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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 17, Iss. 21 — Oct. 12, 2009
  • pp: 19102–19112
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Timing jitter measurement of transmitted laser pulse relative to the reference using type II second harmonic generation in two nonlinear crystals

Xuelian Ma, Lu Liu, and Junxiong Tang  »View Author Affiliations


Optics Express, Vol. 17, Issue 21, pp. 19102-19112 (2009)
http://dx.doi.org/10.1364/OE.17.019102


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Abstract

A new method is proposed and analyzed for measuring the timing jitter of the transmitted pulse relative to the reference pulse using two type II phase-matched nonlinear crystals for second harmonic generation (SHG). The polarizations of the two pulses are exchanged in two crystals and the difference between two detected second harmonic signals can reflect the transmitted jitter. This new method provides a high sensitivity and timing resolution compared with the conventional RF (radio frequency) method. Since the overlapping levels in the two crystals are the same, the final output is zero when there is no time delay between the two pulses. Thus no offset is necessary to be subtracted from the final output and no time delay adjustment is required between the two pulses, compared with the previous optical method using one crystal and two dichroic beamsplitters. The jitter measuring performance is studied theoretically using non-stationary nonlinear wave-coupled equations for type II SHG of two pulses. The theoretical computation and analysis show that the sensitivity and the dynamic range of this new method depend on pulse width, crystal pulses and group velocity difference between two fundamental pulses.

© 2009 OSA

1. Introduction

In the case of the frequency comb as a reference [7

7. S. N. Bagayev, S. V. Chepurov, V. I. Denisov, A. K. Dmitriyev, A. S. Dychkov, V. M. Klementyev, D. B. Kolker, I. I. Korel, S. A. Kuznetsov, Y. A. Matyugin, M. V. Okhapkin, V. S. Pivtsov, M. N. Skvortsov, V. F. Zakharyash, T. A. Birks, W. J. Wadsworth, and P. S. J. Russell, “Femtosecond optical clock with the use of a frequency comb,” Proc. SPIE 4900, 125–131 (2002). [CrossRef]

], the timing jitter of transmitted pulse relative to the reference pulse is required. One conventional RF method of jitter measuring uses two photodetectors and a mixer to obtain the phase noise and timing jitter [3

3. F. O. Ilday, A. Winter, J. W. Kim, J. Chen, P. Schmuser, H. Schlarb, and F. X. Kartner, “Ultra-low timing-jitter passively mode-locked fiber lasers for long-distance timing synchronization,” Proc. SPIE 6389, 63890L (2006). [CrossRef]

6

6. K. W. Holman, D. D. Hudson, J. Ye, and D. J. Jones, “Remote transfer of a high-stability and ultralow-jitter timing signal,” Opt. Lett. 30(10), 1225–1227 (2005). [CrossRef] [PubMed]

]. Recently, a balanced optical cross correlator based on a type II SHG crystal and two dichroic beamsplitters is used for jitter measurement in timing distribution system [8

8. 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]

]. The sensitivity of this optical method is higher than that of the conventional one. However, since the output of this method is not zero when the two pulses are overlapped entirely (no time delay between them) before entering into the crystal, the transmitted pulse is required to be adjusted to have a fixed time delay relative to the reference before measuring, or an offset signal should be subtracted from the output.

In this paper, a new method is proposed for measuring the jitter between two pulses (one is the transmitted pulse, while the other is the original pulse as a reference) using two type II phase-matched nonlinear crystals for SHG. Compared with the method in Ref. 8

8. 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]

, when the two pulses are overlapped entirely (no time delay) before entering into the crystals, the overlapping levels in the two crystals are the same and the final output is zero for this new measuring method, so there is no need to subtract an offset from the final output and no time delay adjustment is required between the two pulses. Furthermore, this new method provides a high sensitivity compared with the conventional RF method.

The principle of this new method is demonstrated in section 2 to show how the final output reflects the jitter. The mathematical model and computation originated from non-stationary nonlinear wave-coupled equations are given in section 3. The performance of jitter measurement including sensitivity and dynamic range are analyzed theoretically in section 4. Conclusions are given finally in section 5.

2. The measuring principle

Figure 1
Fig. 1 (Color online) Schematic diagram for the jitter measurement of the transmitted pulse relative to the reference pulse using type II phase-matched SHG. The blue solid pulse represents the reference pulse and the red dashed pulse represents the transmitted pulse. The green solid pulse represents the second harmonic field. The dark dashed line represents the electric signal detected by PMD.
shows the schematic diagram for the proposed method of jitter measurement between the transmitted pulse and the reference pulse using type II phase-matched SHG. Two fundamental pulses in one nonlinear crystal are required to be perpendicular polarized in the case of type II SHG [9

9. S. Qian and R. Zhu, Nonlinear Optics (Fudan University Press, Shanghai, 2005), Chap. 3.

]. The polarization of the reference pulse is adjusted by a half wave plate (HWP) and split by a polarizing beam splitter (PBS) to obtain two perpendicular polarized parts with equal intensity. In this diagram, we assume the group velocity of the part transmitted by PBS is larger than the one reflected by PBS when they are propagated in the crystals. Therefore, the transmitted part can be called slow part and the reflected one can be called fast part. The transmitted pulse is adjusted and split in the same situation with the reference, except that the incident direction on the PBS is orthogonal to that of the reference. The fast part of the reference pulse and the slow part of the transmitted pulse are collinearly propagated in the first crystal (crystal-1) to generate a second harmonic signal (SH signal-1); meanwhile the slow part of the reference pulse and the fast part of the transmitted pulse is collinearly propagated in the second crystal (crystal 2) to generate another second harmonic signal (SH signal-2). Each second harmonic signal is detected by a photomultiplier detector (PMD). The final output is the difference between the two converted electric signals.

The intensity of the second harmonic field at the crystal end depends on the intensities of the two perpendicular polarized fundamental pulses and the time overlapping level between them. Because of equal splitting for both the reference and the transmitted pulse, the intensity difference between SH signal-1 and SH signal-2 depends only on pulse overlapping level in the two crystals.

The transmitted pulse and the reference will be overlapped entirely all the time before entering into the crystal if there is no time delay (no jitter) between them. In this case, when the two pulses are split by PBS and propagated in the two crystals, the fast part becomes ahead of the slow part gradually in the crystal-1, and the same situation happens in crystal-2. Since the transmitted pulse has the same wavelength with the reference, their slow parts have the same group velocity and so do the fast parts. In this case, the pulse overlapping levels keep the same in two crystals, so there is no difference between two second harmonic signals and the final output is zero.

The transmitted pulse will have a variable delay time relative to the reference before their incidence to the crystal if there is jitter between them. When the transmitted pulse is delayed relative to the reference, the average overlapping level in crystal-2 is larger than in crystal-1, so the SH signal-2 is larger than the SH signal-1 and the final output becomes positive. On the other hand, when the transmitted pulse is advanced relative to the reference, the final output becomes negative. Furthermore, the larger the delayed or advanced time is, the bigger the difference between the two second harmonic signals becomes. Therefore, the final output increases as the delay time increases. The sign of the final output indicates the jitter direction (the transmitted pulse is delayed or advanced relative to the reference), and the amplitude of the final output represents the jitter value. Accordingly the final output can reflect the timing jitter of the transmitted pulse relative to the reference.

3. Mathematical model and computation

The non-stationary nonlinear wave-coupled equations are used to study the proposed jitter measuring method. The variation of field amplitude with time must be considered in the nonlinear equations in the case of pulse SHG [10

10. J.-C. Diels and W. Rudolph, Ultrashort Laser Pulse Phenomena: Fundamentals, Techniques, and Applications on a Femtosecond Time Scale, 2nd ed. (Academic Press, Burlington, 2006), Chap. 3.

]. First the intensity of second harmonic field of type II SHG for two fundamental pulses with no jitter is derived, and then the relation between the final output and the timing jitter for this measuring method is given.

3.1The intensity of the second harmonic field

The two fundamental pulses are required to be perpendicular polarized in a nonlinear crystal for type II phase-matched SHG [9

9. S. Qian and R. Zhu, Nonlinear Optics (Fudan University Press, Shanghai, 2005), Chap. 3.

]. One polarized pulse propagates slowly and the other propagates fast in the crystal. The generated second harmonic field should have the same polarization with the fast fundamental field so as to satisfy phase matching [9

9. S. Qian and R. Zhu, Nonlinear Optics (Fudan University Press, Shanghai, 2005), Chap. 3.

,11

11. M. V. Hobden, “Phase-matched second-harmonic generation in biaxial crystals,” J. Appl. Phys. 38(11), 4365–4372 (1967). [CrossRef]

]. The two fundamental slow polarized field, fast polarized field and the second harmonic field propagated along the z direction, denoted by E 1 s(z,t), E 1 f(z,t) and E2(z,t) can be expressed in the complex form
E1s(z,t)={12U1s(z,t)exp[i(k1szω1t)]+c.c.}e1s
(1a)
E1f(z,t)={12U1f(z,t)exp[i(k1fzω1t)]+c.c.}e1f
(1b)
E2(z,t)={12U2(z,t)exp[i(k2zω1t)]+c.c.}e2
(1c)
where e 1 s, e 1 f and e 2 are the unit polarization vectors; k 1 s, k 1 f and k 2 are the wave numbers; ω 1 and ω 2 are the fundamental and second harmonic angular frequency respectively (ω 1 = ω 2); U 1 s(z,t), U 1 f(z,t) and U 2(z,t) are the pulse envelopes, which represent the slowly varying complex amplitudes.

Considering the case of no absorption in the crystal, if the phase is totally matched (Δk = k 2-k 1 s-k 1 f), the nonlinear wave-coupled equations of the two fundamental fields and the second harmonic field can be expressed as [9

9. S. Qian and R. Zhu, Nonlinear Optics (Fudan University Press, Shanghai, 2005), Chap. 3.

,10

10. J.-C. Diels and W. Rudolph, Ultrashort Laser Pulse Phenomena: Fundamentals, Techniques, and Applications on a Femtosecond Time Scale, 2nd ed. (Academic Press, Burlington, 2006), Chap. 3.

]
U1s(z,t)z+1υg1sU1s(z,t)t=iω1deffcn1sU2(z,t)[U1f(z,t)]
(2a)
U1f(z,t)z+1υg1fU1f(z,t)t=iω1deffcn1fU2(z,t)[U1s(z,t)]
(2b)
U2(z,t)z+1υg2U2(z,t)t=2iω1deffcn2U1s(z,t)U1f(z,t)
(2c)
where υg 1 s, υg 1 f and υg 2 are the group velocities of fundamental slow pulse, fundamental fast pulse and second harmonic pulse respectively, n 1 s, n 1 f and n 2 are the refractive indexes in the crystal, deff is the effective nonlinear coefficient, c is the light velocity.

In the situation of the fundamental fields being low depleted and approximately unchanged through the whole crystal, the amplitudes of the fundamental fields can be defined as
U1s(z,t)=U1s(0,tzυg1s)U1s(tzυg1s)
(3a)
U1f(z,t)=U1f(0,tzυg1f)U1f(tzυg1f)
(3b)
When the variation t in the amplitude of second harmonic field is replaced by t′ = t-z/υg 2, Eq. (2)c) becomes
U2(z,t)z=2iω1deffcn2U1s(t+zυg2zυg1s)U1f(t+zυg2zυg1f)
(4)
With Eq. (4) being integrated in the range of the whole crystal length, the amplitude of the output second harmonic field at the crystal end can be obtained as
U2(L,t)=2iω1deffcn20LU1s(t+βzαsz)U1f(t+βzαfz)dz
(5)
where L is the crystal length, αs and αf are the reciprocals of group velocities of the fundamental slow pulse and the fast pulse respectively, β is the reciprocal of group velocity of the second harmonic pulse, αs = (υg 1 s)−1, αf = (υg 1 f)−1, β = (υg 2)−1.

By substituting the variation t′ for the original variation t at the crystal end z = L, we can express Eq. (5) as
U2(L,t)=2iω1deffcn21(βαs)αsLβLU1s(tt1)U1f[tγt1+(γ1)βL]dt1
(6)
where t 1 = βL-(β-αs)z, γ = (β-αf)/(β-αs). When the pulse half-width is larger than (αs-αf)L/4, the overlapping level of the two fundamental pulses in a crystal varies slowly in the process of propagation, which means the term U 1 s(t-t 1)U 1 f[t-γt1 + (γ-1)βL] is slowing varying over the range of integration, therefore this term in the integral of Eq. (6) can be removed from the integral and replaced by its value at midpoint t 1 = (β + αs)L/2, then Eq. (6) can be solved as
U2(t)=2iω1deffcn2LU1s[t(β+αs)L2]U1f[tγ(β+αs)L2+(γ1)βL]
(7)
The field intensity can be expressed as
Ij(t)=12ε0cnj|Uj(t)|2
(8)
where j = 1, 2 represent the fundamental and second harmonic field respectively, ε 0 is the vacuum dielectric constant. The intensity of second harmonic fields at the crystal end I 2(t) can be written as
I2(t)=8ω12deff2L2ε0c3n1sn1fn2I1s[t(β+αs)L2]I1f[tγ(β+αs)L2+(γ1)βL]
(9)
where I 1 s(t) and I 1 f(t) are the intensities of fundamental slow pulse and fast pulse respectively. Equation (9) shows how the intensity of second harmonic field at the crystal end varies with time.

3.2 The relation between the final output and the timing jitter

When the transmitted pulse has timing jitter relative to the reference pulse, delay time τ between the two pulses varies with time continually. We assume τ > 0 when the transmitted pulse is delayed relative to the reference and τ < 0 when the transmitted pulse is advanced relative to the reference. With consideration of delay time τ between the two fundamental pulses, the intensities of the two second harmonic signals at the end of crystal-1 and crystal-2, denoted by I 2-1 and I 2-2, can be written as
I21(t,τ)=AL2Itrs(ttaτ)Ireff(ttb)
(10a)
I22(t,τ)=AL2Irefs(tta)Itrf(ttbτ)
(10b)
where Iref-s and Iref-f are the intensities of fundamental slow part and fast part of the reference pulse, Itr-s and Itr-f are the intensities of fundamental slow part and fast part of the transmitted pulse; the parameters A, t a and t b are defined as A = (8ω 1 2 deff 2)/(ε 0 c 3 n 1 sn 1 fn 2), t a = (β + αs)L/2 and t b = (β + αf)L/2. The signals detected by PMD1 and PMD2, denoted by SPD1 and SPD1, can be expressed as
SPD1(τ)=ηI21(t,τ)dt=ηAL2Itrs(ttaτ)Ireff(ttb)dt
(11a)
SPD2(τ)=ηI22(t,τ)dt=ηAL2Irefs(tta)Itrf(ttbτ)dt
(11b)
where η is the photoelectric conversion efficiency.

We assume that the pulse widths of the fundamental pulses are identical. We investigate two cases of the pulse type: Gaussian pulses and hyperbolic secant pulses. Since both reference pulse and transmitted pulse are equally split by the PBS as we can see in Fig. 1, the pulse intensities of the Gaussian type can be expressed as
Irefs(t)=Ireff(t)=I01e(t/TP)2
(12a)
Itrs(t)=Itrf(t)=I02e(t/TP)2
(12b)
and the intensities of the hyperbolic secant pulse can be expressed as
Irefs(t)=Ireff(t)=I01sech(2t/Tp)
(12c)
Itrs(t)=Itrf(t)=I02sech(2t/Tp)
(12d)
where I 01 is half peak intensity of the referenced pulse and I 02 is half peak intensity of the transmitted pulse, and TP is the pulse half-width. Inserting the above intensity expressions Eq. (12)a) and Eq. (12)b) into Eq. (11)a) and Eq. (11)b), by using the method of Fourier transform, we can obtain the expressions for SPD1 and SPD2 for Gaussian pulse type as follows
SPD1(τ)=πηAL2I01I02TPe(τ+tatb)2/(2TP2)
(13a)
SPD2(τ)=πηAL2I01I02TPe(τta+tb)2/(2TP2)
(13b)
Inserting the above intensity expressions Eq. (12)c) and Eq. (12)d) into Eq. (11)a) and Eq. (11)b), by using the method of Fourier transform, we can obtain the expressions for SPD1 and SPD2 for hyperbolic secant pulse type as follows
SPD1(τ)=4ηAL2I01I02TP(τ+tatb)/TPe(τ+tatb)/TPe(τ+tatb)/TP
(13c)
SPD2(τ)=4ηAL2I01I02TP(τta+tb)/TPe(τta+tb)/TPe(τta+tb)/TP
(13d)
The difference between the two detected electric signals, denoted by Sdiff, can be written as
Sdiff(τ)=SPD2(τ)SPD1(τ)=πηAL2I01I02TP[e(τξ/2)2/(2TP2)e(τ+ξ/2)2/(2TP2)]
(14a)
for the Gaussian pulse case and
Sdiff(τ)=4ηAL2I01I02TP[(τξ/2)/TPe(τξ/2)/TPe(τξ/2)/TP(τ+ξ/2)/TPe(τ+ξ/2)/TPe(τ+ξ/2)/TP]
(14b)
for the hyperbolic secant pulse case, where ξ is an important parameter defined as ξ = (αs-αf)L.

It is worth noticing that ξ is the time difference of the fundamental slow and fast pulse propagating through the whole crystal length, which describes the walk-off between the two fundamental pulses owing to the different group velocities.

The differential value Sdiff is the final output. Equation (14) shows how the final output varies with the delay time of the transmitted pulse relative to the reference. This expression reflects the relation between the final output and the timing jitter of the two pulses.

4. Theoretical analysis of the performance for timing jitter measurement

Figure 2
Fig. 2 Final output Sdiff as a function of delay time τ between two pulses for the two pulse cases: (a) Gaussian pulse type; (b) hyperbolic secant pulse type.
shows the final output signal Sdiff varying with the delay time τ of the transmitted pulse relative to the reference in the case of ξ/TP = 2. The two cases of Gaussian pulse type and hyperbolic secant pulse type are given in Fig. 2(a) and Fig. 2(b) respectively. The approximate linear region blocked by the dashed line in the middle of the curve is applicable to the measurement. Since final output is proportional to the delay time, the final output can reflect the timing jitter in this linear region. The gradient of this linear region KS represents sensitivity of jitter measurement and the width Δτ represents dynamic range of jitter measurement. The sensitivity and the dynamic range indicate jitter measuring performance. As we can see, the properties of the two pulse types are similar, with only the numerical difference, therefore we can investigate the performance of the jitter measurement using one of the two pulse type, and the Gaussian pulse type is used in the following analysis.

In the linear region, the expression of the sensitivity KS can be obtained approximately as following through calculating the derivative of Eq. (14)a) at τ = 0
KSπηAL2I01I02ξTPexp[18(ξTP)2]
(15)
The sensitivity KS has a maximum value of 1.2π1/2 ηAL 2 I 01 I 02 at ξ/TP = 2 as we can compute from the above expression. The effect of the parameter ξ/TP on the sensitivity KS and the dynamic range Δτ is studied in the case of ξ/TP ≤ 4.

Figure 3
Fig. 3 Sensitivity KS and Dynamic range Δτ as a function of ξ/TP.
shows KS and Δτ vary as a function of ξ/TP. For a certain value of L, Ks increases with ξ/TP in the range of ξ/TP ≤ 2 and decreases in the range of 2 ≤ ξ/TP ≤ 4. Ks arrives at its maximum value at ξ/TP = 2, which can be obtained from Eq. (15). To enhance the sensitivity, the value of ξ/TP is expected to be close to 2. Furthermore, the sensitivity is proportional to the square of the crystal length L with the value of ξ/TP fixed. From this angle we hope the crystal is long and the value of αs-αf is small. For a given value of TP, the dynamic range Δτ has an increasing trend in the whole extent of ξ/TP ≤ 4. The maximum value of Δτ is 4TP at ξ/TP = 4.

The parameter ξ/TP is the normalized time difference between two fundamental pulses propagating through the crystal length. If the group velocities of the two pulses are the same and so there is no time difference (ξ/TP = 0), the final output is always zero and the delay time cannot be measured. In this case, the sensitivity and the dynamic range are both zero. With the increasing of this normalized time difference, the original time delay between the two pulses before entering into the crystal can increase without the loss of pulse overlapping level in the crystal; hence the larger value for ξ/TP leads to the bigger value for the dynamic range. However, when ξ/TP is fixed, the dynamic range is proportional to the pulse half-width TP. As regard to a mode-locked laser with pulse width at hundreds of femtoseconds level, the dynamic range of this jitter measurement method can achieve hundreds of femtoseconds to several picoseconds.

It is worth noticing that all the above curves and results are given in the case of T P≥(αs-αf)L/4, which means L≤4T P/(αs-αf). Further analysis for this measuring method shows that the sensitivity and the dynamic range for L>4T P/(αs-αf) are the same as L = 4T P/(αs-αf), therefore increasing the crystal length further will not improve the jitter measuring performance any more.

In the process of the pulse transmitting through the fiber link, both phase noise and amplitude noise can be introduced. In this paper, only phase noise (timing jitter) is required to be measured. From the above analysis, we can see that in the dynamic range of this jitter measurement, the final output, which represents the intensity difference of the two second harmonic signals, is nearly proportional to the delay time between the two pulses. In the vicinity of zero timing offset, the amplitude noise makes the same contribution to the two second harmonic signals in Fig. 1, so the intensity difference of the two signals (the final output) is not affected by the amplitude noise and only affected by phase noise. Hence, the timing jitter measurement can be robust against the amplitude noise.

5. Conclusion

References and links

1.

S. M. Foreman, K. W. Holman, D. D. Hudson, D. J. Jones, and J. Ye, “Remote transfer of ultrastable frequency references via fiber networks,” Rev. Sci. Instrum. 78(2), 021101 (2007). [CrossRef] [PubMed]

2.

F. Narbonneau, M. Lours, S. Bize, A. Clairon, G. Santarelli, O. Lopez, C. Daussy, A. Amy-Klein, and C. Chardonnet, “High resolution frequency standard dissemination via optical fiber metropolitan network,” Rev. Sci. Instrum. 77(6), 064701 (2006). [CrossRef]

3.

F. O. Ilday, A. Winter, J. W. Kim, J. Chen, P. Schmuser, H. Schlarb, and F. X. Kartner, “Ultra-low timing-jitter passively mode-locked fiber lasers for long-distance timing synchronization,” Proc. SPIE 6389, 63890L (2006). [CrossRef]

4.

M. Calhoun, S. Huang, and R. L. Tjoelker, “Stable photonic links for frequency and time transfer in the deep-space network and antenna arrays,” Proc. IEEE 95(10), 1931–1946 (2007). [CrossRef]

5.

K. W. Holman, D. J. Jones, D. D. Hudson, and J. Ye, “Precise frequency transfer through a fiber network by use of 1.5-microm mode-locked sources,” Opt. Lett. 29(13), 1554–1556 (2004). [CrossRef] [PubMed]

6.

K. W. Holman, D. D. Hudson, J. Ye, and D. J. Jones, “Remote transfer of a high-stability and ultralow-jitter timing signal,” Opt. Lett. 30(10), 1225–1227 (2005). [CrossRef] [PubMed]

7.

S. N. Bagayev, S. V. Chepurov, V. I. Denisov, A. K. Dmitriyev, A. S. Dychkov, V. M. Klementyev, D. B. Kolker, I. I. Korel, S. A. Kuznetsov, Y. A. Matyugin, M. V. Okhapkin, V. S. Pivtsov, M. N. Skvortsov, V. F. Zakharyash, T. A. Birks, W. J. Wadsworth, and P. S. J. Russell, “Femtosecond optical clock with the use of a frequency comb,” Proc. SPIE 4900, 125–131 (2002). [CrossRef]

8.

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]

9.

S. Qian and R. Zhu, Nonlinear Optics (Fudan University Press, Shanghai, 2005), Chap. 3.

10.

J.-C. Diels and W. Rudolph, Ultrashort Laser Pulse Phenomena: Fundamentals, Techniques, and Applications on a Femtosecond Time Scale, 2nd ed. (Academic Press, Burlington, 2006), Chap. 3.

11.

M. V. Hobden, “Phase-matched second-harmonic generation in biaxial crystals,” J. Appl. Phys. 38(11), 4365–4372 (1967). [CrossRef]

OCIS Codes
(190.0190) Nonlinear optics : Nonlinear optics
(190.2620) Nonlinear optics : Harmonic generation and mixing
(190.4360) Nonlinear optics : Nonlinear optics, devices
(320.0320) Ultrafast optics : Ultrafast optics
(320.7080) Ultrafast optics : Ultrafast devices
(320.7100) Ultrafast optics : Ultrafast measurements
(320.7110) Ultrafast optics : Ultrafast nonlinear optics

ToC Category:
Ultrafast Optics

History
Original Manuscript: June 1, 2009
Revised Manuscript: August 20, 2009
Manuscript Accepted: August 21, 2009
Published: October 8, 2009

Citation
Xuelian Ma, Lu Liu, and Junxiong Tang, "Timing jitter measurement of transmitted laser pulse relative to the reference using type II second harmonic generation in two nonlinear crystals," Opt. Express 17, 19102-19112 (2009)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-21-19102


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References

  1. S. M. Foreman, K. W. Holman, D. D. Hudson, D. J. Jones, and J. Ye, “Remote transfer of ultrastable frequency references via fiber networks,” Rev. Sci. Instrum. 78(2), 021101 (2007). [CrossRef] [PubMed]
  2. F. Narbonneau, M. Lours, S. Bize, A. Clairon, G. Santarelli, O. Lopez, C. Daussy, A. Amy-Klein, and C. Chardonnet, “High resolution frequency standard dissemination via optical fiber metropolitan network,” Rev. Sci. Instrum. 77(6), 064701 (2006). [CrossRef]
  3. F. O. Ilday, A. Winter, J. W. Kim, J. Chen, P. Schmuser, H. Schlarb, and F. X. Kartner, “Ultra-low timing-jitter passively mode-locked fiber lasers for long-distance timing synchronization,” Proc. SPIE 6389, 63890L (2006). [CrossRef]
  4. M. Calhoun, S. Huang, and R. L. Tjoelker, “Stable photonic links for frequency and time transfer in the deep-space network and antenna arrays,” Proc. IEEE 95(10), 1931–1946 (2007). [CrossRef]
  5. K. W. Holman, D. J. Jones, D. D. Hudson, and J. Ye, “Precise frequency transfer through a fiber network by use of 1.5-microm mode-locked sources,” Opt. Lett. 29(13), 1554–1556 (2004). [CrossRef] [PubMed]
  6. K. W. Holman, D. D. Hudson, J. Ye, and D. J. Jones, “Remote transfer of a high-stability and ultralow-jitter timing signal,” Opt. Lett. 30(10), 1225–1227 (2005). [CrossRef] [PubMed]
  7. S. N. Bagayev, S. V. Chepurov, V. I. Denisov, A. K. Dmitriyev, A. S. Dychkov, V. M. Klementyev, D. B. Kolker, I. I. Korel, S. A. Kuznetsov, Y. A. Matyugin, M. V. Okhapkin, V. S. Pivtsov, M. N. Skvortsov, V. F. Zakharyash, T. A. Birks, W. J. Wadsworth, and P. S. J. Russell, “Femtosecond optical clock with the use of a frequency comb,” Proc. SPIE 4900, 125–131 (2002). [CrossRef]
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