Abstract

We investigate the fine structure of the optical spectrum of a broad-area laser diode with approximately 30-MHz resolution using spatially-resolved self-heterodyning technique. We show that this method is capable of measuring the relative powers and spacings of the individual lateral modes.

© 2014 Optical Society of America

1. Introduction

Broad-area laser diodes (BALDs) are widely used today as pump sources for fiber and solid-state lasers [13]. The efficient coupling between the BALD pumps and gain media depends on the structure and long-term stability of the BALD lateral modes [2, 4]. A precise knowledge of the relative intensities and wavelength positions of the modes is necessary to assure the optimum light delivery to the active material. Moreover, an observed change in the spatially-resolved BALD spectrum may reflect a growing bulk defect that initially impacts only a subset of modes [5], and can be used in predicting laser failure [6]. At the initial stage of laser evolution toward failure the future site of the catastrophic optical damage slowly (over the years of operation) heats up and eventually reaches a critical temperature beyond which a rapid thermal runaway, melting, and vast expansion of the damage area occur on nanosecond scale [7]. Thus, monitoring the changes in the lateral mode spectra may assist in laser defectoscopy and failure analysis and prediction. Mode intensity monitoring devices based on traditional optical instruments such as gratings [8, 9] and interferometers [10] require significant space. This is especially true for modern long-cavity BALDs, where the frequency spacing between the longitudinal modes is on the order of several GHz. The alternative way of mode monitoring by coherent heterodyne detection requires external optical sources which also add complexity and require additional space.

In this paper we propose and demonstrate the method for investigating the fine structure of BALD optical spectra using optical self-heterodyning between BALD lateral modes. The mode frequency beatings in radio frequency (RF) domain are detected with a fast photodetector followed by a high-sensitivity RF spectrum analyzer. Small size of the photodetector breaks the spatial orthogonality between the modes and enables the measurement of the corresponding frequency signals with sufficient spatial resolution in both near- and far-field planes. The resulting analysis fully identifies the modes’ relative powers and their wavelengths with frequency resolution limited only by the intrinsic linewidth of the laser (~1–30MHz). We have shown some of the preliminary results at a conference [11], demonstrating the identification of relative mode frequencies. The present paper provides the detailed and comprehensive account of our work and describes procedures for recovery of both frequencies and optical powers of various lateral modes.

The paper is organized as follows. In Section 2 we introduce a simple box model of the BALD modes and the relation between the optical and self-heterodyned RF spectra. In Section 3 we show how the mode powers and wavelengths can be obtained from the near-field measurements. In Section 4 we show a similar procedure for far-field data. In Section 5 we present and analyze the experimental measurements for both near and far fields, and in Section 6 we summarize the results. A detailed derivation of the relationship between the optical and self-heterodyned RF spectra is provided in the Appendix.

2. Background

The simplest and, surprisingly, reasonably accurate description of the longitudinal and lateral BALD modes, confirmed by our previous investigations of BALDs [8, 12, 13] and broad-area quantum cascade lasers [14], comes from assuming the laser cavity in the form of a rectangular box of width W and length L with perfectly reflecting boundaries [15]. The output electric field is given by

Emp(x,t)=Amp(t)eiωmptcos[πpxW+π2(p1)]=Amp(t)eiωmptΨp(x),
where m and p are the longitudinal and lateral mode indices, respectively, Amp(t) = |Amp(t)|exp[iφmp(t)] is the slowly-varying complex mode amplitude, φmp(t) is a random phase, Ψp(x) is the profile of the mode with lateral index p, x is the lateral coordinate ranging from –W/2 to W/2, and t is the time. For p = 1, 3, 5, ..., Eq. (1) yields even (cosine) function of x; for p = 2, 4, 6, ..., Eq. (1) yields odd (sine) function of x. The field outside of the laser facet (|x| > W/2) is assumed to be zero. The frequency ωmp of the pth lateral mode of the mth longitudinal order is given by [8]:
ωmp=mπcLnph(λ)1+(LpWm)2,
where nph(λ) is the phase refractive index and c is the speed of light in vacuum. With notation
ωm0=mπcLnph(λm),
where λm = 2πcm0 is the wavelength of the mode with indices m ≠ 0 and p = 0, we can write Taylor series expansion to approximate Eq. (2) for p << 2Wnph0 (where λ0 is the center wavelength of the BALD spectrum) as
ωmpωm0+πcλm4nph2(λm)W2p2,
which demonstrates the characteristic parabolic dependence of frequency upon the lateral mode index p. For most purposes, the mode wavelength λm in Eq. (4) can be approximated by the center wavelength λ0.

Our earlier experiments [8, 1214] have shown that the description provided by Eqs. (1) and (2) remains acceptable up to the power levels close to the point of catastrophic degradation of the diode.

Each mode has its own gain that determines its output power. The total number of lateral modes Pmax theoretically permitted by the ideal box model of Eqs. (1) and (2) can be more than several hundred (Pmax = 2Wnph0). However, in real BALDs the anti-guiding nature of the lateral confinement introduces higher losses for high-order lateral modes and, therefore, restricts the number P of actual “illuminated” (i.e., containing noticeable amount of light energy) lateral modes to just a few. This number P varies with the pump current and is typically less than 20. Due to inhomogeneous gain depletion, the detailed calculations of the modal gain and intensity are quite complex and require numerical simulations [4, 15].

When these modes fall onto a photodetector, the fields of all of the modes interfere. The total field is a superposition of all longitudinal and lateral modes:

E(x,t)=m=m0M/2m0+M/2p=1PEmp(x,t).

Spectral profile of the gain limits the number of “illuminated” longitudinal mode orders to number M << 2Lnph0. Therefore, the summation in Eq. (5) can be done over a narrow range of m and p. Typically, at the pumping level of 3–5 times over the threshold for a BALD, the emission is concentrated in 20–30 longitudinal groups of 5–20 lateral modes, which contain more than 95% of the total output power.

Because all modes have different frequencies and are phase independent, the random phase terms must be properly averaged to accurately describe the mode beatings at the detector. A photodetector generates current proportional to the instantaneous power arriving at the detector (within photodetector’s RF bandwidth). If the lateral size of the photodetector d is considerably smaller than W/P, the mode orthogonality condition is equally broken for all lateral modes. In this case, the lateral dependence of the RF beat signal is the product of optical lateral profiles of the beating modes. It is important to note that the amplitude of the RF beat signal averaged over multiple longitudinal orders (with random phases) equals zero. Therefore, we should instead look for the averaged power of the RF signal from the photodetector. Moreover, we will assume that the frequency beatings between any two modes from different longitudinal groups will produce RF frequencies outside the photodetector bandwidth (i.e., they are >> 10 GHz). In such a case, the summation over the index m can be realized as a summation of RF powers generated by each individual longitudinal group. A detailed derivation described in the Appendix leads to the following relationship between the spatially-varying optical and RF spectra in the case of a finite number of isolated modes:

S2RF(x,ω)=(2Rd/W)2|H(ω)|2pp|Ψp(x)|2|Ψp(x)|2m{Smpopt(ω)Smpopt(ωωpp)},
where R is the photodetector’s responsivity in Amperes per Watt, H(ω) is its frequency response (including amplifiers), and the part of Eq. (6) in figure brackets represents the convolution of modes’ optical spectra (power spectral density) Smpopt(ω) and Smpopt(ω), shifted by the beat frequency ωpp = (ωmp – ωmp) from DC. Thus, in order to find the lateral profile of the intermodal frequency beating powers, one needs to calculate the convolution of optical spectra of the beating modes and multiply it by the product of corresponding lateral profiles. Due to the simple form of Ψp(x) given by Eq. (1), S2RF(x,ω) can be calculated analytically.

Equation (6) describes two-sided RF spectrum, whereas a typical RF spectrum analyzer shows one-sided spectrum, which, after several simplifying assumptions detailed in the Appendix, can be represented in the following form:

SRF(x,ω)=ξ|H(ω)|2p=2Pp=1p|Ψp(x)|2|Ψp(x)|2g[(ωωpp)/Δω]g(0)SpoptSpopt,
where ξ is a calibration constant proportional to square of the detector responsivity (see Appendix for its definition), Spopt is the total (i.e., summed over all longitudinal orders m) laser power contained in the lateral mode with index p, ωpp=ωmpωmpπcλ04nph2(λ0)W2(p2p2) is the beat frequency between the modes with lateral indices p′ and p from the same longitudinal order [obtained from Eq. (4), assuming λm ≈λ0], and g′(ω/Δω′) is the RF lineshape with full width at half-maximum (FWHM) Δω′, obtained by the convolution of the optical lineshapes of the two beating modes.

3. Self-heterodyned RF spectra of BALD in the near-field optical region

Let us assume that we use a small-diameter photodetector that can be moved along the lateral coordinate of BALD in very close proximity to the radiating facet. The number of harmonics in the corresponding analytical expression of Eq. (7) is proportional to the value of P(P–1)/2 [11]. If the laser emission contains 7 lateral modes (P = 7), the RF spectrum will contain 21 frequencies. Some beating frequencies might be degenerate, hence, in reality, fewer than P(P–1)/2 RF tones are present. The simplified trigonometric expression derived by substituting the lateral profile Ψp(x) from Eq. (1) into Eq. (7) yields the following near-field RF spectrum:

SRF,NF(x,ω)=ξ4|H(ω)|2p=2Pp=1pSpoptSpoptg[(ωωpp)/Δω]g(0)×|cos[(pp)πxW+πpp2]cos[(p+p)πxW+πp+p2]|2.

Equation (8) describes the signal detected by RF spectrum analyzer as a function of beat frequency ω and lateral displacement of the detector x, and is illustrated in Fig. 1.

 

Fig. 1 Simulated pictures of spatially-resolved RF spectra of a BALD (near field). Left: Spatial profiles of RF power corresponding to mode beating frequencies. The vertical axis of the graph has both RF frequency and intensity meanings. Number of shown lateral modes is P = 5. The corresponding number of the resulting beat frequencies is P(P–1)/2 = 10. Right: Same as picture on the left, but with intensity shown by color scale (i.e., vertical scale represents RF frequency only). The double-script notation of each plot indicates the corresponding indices of the lateral modes contributing to the given RF component.

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If the optical spectrum of lateral modes deviates from the ideal parabolic behavior given by Eq. (4), the RF spectrum will differ from picture in Fig. 1. In Section 5 it will be shown that the accurate measurement of the RF spectrum allows to clearly observe the shift of spectral mode positions from the predictions of the ideal box model.

For a small number of lateral modes (less than 10–20), the RF spectra of different mode beatings do not overlap, and one can develop the inverse procedure to recover the modes’ optical powers and wavelength positions from the RF data. Equation (8) can be re-written as

SRF,NF(x,ω)=p=2Pp=1pSppRF|H(ω)|2|H(ωpp)|2g[(ωωpp)/Δω]g(0),
where
SppRF=maxW2xW2[SRF,NF(x,ωpp)]=ξ|H(ωpp)|2μppSpoptSpopt
is the peak (i.e., maximum in both spectral and spatial domains) of the RF spectral density of beating between the modes with lateral indices p and p′, and
μpp=maxW2xW2{14|cos[(pp)πxW+πpp2]cos[(p+p)πxW+πp+p2]|2}.
A set of Eqs. (10) for various combinations of p and p′ forms the rules of transformation between the modes’ optical powers and the measured RF powers. If the photodiode frequency response function is known, the optical power of the pth lateral mode can be easily found as
Spopt=1ξSppRFSppRFSppRFμppμppμpp|H(ωpp)|2|H(ωpp)|2|H(ωpp)|2.
From practical point of view, the knowledge of the normalization constant ξ and absolute values of SppRF is not essential (only the relative values of SppRF are important), because the absolute optical powers of the lateral modes can always be recovered from their relative powers by using the fact that their sum is the total optical power: p=1PSpopt=Stotalopt.

In the case of an infinite bandwidth photodiode [H(ω) = const], the system of Eqs. (10) is overdetermined. Analyzing the spatial distribution of the RF signal, it is possible to do a visual selection of P lines in RF spectrum with best signal-to-noise ratios and algebraically solve the chosen system of P equations for P mode powers. Alternatively, if the signal-to-noise ratio and individual distinguishability of beat signals are good, it is possible to use all P(P–1)/2 equations to determine P mode powers and extract P(P–3)/2 points of the RF response of the photodetector.

The recovery of optical frequency position inside of mth longitudinal group (ωmp) can be done using formula ωmp=ωm1+ω1p, where ωm1 is the optical frequency of the fundamental lateral mode (p = 1) in the mth longitudinal group (it must be known from optical measurements) and ω1p is the RF frequency beat between fundamental and pth lateral modes of the same longitudinal group.

In the case of a large number of lateral modes, the equation system may have hundreds of frequency components. In that case, the mode power and the wavelength recovery process from the RF spectra measured in the near field may become difficult due to the mode frequency degeneracies and interference of modes belonging to different longitudinal orders.

This situation can be significantly improved if the self-heterodyning is realized at a large distance from the laser facet (i.e., in the far field), as described in the next Section.

4. Self-heterodyned RF spectra of BALD in far-field optical region

The far-field pattern has a different nature of lateral mode overlaps. Instead of P peaks in near-field, the far field of lateral mode p of BALD has only two lobes and has a significant overlap only with the adjacent modes p + 1 and p–1. This structure has a lower number of the resulting beat frequencies with significant RF power. By observing the RF beats in the far-field plane, we can reduce the number of RF frequencies from P(P–1)/2 to (P–1). The corresponding simplified expression can be obtained similarly to Eq. (8) and is given by:

SRF,FF(kx,ω)=ξ|H(ω)|2p=1P1|2π2W2p(p+1)sinkxW(π2p2kx2W2)[π2(p+1)2kx2W2]|2×g[ω(2p+1)δωΔω]g(0)SpoptSp+1opt,
where ξ′ is a calibration constant defined in the Appendix, H(ω) is the photodiode’s transfer function, kx is the lateral wavenumber [kx = (2π/λ) sin α], α is the angle between the direction of observation and the axis of laser emission, and beat frequency between the adjacent lateral modes ωp,(p+1) ≈(2p + 1)δω, with δωπcλ04nph2(λ0)W2. Figure 2 plots Eq. (13) for P = 5.

 

Fig. 2 Simulated picture of angularly-resolved RF spectra of a BALD (far field). Left: Spatial profiles of RF power corresponding to mode beating frequencies. The vertical axis of the graph has both RF frequency and intensity meanings. Number of shown lateral modes P is 5. The corresponding number of resulting strong beat frequencies is P–1 = 4 (shown in the bottom half of the picture). Right: Same as picture on the left, but with intensity shown by color scale (i.e., vertical scale represents RF frequency only). The double-script notation of each plot indicates the corresponding indices of the lateral modes contributing to the given RF component.

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The direct relationship between the modes’ optical powers and the corresponding RF powers at the beat frequencies can be summarized in the following set of P equations, which is sufficient for the recovery of all mode powers Soptp, assuming that the total optical power Sopttotal is known:

ξ|H(ω12)|2μ12S1optS2opt=S12RF,ξ|H(ω23)|2μ23S2optS3opt=S23RF,ξ|H(ω34)|2μ34S3optS4opt=S34RF,........ξ|H(ωP1,P)|2μ(P1),PSP1optSPopt=S(P1),PRF,p=1PSpopt=Stotalopt,
where Sp,(p+1)RF is the peak (i.e., maximum in both spectral and angular domains) of the RF spectral density of beating between the modes with lateral indices p and (p + 1), and
μp,(p+1)=max2πλkx2πλ{|2π2W2p(p+1)sinkxW(π2p2kx2W2)[π2(p+1)2kx2W2]|2}.
The recovery of the frequency position is done similarly to the near-field case. The frequency of pth lateral mode in mth longitudinal group is
ωmp=ωm1+p=1p1ωp,(p+1),
where ωm1 is the optical frequency of the fundamental lateral mode (p = 1) in the mth longitudinal group (it must be known from optical measurements) and ωp,(p+1) is the RF beat frequency between pth and (p + 1)th lateral modes of the same group. The frequency (or wavelength) information of mode can be determined by simple addition of the corresponding RF beat frequencies.

The number of variables in Eq. (14) is equal to the number of equations. There are no extra equations that can be used for the photodiode frequency response calibration, i.e., for far-field measurements the frequency response of the RF receiver has to be well known.

The far-field RF measurements are particularly useful in dealing with BALDs operating at high pumping levels (where the number of lateral modes is so large that within the same longitudinal order there are different mode pairs with the same frequency differences ωmp – ωmp = ωmp′″ – ωmp″″, i.e., p2p2 = p″′2p″″2, e.g., p′ = 7, p″ = p″′ = 5, p″″ = 1) and with long BALDs (where the spectra of the different longitudinal groups overlap, and different lateral modes of different longitudinal groups fall onto the same optical frequency ωmp = ωmp, as discussed in great detail in Ref [14].). In the former case, the modes’ optical frequencies remain non-degenerate, and only their RF beat frequencies are degenerate; in the latter case, the optical frequencies of the modes become degenerate. Even an approximate degeneracy can result in the RF beat spectra of different mode pairs overlapping in the near field, which makes the near-field identification of lateral modes difficult, or even impossible. This difficulty can be overcome by performing the measurements in the far field: here the degeneracy is completely lifted in the case of a BALD at high pumping level, whereas in the case of a long BALD the different beating mode pairs comprising each degenerate RF tone can be matched to the distinct pairs of the far-field spots at the same RF frequency. This permits unambiguous identification of the lateral modes by the far-field RF heterodyning. Although Eqs. (8) and (13) are valid only for the short BALDs, where the spectra of the adjacent longitudinal orders do not overlap, their equivalents for the long BALDs can be straightforwardly obtained by a derivation similar to that in the Appendix, but without simplifying Eq. (29) by assumption m = m′.

5. Experiment

The experimental verification of the techniques described in Sections 3 and 4 has been realized in the spatially-resolving RF spectrum measurement setup shown in Fig. 3. The output facet of a BALD is imaged onto the plane of a fast small-diameter (~50 μm) silicon photodiode with the bandwidth > 2 GHz by a 1:5 telescope made of two achromatic lenses with 40-mm and 200-mm focal distances. The intensity oscillations of the multimode output are detected by the photodiode, amplified, and delivered to the RF spectrum analyzer (HP8563A). By moving the photodiode along the output facet’s image, the RF spectra are measured for different lateral coordinates. The lateral variations of the RF signal assure the reliable identification of the beating modes. The translation stage moves and the RF spectrum acquisition are synchronized by a computer. Measurements of the far-field RF spectra are performed directly by the photodiode placed at 480 mm distance from the laser facet.

 

Fig. 3 RF spectra measurement setup. “Near-field optical system” is a 1:5 telescope made of two achromatic doublet lenses with 40 mm and 200 mm focal distances. “Far-field optical system” is simply a free-space propagation distance of 480 mm.

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In the first experiment, the RF spectrum measurements are realized in the near field, and their results are shown in Fig. 4. We use HPD BALD with width W = 96 µm and length L = 1 mm, operating at wavelength λ0 = 980 nm. Laser spectra shown in Fig. 4 are measured at 300 mA pump current and consist of 19–20 longitudinal mode groups spaced by 0.128 nm (or 40.0 GHz). The total laser output power is about 200 mW. Figure 4(a) shows the optical spectra of two consecutive longitudinal mode groups, obtained by the high-resolution imaging spectrometer described in Ref [8]. The relative powers of the modes, estimated from the optical measurements, are: S1opt = 1, S2opt = 1.3, S3opt = 2.35, S4opt = 1.7, S5opt = 0.32. Even though the resolution of the direct optical measurement is ~1 GHz, the high signal-to-noise ratio allows us to estimate the mode frequency spacings fpp: f12 = 1.47 GHz, f13 = 3.53 GHz, f14 = 5.88 GHz, f15 = 9.12 GHz.

 

Fig. 4 (a) High-resolution detail of the optical spectrum of a 1-mm-long, 96-μm-wide BALD at 300 mA pump current. (b)–(d) Spatially-resolved near-field RF spectra of the same BALD. (b) Box model simulation. (c) Experimental results. (d) Box model modified by lowering the fundamental mode’s frequency by 370 MHz. Note that the horizontal scale in (c) is 5 × of that in (b) and (d), owing to the telescope magnification. The low-frequency (< 1 GHz) RF components in c) are most likely due to relaxation oscillations / mode partitioning noise.

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The frequency position of the fundamental (p = 1) mode in Fig. 4(a) is farther away from the higher-order modes than the position described by the box model of Eq. (4); the resulting measured frequency spacings differ by 0.3–0.4 GHz from those predicted by the box model, as shown in Fig. 5. However, a small phenomenological modification of the box model by lowering the frequency of the fundamental lateral mode in Eq. (4) by 370 MHz (with no frequency changes for other modes) brings the model into a very good agreement with the experiment. Such frequency shift puts the fundamental mode (p = 1) approximately into a position described by Eq. (4) for a fictitious mode with index p = 0.

 

Fig. 5 Comparison of the relative mode frequencies predicted by the box model of Eq. (4) (solid black line) and observed in optical (blue circles) and RF (green diamonds) measurements. Purple line shows the box model modified by lowering the fundamental mode’s frequency by 370 MHz.

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Figure 4(b) shows box-model simulation of spatially resolved beat frequencies corresponding to four-mode optical spectra described by Eq. (4) and Eqs. (9)(11). These predictions are compared with the experimentally measured spatially-resolved RF spectra of the self-heterodyned signal detected by the photodiode in the near field of the BALD, shown in Fig. 4(c). We find good agreement between the standard box model and the experiment for mode beatings that do not include the fundamental mode (p = 1). On the other hand, the box model with fundamental mode’s frequency modified as described above, shown in Fig. 4(d), agrees very well with the experimental RF results for all modes, including the fundamental. In our previous investigations of other BALDs [8, 12, 13] and broad-area quantum cascade lasers [14] we did not observe the frequency shift of the fundamental mode, which indicates that this shift is specific to the BALD studied in the present paper. Since such mode-specific frequency shift is related to some spatially-varying cavity perturbation (e.g., a defect), its clear observation indicates the suitability of the RF self-heterodyning technique for monitoring the evolution of defects and predicting possible BALD failure.

The modes depicted in Fig. 4(c) are easily recognizable, and the recovery of their frequencies is relatively simple, as discussed in Section 3. The peak RF powers, normalized by the photodiode’s frequency response, and the RF peak frequencies are listed in Table 1.

Tables Icon

Table 1. Results of RF Measurements

Solving the system (10) in the form of 5 equations (S1optS2optS12RF, S2optS3optS23RF, S3optS4optS34RF, S1optS3optS13RF, S2optS4optS24RF, assuming that component S45RF is negligible) by using Eq. (12) yields the relative mode powers presented in Table 2, which also compares them with the relative mode powers and frequencies observed in the direct optical spectrum measurements of Fig. 4(a). Table 2 shows that both the relative powers and frequency positions obtained from the RF spectrum are in a good agreement with the data obtained from the optical spectrum. The mode frequencies recovered from the RF measurements are also shown in Fig. 5 (green diamonds) and exhibit an excellent match with the predictions of the modified box model.

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Table 2. Comparison of the Results Obtained from RF and Direct Optical Measurements

These experimental data show that the near-field self-heterodyning measurement technique is capable of measuring mode beating patterns from laser diodes with a spatial resolution of ~10 µm (determined by the size of the photodiode, divided by the magnification factor) and a spectral resolution of several tens of MHz.

At high pump current, the number of modes measured in the RF spectra in the near field increases and their overlapping makes mode identification difficult. As suggested in Section 4, the beat frequency spectra can be measured instead in the far field, where only the neighboring modes spatially overlap. The total number of peaks present in the angularly-resolved RF-beat spectra is significantly reduced.

The RF spectrum measurements in the far field are conducted in the experimental setup of Fig. 3 with 480 mm of free space between the laser and photodiode. The displacement of the detector in the far field produces angular scan of the RF spectra of laser diode radiation. Figure 6 shows good agreement between the measurements [Fig. 6(a)] and the corresponding simulation of the modified box model [Fig. 6(b)]. The pumping conditions are the same as those for near-field measurements: 300 mA pump current, temperature T = 20° C.

 

Fig. 6 Experimental measurements (a), (c) and theoretical simulations (b), (d) of the angularly-resolved RF spectra (far field) for a 1-mm-long, 96-μm-wide BALD at: (a), (b) 300 mA and (c), (d) 450 mA pump currents. The simulations in (b), (d) represent the box model modified by lowering the fundamental mode’s frequency by 370 MHz. The low-frequency (< 1 GHz) RF components in a) and c) are most likely due to relaxation oscillations / mode partitioning noise.

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The RF spectra measured in the far field at a higher pump current of 450 mA [Fig. 6(c)] show a qualitative agreement with the modified box model of Fig. 6(d). Quantitatively, however, they show noticeable deviations of the frequencies of low-lateral-index modes from the p2 trendline of the box model. Even in the presence of large deviations from the box model, one can monitor the health of the BALD by periodically checking the changes in the far-field RF spectra compared to the spectra taken at the beginning of the BALD’s life.

Results of Fig. 6 prove the feasibility of measuring the angularly-resolved RF spectra in the far-field plane, which can be very useful in the case of a large number of lateral modes.

6. Conclusions

We have described and experimentally demonstrated the near- and far-field self-heterodyning techniques that yield precise information about the lateral mode structure of a broad-area laser diode with resolution limited only by the natural linewidth of the laser modes. They can serve as powerful tools for monitoring the mode positions, detection of coherence spikes and other nonlinear mode interactions at high power levels, and can provide important input for laser failure prediction and analysis. Both near- and far-field self-heterodyning methods give complete information about the mode powers and relative frequencies. By monitoring the modes’ frequency deviations from the box model, as well as evolutions of these deviations and relative mode powers over the BALD’s working life, one can gain invaluable information about the presence and growth of intra-cavity defects and, possibly, predict the laser failure.

For a low number of lateral modes, the near-field self-heterodyning is more advantageous, since it can eliminate the problem of detector’s RF response calibration. Monitoring mode spectra by the near-field self-heterodyning method can be straightforwardly implemented directly on the laser chip in the form of a photodiode array at the laser’s back facet.

On the other hand, for wide-spectrum laser diode structures, the dispersion of the refractive index makes the lateral-mode spacing dependent on the longitudinal order, which broadens the RF spectrum of each pair of beating lateral modes. In addition, for long lasers, the longitudinal free spectral range can become very small (less than 0.1 nm). Both of these effects, as well as the increase in the number of illuminated lateral modes with higher pumping levels, can lead to the overlap of the beatings of many lateral mode pairs in the near field, which makes mode identification difficult. In that case the far-field self-heterodyning becomes preferable.

Appendix: Relation between optical and RF-beat spectra of broad-area laser diodes

Total electrical field emitted by a broad-area laser diode is

ε(x,y,t)=Φ(y)m,pAmp(t)eiωmptΨp(x)=Φ(y)E(x,t),
where
E(x,t)=m,pAmp(t)eiωmptΨp(x)=m,pEmp(x,t),
Φ(y) is normalized so that
|Φ(y)|2dy=hy,
hy is the effective vertical spread of the field, and
Amp(t)=|Amp(t)|eiφmp(t)
are the slowly-variable mode amplitudes. First, let us calculate the spectral density of optical power contained in the radiation pattern above. Since the intensity in free space is equal to I=(1/2)cε0|ε(x,y,t)|2, the optical power falling onto a photodetector with x- and y-dimensions d and hdet, respectively, and vertically centered on the laser radiation axis, is:
P(x,t)=12cε0xd/2x+d/2dxhdet/2hdet/2dy|ε(x,y,t)|2,
where c is the speed of light in vacuum and ε0 = 8.85 × 10−12 F/m is the vacuum permittivity. Assuming that the vertical size of the detector hdet is considerably greater than the vertical spread hy of the radiation field and that the horizontal size of detector d is small enough to neglect the horizontal field variation across the detector, the power can be written as follows:
P(x,t)=(1/2)cε0hyd|E(x,t)|2=γ|E(x,t)|2,
where γ = cε0hyd/2. From Wiener-Khinchin theorem, the spectral density of optical power Sopt(x,ω) is the Fourier transform of the field autocorrelation function:
Sopt(x,ω)=γE*(x,t)E(x,t+τ)eiωτdτ.
Optical power spectral density can be re-written as
Sopt(x,ω)=2dWm,p|Ψp(x)|2Smpopt(ωωmp),
where
Smpopt(ω)=γW2dAmp*(t)Amp(t+τ)eiωτdτ=γW2dAmp*(0)Amp(τ)eiωτdτ
is the optical power spectral density of mode m, p, integrated over its entire horizontal profile. Power spectral density Smpopt(ω) is assumed to be centered at zero frequency. In other words, the expression of Eq. (24) shows the horizontally-varying optical spectrum (e.g., that measured by an ideal spectrometer collecting light from area hdet × d) as a sum of horizontally-varying spectra of individual modes. Factor W/(2d) in Eqs. (24) and (25) is the ratio of the effective mode width |Ψmp(x)|2dx=W/2and the detector width d. Typically, the individual mode spectra Smpopt(ωωmp) are narrow lines centered at the mode frequencies ωmp. Their narrow but finite linewidths reflect the amplitude and phase fluctuations of the modes.

Let us now replace the spectrometer by a fast photodiode collecting the optical power from the same small area hdet × d. The close optical frequencies will generate photocurrent beatings that fit within the bandwidth of the photodiode and hence can be investigated by an RF spectrum analyzer. Below, we will calculate the RF spectral density of the photocurrent for two possibilities of photodiode placement: a) in the near-field plane (at laser diode surface) or b) in the far field (distance of observation is much greater than W 2 / λ0).

Let us consider the near-field case first. It can be realized, e.g., by scanning a small photodiode along the emission surface of the laser in x-direction. As mentioned above, the y-dependence of emission is eliminated by using vertical size hdet of the photodiode greater than the vertical size hy of the radiated field. We consider horizontal size d of the photodiode small enough to neglect field variation across the photodiode. The detected photocurrent is given by

i(x,t)=RP(x,t),
where R = ηeλ0/(hc) is the photodiode responsivity in Amperes per Watt (η is its quantum efficiency, e is the charge of an electron, λ0 is the central wavelength of the laser, and h is the Planck’s constant) and the optical power P(x,t) is given by Eq. (22).

According to the Wiener-Khinchin theorem, the two-sided (i.e., covering both negative and positive frequencies) power spectral density SRF(x,ω) of the photocurrent is given by the Fourier Transform of the photocurrent autocorrelation function:

S2RF(x,ω)=i(x,t)i(x,t+τ)eiωτdτ,
and the autocorrelation function does not depend on t when the laser operates in a steady state:
i(x,t)i(x,t+τ)=i(x,0)i(x,τ).
Substituting Eqs. (18) and (22) into (26), we obtain:
i(x,t)=Rγm,m,p,pAmp(t)Amp*(t)ei(ωmpωmp)tΨp(x)Ψp*(x).
Assuming that the beat frequencies (ωmp–ωmp) between any two modes belonging to different longitudinal orders mm′ are outside of the detector’s bandwidth, we can simplify Eq. (29) by keeping only terms with m = m′. Substituting the simplified Eq. (29) into the photocurrent autocorrelation function, we obtain:
i(x,0)i(x,τ)=R2γ2m,p,pm,p,p[ei(ωmpωmp)τΨp(x)Ψp*(x)Ψp(x)Ψp*(x)×Amp(0)Amp*(0)Amp(τ)Amp*(τ)].
Even though the summation in Eq. (30) is carried over 6 mode indices, most of the terms in that sum are zero. This is because all laser mode amplitudes have independent random phases, which makes the 4th-order correlation function of the electric field in Eq. (30), i.e.,
|Amp(0)Amp*(0)Amp(τ)Amp*(τ)|ei[φmp(0)φmp(0)+φmp(τ)φmp(τ)],
to be zero for all mode combinations except two special cases: a) p = p′, p″ = p″′ or b) m = m″, p = p″′, p′ = p″, but pp′ [otherwise it becomes a particular realization of case a)]. Case a) represents the DC component of the photocurrent spectrum, whereas case b) represents non-zero RF components. Hence, the photocurrent autocorrelation function can be reduced to
i(x,0)i(x,τ)=R2γ2[m,m,p,p|Amp(0)|2|Amp(τ)|2|Ψp(x)|2|Ψp(x)|2++m,p,p(pp)Amp(0)Amp*(τ)Amp*(0)Amp(τ)|Ψp(x)|2|Ψp(x)|2ei(ωmpωmp)τ].
It is the RF part (second line) of Eq. (32) that is of interest to us, and it can be rewritten as
i(x,0)i(x,τ)RF==R2γ2m,ppAmp(0)Amp*(τ)Amp*(0)Amp(τ)|Ψp(x)|2|Ψp(x)|2ei(ωmpωmp)τ
under assumption that the mode powers are either constant or their fluctuations do not noticeably correlate between two different modes, which allows us to factorize the 4th-order correlation function into a product of two 2nd-order correlation functions. By taking the inverse Fourier transform of Eq. (25), we have
Amp*(0)Amp(τ)=2dγWSmpopt(ω)eiωτdω2π.
Thus, the photocurrent spectrum of Eq. (27), after we omit its DC part, takes form of:
S2RF(x,ω)==(2RdW)2m,pp|Ψp(x)|2|Ψp(x)|2Smpopt(ω)Smpopt(ω)ei(ωω+ωmpωmp+ω)τdωdω4π2dτ=(2Rd/W)2m,pp|Ψp(x)|2|Ψp(x)|2Smpopt(ω)Smpopt(ω+ωmpωmp+ω)dω2π=(2Rd/W)2m,pp|Ψp(x)|2|Ψp(x)|2Smpopt(ωω)Smpopt[ω(ωmpωmp)]dω2π=(2Rd/W)2m,pp|Ψp(x)|2|Ψp(x)|2{Smpopt(ω)Smpopt[ω(ωmpωmp)]},
where asterisk “*” stands for the convolution. If the spectra Smpopt(ω) are even functions of frequency, then the part of Eq. (35) in figure brackets represents the convolution of modes’ optical spectra Smpopt(ω) and Smpopt(ω), shifted by (ωmp – ωmp) from DC. For photodiode with finite bandwidth determined by the response function H(ω), Eq. (35) becomes:

S2RF(x,ω)=(2RdW)2|H(ω)|2pp|Ψp(x)|2|Ψp(x)|2m{Smpopt(ω)Smpopt[ω(ωmpωmp)]}.

In order to compare the theory with experiment, we can assume that

Smpopt(ω)=Spoptg(ω/Δω)Fm,
where Spopt is the total (i.e., summed over all longitudinal orders m) laser power contained in the lateral mode with index p, the optical lineshape g(ω/Δω) has full width at half-maximum (FWHM) Δω and is normalized so that
g(ω/Δω)dω2π=1,
and Fm is the power distribution over longitudinal orders, normalized so that
mFm=1.
Let us introduce the notation
g(ω/Δω)=g(ω/Δω)g(ω/Δω)
for the convolution of the optical lineshape with inverted version of itself, where function g′(ω/Δω′) has the FWHM of Δω′ and its integral is automatically normalized to 1, similarly to Eq. (38). For Lorentzian optical lineshape g(ω/Δω), the lineshape of Eq. (40) is g′(ω/Δω′) = g[ω/(2Δω)], and for Gaussian optical lineshape g(ω/Δω), the lineshape of Eq. (40) is g′(ω/Δω′) = g[ω/(21/2Δω)]. With notations of Eqs. (37)(40), Eq. (36) becomes
S2RF(x,ω)=g(0)M(2RdW)2|H(ω)|2pp|Ψp(x)|2|Ψp(x)|2g[(ωωpp)/Δω]g(0)SpoptSpopt,
where ωpp = ωmp – ωmp and M′ is the effective number of longitudinal modes:
m=m0M/2m0+M/2Fm2=1M
(note that for flat, or rectangular, longitudinal spectrum, the effective number of modes M ′ and the actual number of longitudinal modes M coincide).

In the experiment, RF spectrum analyzer shows one-sided RF spectrum

SRF(x,ω)=S2RF(x,ω)+S2RF(x,ω)=2S2RF(x,ω)
for ω > 0, which can be found from Eq. (A25) to be given by
SRF(x,ω)=ξ|H(ω)|2p=2Pp=1p|Ψp(x)|2|Ψp(x)|2g[(ωωpp)/Δω]g(0)SpoptSpopt,
where ξ = [8g′(0)/M′] × (Rd/W)2 is a constant and P is the total number of lateral modes.

For the mode shapes given by the box model of Eq. (1), Ψp(x) = cos[πpx/W+π(p–1)/2] and ωpp = ωmp – ωmp ≈ (p2p2) δω, where δωπcλ04nph2(λ0)W2. Thus, in the near field Eq. (44) takes form of

SRF,NF(x,ω)=ξ4|H(ω)|2p=2Pp=1pSpoptSpoptg[(ωωpp)/Δω]g(0)×|cos[(pp)πxW+πpp2]cos[(p+p)πxW+πp+p2]|2.

In the paraxial approximation, the far-field mode pattern at distance l >> W20 from the laser is the spatial Fourier transform Ψ˜p(kx) of the near-field pattern Ψp(x). Thus, for near-field profile of Eq. (1) from the box model, the far field is given by

Ψ˜p(kx)=2πpWeiπ2(p1)π2p2kx2W2sin(kxW+πp2).
In the far field, only the neighboring modes p′ and p = p′ – 1 have any noticeable overlap, hence the double sum in Eq. (44) is reduced to a single sum:
SRF,FF(kx,ω)=ξ|H(ω)|2p=1P1|Ψ˜p(kx)|2|Ψ˜p+1(kx)|2g[(ωωp,(p+1))/Δω]g(0)SpoptSp+1opt=ξ|H(ω)|2p=1P1|2π2W2p(p+1)sinkxW(π2p2kx2W2)[π2(p+1)2kx2W2]|2×g[ω(2p+1)δωΔω]g(0)SpoptSp+1opt,
where ξ′ = [8g′(0)/M′] × [Rd/(Wλ0l)]2, ωp,(p+1) ≈(2p + 1)δω, and, similarly to the near-field case, we have assumed that the photodetector captures the entire vertical (y-dimension) extent of the far-field pattern whereas the detector’s horizontal size d is sufficiently small to resolve all features of the far-field pattern in the horizontal (x) dimension.

Acknowledgments

This work was supported in part by DARPA Contract No. HR0011-08-1-0063 and by the Texas Higher Education Coordinating Board Advanced Research Program.

References

1. H. G. Treusch, A. Ovtchinnikov, X. He, M. Kanskar, J. Mott, and S. Yang, “High-brightness semiconductor laser sources for materials processing: Stacking, beam shaping, and bars,” IEEE J. Sel. Top. Quantum Electron. 6(4), 601–614 (2000). [CrossRef]  

2. D. F. Welch, “A brief history of high-power semiconductor lasers,” J. Sel. Top. Quantum Electron. 6(6), 1470–1477 (2000). [CrossRef]  

3. L. F. Lester, V. Kovanis, E. P. O’Reilly, and Y. Tohmori, “Editorial: Introduction to the Issue on Semiconductor Lasers—Part 2,” IEEE J. Sel. Top. Quantum Electron. 17(6), 1468–1469 (2011). [CrossRef]  

4. H. Wenzel, “Basic aspects of high-power semiconductor laser simulation,” J. Sel. Top. Quantum Electron. 19(5), 1502913 (2013). [CrossRef]  

5. M. Achtenhagen, A. A. Hardy, and C. S. Harder, “Coherent kinks in high-power ridge waveguide laser diodes,” J. Lightwave Technol. 24(5), 2225–2232 (2006). [CrossRef]  

6. L. Bao, J. Wang, M. DeVito, D. Xu, D. Wise, P. Leisher, M. Grimshaw, W. Dong, S. Zhang, K. Price, D. Li, C. Bai, S. Patterson, and R. Martinsen, “Reliability of high performance 9xx-nm single emitter diode lasers,” Proc. SPIE 7583, 758302 (2010). [CrossRef]  

7. J. W. Tomm, M. Ziegler, M. Hempel, and T. Elsaesser, “Mechanisms and fast kinetics of the catastrophic optical damage (COD) in GaAs-based diode lasers,” Laser & Photon. Rev. 5(3), 422–441 (2011). [CrossRef]  

8. N. Stelmakh and M. Flowers, “Measurement of spatial modes of broad-area diode lasers with 1-GHz resolution grating spectrometer,” IEEE Photon. Technol. Lett. 18(15), 1618–1620 (2006). [CrossRef]  

9. H. Partanen, J. Tervo, and J. Turunen, “Spatial coherence of broad-area laser diodes,” Appl. Opt. 52(14), 3221–3228 (2013). [CrossRef]   [PubMed]  

10. L. Büttner and J. Czarske, “Investigation of the influence of spatial coherence of a broad-area laser diode on the interference fringe system of a Mach-Zehnder interferometer for highly spatially resolved velocity measurements,” Appl. Opt. 44(9), 1582–1590 (2005). [CrossRef]   [PubMed]  

11. N. Stelmakh and M. Vasilyev, “Spatially resolved spectroscopy of lateral modes of broad-area laser diodes by self-heterodyning,” in Conference on Lasers and Electro-Optics, OSA Technical Digest (CD) (Optical Society of America, 2008), May 2008, San Jose, CA, paper CMN6. [CrossRef]  

12. N. Stelmakh, “Harnessing multimode broad-area laser-diode emission into a single-lobe diffraction-limited spot,” IEEE Photon. Technol. Lett. 19(18), 1392–1394 (2007). [CrossRef]  

13. N. Stelmakh and M. Vasilyev, “Mode harnessing for laser diodes,” Opt. and Photon. News 21(4), 20–25 (2010). [CrossRef]  

14. N. Stelmakh, M. Vasilyev, F. Toor, and C. Gmachl, “Degenerate and nondegenerate lateral-mode patterns in quantum cascade lasers,” Appl. Phys. Lett. 94(1), 013501 (2009). [CrossRef]  

15. J. K. Butler and H. Kressel, Semiconductor Lasers and Heterojunction LEDs (Academic Press, 1977), Ch. 6 and Ch. 7.

References

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  1. H. G. Treusch, A. Ovtchinnikov, X. He, M. Kanskar, J. Mott, S. Yang, “High-brightness semiconductor laser sources for materials processing: Stacking, beam shaping, and bars,” IEEE J. Sel. Top. Quantum Electron. 6(4), 601–614 (2000).
    [CrossRef]
  2. D. F. Welch, “A brief history of high-power semiconductor lasers,” J. Sel. Top. Quantum Electron. 6(6), 1470–1477 (2000).
    [CrossRef]
  3. L. F. Lester, V. Kovanis, E. P. O’Reilly, Y. Tohmori, “Editorial: Introduction to the Issue on Semiconductor Lasers—Part 2,” IEEE J. Sel. Top. Quantum Electron. 17(6), 1468–1469 (2011).
    [CrossRef]
  4. H. Wenzel, “Basic aspects of high-power semiconductor laser simulation,” J. Sel. Top. Quantum Electron. 19(5), 1502913 (2013).
    [CrossRef]
  5. M. Achtenhagen, A. A. Hardy, C. S. Harder, “Coherent kinks in high-power ridge waveguide laser diodes,” J. Lightwave Technol. 24(5), 2225–2232 (2006).
    [CrossRef]
  6. L. Bao, J. Wang, M. DeVito, D. Xu, D. Wise, P. Leisher, M. Grimshaw, W. Dong, S. Zhang, K. Price, D. Li, C. Bai, S. Patterson, R. Martinsen, “Reliability of high performance 9xx-nm single emitter diode lasers,” Proc. SPIE 7583, 758302 (2010).
    [CrossRef]
  7. J. W. Tomm, M. Ziegler, M. Hempel, T. Elsaesser, “Mechanisms and fast kinetics of the catastrophic optical damage (COD) in GaAs-based diode lasers,” Laser & Photon. Rev. 5(3), 422–441 (2011).
    [CrossRef]
  8. N. Stelmakh, M. Flowers, “Measurement of spatial modes of broad-area diode lasers with 1-GHz resolution grating spectrometer,” IEEE Photon. Technol. Lett. 18(15), 1618–1620 (2006).
    [CrossRef]
  9. H. Partanen, J. Tervo, J. Turunen, “Spatial coherence of broad-area laser diodes,” Appl. Opt. 52(14), 3221–3228 (2013).
    [CrossRef] [PubMed]
  10. L. Büttner, J. Czarske, “Investigation of the influence of spatial coherence of a broad-area laser diode on the interference fringe system of a Mach-Zehnder interferometer for highly spatially resolved velocity measurements,” Appl. Opt. 44(9), 1582–1590 (2005).
    [CrossRef] [PubMed]
  11. N. Stelmakh and M. Vasilyev, “Spatially resolved spectroscopy of lateral modes of broad-area laser diodes by self-heterodyning,” in Conference on Lasers and Electro-Optics, OSA Technical Digest (CD) (Optical Society of America, 2008), May 2008, San Jose, CA, paper CMN6.
    [CrossRef]
  12. N. Stelmakh, “Harnessing multimode broad-area laser-diode emission into a single-lobe diffraction-limited spot,” IEEE Photon. Technol. Lett. 19(18), 1392–1394 (2007).
    [CrossRef]
  13. N. Stelmakh, M. Vasilyev, “Mode harnessing for laser diodes,” Opt. and Photon. News 21(4), 20–25 (2010).
    [CrossRef]
  14. N. Stelmakh, M. Vasilyev, F. Toor, C. Gmachl, “Degenerate and nondegenerate lateral-mode patterns in quantum cascade lasers,” Appl. Phys. Lett. 94(1), 013501 (2009).
    [CrossRef]
  15. J. K. Butler and H. Kressel, Semiconductor Lasers and Heterojunction LEDs (Academic Press, 1977), Ch. 6 and Ch. 7.

2013 (2)

H. Wenzel, “Basic aspects of high-power semiconductor laser simulation,” J. Sel. Top. Quantum Electron. 19(5), 1502913 (2013).
[CrossRef]

H. Partanen, J. Tervo, J. Turunen, “Spatial coherence of broad-area laser diodes,” Appl. Opt. 52(14), 3221–3228 (2013).
[CrossRef] [PubMed]

2011 (2)

L. F. Lester, V. Kovanis, E. P. O’Reilly, Y. Tohmori, “Editorial: Introduction to the Issue on Semiconductor Lasers—Part 2,” IEEE J. Sel. Top. Quantum Electron. 17(6), 1468–1469 (2011).
[CrossRef]

J. W. Tomm, M. Ziegler, M. Hempel, T. Elsaesser, “Mechanisms and fast kinetics of the catastrophic optical damage (COD) in GaAs-based diode lasers,” Laser & Photon. Rev. 5(3), 422–441 (2011).
[CrossRef]

2010 (2)

L. Bao, J. Wang, M. DeVito, D. Xu, D. Wise, P. Leisher, M. Grimshaw, W. Dong, S. Zhang, K. Price, D. Li, C. Bai, S. Patterson, R. Martinsen, “Reliability of high performance 9xx-nm single emitter diode lasers,” Proc. SPIE 7583, 758302 (2010).
[CrossRef]

N. Stelmakh, M. Vasilyev, “Mode harnessing for laser diodes,” Opt. and Photon. News 21(4), 20–25 (2010).
[CrossRef]

2009 (1)

N. Stelmakh, M. Vasilyev, F. Toor, C. Gmachl, “Degenerate and nondegenerate lateral-mode patterns in quantum cascade lasers,” Appl. Phys. Lett. 94(1), 013501 (2009).
[CrossRef]

2007 (1)

N. Stelmakh, “Harnessing multimode broad-area laser-diode emission into a single-lobe diffraction-limited spot,” IEEE Photon. Technol. Lett. 19(18), 1392–1394 (2007).
[CrossRef]

2006 (2)

N. Stelmakh, M. Flowers, “Measurement of spatial modes of broad-area diode lasers with 1-GHz resolution grating spectrometer,” IEEE Photon. Technol. Lett. 18(15), 1618–1620 (2006).
[CrossRef]

M. Achtenhagen, A. A. Hardy, C. S. Harder, “Coherent kinks in high-power ridge waveguide laser diodes,” J. Lightwave Technol. 24(5), 2225–2232 (2006).
[CrossRef]

2005 (1)

2000 (2)

H. G. Treusch, A. Ovtchinnikov, X. He, M. Kanskar, J. Mott, S. Yang, “High-brightness semiconductor laser sources for materials processing: Stacking, beam shaping, and bars,” IEEE J. Sel. Top. Quantum Electron. 6(4), 601–614 (2000).
[CrossRef]

D. F. Welch, “A brief history of high-power semiconductor lasers,” J. Sel. Top. Quantum Electron. 6(6), 1470–1477 (2000).
[CrossRef]

Achtenhagen, M.

Bai, C.

L. Bao, J. Wang, M. DeVito, D. Xu, D. Wise, P. Leisher, M. Grimshaw, W. Dong, S. Zhang, K. Price, D. Li, C. Bai, S. Patterson, R. Martinsen, “Reliability of high performance 9xx-nm single emitter diode lasers,” Proc. SPIE 7583, 758302 (2010).
[CrossRef]

Bao, L.

L. Bao, J. Wang, M. DeVito, D. Xu, D. Wise, P. Leisher, M. Grimshaw, W. Dong, S. Zhang, K. Price, D. Li, C. Bai, S. Patterson, R. Martinsen, “Reliability of high performance 9xx-nm single emitter diode lasers,” Proc. SPIE 7583, 758302 (2010).
[CrossRef]

Büttner, L.

Czarske, J.

DeVito, M.

L. Bao, J. Wang, M. DeVito, D. Xu, D. Wise, P. Leisher, M. Grimshaw, W. Dong, S. Zhang, K. Price, D. Li, C. Bai, S. Patterson, R. Martinsen, “Reliability of high performance 9xx-nm single emitter diode lasers,” Proc. SPIE 7583, 758302 (2010).
[CrossRef]

Dong, W.

L. Bao, J. Wang, M. DeVito, D. Xu, D. Wise, P. Leisher, M. Grimshaw, W. Dong, S. Zhang, K. Price, D. Li, C. Bai, S. Patterson, R. Martinsen, “Reliability of high performance 9xx-nm single emitter diode lasers,” Proc. SPIE 7583, 758302 (2010).
[CrossRef]

Elsaesser, T.

J. W. Tomm, M. Ziegler, M. Hempel, T. Elsaesser, “Mechanisms and fast kinetics of the catastrophic optical damage (COD) in GaAs-based diode lasers,” Laser & Photon. Rev. 5(3), 422–441 (2011).
[CrossRef]

Flowers, M.

N. Stelmakh, M. Flowers, “Measurement of spatial modes of broad-area diode lasers with 1-GHz resolution grating spectrometer,” IEEE Photon. Technol. Lett. 18(15), 1618–1620 (2006).
[CrossRef]

Gmachl, C.

N. Stelmakh, M. Vasilyev, F. Toor, C. Gmachl, “Degenerate and nondegenerate lateral-mode patterns in quantum cascade lasers,” Appl. Phys. Lett. 94(1), 013501 (2009).
[CrossRef]

Grimshaw, M.

L. Bao, J. Wang, M. DeVito, D. Xu, D. Wise, P. Leisher, M. Grimshaw, W. Dong, S. Zhang, K. Price, D. Li, C. Bai, S. Patterson, R. Martinsen, “Reliability of high performance 9xx-nm single emitter diode lasers,” Proc. SPIE 7583, 758302 (2010).
[CrossRef]

Harder, C. S.

Hardy, A. A.

He, X.

H. G. Treusch, A. Ovtchinnikov, X. He, M. Kanskar, J. Mott, S. Yang, “High-brightness semiconductor laser sources for materials processing: Stacking, beam shaping, and bars,” IEEE J. Sel. Top. Quantum Electron. 6(4), 601–614 (2000).
[CrossRef]

Hempel, M.

J. W. Tomm, M. Ziegler, M. Hempel, T. Elsaesser, “Mechanisms and fast kinetics of the catastrophic optical damage (COD) in GaAs-based diode lasers,” Laser & Photon. Rev. 5(3), 422–441 (2011).
[CrossRef]

Kanskar, M.

H. G. Treusch, A. Ovtchinnikov, X. He, M. Kanskar, J. Mott, S. Yang, “High-brightness semiconductor laser sources for materials processing: Stacking, beam shaping, and bars,” IEEE J. Sel. Top. Quantum Electron. 6(4), 601–614 (2000).
[CrossRef]

Kovanis, V.

L. F. Lester, V. Kovanis, E. P. O’Reilly, Y. Tohmori, “Editorial: Introduction to the Issue on Semiconductor Lasers—Part 2,” IEEE J. Sel. Top. Quantum Electron. 17(6), 1468–1469 (2011).
[CrossRef]

Leisher, P.

L. Bao, J. Wang, M. DeVito, D. Xu, D. Wise, P. Leisher, M. Grimshaw, W. Dong, S. Zhang, K. Price, D. Li, C. Bai, S. Patterson, R. Martinsen, “Reliability of high performance 9xx-nm single emitter diode lasers,” Proc. SPIE 7583, 758302 (2010).
[CrossRef]

Lester, L. F.

L. F. Lester, V. Kovanis, E. P. O’Reilly, Y. Tohmori, “Editorial: Introduction to the Issue on Semiconductor Lasers—Part 2,” IEEE J. Sel. Top. Quantum Electron. 17(6), 1468–1469 (2011).
[CrossRef]

Li, D.

L. Bao, J. Wang, M. DeVito, D. Xu, D. Wise, P. Leisher, M. Grimshaw, W. Dong, S. Zhang, K. Price, D. Li, C. Bai, S. Patterson, R. Martinsen, “Reliability of high performance 9xx-nm single emitter diode lasers,” Proc. SPIE 7583, 758302 (2010).
[CrossRef]

Martinsen, R.

L. Bao, J. Wang, M. DeVito, D. Xu, D. Wise, P. Leisher, M. Grimshaw, W. Dong, S. Zhang, K. Price, D. Li, C. Bai, S. Patterson, R. Martinsen, “Reliability of high performance 9xx-nm single emitter diode lasers,” Proc. SPIE 7583, 758302 (2010).
[CrossRef]

Mott, J.

H. G. Treusch, A. Ovtchinnikov, X. He, M. Kanskar, J. Mott, S. Yang, “High-brightness semiconductor laser sources for materials processing: Stacking, beam shaping, and bars,” IEEE J. Sel. Top. Quantum Electron. 6(4), 601–614 (2000).
[CrossRef]

O’Reilly, E. P.

L. F. Lester, V. Kovanis, E. P. O’Reilly, Y. Tohmori, “Editorial: Introduction to the Issue on Semiconductor Lasers—Part 2,” IEEE J. Sel. Top. Quantum Electron. 17(6), 1468–1469 (2011).
[CrossRef]

Ovtchinnikov, A.

H. G. Treusch, A. Ovtchinnikov, X. He, M. Kanskar, J. Mott, S. Yang, “High-brightness semiconductor laser sources for materials processing: Stacking, beam shaping, and bars,” IEEE J. Sel. Top. Quantum Electron. 6(4), 601–614 (2000).
[CrossRef]

Partanen, H.

Patterson, S.

L. Bao, J. Wang, M. DeVito, D. Xu, D. Wise, P. Leisher, M. Grimshaw, W. Dong, S. Zhang, K. Price, D. Li, C. Bai, S. Patterson, R. Martinsen, “Reliability of high performance 9xx-nm single emitter diode lasers,” Proc. SPIE 7583, 758302 (2010).
[CrossRef]

Price, K.

L. Bao, J. Wang, M. DeVito, D. Xu, D. Wise, P. Leisher, M. Grimshaw, W. Dong, S. Zhang, K. Price, D. Li, C. Bai, S. Patterson, R. Martinsen, “Reliability of high performance 9xx-nm single emitter diode lasers,” Proc. SPIE 7583, 758302 (2010).
[CrossRef]

Stelmakh, N.

N. Stelmakh, M. Vasilyev, “Mode harnessing for laser diodes,” Opt. and Photon. News 21(4), 20–25 (2010).
[CrossRef]

N. Stelmakh, M. Vasilyev, F. Toor, C. Gmachl, “Degenerate and nondegenerate lateral-mode patterns in quantum cascade lasers,” Appl. Phys. Lett. 94(1), 013501 (2009).
[CrossRef]

N. Stelmakh, “Harnessing multimode broad-area laser-diode emission into a single-lobe diffraction-limited spot,” IEEE Photon. Technol. Lett. 19(18), 1392–1394 (2007).
[CrossRef]

N. Stelmakh, M. Flowers, “Measurement of spatial modes of broad-area diode lasers with 1-GHz resolution grating spectrometer,” IEEE Photon. Technol. Lett. 18(15), 1618–1620 (2006).
[CrossRef]

Tervo, J.

Tohmori, Y.

L. F. Lester, V. Kovanis, E. P. O’Reilly, Y. Tohmori, “Editorial: Introduction to the Issue on Semiconductor Lasers—Part 2,” IEEE J. Sel. Top. Quantum Electron. 17(6), 1468–1469 (2011).
[CrossRef]

Tomm, J. W.

J. W. Tomm, M. Ziegler, M. Hempel, T. Elsaesser, “Mechanisms and fast kinetics of the catastrophic optical damage (COD) in GaAs-based diode lasers,” Laser & Photon. Rev. 5(3), 422–441 (2011).
[CrossRef]

Toor, F.

N. Stelmakh, M. Vasilyev, F. Toor, C. Gmachl, “Degenerate and nondegenerate lateral-mode patterns in quantum cascade lasers,” Appl. Phys. Lett. 94(1), 013501 (2009).
[CrossRef]

Treusch, H. G.

H. G. Treusch, A. Ovtchinnikov, X. He, M. Kanskar, J. Mott, S. Yang, “High-brightness semiconductor laser sources for materials processing: Stacking, beam shaping, and bars,” IEEE J. Sel. Top. Quantum Electron. 6(4), 601–614 (2000).
[CrossRef]

Turunen, J.

Vasilyev, M.

N. Stelmakh, M. Vasilyev, “Mode harnessing for laser diodes,” Opt. and Photon. News 21(4), 20–25 (2010).
[CrossRef]

N. Stelmakh, M. Vasilyev, F. Toor, C. Gmachl, “Degenerate and nondegenerate lateral-mode patterns in quantum cascade lasers,” Appl. Phys. Lett. 94(1), 013501 (2009).
[CrossRef]

Wang, J.

L. Bao, J. Wang, M. DeVito, D. Xu, D. Wise, P. Leisher, M. Grimshaw, W. Dong, S. Zhang, K. Price, D. Li, C. Bai, S. Patterson, R. Martinsen, “Reliability of high performance 9xx-nm single emitter diode lasers,” Proc. SPIE 7583, 758302 (2010).
[CrossRef]

Welch, D. F.

D. F. Welch, “A brief history of high-power semiconductor lasers,” J. Sel. Top. Quantum Electron. 6(6), 1470–1477 (2000).
[CrossRef]

Wenzel, H.

H. Wenzel, “Basic aspects of high-power semiconductor laser simulation,” J. Sel. Top. Quantum Electron. 19(5), 1502913 (2013).
[CrossRef]

Wise, D.

L. Bao, J. Wang, M. DeVito, D. Xu, D. Wise, P. Leisher, M. Grimshaw, W. Dong, S. Zhang, K. Price, D. Li, C. Bai, S. Patterson, R. Martinsen, “Reliability of high performance 9xx-nm single emitter diode lasers,” Proc. SPIE 7583, 758302 (2010).
[CrossRef]

Xu, D.

L. Bao, J. Wang, M. DeVito, D. Xu, D. Wise, P. Leisher, M. Grimshaw, W. Dong, S. Zhang, K. Price, D. Li, C. Bai, S. Patterson, R. Martinsen, “Reliability of high performance 9xx-nm single emitter diode lasers,” Proc. SPIE 7583, 758302 (2010).
[CrossRef]

Yang, S.

H. G. Treusch, A. Ovtchinnikov, X. He, M. Kanskar, J. Mott, S. Yang, “High-brightness semiconductor laser sources for materials processing: Stacking, beam shaping, and bars,” IEEE J. Sel. Top. Quantum Electron. 6(4), 601–614 (2000).
[CrossRef]

Zhang, S.

L. Bao, J. Wang, M. DeVito, D. Xu, D. Wise, P. Leisher, M. Grimshaw, W. Dong, S. Zhang, K. Price, D. Li, C. Bai, S. Patterson, R. Martinsen, “Reliability of high performance 9xx-nm single emitter diode lasers,” Proc. SPIE 7583, 758302 (2010).
[CrossRef]

Ziegler, M.

J. W. Tomm, M. Ziegler, M. Hempel, T. Elsaesser, “Mechanisms and fast kinetics of the catastrophic optical damage (COD) in GaAs-based diode lasers,” Laser & Photon. Rev. 5(3), 422–441 (2011).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. Lett. (1)

N. Stelmakh, M. Vasilyev, F. Toor, C. Gmachl, “Degenerate and nondegenerate lateral-mode patterns in quantum cascade lasers,” Appl. Phys. Lett. 94(1), 013501 (2009).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (2)

H. G. Treusch, A. Ovtchinnikov, X. He, M. Kanskar, J. Mott, S. Yang, “High-brightness semiconductor laser sources for materials processing: Stacking, beam shaping, and bars,” IEEE J. Sel. Top. Quantum Electron. 6(4), 601–614 (2000).
[CrossRef]

L. F. Lester, V. Kovanis, E. P. O’Reilly, Y. Tohmori, “Editorial: Introduction to the Issue on Semiconductor Lasers—Part 2,” IEEE J. Sel. Top. Quantum Electron. 17(6), 1468–1469 (2011).
[CrossRef]

IEEE Photon. Technol. Lett. (2)

N. Stelmakh, M. Flowers, “Measurement of spatial modes of broad-area diode lasers with 1-GHz resolution grating spectrometer,” IEEE Photon. Technol. Lett. 18(15), 1618–1620 (2006).
[CrossRef]

N. Stelmakh, “Harnessing multimode broad-area laser-diode emission into a single-lobe diffraction-limited spot,” IEEE Photon. Technol. Lett. 19(18), 1392–1394 (2007).
[CrossRef]

J. Lightwave Technol. (1)

J. Sel. Top. Quantum Electron. (2)

H. Wenzel, “Basic aspects of high-power semiconductor laser simulation,” J. Sel. Top. Quantum Electron. 19(5), 1502913 (2013).
[CrossRef]

D. F. Welch, “A brief history of high-power semiconductor lasers,” J. Sel. Top. Quantum Electron. 6(6), 1470–1477 (2000).
[CrossRef]

Laser & Photon. Rev. (1)

J. W. Tomm, M. Ziegler, M. Hempel, T. Elsaesser, “Mechanisms and fast kinetics of the catastrophic optical damage (COD) in GaAs-based diode lasers,” Laser & Photon. Rev. 5(3), 422–441 (2011).
[CrossRef]

Opt. and Photon. News (1)

N. Stelmakh, M. Vasilyev, “Mode harnessing for laser diodes,” Opt. and Photon. News 21(4), 20–25 (2010).
[CrossRef]

Proc. SPIE (1)

L. Bao, J. Wang, M. DeVito, D. Xu, D. Wise, P. Leisher, M. Grimshaw, W. Dong, S. Zhang, K. Price, D. Li, C. Bai, S. Patterson, R. Martinsen, “Reliability of high performance 9xx-nm single emitter diode lasers,” Proc. SPIE 7583, 758302 (2010).
[CrossRef]

Other (2)

N. Stelmakh and M. Vasilyev, “Spatially resolved spectroscopy of lateral modes of broad-area laser diodes by self-heterodyning,” in Conference on Lasers and Electro-Optics, OSA Technical Digest (CD) (Optical Society of America, 2008), May 2008, San Jose, CA, paper CMN6.
[CrossRef]

J. K. Butler and H. Kressel, Semiconductor Lasers and Heterojunction LEDs (Academic Press, 1977), Ch. 6 and Ch. 7.

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Figures (6)

Fig. 1
Fig. 1

Simulated pictures of spatially-resolved RF spectra of a BALD (near field). Left: Spatial profiles of RF power corresponding to mode beating frequencies. The vertical axis of the graph has both RF frequency and intensity meanings. Number of shown lateral modes is P = 5. The corresponding number of the resulting beat frequencies is P(P–1)/2 = 10. Right: Same as picture on the left, but with intensity shown by color scale (i.e., vertical scale represents RF frequency only). The double-script notation of each plot indicates the corresponding indices of the lateral modes contributing to the given RF component.

Fig. 2
Fig. 2

Simulated picture of angularly-resolved RF spectra of a BALD (far field). Left: Spatial profiles of RF power corresponding to mode beating frequencies. The vertical axis of the graph has both RF frequency and intensity meanings. Number of shown lateral modes P is 5. The corresponding number of resulting strong beat frequencies is P–1 = 4 (shown in the bottom half of the picture). Right: Same as picture on the left, but with intensity shown by color scale (i.e., vertical scale represents RF frequency only). The double-script notation of each plot indicates the corresponding indices of the lateral modes contributing to the given RF component.

Fig. 3
Fig. 3

RF spectra measurement setup. “Near-field optical system” is a 1:5 telescope made of two achromatic doublet lenses with 40 mm and 200 mm focal distances. “Far-field optical system” is simply a free-space propagation distance of 480 mm.

Fig. 4
Fig. 4

(a) High-resolution detail of the optical spectrum of a 1-mm-long, 96-μm-wide BALD at 300 mA pump current. (b)–(d) Spatially-resolved near-field RF spectra of the same BALD. (b) Box model simulation. (c) Experimental results. (d) Box model modified by lowering the fundamental mode’s frequency by 370 MHz. Note that the horizontal scale in (c) is 5 × of that in (b) and (d), owing to the telescope magnification. The low-frequency (< 1 GHz) RF components in c) are most likely due to relaxation oscillations / mode partitioning noise.

Fig. 5
Fig. 5

Comparison of the relative mode frequencies predicted by the box model of Eq. (4) (solid black line) and observed in optical (blue circles) and RF (green diamonds) measurements. Purple line shows the box model modified by lowering the fundamental mode’s frequency by 370 MHz.

Fig. 6
Fig. 6

Experimental measurements (a), (c) and theoretical simulations (b), (d) of the angularly-resolved RF spectra (far field) for a 1-mm-long, 96-μm-wide BALD at: (a), (b) 300 mA and (c), (d) 450 mA pump currents. The simulations in (b), (d) represent the box model modified by lowering the fundamental mode’s frequency by 370 MHz. The low-frequency (< 1 GHz) RF components in a) and c) are most likely due to relaxation oscillations / mode partitioning noise.

Tables (2)

Tables Icon

Table 1 Results of RF Measurements

Tables Icon

Table 2 Comparison of the Results Obtained from RF and Direct Optical Measurements

Equations (47)

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E mp (x,t)= A mp (t) e i ω mp t cos[ πpx W + π 2 (p1) ]= A mp (t) e i ω mp t Ψ p (x),
ω mp =m πc L n ph (λ) 1+ ( Lp Wm ) 2 ,
ω m0 =m πc L n ph ( λ m ) ,
ω mp ω m0 + πc λ m 4 n ph 2 ( λ m ) W 2 p 2 ,
E(x,t)= m= m 0 M/2 m 0 +M/2 p=1 P E mp (x,t).
S 2 RF (x,ω)= (2Rd/W) 2 | H(ω) | 2 p p | Ψ p (x) | 2 | Ψ p (x) | 2 m { S mp opt (ω) S m p opt (ω ω p p ) } ,
S RF (x,ω)=ξ | H(ω) | 2 p =2 P p=1 p | Ψ p (x) | 2 | Ψ p (x) | 2 g [(ω ω p p )/Δ ω ] g (0) S p opt S p opt ,
S RF,NF (x,ω)= ξ 4 | H(ω) | 2 p =2 P p=1 p S p opt S p opt g [(ω ω p p )/Δ ω ] g (0) × | cos[ ( p p) πx W +π p p 2 ]cos[ ( p +p) πx W +π p +p 2 ] | 2 .
S RF,NF (x,ω)= p =2 P p=1 p S p p RF | H(ω) | 2 | H( ω p p ) | 2 g [(ω ω p p )/Δ ω ] g (0) ,
S p p RF = max W 2 x W 2 [ S RF,NF (x, ω p p ) ]=ξ | H( ω p p ) | 2 μ p p S p opt S p opt
μ p p = max W 2 x W 2 { 1 4 | cos[ ( p p) πx W +π p p 2 ]cos[ ( p +p) πx W +π p +p 2 ] | 2 }.
S p opt = 1 ξ S p p RF S p p RF S p p RF μ p p μ p p μ p p |H( ω p p ) | 2 |H( ω p p ) | 2 |H( ω p p ) | 2 .
S RF,FF ( k x ,ω)= ξ | H(ω) | 2 p=1 P1 | 2 π 2 W 2 p(p+1)sin k x W ( π 2 p 2 k x 2 W 2 )[ π 2 (p+1) 2 k x 2 W 2 ] | 2 × g [ ω(2p+1)δω Δ ω ] g (0) S p opt S p+1 opt ,
ξ | H( ω 12 ) | 2 μ 12 S 1 opt S 2 opt = S 12 RF , ξ | H( ω 23 ) | 2 μ 23 S 2 opt S 3 opt = S 23 RF , ξ | H( ω 34 ) | 2 μ 34 S 3 opt S 4 opt = S 34 RF , ........ ξ | H( ω P1,P ) | 2 μ (P1),P S P1 opt S P opt = S (P1),P RF , p=1 P S p opt = S total opt ,
μ p,(p+1) = max 2π λ k x 2π λ { | 2 π 2 W 2 p(p+1)sin k x W ( π 2 p 2 k x 2 W 2 )[ π 2 (p+1) 2 k x 2 W 2 ] | 2 }.
ω mp = ω m1 + p =1 p1 ω p ,( p +1) ,
ε(x,y,t)=Φ(y) m,p A mp (t) e i ω mp t Ψ p (x)=Φ(y)E(x,t),
E(x,t)= m,p A mp (t) e i ω mp t Ψ p (x)= m,p E mp (x,t) ,
| Φ(y) | 2 dy = h y ,
A mp (t)=| A mp (t) | e i φ mp (t)
P(x,t)= 1 2 c ε 0 xd/2 x+d/2 d x h det /2 h det /2 dy | ε( x ,y,t) | 2 ,
P(x,t)=(1/2)c ε 0 h y d | E(x,t) | 2 =γ | E(x,t) | 2 ,
S opt (x,ω)=γ E * (x,t)E(x,t+τ) e iωτ dτ.
S opt (x,ω)= 2d W m,p | Ψ p (x) | 2 S mp opt (ω ω mp ) ,
S mp opt (ω)= γW 2d A mp * (t) A mp (t+τ) e iωτ dτ = γW 2d A mp * (0) A mp (τ) e iωτ dτ
i(x,t)=RP(x,t),
S 2 RF (x,ω)= i(x,t)i(x,t+τ) e iωτ dτ,
i(x,t)i(x,t+τ) = i(x,0)i(x,τ) .
i(x,t)=Rγ m, m ,p, p A mp (t) A m p * (t) e i( ω mp ω m p )t Ψ p (x) Ψ p * (x) .
i(x,0)i(x,τ) = R 2 γ 2 m,p, p m , p , p [ e i( ω m p ω m p )τ Ψ p (x) Ψ p * (x) Ψ p (x) Ψ p * (x) × A mp (0) A m p * (0) A m p (τ) A m p * (τ) ].
| A mp (0) A m p * (0) A m p (τ) A m p * (τ) | e i[ φ mp (0) φ m p (0)+ φ m p (τ) φ m p (τ) ] ,
i(x,0)i(x,τ) = R 2 γ 2 [ m, m ,p, p | A mp (0) | 2 | A m p (τ) | 2 | Ψ p (x) | 2 | Ψ p (x) | 2 + + m,p, p (p p ) A mp (0) A mp * (τ) A m p * (0) A m p (τ) | Ψ p (x) | 2 | Ψ p (x) | 2 e i( ω m p ω mp )τ ].
i(x,0)i(x,τ) RF = = R 2 γ 2 m,p p A mp (0) A mp * (τ) A m p * (0) A m p (τ) | Ψ p (x) | 2 | Ψ p (x) | 2 e i( ω m p ω mp )τ
A mp * (0) A mp (τ) = 2d γW S mp opt (ω) e iωτ dω 2π .
S 2 RF (x,ω)= = ( 2Rd W ) 2 m,p p | Ψ p (x) | 2 | Ψ p (x) | 2 S mp opt ( ω ) S m p opt ( ω ) e i( ω ω + ω mp ω m p +ω)τ d ω d ω 4 π 2 dτ = (2Rd/W) 2 m,p p | Ψ p (x) | 2 | Ψ p (x) | 2 S mp opt ( ω ) S m p opt ( ω + ω mp ω m p +ω) d ω 2π = (2Rd/W) 2 m,p p | Ψ p (x) | 2 | Ψ p (x) | 2 S mp opt ( ω ω) S m p opt [ ω ( ω m p ω mp )] d ω 2π = (2Rd/W) 2 m,p p | Ψ p (x) | 2 | Ψ p (x) | 2 { S mp opt (ω) S m p opt [ω( ω m p ω mp )] },
S 2 RF (x,ω)= ( 2Rd W ) 2 | H(ω) | 2 p p | Ψ p (x) | 2 | Ψ p (x) | 2 m { S mp opt (ω) S m p opt [ω( ω m p ω mp )] } .
S mp opt (ω)= S p opt g(ω/Δω) F m ,
g(ω/Δω) dω 2π =1,
m F m =1.
g (ω/Δ ω )=g(ω/Δω)g(ω/Δω)
S 2 RF (x,ω)= g (0) M ( 2Rd W ) 2 | H(ω) | 2 p p | Ψ p (x) | 2 | Ψ p (x) | 2 g [(ω ω p p )/Δ ω ] g (0) S p opt S p opt ,
m= m 0 M/2 m 0 +M/2 F m 2 = 1 M
S RF (x,ω)= S 2 RF (x,ω)+ S 2 RF (x,ω)=2 S 2 RF (x,ω)
S RF (x,ω)=ξ | H(ω) | 2 p =2 P p=1 p | Ψ p (x) | 2 | Ψ p (x) | 2 g [(ω ω p p )/Δ ω ] g (0) S p opt S p opt ,
S RF,NF (x,ω)= ξ 4 | H(ω) | 2 p =2 P p=1 p S p opt S p opt g [(ω ω p p )/Δ ω ] g (0) × | cos[ ( p p) πx W +π p p 2 ]cos[ ( p +p) πx W +π p +p 2 ] | 2 .
Ψ ˜ p ( k x )= 2πpW e i π 2 (p1) π 2 p 2 k x 2 W 2 sin( k x W+πp 2 ).
S RF,FF ( k x ,ω)= ξ | H(ω) | 2 p=1 P1 | Ψ ˜ p ( k x ) | 2 | Ψ ˜ p+1 ( k x ) | 2 g [(ω ω p,(p+1) )/Δ ω ] g (0) S p opt S p+1 opt = ξ | H(ω) | 2 p=1 P1 | 2 π 2 W 2 p(p+1)sin k x W ( π 2 p 2 k x 2 W 2 )[ π 2 (p+1) 2 k x 2 W 2 ] | 2 × g [ ω(2p+1)δω Δ ω ] g (0) S p opt S p+1 opt ,

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