Abstract

We present a fiber-coupled balanced optical cross-correlator using waveguides in periodically-poled KTiOPO4 (PPKTP). The normalized conversion efficiency of the waveguide device is measured to be η0 = 1.02% / [W·cm2], which agrees well with theory and simulation. This result represents an expected improvement of a factor of 20 over previous bulk-optic devices. The sensitivity of the cross-correlator is characterized and shown to be comparable to the free-space bulk-optic version, with the potential for significant performance enhancements in the future.

© 2014 Optical Society of America

1. Introduction

Mode-locked lasers are capable of generating optical pulse trains with exceptionally low timing jitter, enabling synchronization of optical sources in the few-attosecond regime [1,2]. These low-noise sources have been used to develop femtosecond timing distribution systems for next generation x-ray free-electron lasers (X-FEL) [3]. Such systems have been implemented at VUV FELs such as FLASH in Hamburg, at FERMI in Trieste and an implementation in the Linac Coherent Light Source (LCLS) in Stanford is in progress. The systems typically achieve stable synchronization and diagnostics with about 10-fs accuracy, corresponding to length stabilization on the order of 1 µm over a distance of about 1 km. Meanwhile, pulse durations of X-FELs have been reduced to the few-femtosecond level, when operating in the low charge mode (LCLS). To exploit these short X-ray pulse durations for the benefit of advancing our understanding of structure and function of matter on this time scale, next generation X-FELs will aim for timing and synchronization of the optical laser seed and pump-probe laser sources at the sub-femtosecond level, requiring 100-nm length stabilization. To achieve such precision, an all-fiber implementation of the timing distribution systems [3], including temperature-stabilized integrated fiber-coupled balanced optical cross-correlators (BOC), is highly desirable. The robustness and ease of implementation of the current 10-fs level timing distribution systems will also be greatly improved by such devices. Current femtosecond timing distribution systems make use of type-II second-harmonic generation (SHG) in periodically-poled potassium titanyl phosphate (PPKTP) bulk crystals to perform the cross-correlation [4]. More recently, the use of waveguides in PPKTP has led to dramatic improvements in SHG conversion efficiency [57], and preliminary operation of a fiber-coupled device showed the promise of this approach [8]. In this paper, we present a robust fiber-coupled balanced optical cross-correlator and characterize its performance in detail. The current implementation of the integrated device yields a sensitivity comparable to that achieved with bulk crystals, with opportunities for significant improvement in the future.

2. Theory and principle of operation

Optical cross-correlation has been proposed as a sensitive method for measuring timing jitter of mode locked lasers [9], but early attempts to achieve high precision measurements were limited by optical intensity noise contaminating the error signal. To combat this problem, a balanced cross-correlation scheme was introduced, in which optical intensity noise is canceled to first order [10]. An illustration of this technique is shown in Fig. 1. In this scheme, two pulses whose relative jitter is to be measured are projected onto orthogonal polarizations, and then launched into a nonlinear crystal using type-II phase matching. As the pulses propagate through the crystal, they walk through each other in time, due the difference in group velocity for the two polarizations. The generated signal at the second harmonic will then be a functionof the initial time delay between the pulses as they enter the crystal. A dichroic mirror at the output discriminates the fundamental harmonic (FH) from the second harmonic (SH). The forward SH signal is measured by the first detector of a balanced receiver, and the reflected FH fields travel backwards and generate a new SH signal that will be measured by the second detector of the receiver. When the two FH pulses are exactly overlapped at the end of the crystal, the output voltage of the balanced receiver will be zero. In the presence of timing jitter, the varying electrical signal from the receiver provides the jitter statistics at high precision, since the balanced scheme by nature suppresses effects of amplitude fluctuations from the measured electrical signal. The use of type-II phase-matching is also advantageous for providing a SH signal that is background-free to first order.

 figure: Fig. 1

Fig. 1 An illustration of the principle of balanced optical cross-correlation. Pulses at the fundamental (represented in green) propagate through the nonlinear crystal (represented in blue) at different speeds, generating different amounts of SH light (represented as orange pulses) on the forward and reverse paths. The difference in detected SH power then gives the output voltage error curve, in which the voltage is proportional to the time separation of the pulses. Abbreviations are as follows: DM – dichroic mirror, PD – photodetector, WDM – wavelength division multiplexer.

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2.1 Second-harmonic generation in waveguides

One of the principal advantages of using waveguides in PPKTP instead of bulk crystals is the increased second-harmonic conversion efficiency. In this section we review the theory of second harmonic generation in order to derive an expression for the waveguide SHG conversion efficiency for a continuous-wave (CW) input at a single frequency. First we define the transverse fields in terms of a linear combination of spatial modes in the waveguide:

Et(x,y,z)=mAm(x)Etm(y,z)ejβmx
Ht(x,y,z)=mAm(x)Htm(y,z)ejβmx
where Am are the mode amplitudes, βm are the mode propagation constants, and the modes are normalized for unity power:
12Re{Etm×Htn*}dydz=δmn
The power in each mode is then simply |Am|2. Type-II quasi-phase-matched SHG in the waveguide can be modeled using coupled-mode theory as applied to sum-frequency generation [11]:
dA1/dx=jκ1A3(x)A2*(x)ej2Δx
dA2/dx=jκ2A3(x)A1*(x)ej2Δx
dA3/dx=jκ3A1(x)A2(x)e+j2Δx
where subscripts 1 and 2 refer to the FH y-polarized (TE) and z-polarized (TM) modes, and subscript 3 refers to the SH TE mode. The material loss is assumed to be negligible. The phase mismatch parameter is given by: 2Δ = β3 - (β1 + β2) + 2π/Λ, where Λ is the poling period for nonlinear domain inversion. The mode coupling coefficients κm are given by:
κ1=κ2=(ωε0deff/2)E1*E2*E3dydz
κ3=(ωε0deff)E1E2E3*dydz
where ω is the angular FH frequency and deff is the effective nonlinear tensor, which for our case can be represented as a scalar, deff = (2/π)d24, since the direction of propagation in the waveguides is oriented along the crystal x axis and the input fields are polarized along the crystal y and z axes. For simplicity, we neglect pump depletion, such that A1(x) = A10, A2(x) = A20. We then arrive at the SH mode amplitude after integrating Eq. (3c) over a length L of the crystal:
A3(L)=jκ3A10A20Le+jΔLsinc(ΔL)
Assuming equal un-depleted amplitudes for the two fundamental modes, A10 = A20 = A0, we define the normalized CW conversion efficiency η0 in terms of the power generated at the second harmonic:
PSH=η0L2PFH2|A3(L)|2=η0L2(|A10|2+|A20|2)2=4η0L2A04
η0=κ32sinc2(ΔL)/4
where PFH is the total input power at the fundamental frequency, PFH = P1 + P2, and η0 has units of [W·cm2]−1.

2.2 Balanced optical cross-correlator

Since the purpose of the balanced cross-correlator is to measure minute time differences between two input pulses, its principle figure of merit is the sensitivity, which we typically report in units of [mV/fs]. If the time difference at the input of the BOC is swept across a range of values, the output voltage will trace out an error curve that crosses zero for the point at which the input pulses are exactly overlapped. The sensitivity is then given by the slope of the voltage curve at this zero-crossing. To develop an analytical expression for the sensitivity, we consider two orthogonally polarized pulses traveling in the waveguide at different group velocities. We neglect dispersive broadening in the crystal (typical sample lengths are on the order of a centimeter) and assume the two input pulses have a Gaussian shape, such that the FH mode amplitudes are given by:

A1(x,t)=A10e(tζ1xΔT/2)2/τ2
A2(x,t)=A20e(tζ2x+ΔT/2)2/τ2
ζi=(vgi1vg31)
Here we have shifted to the rest frame of the SH pulse, where vgi are the FH group velocities and ζi are the group-velocity mismatch parameters. The pulse-width is given by τ and ΔT is the time separation between the two FH pulses as they enter the waveguide. To obtain the SH mode amplitude, we substitute Eq. (7a) and (7b) into Eq. (3c) and integrate over a length L of the waveguide. Assuming perfect phase-matching, we arrive at the expression:
A3(L,t)=jπκ3A10A202ce(at+bΔT2)2[erf(cL+aΔT2bt)erf(aΔT2bt)]
a2=(ζ1ζ2)2τ2(ζ12+ζ22),b2=(ζ1+ζ2)2τ2(ζ12+ζ22),c2=(ζ12+ζ22)τ2
For simplicity, we assume the group velocities of the three modes are very close to one another, such that in the SH rest frame the two FH pulses travel with equal and opposite speed, and the mismatch parameters are ζ1 ≈– ζ2 = ζ. Then Eq. (8a) simplifies to:
A3(L,t)=jπκ3A10A20τ8ζe2t2/τ2[erf(2ζL+ΔT2τ)erf(ΔT2τ)]
If the inputs are pulse trains from a mode-locked laser, then the photocurrent generated on the forward pass in the waveguide will be proportional to the power averaged over a single pulse:
J=αfrep|A3(L,t)|2dt
where α is the responsivity of the photodiode and frep is the repetition rate of the mode-locked laser, which is assumed to be the same for both input pulse trains. Similarly, we can obtain the photocurrent generated on the reverse pass and take the difference, such that the output voltage from the balanced detector will be given by:
ΔV(ΔT')=Gαfrepκ32A102A202(πτ)316ζ2[[erf(2ζL+ΔT'2τ)erf(ΔT'2τ)]2[erf(ΔT'2τ)erf(ΔT'2ζL2τ)]2]
where G is the trans-impedance gain of the detector, and we have defined ΔT ' = (ΔT +L) so that ΔV = 0 for ΔT ' = 0. Finally, taking the slope of the output voltage at ΔT ' = 0 gives the expression for the sensitivity of the BOC:
K=Gακ32Pavg,FH242frepζ2erf(2ζL2τ)[e(2ζL/2τ)21]
where Pavg,FH is the total average input power at the fundamental, and we have assumed equal mode amplitudes for the two FH modes. For a sufficiently long crystal, such that2ζL2τ, Eq. (12) simplifies toK0=Gαη0Pavg,FH2/2frepζ2, where η0 is the CW conversion efficiency derived earlier. From this expression, we see that the sensitivity has a quadratic dependence on the FH input power, as expected from the behavior of SHG. The group-velocity mismatch parameter functions as an effective interaction length; with a smaller mismatch the FH pulses are overlapped for a greater distance in the crystal, thus generating more SH power. Finally, we can see that the waveguide design is critical, as optimization of the mode overlaps and group velocities can have a dramatic impact on the device performance.

3. Simulation of waveguide conversion efficiency

In order to calculate the mode profiles in the waveguides, we used a MATLAB implementation of the anisotropic mode-solver described in [12], which is ideally suited for crystals like KTP. The 1-cm long crystal sample was periodically poled using electric-field-induced domain inversion, and the sample was poled over its entire length. The average poling period was Λ = 122.0 ± 0.2 µm, with uncertainty due to variability in the length of each poling section ( ± 5µm). The waveguides were fabricated by AdvR using Rb+ ion exchange, with the direction of propagation along the crystal x axis. Since the ion diffusion rate along the KTP z axis is significantly higher than that of the other crystal axes, the waveguides have well-defined edges with respect to the y axis; we approximated the refractive index profile as a step function in this direction. As for the depth profile, this has been shown to be well approximated by the complementary error function [13]. We modeled the index change in the exchanged region as follows, according to typical observed values from AdvR:

Δni(λ)=Ai+Biλ+Ciλ2+Fie(λ350/Gi),
ni(λ,y,z)=n0,i(λ)+Δni(λ)rect(y/w)erfc(z/h)
where the subscript i refers to (y,z), n0,i is the bulk index of KTP, Ay = 2.90816 × 10−3, By = −6.5850 × 10−6, Cy = 2.13894 × 10−9, Fy = 9.60547 × 10−3, Gy = 39.20047, Az = 2.67947 × 10−3, Bz = −1.09737 × 10−5, Cz = 2.29268 × 10−9, Fz = 2.24565 × 10−2, Gz = 44.62477, and λ has units of millimeters. The width of the waveguides was w = 3 µm and the depth parameter was h = 5.8 µm. There are a large number of Sellmeier fits reported in the literature for the bulk indices of KTP. An extensive investigation of type-II phase-matching for FH wavelengthsbetween 1520 and 1630 nm was carried out in [14], in which a new modified Sellmeier fit for n0,y was reported. We found that our simulation predicted the measured peak phase-matching wavelength for our waveguides (1560 nm) most accurately when, as is done in [14], we used the equation in [15] for n0,z and the equation in [14] for n0,y. For nx we used the equation reported in [16]. Figure 2 shows the resulting electric field profiles of the modes of interest for type-II phase matching, at 1560 nm input wavelength. A visual representation of the waveguide corresponding to the index profile described by Eq. (12b) can be seen at the bottom right of the figure.

 figure: Fig. 2

Fig. 2 Calculated electric field distributions in the waveguide for the TE and TM modes at the fundamental frequency and the TE mode at the second harmonic. The refractive index profile for ny is illustrated at bottom right. Here again we refer to the y-polarized mode as the TE mode. Mode profiles were calculated using MATLAB and are normalized for unity power.

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With these field distributions, and using d24 ≈3.6 pm/V [17], we can calculate the expected value of the mode-coupling coefficient, κ3 = 0.208 [W1/2·cm]−1. Assuming perfect quasi-phase-matching for this wavelength, this corresponds to an expected value for the normalized conversion efficiency of η0 = 1.08% / [W·cm2]. Using the calculated mode indices as a function of wavelength, we also find the group-velocity mismatch parameters to be ζ1 = 81.88 fs/mm and ζ2 = 184.9 fs/mm. With such a large difference between the two values, the assumption made to obtain the simplified form for Eq. (9) will no longer be accurate, and the effective interaction length for the pulses as they propagate in the crystal will be reduced. We can therefore expect the sensitivity for this cross-correlator to be smaller than that predicted by Eq. (12). It is also worth noting that the waveguides support multiple spatial modes at the second harmonic. We might therefore expect that during the SHG process some FH light may be converted into these higher-order SH modes, which could then interfere with the cross-correlation signal. However, for modes with many spatial oscillations in the field distribution, it is clear that the mode-coupling coefficient κ will be very small, and these modes can be safely ignored. Further, although some of the lower-order SH modes have a non-trivial mode overlap with the FH modes, the phase mismatch parameter is dramatically increased, resulting in a vanishingly small conversion efficiency in the wavelength range of interest.

4. Experimental results and discussion

4.1 Second-harmonic conversion efficiency

To make a robust cross-correlator device, we mounted the PPKTP waveguide chip in a fiber-coupled package with internal temperature control. A picture of the packaged module is shown in Fig. 3(a). A dichroic coating was deposited on the end facet of the KTP chip, to reflect the FH light back into the waveguide. The coating was anti-reflective at 775 ± 25 nm and highly reflective at 1550 ± 50 nm. Also, a second coating was deposited on the entrance facet to the waveguide that was anti-reflective for both wavelength bands of interest. The input fiber was a standard polarization-maintaining fiber whose fast and slow axes were aligned with the principal crystal axes. The forward-generated SH light was collected into a multimode fiber, and the reverse-generated SH light was coupled back into the input fiber. The experimental setup for measuring the conversion efficiency of the waveguides is shown in Fig. 3(b). The output of a tunable laser was amplified through an erbium-doped fiberamplifier, passed through a fiber paddle-type polarization controller and a fiber circulator, and coupled into the PPKTP waveguide. The reflected FH light was measured through the return port of the circulator in order to calibrate for the coupling loss between the input fiber and the entrance to the waveguide. The coupling loss was measured to be approximately 3 dB, and the multimode fiber captured approximately 90% of the forward-generated SH light. Figure 4(a) shows the SH power generated at the exit of waveguide as a function of FH power at the entrance to the waveguide, after calibrating out the coupling losses. A quadratic fit to the data shows a normalized conversion efficiency of η0 = 1.02% / [W·cm2] for 1560 nm input wavelength. This value for the conversion efficiency agrees well with the result predicted from the simulation, and is well within the uncertainties in the measurement and in the waveguide fabrication parameters. The measurement of the wavelength dependence is shown in Fig. 4(b).

 figure: Fig. 3

Fig. 3 (a) Photograph of a packaged fiber-coupled BOC module. (b) Diagram of experimental setup for measuring conversion efficiency. Abbreviations are as follows: CW – continuous-wave laser, EDFA – erbium-doped fiber amplifier, PC – polarization controller, PD – photodetector.

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 figure: Fig. 4

Fig. 4 (a) Second harmonic power generated in the waveguide as a function of input power (after coupling calibration). Quadratic fit to the data shown as solid black line. (b) Second harmonic power as a function of input wavelength.

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4.2 Fiber-coupled cross-correlator performance

Next we investigated the performance of the fiber-coupled modules in balanced cross-correlator operation. Two new PPKTP waveguide chips were fabricated and mounted in fiber-coupled packages. The waveguide parameters for the new chips were similar to the chip described in the previous section, but the poling periods were adjusted slightly in order to have peak phase-matching at 1553 nm, to match the mode-locked laser we used for testing. The experimental setup for characterizing the cross-correlator performance is illustrated in Fig. 5(a). Pulses from the mode-locked laser were split onto orthogonal polarizations and delayed with respect to one another with a motorized delay stage. The pulses were then focused into a collimator and into the WDM, which was a custom fiber-coupled dichroic beam-splitter cube; the SH return path was coupled into a multimode fiber and fed to one port of the balanced detector. The output of the detector was then measured as a function of the delay between the input pulses.

 figure: Fig. 5

Fig. 5 (a) Diagram of experimental setup for measuring cross-correlation traces with the fiber-coupled BOCs. Abbreviations are as follows: MLL – mode-locked laser, HWP – half-wave plate, QWP – quarter-wave plate, PBS – polarization beam-splitter, COLL – collimator, WDM – wavelength-division multiplexer, BPD – balanced photodetector, CTRL – controller. (b) Typical balanced cross-correlation trace from fiber-coupled BOC. Detector output voltage recorded as a function of the time delay between the two FH input pulses.

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Due to the mode mismatch between the input fiber and the SH TE mode, we observed significant excess coupling loss for the reverse-generated SH light, approximately 10 dB less SH light was collected on the reverse path relative to the forward path. To symmetrize the balanced cross-correlation curve, we inserted a 10 dB optical attenuator in the forward path as shown in the diagram. A typical balanced cross-correlation trace for one of the fiber-coupled modules is shown in Fig. 5(b), in which the sensitivity is 9.8 mV/fs. The total average FH power at the input of the WDM was 7 mW, the repetition rate of the mode-locked laser was 216.67 MHz, and the pulse-width was 172 fs. The balanced photo-detector had a trans-impedance gain of 2 MΩ, a bandwidth of 150 kHz and a responsivity of 0.5 A/W. The WDM had an insertion loss of approximately 1.5 dB. If we were able to capture all of the reverse SH light, we would immediately gain a factor of 10 in sensitivity for the device, giving an optimum sensitivity of 98 mV/fs. To estimate the expected value for the sensitivity, we canapproximate the group-velocity mismatch parameter as the average of ζ1 and ζ2, which gives ζ = 134 fs/mm. From Eq. (12), for these operating parameters we should expect to observe a sensitivity of K0 ≈118 mV/fs, after calibrating out the fiber coupling losses and WDM insertion loss. Due to the large discrepancy between ζ1 and ζ2, the effective interaction length in the waveguide will be reduced and therefore the effective group-velocity mismatch will be increased. An increase of 10% in the effective value of the mismatch parameter (ζeff = 147 fs/mm) would account for the difference between the measured sensitivity and that predicted by Eq. (12).

Using the internal thermoelectric heating element (TEC), we also characterized the temperature dependence of the device sensitivity by measuring the slope of the cross-correlation curve at various TEC settings. The results showed very little dependence on applied temperature around the typical operating point of 25°C, as can be seen in Fig. 6(a), which indicates that the waveguide performance and fiber coupling efficiency is very robust against temperature fluctuations in the environment. As the temperature is increased to 40°C the sensitivities shift slightly, which most likely reflects a small change in the phase matching between the modes in the waveguide. To compare the performance of the new fiber-coupled module against the free-space bulk-optics BOCs that we have used in the past, we measured the sensitivity of each as a function of average FH power at the input to the WDM. Since the current implementation of the fiber-coupled BOC requires the use of the external WDM, theyshould be considered together as components of the same fiber-coupled module. In [4], a conversion efficiency of 0.4% is reported for a laser with 200 fs pulse-width, 200 MHz repetition rate and 15 mW of total average input power. Assuming similar laser parameters and using the effective mismatch parameter discussed earlier, our waveguides would deliver an expected conversion efficiency of 7.38%, an improvement of a factor of 20 over the bulk-optic result. Due to the SH coupling loss mentioned previously, the fiber-coupled devices were not able to achieve their full potential as expected from the conversion efficiency improvements, but they nonetheless exhibit performance comparable to the bulk-optic version, as can be seen in Fig. 6(b). We expect that further integration of the WDM coupler onto the KTP chip would solve the coupling problem and allow for significantly higher device sensitivities.

 figure: Fig. 6

Fig. 6 Fiber-coupled BOC sensitivity as a function of (a) temperature and (b) total average FH power at the input of the WDM. Red squares and blue diamonds denote the two fiber-coupled modules, green triangles denote a typical bulk-optic BOC for comparison. Sensitivity was determined by a linear regression fit to the linear region at the zero-crossing of the cross-correlation curve. Error bars in (a) denote the confidence interval of the fit.

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5. Conclusion

We have presented a fiber-coupled balanced optical cross-correlator using second-harmonic generation in PPKTP waveguides. The normalized SH conversion efficiency of the waveguide was measured to be η0 = 1.02% / [W·cm2], which is a significant improvement over bulk-optic crystals. Simulation of the waveguide modes using an anisotropic mode-solver gave a predicted value of η0 = 1.08% / [W·cm2], which agrees well with the measured result. This promises to be a powerful design tool for future waveguide optimization. A simplified analytical expression for the BOC sensitivity was derived and shown to match the measurement fairly well for typical operating parameters. The performance of the fiber-coupled module was characterized and shown to be robust against temperature fluctuations. The current implementation of the fiber-coupled BOC is limited by excess SH coupling loss, but still delivers a sensitivity comparable to the bulk-optic BOCs. The next generation device will include an integrated WDM coupler to eliminate the coupling problem, thereby increasing the performance by an order of magnitude.

Acknowledgments

The authors acknowledge financial support by the United States Department of Energy under contract DE-SC0005262, the Air Force Office of Scientific Research under grant AFOSR FA9550-12-1-0499, and the Center for Free-Electron Science at Deutsches Elektronen-Synchrotron in Hamburg, a research center of the Helmholtz Association in Germany.

References and links

1. A. Benedick, J. G. Fujimoto, and F. X. Kärtner, “Optical flywheels with attosecond jitter,” Nat. Photonics 6(2), 97–100 (2012). [CrossRef]  

2. J. Kim, J. A. Cox, J. Chen, and F. X. Kärtner, “Drift-free femtosecond timing synchronization of remote optical and microwave sources,” Nat. Photonics 2(12), 733–736 (2008). [CrossRef]  

3. M. Y. Peng, P. T. Callahan, A. H. Nejadmalayeri, S. Valente, M. Xin, L. Grüner-Nielsen, E. M. Monberg, M. Yan, J. M. Fini, and F. X. Kärtner, “Long-term stable, sub-femtosecond timing distribution via a 1.2-km polarization-maintaining timing link: approaching 10−21 link stability,” Opt. Express 21(17), 19982–19989 (2013). [CrossRef]   [PubMed]  

4. J. Kim, J. Chen, Z. Zhang, F. N. C. Wong, F. X. Kärtner, F. Loehl, and H. Schlarb, “Long-term femtosecond timing link stabilization using a single-crystal balanced cross correlator,” Opt. Lett. 32(9), 1044–1046 (2007). [CrossRef]   [PubMed]  

5. A. H. Nejadmalayeri, F. N. C. Wong, T. D. Roberts, P. Battle, and F. X. Kärtner, “Guided wave optics in periodically poled KTP: quadratic nonlinearity and prospects for attosecond jitter characterization,” Opt. Lett. 34(16), 2522–2524 (2009). [CrossRef]   [PubMed]  

6. A. H. Nejadmalayeri, J. A. Cox, J. Kim, F. N. C. Wong, T. D. Roberts, P. Battle, and F. X. Kärtner, “Phase noise measurement of mode locked lasers using guided wave PPKTP balanced cross correlators,” in Conference on Lasers and Electro-Optics 2010, OSA Technical Digest (Optical Society of America, 2010), paper JTuD75. [CrossRef]  

7. R. Machulka, J. Svozilík, J. Soubusta, J. Peřina Jr, and O. Haderka, “Spatial and spectral properties of fields generated by pulsed second-harmonic generation in a periodically poled potassium-titanyl-phosphate waveguide,” Phys. Rev. A 87(1), 013836 (2013). [CrossRef]  

8. P. T. Callahan, T. D. Roberts, P. Battle, A. H. Nejadmalayeri, and F. X. Kärtner, “A hybrid integrated balanced optical cross-correlator using PPKTP waveguides,” in Conference on Lasers and Electro-Optics 2013, OSA Technical Digest (Optical Society of America, 2013), paper CW3B.6. [CrossRef]  

9. L. A. Jiang, S. T. Wong, M. E. Grein, E. P. Ippen, and H. A. Haus, “Measuring timing jitter with optical cross correlations,” IEEE J. Quantum Electron. 38(8), 1047–1052 (2002). [CrossRef]  

10. T. R. Schibli, J. Kim, O. Kuzucu, J. T. Gopinath, S. N. Tandon, G. S. Petrich, L. A. Kolodziejski, J. G. Fujimoto, E. P. Ippen, and F. X. Kaertner, “Attosecond active synchronization of passively mode-locked lasers by balanced cross correlation,” Opt. Lett. 28(11), 947–949 (2003). [CrossRef]   [PubMed]  

11. T. Suhara and M. Fujimura, Waveguide Nonlinear-Optic Devices, (Springer, 2003). Chap. 3.

12. A. B. Fallahkhair, K. S. Li, and T. E. Murphy, “Vector Finite Difference Modesolver for Anisotropic Dielectric Waveguides,” J. Lightwave Technol. 26(11), 1423–1431 (2008). [CrossRef]  

13. M. G. Roelofs, A. Suna, W. Bindloss, and J. D. Bierlein, “Characterization of optical waveguides in KTiOPO4 by second-harmonic spectroscopy,” J. Appl. Phys. 76(9), 4999–5006 (1994). [CrossRef]  

14. F. König and F. N. C. Wong, “Extended phase matching of second-harmonic generation in periodically poled KTiOPO4 with zero group-velocity mismatch,” Appl. Phys. Lett. 84(10), 1644–1646 (2004). [CrossRef]  

15. K. Fradkin, A. Arie, A. Skliar, and G. Rosenman, “Tunable midinfrared source by difference frequency generation in bulk periodically poled KTiOPO4,” Appl. Phys. Lett. 74(7), 914–916 (1999). [CrossRef]  

16. K. Kato and E. Takaoka, “Sellmeier and thermo-optic dispersion formulas for KTP,” Appl. Opt. 41(24), 5040–5044 (2002). [CrossRef]   [PubMed]  

17. H. Vanherzeele and J. D. Bierlein, “Magnitude of the nonlinear-optic coefficients of KTiOPO4,” Opt. Lett. 17(14), 982–984 (1992). [CrossRef]   [PubMed]  

References

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  1. A. Benedick, J. G. Fujimoto, and F. X. Kärtner, “Optical flywheels with attosecond jitter,” Nat. Photonics 6(2), 97–100 (2012).
    [Crossref]
  2. J. Kim, J. A. Cox, J. Chen, and F. X. Kärtner, “Drift-free femtosecond timing synchronization of remote optical and microwave sources,” Nat. Photonics 2(12), 733–736 (2008).
    [Crossref]
  3. M. Y. Peng, P. T. Callahan, A. H. Nejadmalayeri, S. Valente, M. Xin, L. Grüner-Nielsen, E. M. Monberg, M. Yan, J. M. Fini, and F. X. Kärtner, “Long-term stable, sub-femtosecond timing distribution via a 1.2-km polarization-maintaining timing link: approaching 10−21 link stability,” Opt. Express 21(17), 19982–19989 (2013).
    [Crossref] [PubMed]
  4. J. Kim, J. Chen, Z. Zhang, F. N. C. Wong, F. X. Kärtner, F. Loehl, and H. Schlarb, “Long-term femtosecond timing link stabilization using a single-crystal balanced cross correlator,” Opt. Lett. 32(9), 1044–1046 (2007).
    [Crossref] [PubMed]
  5. A. H. Nejadmalayeri, F. N. C. Wong, T. D. Roberts, P. Battle, and F. X. Kärtner, “Guided wave optics in periodically poled KTP: quadratic nonlinearity and prospects for attosecond jitter characterization,” Opt. Lett. 34(16), 2522–2524 (2009).
    [Crossref] [PubMed]
  6. A. H. Nejadmalayeri, J. A. Cox, J. Kim, F. N. C. Wong, T. D. Roberts, P. Battle, and F. X. Kärtner, “Phase noise measurement of mode locked lasers using guided wave PPKTP balanced cross correlators,” in Conference on Lasers and Electro-Optics 2010, OSA Technical Digest (Optical Society of America, 2010), paper JTuD75.
    [Crossref]
  7. R. Machulka, J. Svozilík, J. Soubusta, J. Peřina, and O. Haderka, “Spatial and spectral properties of fields generated by pulsed second-harmonic generation in a periodically poled potassium-titanyl-phosphate waveguide,” Phys. Rev. A 87(1), 013836 (2013).
    [Crossref]
  8. P. T. Callahan, T. D. Roberts, P. Battle, A. H. Nejadmalayeri, and F. X. Kärtner, “A hybrid integrated balanced optical cross-correlator using PPKTP waveguides,” in Conference on Lasers and Electro-Optics 2013, OSA Technical Digest (Optical Society of America, 2013), paper CW3B.6.
    [Crossref]
  9. L. A. Jiang, S. T. Wong, M. E. Grein, E. P. Ippen, and H. A. Haus, “Measuring timing jitter with optical cross correlations,” IEEE J. Quantum Electron. 38(8), 1047–1052 (2002).
    [Crossref]
  10. T. R. Schibli, J. Kim, O. Kuzucu, J. T. Gopinath, S. N. Tandon, G. S. Petrich, L. A. Kolodziejski, J. G. Fujimoto, E. P. Ippen, and F. X. Kaertner, “Attosecond active synchronization of passively mode-locked lasers by balanced cross correlation,” Opt. Lett. 28(11), 947–949 (2003).
    [Crossref] [PubMed]
  11. T. Suhara and M. Fujimura, Waveguide Nonlinear-Optic Devices, (Springer, 2003). Chap. 3.
  12. A. B. Fallahkhair, K. S. Li, and T. E. Murphy, “Vector Finite Difference Modesolver for Anisotropic Dielectric Waveguides,” J. Lightwave Technol. 26(11), 1423–1431 (2008).
    [Crossref]
  13. M. G. Roelofs, A. Suna, W. Bindloss, and J. D. Bierlein, “Characterization of optical waveguides in KTiOPO4 by second-harmonic spectroscopy,” J. Appl. Phys. 76(9), 4999–5006 (1994).
    [Crossref]
  14. F. König and F. N. C. Wong, “Extended phase matching of second-harmonic generation in periodically poled KTiOPO4 with zero group-velocity mismatch,” Appl. Phys. Lett. 84(10), 1644–1646 (2004).
    [Crossref]
  15. K. Fradkin, A. Arie, A. Skliar, and G. Rosenman, “Tunable midinfrared source by difference frequency generation in bulk periodically poled KTiOPO4,” Appl. Phys. Lett. 74(7), 914–916 (1999).
    [Crossref]
  16. K. Kato and E. Takaoka, “Sellmeier and thermo-optic dispersion formulas for KTP,” Appl. Opt. 41(24), 5040–5044 (2002).
    [Crossref] [PubMed]
  17. H. Vanherzeele and J. D. Bierlein, “Magnitude of the nonlinear-optic coefficients of KTiOPO4,” Opt. Lett. 17(14), 982–984 (1992).
    [Crossref] [PubMed]

2013 (2)

M. Y. Peng, P. T. Callahan, A. H. Nejadmalayeri, S. Valente, M. Xin, L. Grüner-Nielsen, E. M. Monberg, M. Yan, J. M. Fini, and F. X. Kärtner, “Long-term stable, sub-femtosecond timing distribution via a 1.2-km polarization-maintaining timing link: approaching 10−21 link stability,” Opt. Express 21(17), 19982–19989 (2013).
[Crossref] [PubMed]

R. Machulka, J. Svozilík, J. Soubusta, J. Peřina, and O. Haderka, “Spatial and spectral properties of fields generated by pulsed second-harmonic generation in a periodically poled potassium-titanyl-phosphate waveguide,” Phys. Rev. A 87(1), 013836 (2013).
[Crossref]

2012 (1)

A. Benedick, J. G. Fujimoto, and F. X. Kärtner, “Optical flywheels with attosecond jitter,” Nat. Photonics 6(2), 97–100 (2012).
[Crossref]

2009 (1)

2008 (2)

J. Kim, J. A. Cox, J. Chen, and F. X. Kärtner, “Drift-free femtosecond timing synchronization of remote optical and microwave sources,” Nat. Photonics 2(12), 733–736 (2008).
[Crossref]

A. B. Fallahkhair, K. S. Li, and T. E. Murphy, “Vector Finite Difference Modesolver for Anisotropic Dielectric Waveguides,” J. Lightwave Technol. 26(11), 1423–1431 (2008).
[Crossref]

2007 (1)

2004 (1)

F. König and F. N. C. Wong, “Extended phase matching of second-harmonic generation in periodically poled KTiOPO4 with zero group-velocity mismatch,” Appl. Phys. Lett. 84(10), 1644–1646 (2004).
[Crossref]

2003 (1)

2002 (2)

L. A. Jiang, S. T. Wong, M. E. Grein, E. P. Ippen, and H. A. Haus, “Measuring timing jitter with optical cross correlations,” IEEE J. Quantum Electron. 38(8), 1047–1052 (2002).
[Crossref]

K. Kato and E. Takaoka, “Sellmeier and thermo-optic dispersion formulas for KTP,” Appl. Opt. 41(24), 5040–5044 (2002).
[Crossref] [PubMed]

1999 (1)

K. Fradkin, A. Arie, A. Skliar, and G. Rosenman, “Tunable midinfrared source by difference frequency generation in bulk periodically poled KTiOPO4,” Appl. Phys. Lett. 74(7), 914–916 (1999).
[Crossref]

1994 (1)

M. G. Roelofs, A. Suna, W. Bindloss, and J. D. Bierlein, “Characterization of optical waveguides in KTiOPO4 by second-harmonic spectroscopy,” J. Appl. Phys. 76(9), 4999–5006 (1994).
[Crossref]

1992 (1)

Arie, A.

K. Fradkin, A. Arie, A. Skliar, and G. Rosenman, “Tunable midinfrared source by difference frequency generation in bulk periodically poled KTiOPO4,” Appl. Phys. Lett. 74(7), 914–916 (1999).
[Crossref]

Battle, P.

Benedick, A.

A. Benedick, J. G. Fujimoto, and F. X. Kärtner, “Optical flywheels with attosecond jitter,” Nat. Photonics 6(2), 97–100 (2012).
[Crossref]

Bierlein, J. D.

M. G. Roelofs, A. Suna, W. Bindloss, and J. D. Bierlein, “Characterization of optical waveguides in KTiOPO4 by second-harmonic spectroscopy,” J. Appl. Phys. 76(9), 4999–5006 (1994).
[Crossref]

H. Vanherzeele and J. D. Bierlein, “Magnitude of the nonlinear-optic coefficients of KTiOPO4,” Opt. Lett. 17(14), 982–984 (1992).
[Crossref] [PubMed]

Bindloss, W.

M. G. Roelofs, A. Suna, W. Bindloss, and J. D. Bierlein, “Characterization of optical waveguides in KTiOPO4 by second-harmonic spectroscopy,” J. Appl. Phys. 76(9), 4999–5006 (1994).
[Crossref]

Callahan, P. T.

Chen, J.

J. Kim, J. A. Cox, J. Chen, and F. X. Kärtner, “Drift-free femtosecond timing synchronization of remote optical and microwave sources,” Nat. Photonics 2(12), 733–736 (2008).
[Crossref]

J. Kim, J. Chen, Z. Zhang, F. N. C. Wong, F. X. Kärtner, F. Loehl, and H. Schlarb, “Long-term femtosecond timing link stabilization using a single-crystal balanced cross correlator,” Opt. Lett. 32(9), 1044–1046 (2007).
[Crossref] [PubMed]

Cox, J. A.

J. Kim, J. A. Cox, J. Chen, and F. X. Kärtner, “Drift-free femtosecond timing synchronization of remote optical and microwave sources,” Nat. Photonics 2(12), 733–736 (2008).
[Crossref]

Fallahkhair, A. B.

Fini, J. M.

Fradkin, K.

K. Fradkin, A. Arie, A. Skliar, and G. Rosenman, “Tunable midinfrared source by difference frequency generation in bulk periodically poled KTiOPO4,” Appl. Phys. Lett. 74(7), 914–916 (1999).
[Crossref]

Fujimoto, J. G.

Gopinath, J. T.

Grein, M. E.

L. A. Jiang, S. T. Wong, M. E. Grein, E. P. Ippen, and H. A. Haus, “Measuring timing jitter with optical cross correlations,” IEEE J. Quantum Electron. 38(8), 1047–1052 (2002).
[Crossref]

Grüner-Nielsen, L.

Haderka, O.

R. Machulka, J. Svozilík, J. Soubusta, J. Peřina, and O. Haderka, “Spatial and spectral properties of fields generated by pulsed second-harmonic generation in a periodically poled potassium-titanyl-phosphate waveguide,” Phys. Rev. A 87(1), 013836 (2013).
[Crossref]

Haus, H. A.

L. A. Jiang, S. T. Wong, M. E. Grein, E. P. Ippen, and H. A. Haus, “Measuring timing jitter with optical cross correlations,” IEEE J. Quantum Electron. 38(8), 1047–1052 (2002).
[Crossref]

Ippen, E. P.

Jiang, L. A.

L. A. Jiang, S. T. Wong, M. E. Grein, E. P. Ippen, and H. A. Haus, “Measuring timing jitter with optical cross correlations,” IEEE J. Quantum Electron. 38(8), 1047–1052 (2002).
[Crossref]

Kaertner, F. X.

Kärtner, F. X.

Kato, K.

Kim, J.

Kolodziejski, L. A.

König, F.

F. König and F. N. C. Wong, “Extended phase matching of second-harmonic generation in periodically poled KTiOPO4 with zero group-velocity mismatch,” Appl. Phys. Lett. 84(10), 1644–1646 (2004).
[Crossref]

Kuzucu, O.

Li, K. S.

Loehl, F.

Machulka, R.

R. Machulka, J. Svozilík, J. Soubusta, J. Peřina, and O. Haderka, “Spatial and spectral properties of fields generated by pulsed second-harmonic generation in a periodically poled potassium-titanyl-phosphate waveguide,” Phys. Rev. A 87(1), 013836 (2013).
[Crossref]

Monberg, E. M.

Murphy, T. E.

Nejadmalayeri, A. H.

Peng, M. Y.

Perina, J.

R. Machulka, J. Svozilík, J. Soubusta, J. Peřina, and O. Haderka, “Spatial and spectral properties of fields generated by pulsed second-harmonic generation in a periodically poled potassium-titanyl-phosphate waveguide,” Phys. Rev. A 87(1), 013836 (2013).
[Crossref]

Petrich, G. S.

Roberts, T. D.

Roelofs, M. G.

M. G. Roelofs, A. Suna, W. Bindloss, and J. D. Bierlein, “Characterization of optical waveguides in KTiOPO4 by second-harmonic spectroscopy,” J. Appl. Phys. 76(9), 4999–5006 (1994).
[Crossref]

Rosenman, G.

K. Fradkin, A. Arie, A. Skliar, and G. Rosenman, “Tunable midinfrared source by difference frequency generation in bulk periodically poled KTiOPO4,” Appl. Phys. Lett. 74(7), 914–916 (1999).
[Crossref]

Schibli, T. R.

Schlarb, H.

Skliar, A.

K. Fradkin, A. Arie, A. Skliar, and G. Rosenman, “Tunable midinfrared source by difference frequency generation in bulk periodically poled KTiOPO4,” Appl. Phys. Lett. 74(7), 914–916 (1999).
[Crossref]

Soubusta, J.

R. Machulka, J. Svozilík, J. Soubusta, J. Peřina, and O. Haderka, “Spatial and spectral properties of fields generated by pulsed second-harmonic generation in a periodically poled potassium-titanyl-phosphate waveguide,” Phys. Rev. A 87(1), 013836 (2013).
[Crossref]

Suna, A.

M. G. Roelofs, A. Suna, W. Bindloss, and J. D. Bierlein, “Characterization of optical waveguides in KTiOPO4 by second-harmonic spectroscopy,” J. Appl. Phys. 76(9), 4999–5006 (1994).
[Crossref]

Svozilík, J.

R. Machulka, J. Svozilík, J. Soubusta, J. Peřina, and O. Haderka, “Spatial and spectral properties of fields generated by pulsed second-harmonic generation in a periodically poled potassium-titanyl-phosphate waveguide,” Phys. Rev. A 87(1), 013836 (2013).
[Crossref]

Takaoka, E.

Tandon, S. N.

Valente, S.

Vanherzeele, H.

Wong, F. N. C.

Wong, S. T.

L. A. Jiang, S. T. Wong, M. E. Grein, E. P. Ippen, and H. A. Haus, “Measuring timing jitter with optical cross correlations,” IEEE J. Quantum Electron. 38(8), 1047–1052 (2002).
[Crossref]

Xin, M.

Yan, M.

Zhang, Z.

Appl. Opt. (1)

Appl. Phys. Lett. (2)

F. König and F. N. C. Wong, “Extended phase matching of second-harmonic generation in periodically poled KTiOPO4 with zero group-velocity mismatch,” Appl. Phys. Lett. 84(10), 1644–1646 (2004).
[Crossref]

K. Fradkin, A. Arie, A. Skliar, and G. Rosenman, “Tunable midinfrared source by difference frequency generation in bulk periodically poled KTiOPO4,” Appl. Phys. Lett. 74(7), 914–916 (1999).
[Crossref]

IEEE J. Quantum Electron. (1)

L. A. Jiang, S. T. Wong, M. E. Grein, E. P. Ippen, and H. A. Haus, “Measuring timing jitter with optical cross correlations,” IEEE J. Quantum Electron. 38(8), 1047–1052 (2002).
[Crossref]

J. Appl. Phys. (1)

M. G. Roelofs, A. Suna, W. Bindloss, and J. D. Bierlein, “Characterization of optical waveguides in KTiOPO4 by second-harmonic spectroscopy,” J. Appl. Phys. 76(9), 4999–5006 (1994).
[Crossref]

J. Lightwave Technol. (1)

Nat. Photonics (2)

A. Benedick, J. G. Fujimoto, and F. X. Kärtner, “Optical flywheels with attosecond jitter,” Nat. Photonics 6(2), 97–100 (2012).
[Crossref]

J. Kim, J. A. Cox, J. Chen, and F. X. Kärtner, “Drift-free femtosecond timing synchronization of remote optical and microwave sources,” Nat. Photonics 2(12), 733–736 (2008).
[Crossref]

Opt. Express (1)

Opt. Lett. (4)

Phys. Rev. A (1)

R. Machulka, J. Svozilík, J. Soubusta, J. Peřina, and O. Haderka, “Spatial and spectral properties of fields generated by pulsed second-harmonic generation in a periodically poled potassium-titanyl-phosphate waveguide,” Phys. Rev. A 87(1), 013836 (2013).
[Crossref]

Other (3)

P. T. Callahan, T. D. Roberts, P. Battle, A. H. Nejadmalayeri, and F. X. Kärtner, “A hybrid integrated balanced optical cross-correlator using PPKTP waveguides,” in Conference on Lasers and Electro-Optics 2013, OSA Technical Digest (Optical Society of America, 2013), paper CW3B.6.
[Crossref]

T. Suhara and M. Fujimura, Waveguide Nonlinear-Optic Devices, (Springer, 2003). Chap. 3.

A. H. Nejadmalayeri, J. A. Cox, J. Kim, F. N. C. Wong, T. D. Roberts, P. Battle, and F. X. Kärtner, “Phase noise measurement of mode locked lasers using guided wave PPKTP balanced cross correlators,” in Conference on Lasers and Electro-Optics 2010, OSA Technical Digest (Optical Society of America, 2010), paper JTuD75.
[Crossref]

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Figures (6)

Fig. 1
Fig. 1 An illustration of the principle of balanced optical cross-correlation. Pulses at the fundamental (represented in green) propagate through the nonlinear crystal (represented in blue) at different speeds, generating different amounts of SH light (represented as orange pulses) on the forward and reverse paths. The difference in detected SH power then gives the output voltage error curve, in which the voltage is proportional to the time separation of the pulses. Abbreviations are as follows: DM – dichroic mirror, PD – photodetector, WDM – wavelength division multiplexer.
Fig. 2
Fig. 2 Calculated electric field distributions in the waveguide for the TE and TM modes at the fundamental frequency and the TE mode at the second harmonic. The refractive index profile for ny is illustrated at bottom right. Here again we refer to the y-polarized mode as the TE mode. Mode profiles were calculated using MATLAB and are normalized for unity power.
Fig. 3
Fig. 3 (a) Photograph of a packaged fiber-coupled BOC module. (b) Diagram of experimental setup for measuring conversion efficiency. Abbreviations are as follows: CW – continuous-wave laser, EDFA – erbium-doped fiber amplifier, PC – polarization controller, PD – photodetector.
Fig. 4
Fig. 4 (a) Second harmonic power generated in the waveguide as a function of input power (after coupling calibration). Quadratic fit to the data shown as solid black line. (b) Second harmonic power as a function of input wavelength.
Fig. 5
Fig. 5 (a) Diagram of experimental setup for measuring cross-correlation traces with the fiber-coupled BOCs. Abbreviations are as follows: MLL – mode-locked laser, HWP – half-wave plate, QWP – quarter-wave plate, PBS – polarization beam-splitter, COLL – collimator, WDM – wavelength-division multiplexer, BPD – balanced photodetector, CTRL – controller. (b) Typical balanced cross-correlation trace from fiber-coupled BOC. Detector output voltage recorded as a function of the time delay between the two FH input pulses.
Fig. 6
Fig. 6 Fiber-coupled BOC sensitivity as a function of (a) temperature and (b) total average FH power at the input of the WDM. Red squares and blue diamonds denote the two fiber-coupled modules, green triangles denote a typical bulk-optic BOC for comparison. Sensitivity was determined by a linear regression fit to the linear region at the zero-crossing of the cross-correlation curve. Error bars in (a) denote the confidence interval of the fit.

Equations (22)

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E t ( x,y,z )= m Am( x ) E tm( y,z ) e j β m x
H t ( x,y,z )= m Am( x ) H tm( y,z ) e j β m x
1 2 Re{ E tm × H tn * } dydz= δ mn
d A 1 / dx =j κ 1 A 3 ( x ) A 2 * ( x ) e j2Δx
d A 2 / dx =j κ 2 A 3 ( x ) A 1 * ( x ) e j2Δx
d A 3 / dx =j κ 3 A 1 ( x ) A 2 ( x ) e +j2Δx
κ 1 = κ 2 =( ω ε 0 d eff /2 ) E 1 * E 2 * E 3 dydz
κ 3 =( ω ε 0 d eff ) E 1 E 2 E 3 * dydz
A 3 ( L )=j κ 3 A 10 A 20 L e +jΔL sinc( ΔL )
P SH = η 0 L 2 P FH 2 | A 3 ( L ) | 2 = η 0 L 2 ( | A 10 | 2 + | A 20 | 2 ) 2 =4 η 0 L 2 A 0 4
η 0 = κ 3 2 sin c 2 ( ΔL ) /4
A 1 ( x,t )= A 10 e ( t ζ 1 x ΔT /2 ) 2 / τ 2
A 2 ( x,t )= A 20 e ( t ζ 2 x+ ΔT /2 ) 2 / τ 2
ζ i =( v gi 1 v g3 1 )
A 3 ( L,t )= j π κ 3 A 10 A 20 2c e ( at+ bΔT 2 ) 2 [ erf( cL+ aΔT 2 bt )erf( aΔT 2 bt ) ]
a 2 = ( ζ 1 ζ 2 ) 2 τ 2 ( ζ 1 2 + ζ 2 2 ) , b 2 = ( ζ 1 + ζ 2 ) 2 τ 2 ( ζ 1 2 + ζ 2 2 ) , c 2 = ( ζ 1 2 + ζ 2 2 ) τ 2
A 3 ( L,t )= j π κ 3 A 10 A 20 τ 8 ζ e 2 t 2 / τ 2 [ erf( 2ζL+ΔT 2 τ )erf( ΔT 2 τ ) ]
J=α f rep | A 3 ( L,t ) | 2 dt
ΔV( ΔT' )= Gα f rep κ 3 2 A 10 2 A 20 2 ( π τ ) 3 16 ζ 2 [ [ erf( 2ζL+ΔT' 2 τ )erf( ΔT' 2 τ ) ] 2 [ erf( ΔT' 2 τ )erf( ΔT'2ζL 2 τ ) ] 2 ]
K= Gα κ 3 2 P avg,FH 2 4 2 f rep ζ 2 erf( 2ζL 2 τ )[ e ( 2ζL / 2 τ ) 2 1 ]
Δ n i ( λ )= A i + B i λ+ C i λ 2 + F i e ( λ350 / G i ) ,
n i ( λ,y,z )= n 0,i ( λ )+Δ n i ( λ )rect( y/w )erfc( z/h )

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