =105 GHz, c=30 femtoseconds. For incoherent light, the coherence may be requested as, c = 2/(c) which for =100 Å, and =1.3 m, c ~0.5ps
Figure 3.1 – Interference pattern as a function of path imbalance.
Figure 3.2 --- Interferometric passband and laser mode intensities.
(constant set equal to unity). This formula will be used for determining the coherence time for an incoherent source such as LED. The expected interference pattern for an incoherent source is shown in Figure 3.1. In this case, interference only occurs within a coherence length lc=2/. Outside of this region, the beams do not have a fixed phase relationship and do not form an interference pattern.
For a quasicoherent source such as a multimode laser, the interplay between the Michelson transmission characteristics (for constant L and as a function of ) and laser mode spectrum determine the received signals. Figure 3.2 illustrates the interferometric passband with the peaks separated by each path length imbalance of L = which repeat with a 2 periodicity. The laser modes each have linewidth and are spaced c(2l) where l is the laser cavity length. Each mode, therefore, is transmitted through the Michelson interferometer with different transmission factors. As is described in Reference 3.1, each mode has a fraction i of the total interferometer output power () and the modal power is denoted as
Pi(t) = iP[1 + cos(+2i)+t, ](3.7)
where and t, is the phase fluctuation or noise in a given mode and i is the mode number. Usually, in multimode lasers is the same for all modes and in this analysis will be treated as a constant. The total power is a sum over the individual powers
PT = [1+cos i]
= PT {1+cos [cos(i](3.8)
sin [sin( + i]}.
The mean transmitted power for interfering fields in interferometers may be derived in a manner analogous to Equation 3.4 (see also Section 8.1.4 or Reference 3.1 for details) and is given by
(3.9)
which following Equation 3.6 may be rewritten as
(3.10)
If the time delay is adjusted so that =integer, all the modes possess the same phase and Equation 3.10 can be written as
.(3.11)
where is not an integer, all the terms in the summation of Equation 3.10 have different phases and for reasonably large numbers of modes, the summation approaches zero so that . These effects are illustrated in Figure 3.3 with PT equal to unity. The width of each spike in Figure 3.3 is determined by the spectral width of each mode and there is a “coherence length” for each spike given by lc. For single-mode lasers , the coherence length, again, is approximately given by lc=c() which for =1 MHz, lc ~10 meters.
In interferomatic sensors the path imbalance is important because phase noise is converted to amplitude noise thereby degrading performance. The phase noise output of an interferometer was given in Equation 1.5 as
Figure 3.3 – Total power output of the interferometer.
For a large path imbalance, large intensity noise is induced by the laser linewidth. This same effect can be used to measure and minimize the path imbalance L. This is to accomplish two stages. First, by modulating the laser current to induce a relatively large dv/di, adjustment of L to about 1 mm is possible by monitoring the interferometer output. Next, to obtain shorter L, the fringe visibility is used either with a multimode laser or an LED to obtain L on the order of a few microns.
For the larger L adjustments, the diode frequency is modulated by varying the drive current. Below 10 kHz, both carrier induced laser index charges and thermally induced charges in the index lead to a variation in laser frequency with drive current. At low frequencies, dv/di ranges typically from 2-10 GHz/mA depending on the laser structure. When the Michelson interferometer is in quadrature, the output of the interferometer is gven by dI =(2c) L . However, as discussed in the Primer, varying the current also induces an intensity modulation in the laser output, dIL, so that the total signal out of the interferometer is given by
dIT = dILdI.(3.12)
For large L, dIdIL whereas as one tries to balance the interferometer arms, i.e., L0, then, dI becomes small and dILdI. The in Equation 3.12 arises because the sign of dI depends on which side of the interferometer response curve (separated by ) the interferometer is operating (see Figure 0.31). Using the arrangement of Figure 1.6, if one uses a photodetector for intensity measurements at one output and another photodetector for phase measurements of the other output (with detector outputs made equal when the interferometer is in quadrature and no modulation is present), then common amplitude rejection is possible. Common mode rejection can be made accurate to about 30 dB of cancellation when performed properly. In the case when common mode rejection is not used, then one varies dI by radians so that the signals are added/subtracted from dILwhich is not a function of path length imbalance L. This is illustrated in Figure 3.4. As L becomes on the order of a
Figure 3.4 – Variation of interferometric output with path length difference.
millimeter, phase noise makes the curves less distinct and one switches to either multimode lasers or LEDs. The original path length imbalance L must be less than lc for the technique, illustrated in the figure, to work. Generally, the current modulation amplitude is maintained so that dI only varies radians, i.e., from plus to minus. Using the data in Figure 3.4 and moving only one of the arms of the Michelson interferometer, one will know which arm of the interferometer is longer and which direction it has to move to continue to reduce L.
For L less than 1 millimeter, we will use the characteristics of a laser diode used as an LED. By biasing the laser diode below threshold, it becomes an emitter with a very short coherence length, e.g., for -100nm, lc-5m. By placing the mirror of the elongated arm on a translation stage and mounted on a piezoelectric element, the mirror’s position may be precisely changed. Since for the first part of the experiment, the direction of motion needed to decrease L is known, L is continually decreased until the interference pattern illustrated in Figure 3.1 is seen. By varying the voltage on the piezoelectric, the path length mismatch may be reduced to a few microns. Critical to this experiment is component stability from the components themselves and from air currents. It is therefore important to mount the components rigidly and to shield the interferometer from ambient air currents, if necessary, by placing the setup in a box.
Multimode lasers, likewise, can be used to zero out the path imbalance. In this case, the output is given in Figure 3.3 and for L->0, one has to be able to identify the peak output spike at =0. This procedure is illustrated in Figure 4 of Reference 3.2. Also discussed in this reference is the phase noise performance of a diode laser and the use of a path imbalanced interferometer to measure the noise.
Newport Equipment Required
Assembly ModelDescriptionQuantity
LMA-1Laser mount assembly1
CMCube mount1
CWMCube and waveplate mount1
BSA-IBeam steering assembly1
BSA-IIBeam steering assembly1
M818Silicon detector mount1
02196-02815 Power meter1
05BC16NP.6Nonpolarizing beamsplitter1
F-IRC1Infrared sensor card1
G-LGS-NDGAProtective laser goggles1
Additional Equipment
DescriptionQuantity
Infrared imager (CCD camera and monitor)1
Function generator1
Oscilloscope1
DC power supply1
Ruler1
Index card1
3.2Experimental Setup
The path length mismatch in a Michelson interferometer is to be reduced from a few centimeters into the micrometer range in two steps. First, the light frequency is modulated through current modulation to produce an intensity modulation in the output of the interferometer. The output intensity modulation depth of the interferometer, which is directly proportional to path mismatch, is minimized by adjusting the path length in one arm of the interferometer. Next, coherence length of the laser beam is reduced by reducing the diode current below the lasing threshold so that interference occurs for smaller path length mismatches. The path length, then, is adjusted to obtain maximum visibility.
Read the Safety sections in the Laser Diode Driver Operating Manual and in the laser diode section of Component Handling and Assembly (Appendix A) before proceeding.
Figure 3.5 – Experimental setup for obtaining zero path length difference.
1.Assemble the Michelson interferometer as shown in Figure 3.5. Ensure that the laser is operating single mode by selecting an appropriate operating point above threshold and minimizing reflections. Start with a path length of 2 to 5 centimeters (round trip).
2.Apply a current modulation to the laser so that an adequate signal is obtained from the interferometer. In Project 1 we determined that dv/di for our laser since, with the interferometer adjusted to the quadrature, the output was
where is the induced phase shift in the interferometer and the output modulated signal I=K where K is a function of fringe visibility. Note that the measured interferometer output contains the two terms given in Equation 3.12.
3.Obtain quadrature (maximize visibility and set the interferometer between its minimum and maximum output intensity points) by adjusting mirror tilt and applying a voltage to the piezoelectric transducer block (PZB) in the beam steering assembly (BSA-II). Record the signal modulation depth.
4.Adjust the PZB voltage to operate at quadrature on the other side of the interferometer response curve. The signal should be radians out of phase with that in Step 3, i.e., it should appear inverted (Figure 3.6). Again, record the output modulation depth.
Figure 3.6 – Two quadrature points in the interferometer output.
5.Incrementally decrease the path imbalance L by adjusting the micrometer on the beam steering assembly (BSA-II), repeating Steps 3 and 4 to record data and obtain a plot similar to Figure 3.4. Note the direction to reduce L, especially below 1 mm.
When L is on the order of a millimeter, it will not be possible to further reduce L by this modulation technique because the signal from approaches zero as L0. If inadequate signal is available to obtain
L ~1 mm, increase the drive current modulation, and thus dv/di, somewhat to ensure that the interferometer signal is greater than the detection system noise. At each increment, keep the interferometer in quadrature, making sure that the visibility is high enough to accomodate the intensity modulation. Otherwise, if fringe visibility varies as L is changed, it must be corrected by plotting Figure 3.4.
6.Below about 1 mm, detection system noise will limit accuracy. At this point, reduce the laser drive current below threshold (to about 0.9Ith).
7.Because of the short coherence length of the source when operated as an LED, interference will only occur for a short length (-20-40 m).
Translate the beam steering assembly (BSA-II) as before in the direction of decreasing L. If you do not observe an interference pattern similar to that illustrated in Figure 3.1 within a few millimeters of translation, repeat Steps 1 through 7.
8.From the width of the interference peak, calculate the coherence length of the source.
Steps 9 and 10 are optional.
9.For an “incoherent” source such as a multimode laser, Figure 3.3 may be observed by varying the path length mismatch over multiples of the laser cavity length. For our light source, operating in the multimode regime, these peaks should occur symmetrically around the L = 0 peak for every few millimeters of path imbalance L.
Measure and plot the decay of the envelope as shown in Figure 3.3 to determine the coherence length and compare it with Step 8.
10.From the visibility decay graph, determine the laser cavity length l by measuring the delay time =L/c and =c/(2l), where L is the round trip delay in the Michelson interferometer.
3.3References
3.1)K. Petermann, Laser Diode Modulation and Noise, Kluwer Academic Press, Chapter 8, Tokyo (1988)
3.2)A. Dandridge, “Zero Path Length Difference in Fiber Optic Interferometers,” IEEE/OSA J. Lightwave TechnologyLT-1, p. 514 (1983)