High-Resolution Mode-Spacing Measurement of the Blue-Violet Diode Laser Using

Interference of Felds Created with Time Delays Greater than the Coherence Time

So-Young Baek, Osung Kwon, and Yoon-Ho Kim∗

Department of Physics, Pohang University of Science and Technology (POSTECH), Pohang, 790-784, Korea(To appear in Japanese Journal of Applied Physics (Dec., 2007))

It is observed that the multi-mode cw blue-violet diode laser exhibits revival of interference whenthe interferometric path length difference is much greater than the coherence time of the laser andthat the recurring interference peaks are separated by the same distance. We report that thisunusual interference phenomenon can be used for high-resolution mode spacing measurement of themulti-mode cw blue-violet diode laser without using a high-resolution spectrometer.

INTRODUCTION

Before the development of InGaN laser diodes [1–4], access to blue or shorter wavelength laser lines wasonly possible by using mainframe gas lasers and/orharmonic generations of high-power pulsed lasers, suchas, Q-switched Nd:YAG lasers, Ti:Sapphire lasers, etc.Presently, blue, violet, or even ultraviolet (UV) semi-conductor lasers with the average power of tens of mil-liwatt are available commercially. These blue-violet-UVdiode lasers have strong potential applications (not tomention applications in high density optical data storageand printing) in quantum optics and quantum informa-tion research as compact pump sources (replacing main-frame UV lasers) for generating entangled photons viaspontaneous parametric down-conversion [5, 6].

For many classical and quantum optical applicationsinvolving interferometry, it is important to have accurateknowlege of the spectral properties of the laser beam.Since high-power blue-violet-UV diode lasers availabletoday tend to operate in the multi-mode (longitudinalmodes) regime, the mode spacing measurement of thelasing spectra will tell us a lot about the spectral prop-erties of the laser and, in addition, the laser diode itself.In fact, the mode spacing measurement of the blue-violetdiode laser is important not only for characterizing thelaser diode [1, 2, 7–9] but also for understanding the op-tical gain mechanism in InGaN laser diodes [10, 11].

Direct measurement of the mode spacing (∆λ) ofthe lasing spectra could be trivial if a spectrometer(or a monochromator) with sufficiently high resolution(∼ 10−3 nm or better) were readily available [1–3, 7–9]. This, however, is not the case for most researchers(e.g., in quantum optics or in classical interferometry)who make use of commercial blue-violet diode lasers sincespectrometers (or monochromators) with the requiredresolution tend to be quite bulky and costly.

The other approach would be to indirectly determine∆λ by making use of the relation derived from the reso-

∗Electronic address: yoonho@postech.ac.kr

nant condition of the Fabry-Perot modes [4, 10],

∆λ = λ2

0/2Lneff, (1)

where λ0 is the wavelength of the peak of the lasing spec-trum, L is the laser cavity length, and the effective re-fractive index, neff, of the waveguide at λ0 is given asneff = n − λ0dn/dλ0 [7–11]. Quite naturally, if all theparameters on the right hand side of eq. (1) can be mea-sured accurately, one can readily determine the modespacing ∆λ of the diode laser.

The peak of the lasing spectrum λ0 could bemeasured accurately by using a low cost spectrome-ter/monochromator or via interference. However, the in-formation on the physical characteristics of the commer-cial diode laser, such as L and neff, is often proprietaryand is generally not made available to the users. For thecavity length L, it is nevertheless possible to measure itif one risks the possibility of destroying the diode laserwhich can be rather costly [12]. Accurately measuringneff of the gain region of the laser diode, however, is nottrivial and there are uncertainties regarding neff for theInGaN laser diode, causing so-called the mode spacinganomaly [9–11, 13, 14]. Therefore, indirect determina-tion of the mode spacing ∆λ of the InGaN diode laseris still quite prohibitive as the measurement requires re-sources that are costly and not normally available to theusers of such lasers.

In this paper, we first report on the observation of in-terference revival when the interfrometric path length dif-ference is much greater than the coherence length of themulti-mode blue-violet cw diode laser. We then demon-strate that this unusual interference revival phenomenoncan be used for low-cost high-resolution mode spacingmeasurement of the diode laser without resorting to ahigh-resolution spectrometer nor accurately measuringthe physical characteristics of the laser diode. In addi-tion, the interference revival effect reported in this papershould have quantum optics and quantum informationapplications in generating engineered entangled photonsources.

2

INTERFERENCE REVIVAL

Let us first briefly discuss the physics behind the re-vival of interference when the interferometric path lengthdifference is much greater (by several orders of magni-tude) than the coherence length. Consider the multi-mode cw field Ei(t) at any given point inside the diodelaser cavity as the incoherent sum of multiple longitudi-nal modes,

Ei(t) =∑

m

Em exp[i{2π(ν0 + m∆ν)t + θm}], (2)

where the subscript m is the mode number, Em is theamplitude of a particular field mode, ν0 is the frequencyof the peak mode, ∆ν is the frequency difference betweenthe adjacent modes, and θm is the random phase of eachlongitudinal mode. Since the multi-mode field outsidethe laser cavity Eo(t) is related to Ei(t) with an attenu-ation factor due to the output coupler efficiency only, alltemporal and spectral features of Ei(t) are reflected onEo(t) without any change.

Consider now the instantaneous field amplitude out-side the cavity. The individual field modes have randomphases, so the multi-mode field Eo(t) has a chaotic wave-form [15]. However, the chaotic waveform should repeatitself at every time interval equal to 1/∆ν because of thefact that the mode spacing (in the frequency domain)∆ν is constant. Since 1/∆ν is equivalent to the timeit takes for the cavity field to make one round trip in-side the laser cavity of the length L and of the effectiverefractive index neff, the chaotic multi-mode cw field out-side the cavity should repeat itself at every Tp = 2L/vg,where vg = c/neff is the group velocity of the light in thecavity. Then the round trip time Tp is calculated to be

Tp = 2Lneff/c. (3)

Therefore, by using eq. (1) and eq. (3), we arrive at theuseful relation for determining the mode spacing,

∆λ = λ2

0/cTp = λ2

0/Lp. (4)

As mentioned earlier, λ0 can be easily measured in anumber of different ways. To determine the mode spacingaccurately using eq. (4), one only needs to measure Lp =cTp, the distance traveled by the chaotic out-of-the-cavityfield Eo(t) before it repeats itself.

Alternatively, we may use the frequency-domain anal-ysis to obtain the mode-spacing ∆λ. This way, wecan arrive at the desired result, eq. (4), without usingeq. (1) which is wavelength-dependent. As mentionedearlier, the frequency-domain mode spacing ∆ν is con-stant. Since the chaotic field repeats itself at the timeinterval 1/△ν, ∆ν is directly related to the interferencerevival period Tp, which can be measured accurately, as∆ν = 1/Tp. When desired, the frequency-domain mode

Mirror

τ408 nm cw LD BS

Detector

Iris

FIG. 1: Experimental setup. The path length difference be-tween the two arms of the interferometer is much bigger (byseveral orders of magnitude) than the coherence length of thelaser. BS is the 50/50 beamsplitter.

spacing ∆ν can be converted to ∆λ using the well-knownrelation, ∆λ = ∆νλ2

0/c, thus arriving at eq. (4).Experimentally, the measurement of Lp can be accom-

plished by observing interference of fields created at dif-ferent times, in which the time delay is much greaterthan the coherence time of the laser, by using the un-balanced Michelson interferometer shown in Fig. 1. Toanalyze this phenomenon a bit more in detail, considerthe multi-mode cw diode laser field Eo(t) at the inputto the interferometer shown in Fig. 1. The time aver-aged intensity I at the detector as a function of the pathlength delay τ is then calculated to be,

I ∝ 〈|Eo(t) + Eo(t − τ)|2〉 ∝ 1 + |g(τ)| cos(ω0τ), (5)

where the first order correlation function g(τ) =〈E∗

o (t)Eo(t − τ)〉/〈E∗

o (t)Eo(t)〉 and ω0 = 2πc/λ0.For τ ≈ 0, i.e., around the balanced position of the

Michelson interferometer, eq. (5) predicts the typicalMichelson fringes with a short coherence length lc dueto the multi-mode nature (large bandwidth) of the laser.When τ becomes bigger than the coherence time lc/c, nointerference occurs.

However, if the delay τ is further increased so thatτ ≫ lc/c and becomes equal to Tp (the unbalancedMichelson interferometer), revival of interference shouldoccur since the two superposed fields are now completelyidentical even though they are born at different times.(Two-photon quantum interference can also be observedunder similar conditions, see [16].) Therefore, measure-ment of the path length difference of the interferometerat which the interference revival occurs is equivalent tomeasuring Lp = cTp. Equation (4) can then be used toaccurately determine the mode spacing of the multi-modecw blue-violet diode laser.

EXPERIMENT

To demonstrate the revival of interference just de-scribed and its application in high-resolution mode spac-

3

6

4

2

0

Inte

nsi

ty (

arb. unit

)

408.6408.4408.2408.0407.8407.6407.4

Wavelength [nm]

FIG. 2: The emission spectrum of the commercial blue-violetdiode laser used in this experiment. The peak wavelengthλ0 = 408.20 nm and the FWHM spectral bandwidth (δλ)BW

is approximately 0.5 nm.

ing measurement, we setup an experiment using a com-mercial output-power-stabilized 408 nm diode laser (Co-herent Cube) which is capable of delivering up to 50 mWof cw power, see Fig. 1. The experiment was performedat 40 mW output power setting and the diode tempera-ture was set at 22◦.

Before the interference experiment, we measured thelasing spectrum with a 1/2 m monochromator installedwith a 1200 gr/mm grating blazed at 500 nm. The en-trance and the exit slits were adjusted to 11 µm and thelaser beam was focused at the slit with a 4 cm focus lens.The resolution of the monochromator, measured witha He-Ne laser, was approximately 0.06 nm. The spec-tral measurement is shown in Fig. 2. The peak emissionwavelength was measured to be 408.20 nm with FWHMbandwidth of 0.50 nm. It is clear that our monochroma-tor cannot resolve the mode spacing of the multi-modeblue-violet diode laser.

When the Michelson interferometer shown in Fig. 1is scanned about the zero path length difference po-sition, the usual Michelson fringe which is equivalentto the Fourier transform of the power spectrum of thelight is observed (see Fig. 3). The full width at halfmaximum (FWHM) coherence length is measured to be354.73±3.94 µm and agrees well with the value obtainedwith the simple relation lc ≈ λ2

0/(δλ)BW .

Next, the Michelson interferometer is scanned furtherso that the path length difference is much greater (byseveral orders of magnitude) than the coherence lengthlc: interference disappears at this condition. We, how-ever, observe the revival of interference when the pathlength difference is equal to integer multiples of Lp, seeFig. 4. As explained earlier, this is due to the fact thatthe chaotic multi-mode laser field Eo(t) repeats itself atevery Tp outside the cavity according to eq. (3). Smallerside peaks, which are sufficiently far away from the mainpeak, are due to unwanted reflections at various surfaces,

3.0

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2.0

1.5

1.0

0.5

0.0

In

tensi

ty (

arb. unit

)

-600 -400 -200 0 200 400 600

Path length difference [µ m]

FIG. 3: The Michelson interferometer is scanned about thezero path length difference (balanced) position. The FWHMcoherence length is 354.73 ± 3.94 µm.

including the beamsplitter. This happened because thesome of the optics used for the experiment were not prop-erly coated at 408 nm. The visibility decrease is due toslight misalignment at large path length difference of theinterferometer. The spacing, Lp, nevertheless remainsthe same. We note that the unwanted side peaks and thevisibility decrease at large path length difference can beavoided by using proper optics and judicious alignment,see Fig. 5.

From the measured value of Lp = 4.804 ± 0.017 mm,we are able to determine the mode-spacing of the multi-mode blue-violet cw diode laser. Since λ0 is alreadyknown from Fig. 2, we readily determine that ∆λ =0.0347 nm for the laser (Coherent Cube) used in our ex-periment. Note that this value agrees well with a previousreport on high-power blue-violet diode lasers reported in

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(ar

b.

un

it)

-15 -10 -5 0 5 10 15

Path length difference [mm]

FIG. 4: Revival of interference is observed when the pathlength difference is equal to integer multiples of Lp = 4.804±0.017 mm. Smaller side peaks are due to unwanted surfacereflections and the visibility decrease is due to slight misalign-ment at large path length difference of the interferometer.The spacing, Lp, however remains the same.

4

ref. [17]. This suggests that the InGaN laser diode mod-ule used in our laser could in fact be a commercial prod-uct based on the proprietary InGaN design reported inref. [17].

DISCUSSION

Let us now briefly discuss the resolution of the modespacing measurement. For a value θ which depends ontwo independently measured quantities x and y, the un-certainty δθ of the value θ(x, y) is given as,

δθ =

√

(

∂θ

∂xδx

)2

+

(

∂θ

∂yδy

)2

, (6)

where δx and δy are measurement uncertainties. Sincethe mode spacing ∆λ is determined by independentlymeasuring λ0 and Lp, the resolution is determined bymeasurement uncertainties related to λ0 and Lp measure-ments. Using the measured values of λ0 and Lp as well aseq. (4) and eq. (6), we roughly estimate that the uncer-tainty of the mode spacing can reach on the order of 10−4

nm. In fact, the calculated uncertainty of the mode spac-ing measurement reported in this experiment is 0.0001nm. We therefore conclude that the commercial 408 nmcw diode laser (Coherent Cube at 40 mW) used in thisexperiment has the mode spacing ∆λ = 0.0347 ± 0.0001nm. Note that, if the cavity length L were known, theeffective refractive index neff of the gain region can beaccurately determined with ease.

To further demonstrate the effectiveness of the mode-spacing measurement method described so far, the sametest was performed on a different blue-violet diode lasersystem of the same model (Coherent Cube 405C). Thediode temperature was set at 25◦ and the output powerwas maintained 30 mW. At this condition, the peak emis-sion wavelength was measured to be 407.97 ±0.07 nmwith FWHM bandwidth of 0.51 nm (see Fig. 5). The bal-anced Michelson interferometer data show the coherencelength of 355.81 ±7.34 µm for the diode laser radiation,which is in good agreement with the FWHM bandwidthobtained from the spectral measurement.

When the path length difference is slightly increasedso that the Michelson interferometer becomes unbal-anced, the interference disappears. As expected, how-ever, the periodic interference revival is observed as thepath length difference is increased further (see Fig. 5).The period of interference revival is measured to beLp = 4.806 ± 0.019 mm. With these measurement data,we can readily determine the mode spacing of the blue-violet diode laser as ∆λ = 0.0346±0.0001 nm. Note that,with improved optics and judicious alignment, unwantedside peaks observed in Fig. 4 have disappeared. In ad-dition, the visibility of the recurring interference peaks

3.0

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arb.u

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-10 -5 0 5 10

Path length difference (mm)

8

6

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0

Inte

nsi

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arb. unit

)

408.2408.0407.8407.6407.4407.2

Wavelength [nm]

3.0

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0.0

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nsi

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)

-300 -200 -100 0 100 200 300

Path length difference[µm]

FIG. 5: Experimental data for a different diode laser of thesame model (Coherent Cube). The peak emission wavelengthwas 407.97 nm with FWHM bandwidth of 0.51 nm. Withimproved alignment and optics, interference revival is clearlyobserved without strong side peaks and visibility degradationat large path length difference, see Fig. 4 for comparison.

remains almost the same even at very large path lengthdifference (see Fig. 4 for comparison).

Finally, we note that a mode-locked Ti:Sapphire laseris known to exhibit a similar interference revival effect.However, in this case, the effect is due to the fact thateach ultrashort pulse is coherent (mode locked) to the ad-jacent pulse which is separated by the cavity round-triptime of the pulse. This effect has been used to demon-strate interesting two-photon quantum interference forultrafast laser pumped spontaneous parametric down-conversion and shown to have quantum information ap-plications [18, 19].

The interference (coherence) revival of the chaoticmulti-mode cw blue-violet diode laser reported in thispaper can be exploited in a similar way. When alaser is used to pump the spontaneous parametric down-conversion process to generate entangled photon pairs,the coherence properties (spectral and temporal) of thepump laser are transferred to the two-photon entangledstate [16]. The coherence revival of the chaotic cw pumplaser therefore will be completely transferred to the two-photon entangled state and a properly designed two-photon interference experiment will exhibit the same pe-riodic interference revival effect but in this case the in-terference is of quantum origin, i.e., two-photon quan-tum interference. We therefore believe that the inter-ference revival of a multi-mode cw diode laser reportedin this paper can be exploited to demonstrate interest-ing new two-photon quantum interference and to gener-ate engineered energy-time entangled photon states for

5

multi-mode diode laser pumped spontaneous parametricdown-conversion.

This work was supported, in part, by Korea ResearchFoundation (R08-2004-000-10018-0 and KRF-2005-015-C00116) and Korea Science and Engineering Foundation(R01-2006-000-10354-0).

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