Abstract

The nonlinear interaction of TE and TM optical modes with a microwave by means of optical rectification and the electro-optic effect in a traveling-wave structure is investigated. The phase velocity mismatch between the optical waves leads to effects that are essentially different from those in the single-polarization case. Whereas, for a velocity-matched structure, alternating upconversion and downconversion occur, a slight difference between optical group velocity and microwave phase velocity gives rise to a sinusoidal terahertz signal with a number of cycles that depends on beat length, length of the structure, and velocity mismatch. A structure for on-chip generation of tunable narrow-bandwidth terahertz signals is suggested. Cascading caused by optical rectification and the electro-optic effect can lead to an effective change of beat length and self-induced mode conversion at high optical powers.

© 2002 Optical Society of America

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References

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  1. M. Bass, P. A. Franken, J. F. Ward, and G. Weinreich, “Optical rectification,” Phys. Rev. Lett. 9, 446–448 (1962).
    [CrossRef]
  2. Y. H. Jin and X. C. Zhang, “Terahertz optical rectification,” J. Nonlinear Opt. Phys. Mater. 4, 459–495 (1995).
    [CrossRef]
  3. T. K. Gustafson, J.-P. E. Taran, P. L. Kelley, and R. Y. Chiao, “Self-modulation of picosecond pulses in electro-optic crystals,” Opt. Commun. 2, 17–21 (1970).
    [CrossRef]
  4. C. Bosshard, R. Spreiter, M. Zgonik, and P. Günter, “Kerr nonlinearity via cascaded optical rectification and the linear electro-optic effect,” Phys. Rev. Lett. 74, 2816–2818 (1995).
    [CrossRef] [PubMed]
  5. U. Peschel, K. Bubke, D. C. Hutchings, J. S. Aitchison, and J. M. Arnold, “Optical rectification in a travelling-wave geometry,” Phys. Rev. A 60, 4918–4926 (1999).
    [CrossRef]
  6. S. Graf, H. Sigg, and W. Bächtold, “High-frequency electri-cal pulse generation using optical rectification in bulk GaAs,” Appl. Phys. Lett. 76, 2647–2649 (2000).
    [CrossRef]
  7. M. Börner, R. Müller, R. Schiek, and G. Trommer, Elements of Integrated Optics (Teubner, Stuttgart, Germany, 1990, in German).
  8. P. N. Butcher and D. Cotter, The Elements of Nonlinear Op-tics, Vol. 9 of Cambridge Studies in Modern Optics (Cambridge U. Press, Cambridge, 1990).
  9. G. B. Whitham, Linear and Nonlinear Waves (Academic, New York, 1974).
  10. K. Bubke, U. Peschel, and D. C. Hutchings, “KdV solitons on GaAs transmission lines due to the intrinsic second order nonlinearity,” in Nonlinear Guided Waves and Their Applications, 1999 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1999), pp. 339–341.
  11. H. E. Green, “The numerical solution of some important transmission-line problems,” IEEE Trans. Microwave Theory Tech. 13, 676–692 (1965).
    [CrossRef]
  12. A. P. Ansbro and I. Montrosset, “Vectorial finite difference scheme for isotropic dielectric wave guides: transverse electric field representation,” IEE Proc. Optoelectron. 140, 253–259 (1993).
    [CrossRef]
  13. N. A. F. Jaeger and Z. K. F. Lee, “Slow-wave electrode for use in compound semiconductor electrooptic modulators,” IEEE J. Quantum Electron. 28, 1778–1784 (1992).
    [CrossRef]
  14. Y. S. Lee, T. Meade, V. Perlin, H. Winful, T. B. Norris, and A. Galvanauskas, “Generation of narrow-band terahertz radiation via optical rectification of femtosecond pulses in periodically poled lithium niobate,” Appl. Phys. Lett. 76, 2505–2507 (2000).
    [CrossRef]

2000 (2)

S. Graf, H. Sigg, and W. Bächtold, “High-frequency electri-cal pulse generation using optical rectification in bulk GaAs,” Appl. Phys. Lett. 76, 2647–2649 (2000).
[CrossRef]

Y. S. Lee, T. Meade, V. Perlin, H. Winful, T. B. Norris, and A. Galvanauskas, “Generation of narrow-band terahertz radiation via optical rectification of femtosecond pulses in periodically poled lithium niobate,” Appl. Phys. Lett. 76, 2505–2507 (2000).
[CrossRef]

1999 (1)

U. Peschel, K. Bubke, D. C. Hutchings, J. S. Aitchison, and J. M. Arnold, “Optical rectification in a travelling-wave geometry,” Phys. Rev. A 60, 4918–4926 (1999).
[CrossRef]

1995 (2)

Y. H. Jin and X. C. Zhang, “Terahertz optical rectification,” J. Nonlinear Opt. Phys. Mater. 4, 459–495 (1995).
[CrossRef]

C. Bosshard, R. Spreiter, M. Zgonik, and P. Günter, “Kerr nonlinearity via cascaded optical rectification and the linear electro-optic effect,” Phys. Rev. Lett. 74, 2816–2818 (1995).
[CrossRef] [PubMed]

1993 (1)

A. P. Ansbro and I. Montrosset, “Vectorial finite difference scheme for isotropic dielectric wave guides: transverse electric field representation,” IEE Proc. Optoelectron. 140, 253–259 (1993).
[CrossRef]

1992 (1)

N. A. F. Jaeger and Z. K. F. Lee, “Slow-wave electrode for use in compound semiconductor electrooptic modulators,” IEEE J. Quantum Electron. 28, 1778–1784 (1992).
[CrossRef]

1970 (1)

T. K. Gustafson, J.-P. E. Taran, P. L. Kelley, and R. Y. Chiao, “Self-modulation of picosecond pulses in electro-optic crystals,” Opt. Commun. 2, 17–21 (1970).
[CrossRef]

1965 (1)

H. E. Green, “The numerical solution of some important transmission-line problems,” IEEE Trans. Microwave Theory Tech. 13, 676–692 (1965).
[CrossRef]

1962 (1)

M. Bass, P. A. Franken, J. F. Ward, and G. Weinreich, “Optical rectification,” Phys. Rev. Lett. 9, 446–448 (1962).
[CrossRef]

Aitchison, J. S.

U. Peschel, K. Bubke, D. C. Hutchings, J. S. Aitchison, and J. M. Arnold, “Optical rectification in a travelling-wave geometry,” Phys. Rev. A 60, 4918–4926 (1999).
[CrossRef]

Ansbro, A. P.

A. P. Ansbro and I. Montrosset, “Vectorial finite difference scheme for isotropic dielectric wave guides: transverse electric field representation,” IEE Proc. Optoelectron. 140, 253–259 (1993).
[CrossRef]

Arnold, J. M.

U. Peschel, K. Bubke, D. C. Hutchings, J. S. Aitchison, and J. M. Arnold, “Optical rectification in a travelling-wave geometry,” Phys. Rev. A 60, 4918–4926 (1999).
[CrossRef]

Bächtold, W.

S. Graf, H. Sigg, and W. Bächtold, “High-frequency electri-cal pulse generation using optical rectification in bulk GaAs,” Appl. Phys. Lett. 76, 2647–2649 (2000).
[CrossRef]

Bass, M.

M. Bass, P. A. Franken, J. F. Ward, and G. Weinreich, “Optical rectification,” Phys. Rev. Lett. 9, 446–448 (1962).
[CrossRef]

Bosshard, C.

C. Bosshard, R. Spreiter, M. Zgonik, and P. Günter, “Kerr nonlinearity via cascaded optical rectification and the linear electro-optic effect,” Phys. Rev. Lett. 74, 2816–2818 (1995).
[CrossRef] [PubMed]

Bubke, K.

U. Peschel, K. Bubke, D. C. Hutchings, J. S. Aitchison, and J. M. Arnold, “Optical rectification in a travelling-wave geometry,” Phys. Rev. A 60, 4918–4926 (1999).
[CrossRef]

Chiao, R. Y.

T. K. Gustafson, J.-P. E. Taran, P. L. Kelley, and R. Y. Chiao, “Self-modulation of picosecond pulses in electro-optic crystals,” Opt. Commun. 2, 17–21 (1970).
[CrossRef]

Franken, P. A.

M. Bass, P. A. Franken, J. F. Ward, and G. Weinreich, “Optical rectification,” Phys. Rev. Lett. 9, 446–448 (1962).
[CrossRef]

Galvanauskas, A.

Y. S. Lee, T. Meade, V. Perlin, H. Winful, T. B. Norris, and A. Galvanauskas, “Generation of narrow-band terahertz radiation via optical rectification of femtosecond pulses in periodically poled lithium niobate,” Appl. Phys. Lett. 76, 2505–2507 (2000).
[CrossRef]

Graf, S.

S. Graf, H. Sigg, and W. Bächtold, “High-frequency electri-cal pulse generation using optical rectification in bulk GaAs,” Appl. Phys. Lett. 76, 2647–2649 (2000).
[CrossRef]

Green, H. E.

H. E. Green, “The numerical solution of some important transmission-line problems,” IEEE Trans. Microwave Theory Tech. 13, 676–692 (1965).
[CrossRef]

Günter, P.

C. Bosshard, R. Spreiter, M. Zgonik, and P. Günter, “Kerr nonlinearity via cascaded optical rectification and the linear electro-optic effect,” Phys. Rev. Lett. 74, 2816–2818 (1995).
[CrossRef] [PubMed]

Gustafson, T. K.

T. K. Gustafson, J.-P. E. Taran, P. L. Kelley, and R. Y. Chiao, “Self-modulation of picosecond pulses in electro-optic crystals,” Opt. Commun. 2, 17–21 (1970).
[CrossRef]

Hutchings, D. C.

U. Peschel, K. Bubke, D. C. Hutchings, J. S. Aitchison, and J. M. Arnold, “Optical rectification in a travelling-wave geometry,” Phys. Rev. A 60, 4918–4926 (1999).
[CrossRef]

Jaeger, N. A. F.

N. A. F. Jaeger and Z. K. F. Lee, “Slow-wave electrode for use in compound semiconductor electrooptic modulators,” IEEE J. Quantum Electron. 28, 1778–1784 (1992).
[CrossRef]

Jin, Y. H.

Y. H. Jin and X. C. Zhang, “Terahertz optical rectification,” J. Nonlinear Opt. Phys. Mater. 4, 459–495 (1995).
[CrossRef]

Kelley, P. L.

T. K. Gustafson, J.-P. E. Taran, P. L. Kelley, and R. Y. Chiao, “Self-modulation of picosecond pulses in electro-optic crystals,” Opt. Commun. 2, 17–21 (1970).
[CrossRef]

Lee, Y. S.

Y. S. Lee, T. Meade, V. Perlin, H. Winful, T. B. Norris, and A. Galvanauskas, “Generation of narrow-band terahertz radiation via optical rectification of femtosecond pulses in periodically poled lithium niobate,” Appl. Phys. Lett. 76, 2505–2507 (2000).
[CrossRef]

Lee, Z. K. F.

N. A. F. Jaeger and Z. K. F. Lee, “Slow-wave electrode for use in compound semiconductor electrooptic modulators,” IEEE J. Quantum Electron. 28, 1778–1784 (1992).
[CrossRef]

Meade, T.

Y. S. Lee, T. Meade, V. Perlin, H. Winful, T. B. Norris, and A. Galvanauskas, “Generation of narrow-band terahertz radiation via optical rectification of femtosecond pulses in periodically poled lithium niobate,” Appl. Phys. Lett. 76, 2505–2507 (2000).
[CrossRef]

Montrosset, I.

A. P. Ansbro and I. Montrosset, “Vectorial finite difference scheme for isotropic dielectric wave guides: transverse electric field representation,” IEE Proc. Optoelectron. 140, 253–259 (1993).
[CrossRef]

Norris, T. B.

Y. S. Lee, T. Meade, V. Perlin, H. Winful, T. B. Norris, and A. Galvanauskas, “Generation of narrow-band terahertz radiation via optical rectification of femtosecond pulses in periodically poled lithium niobate,” Appl. Phys. Lett. 76, 2505–2507 (2000).
[CrossRef]

Perlin, V.

Y. S. Lee, T. Meade, V. Perlin, H. Winful, T. B. Norris, and A. Galvanauskas, “Generation of narrow-band terahertz radiation via optical rectification of femtosecond pulses in periodically poled lithium niobate,” Appl. Phys. Lett. 76, 2505–2507 (2000).
[CrossRef]

Peschel, U.

U. Peschel, K. Bubke, D. C. Hutchings, J. S. Aitchison, and J. M. Arnold, “Optical rectification in a travelling-wave geometry,” Phys. Rev. A 60, 4918–4926 (1999).
[CrossRef]

Sigg, H.

S. Graf, H. Sigg, and W. Bächtold, “High-frequency electri-cal pulse generation using optical rectification in bulk GaAs,” Appl. Phys. Lett. 76, 2647–2649 (2000).
[CrossRef]

Spreiter, R.

C. Bosshard, R. Spreiter, M. Zgonik, and P. Günter, “Kerr nonlinearity via cascaded optical rectification and the linear electro-optic effect,” Phys. Rev. Lett. 74, 2816–2818 (1995).
[CrossRef] [PubMed]

Taran, J.-P. E.

T. K. Gustafson, J.-P. E. Taran, P. L. Kelley, and R. Y. Chiao, “Self-modulation of picosecond pulses in electro-optic crystals,” Opt. Commun. 2, 17–21 (1970).
[CrossRef]

Ward, J. F.

M. Bass, P. A. Franken, J. F. Ward, and G. Weinreich, “Optical rectification,” Phys. Rev. Lett. 9, 446–448 (1962).
[CrossRef]

Weinreich, G.

M. Bass, P. A. Franken, J. F. Ward, and G. Weinreich, “Optical rectification,” Phys. Rev. Lett. 9, 446–448 (1962).
[CrossRef]

Winful, H.

Y. S. Lee, T. Meade, V. Perlin, H. Winful, T. B. Norris, and A. Galvanauskas, “Generation of narrow-band terahertz radiation via optical rectification of femtosecond pulses in periodically poled lithium niobate,” Appl. Phys. Lett. 76, 2505–2507 (2000).
[CrossRef]

Zgonik, M.

C. Bosshard, R. Spreiter, M. Zgonik, and P. Günter, “Kerr nonlinearity via cascaded optical rectification and the linear electro-optic effect,” Phys. Rev. Lett. 74, 2816–2818 (1995).
[CrossRef] [PubMed]

Zhang, X. C.

Y. H. Jin and X. C. Zhang, “Terahertz optical rectification,” J. Nonlinear Opt. Phys. Mater. 4, 459–495 (1995).
[CrossRef]

Appl. Phys. Lett. (2)

S. Graf, H. Sigg, and W. Bächtold, “High-frequency electri-cal pulse generation using optical rectification in bulk GaAs,” Appl. Phys. Lett. 76, 2647–2649 (2000).
[CrossRef]

Y. S. Lee, T. Meade, V. Perlin, H. Winful, T. B. Norris, and A. Galvanauskas, “Generation of narrow-band terahertz radiation via optical rectification of femtosecond pulses in periodically poled lithium niobate,” Appl. Phys. Lett. 76, 2505–2507 (2000).
[CrossRef]

IEE Proc. Optoelectron. (1)

A. P. Ansbro and I. Montrosset, “Vectorial finite difference scheme for isotropic dielectric wave guides: transverse electric field representation,” IEE Proc. Optoelectron. 140, 253–259 (1993).
[CrossRef]

IEEE J. Quantum Electron. (1)

N. A. F. Jaeger and Z. K. F. Lee, “Slow-wave electrode for use in compound semiconductor electrooptic modulators,” IEEE J. Quantum Electron. 28, 1778–1784 (1992).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

H. E. Green, “The numerical solution of some important transmission-line problems,” IEEE Trans. Microwave Theory Tech. 13, 676–692 (1965).
[CrossRef]

J. Nonlinear Opt. Phys. Mater. (1)

Y. H. Jin and X. C. Zhang, “Terahertz optical rectification,” J. Nonlinear Opt. Phys. Mater. 4, 459–495 (1995).
[CrossRef]

Opt. Commun. (1)

T. K. Gustafson, J.-P. E. Taran, P. L. Kelley, and R. Y. Chiao, “Self-modulation of picosecond pulses in electro-optic crystals,” Opt. Commun. 2, 17–21 (1970).
[CrossRef]

Phys. Rev. A (1)

U. Peschel, K. Bubke, D. C. Hutchings, J. S. Aitchison, and J. M. Arnold, “Optical rectification in a travelling-wave geometry,” Phys. Rev. A 60, 4918–4926 (1999).
[CrossRef]

Phys. Rev. Lett. (2)

M. Bass, P. A. Franken, J. F. Ward, and G. Weinreich, “Optical rectification,” Phys. Rev. Lett. 9, 446–448 (1962).
[CrossRef]

C. Bosshard, R. Spreiter, M. Zgonik, and P. Günter, “Kerr nonlinearity via cascaded optical rectification and the linear electro-optic effect,” Phys. Rev. Lett. 74, 2816–2818 (1995).
[CrossRef] [PubMed]

Other (4)

M. Börner, R. Müller, R. Schiek, and G. Trommer, Elements of Integrated Optics (Teubner, Stuttgart, Germany, 1990, in German).

P. N. Butcher and D. Cotter, The Elements of Nonlinear Op-tics, Vol. 9 of Cambridge Studies in Modern Optics (Cambridge U. Press, Cambridge, 1990).

G. B. Whitham, Linear and Nonlinear Waves (Academic, New York, 1974).

K. Bubke, U. Peschel, and D. C. Hutchings, “KdV solitons on GaAs transmission lines due to the intrinsic second order nonlinearity,” in Nonlinear Guided Waves and Their Applications, 1999 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1999), pp. 339–341.

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Figures (8)

Fig. 1
Fig. 1

Structures under consideration: (a) a CPW or (b) a CPS is combined with an optical rib waveguide upon a substrate of AlGaAs with the usual [100] growth and [011] cleaving directions. The dashed arrows indicate the field direction of the microwave interacting with the optical TE and TM modes [solid arrows]. An optical pulse is injected into the structure at a polarization angle γ. (c) Suggested structure for tuning the mid-frequency of narrow-bandwidth THz generation. A dc voltage across the microstrip structure tunes the phase-velocity mismatch between TE and TM; a strip line extracts the generated THz signal.

Fig. 2
Fig. 2

Evolution of the microwave signal in a strip line for velocity matching (Δn=0). An optical pulse with a peak power P=1 kW is injected at z=0. Walk-off, dispersion, and attenuation are neglected.

Fig. 3
Fig. 3

Evolution of factor M(z) in Eq. (35) for varying losses versus propagation lengths z. Solid curve, αmic=0; dashed curve, αmic=25 dB/cm; dotted curve, αmic=50 dB/cm.

Fig. 4
Fig. 4

Evolution of a microwave signal in a strip line with velocity mismatch (Δn=-0.17; other parameters as in Fig. 2).

Fig. 5
Fig. 5

Microwave signal in Fig. 4 after propagation length z=2 cm. Also shown (shaded area) is the injected optical pulse at z=0. Solid curve, αmic=0; dashed curve, αmic=5 dB/cm.

Fig. 6
Fig. 6

Amplitude spectra of microwave signals generated with velocity mismatch. Solid curve, Δn=-0.1, αmic=0; dotted curve, Δn=-0.1, αmic=5 dB/cm; dashed curve, Δn=-0.2, αmic=0.

Fig. 7
Fig. 7

Evolution of circular polarization owing to nonlinearly induced phase modulation in the TE mode in a coplanar waveguide (Δn=-0.038). Solid curve, low power; dashed curve, optical peak power P=5 kW, αmic=0; dotted curve, P=5 kW, αmic=10 dB/cm.

Fig. 8
Fig. 8

Self-induced mode conversion owing to cascading of OR and the electro-optic effect in a strip line (Δn=-0.17). Optical peak power, P=5 kW. Solid curve, no losses; dashed curve, αmic=5 dB/cm.

Tables (1)

Tables Icon

Table 1 Characteristic Parameters of Structures Used for Calculationsa

Equations (55)

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×E˜=iωμ0H˜,
×H˜=-iω0rE˜-iωP˜NL,
E˜0=n exp[iβ(ω)z]en(ω, x, y),
H˜0=n exp[iβ(ω)z]hn(ω, x, y),
E˜=n un(z, ω)en(ω, x, y),
H˜=n un(z, ω)hn(ω, x, y).
·(E˜×H˜0*+E˜0*×H˜)=iωP˜NL·E˜0*.
z [E˜×H˜0*+E˜0*×H˜]zdxdy
=iω  P˜NL·E˜0*dxdy.
unz-iβnun=iωpn  en*·P˜NLdxdy,
pn=2 Re  [en×hn*]zdxdy.
βn(ω)βn+ω-ωoptvn+Dn2(ω-ωopt)2,
βn=ωoptnnc=βn(ωopt),1vn=nngrc=βnωω=ωopt,
Dn=2βnω2ω=ωopt.
β(ω)jαmic+1vmicω+jαmicω2+Tmic6ω3,
αmic=Im[β(0)],1vmic=nmicc=Reβmicωω=0,
αmic=2 Im2βmicω2ω=0,
Tmic=Re3βmicω3ω=0.
pel=p02=limω0 Re  [emic×hmic*]zdxdy=V2Z.
z+αmic+1vmic t-αmic 2t2-Tmic6 3t3umic
=-12pel t  emic*·Pmicdxdy,
z-iβn+1vn t+i Dn2 2t2un
=1pn iωopt-t en*·Poptdxdy.
E(x, y, z, t)=[uTE(z, t)exp(-iωoptt)eTE(ωopt)+c.c.]+[uTM(z, t)exp(-iωoptt)eTM(ωopt)+c.c.]+umic(z t)emic(0),
PNL(x, y, z, t)=Popt(x, y, z, t)exp(-iωoptt)+c.c.+Pmic(x, y, z, t).
[PNL]x=-0χ(2)2[E]y2,
[PNL]y=-0χ(2)[E]x[E]y,
[Pmic]x=-0χ(2)2([emic]y2umic2+2[eTE]y2|uTE|2),
[Pmic]y=-0χ(2){[emic]y[emic]xumic2+2 Re([eTE]y[eTM]xuTEuTM*)},
[Popt]x=-0χ(2)[eTE]y[emic]yuTEumic,
[Popt]y=-0χ(2)([eTE]y[emic]xuTEumic+[eTM]x[emic]yuTMumic).
Umic=umicpel,
UTE=uTEpTE exp(-iβTEz),
UTM=uTMpTM exp(-iβTEz).
z+αmic+Δnc t˜-αmic 2t˜2-Tmic6 3t˜3Umic
=χeff 12 t˜|UTE|2+χeff 22 t˜(UTEUTM*+UTE*UTM)+χeff 3 t˜Umic2,
i z-DTE2 2t˜2UTE=χeff 1ω0+i t˜(UTEUmic)+χeff 2ω0+i t˜(UTMUmic),
i z-Δβ-iδvopt t˜-DTM2 2t˜2UTM
=χeff 2ω0+i t˜(UTEUmic),
χeff 1=0  χ(2)(x, y)[emic]x[eTE]y2dxdypmic1/2pTE,
χeff 2=0  χ(2)(x, y)[emic]y[eTE]y[eTM]xdxdy[pmicpTEpTM]1/2,
χeff 3=304  χ(2)(x, y)[emic]x[emic]y2dxdypmic3/2.
Umic(z, t˜)=M(z) χeff 2 sin(2γ)2 t˜|U0(t˜)|2,
M(z)=Δβ sin(Δβz)+αmic[cos(Δβz)-exp(-αmicz)]Δβ2+αmic2.
Umic(z, t˜)=χeff 2 sin(2γ)c2Δn[|U0(t˜)|2 cos(Δβz)-|U0(t˜-Δn/cz)|2×exp(-αmicz)+UTHz(z, t˜)],
UTHz(z, t˜)=-exp(-αmicz)×0z exp(αmicz)[αmic cos(Δβz)-Δβ sin(Δβz)]×|U0[t˜-Δn/c(z-z)]|2dz.
|U0(t˜)|2=P0-ΔT/2t˜ΔT/20otherwise.
UTHz(z, t˜)=2P0 sinΔβΔTc2ΔnsinΔβz-cΔnt˜.
fmic=cΔnoptλΔn.
Lwo=λ2Δnopt.
|U0(t˜)|2=(P0/2)(cos(πt˜/ΔT)+1)-ΔTt˜ΔT0otherwise,
UTHz(z, t˜)=P0 sin(x)1-(x/π)2 sinΔβz-cΔnt˜,
x=ΔβΔTcΔn.
Lwo=λ2Δnopt×0.837.
fmic=c(Δnopt-cχeff 1Vdc)λΔn.

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