## Timing jitter measurement of transmitted laser pulse relative to the reference using type II second harmonic generation in two nonlinear crystals

Optics Express, Vol. 17, Issue 21, pp. 19102-19112 (2009)

http://dx.doi.org/10.1364/OE.17.019102

Acrobat PDF (275 KB)

### 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

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]

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]

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. 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]

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]

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]

## 2. The measuring principle

## 3. Mathematical model and computation

### 3.1The intensity of the second harmonic field

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

*z*direction, denoted by

*E*_{1}

*(*

_{s}*z,t*),

*E*_{1}

*(*

_{f}*z,t*) and

**(**

*E*_{2}*z,t*) can be expressed in the complex form where

*e*_{1}

*,*

_{s}

*e*_{1}

*and*

_{f}

*e*_{2}are the unit polarization vectors;

*k*

_{1}

*,*

_{s}*k*

_{1}

*and*

_{f}*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.

*k*=

*k*

_{2}-

*k*

_{1}

*-*

_{s}*k*

_{1}

*), the nonlinear wave-coupled equations of the two fundamental fields and the second harmonic field can be expressed as [9,10] where*

_{f}*υ*

_{g}_{1}

*,*

_{s}*υ*

_{g}_{1}

*and*

_{f}*υ*

_{g}_{2}are the group velocities of fundamental slow pulse, fundamental fast pulse and second harmonic pulse respectively,

*n*

_{1}

*,*

_{s}*n*

_{1}

*and*

_{f}*n*

_{2}are the refractive indexes in the crystal,

*d*is the effective nonlinear coefficient,

_{eff}*c*is the light velocity.

*t*in the amplitude of second harmonic field is replaced by

*t*′ =

*t*-

*z*/

*υ*

_{g}_{2}, Eq. (2)c) becomesWith 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 aswhere

*L*is the crystal length,

*α*and

_{s}*α*are the reciprocals of group velocities of the fundamental slow pulse and the fast pulse respectively,

_{f}*β*is the reciprocal of group velocity of the second harmonic pulse,

*α*= (

_{s}*υ*

_{g}_{1}

*)*

_{s}^{−1},

*α*= (

_{f}*υ*

_{g}_{1}

*)*

_{f}^{−1},

*β*= (

*υ*

_{g}_{2})

^{−1}.

*t*′ for the original variation

*t*at the crystal end

*z = L*, we can express Eq. (5) aswhere

*t*

_{1}=

*βL-(β-α*, γ = (

_{s})z*β-α*)/(

_{f}*β-α*). When the pulse half-width is larger than (

_{s}*α*-

_{s}*α*)

_{f}*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-γt*+ (

_{1}*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*+

*α*)

_{s}*L*/2, then Eq. (6) can be solved asThe field intensity can be expressed aswhere

*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 aswhere

*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

*I*

_{2-1}and

*I*

_{2-2}, can be written as where

*I*and

_{ref-s}*I*are the intensities of fundamental slow part and fast part of the reference pulse,

_{ref-f}*I*and

_{tr-s}*I*are the intensities of fundamental slow part and fast part of the transmitted pulse; the parameters

_{tr-f}*A*,

*t*

_{a}and

*t*

_{b}are defined as

*A*= (8

*ω*

_{1}

^{2}

*d*

_{eff}^{2})/(

*ε*

_{0}

*c*

^{3}

*n*

_{1}

_{s}n_{1}

_{f}n_{2}),

*t*

_{a}= (

*β*+

*α*)

_{s}*L*/2 and

*t*

_{b}= (

*β*+

*α*)

_{f}*L*/2. The signals detected by PMD1 and PMD2, denoted by SPD1 and SPD1, can be expressed as where η is the photoelectric conversion efficiency.

*I*

_{01}is half peak intensity of the referenced pulse and

*I*

_{02}is half peak intensity of the transmitted pulse, and

*T*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 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 The difference between the two detected electric signals, denoted by

_{P}*S*, can be written asfor the Gaussian pulse case andfor the hyperbolic secant pulse case, where

_{diff}*ξ*is an important parameter defined as ξ = (

*α*-

_{s}*α*)

_{f}*L*.

*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.

*S*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.

_{diff}## 4. Theoretical analysis of the performance for timing jitter measurement

*S*varying with the delay time

_{diff}*τ*of the transmitted pulse relative to the reference in the case of

*ξ*/

*T*= 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

_{P}*K*represents sensitivity of jitter measurement and the width Δ

_{S}*τ*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.

*K*can be obtained approximately as following through calculating the derivative of Eq. (14)a) at

_{S}*τ*= 0The sensitivity

*K*has a maximum value of 1.2π

_{S}^{1/2}

*ηAL*

^{2}

*I*

_{01}

*I*

_{02}at

*ξ*/

*T*= 2 as we can compute from the above expression. The effect of the parameter

_{P}*ξ*/

*T*on the sensitivity

_{P}*K*and the dynamic range Δ

_{S}*τ*is studied in the case of

*ξ*/

*T*≤ 4.

_{P}*K*and Δ

_{S}*τ*vary as a function of

*ξ*/

*T*. For a certain value of

_{P}*L*,

*Ks*increases with

*ξ*/

*T*in the range of

_{P}*ξ*/

*T*≤ 2 and decreases in the range of 2 ≤

_{P}*ξ*/

*T*≤ 4.

_{P}*K*s arrives at its maximum value at

*ξ*/

*T*= 2, which can be obtained from Eq. (15). To enhance the sensitivity, the value of

_{P}*ξ*/

*T*is expected to be close to 2. Furthermore, the sensitivity is proportional to the square of the crystal length

_{P}*L*with the value of

*ξ*/

*T*fixed. From this angle we hope the crystal is long and the value of

_{P}*α*-

_{s}*α*is small. For a given value of

_{f}*T*, the dynamic range Δ

_{P}*τ*has an increasing trend in the whole extent of

*ξ*/

*T*≤ 4. The maximum value of Δ

_{P}*τ*is 4

*T*at

_{P}*ξ*/

*T*= 4.

_{P}*ξ/T*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 (

_{P}*ξ/T*= 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

_{P}*ξ*/

*T*leads to the bigger value for the dynamic range. However, when

_{P}*ξ*/

*T*is fixed, the dynamic range is proportional to the pulse half-width

_{P}*T*. 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.

_{P}*T*

_{P}≥(

*α*-

_{s}*α*)

_{f}*L*/4, which means

*L*≤4

*T*

_{P}/(

*α*-

_{s}*α*). Further analysis for this measuring method shows that the sensitivity and the dynamic range for

_{f}*L*>4

*T*

_{P}/(

*α*-

_{s}*α*) are the same as

_{f}*L*= 4

*T*

_{P}/(

*α*-

_{s}*α*), therefore increasing the crystal length further will not improve the jitter measuring performance any more.

_{f}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]

**32**(9), 1044–1046 (2007). [CrossRef] [PubMed]

## 5. Conclusion

*ξ*/

*T*on the measuring sensitivity and dynamic range is analyzed. For a given crystal length

_{P}*L*, the sensitivity increases in the range of

*ξ*/

*T*≤ 2 and decreases in the range of 2 ≤

_{P}*ξ*/

*T*≤ 4 as

_{P}*ξ*/

*T*increases. The sensitivity has a maximum value of 1.2π

_{P}^{1/2}

*ηAL*

^{2}

*I*

_{01}

*I*

_{02}at

*ξ*/

*T*= 2. With

_{P}*ξ*/

*T*fixed, the sensitivity is proportional to

_{P}*L*

^{2}. For a given mode-locked laser with certain pulse width, the dynamic range has an increasing trend at

*ξ*/

*T*≤ 4 and has a maximum value of 4

_{P}*T*at

_{P}*ξ*/

*T*= 4. With

_{P}*ξ*/

*T*fixed, the dynamic range is proportional to

_{P}*T*. The proposed method has a higher sensitivity compared with the conventional RF method while the dynamic range is limited to the pulse width. The overlapping levels in the two crystals are the same and the final output is zero when there is no time delay between the transmitted pulse and the reference. There is no need to subtract an offset from the final output and no time delay adjustment is required between the two pulses before entering into the crystals compared with the previous optical method. The proposed method can be applied in timing distribution, frequency transfer, and other areas requiring reliable and precise jitter measurement.

_{P}## 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. |

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. |

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 |

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 |

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. |

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. |

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 |

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. |

9. | S. Qian and R. Zhu, |

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. |

**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

- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- S. Qian and R. Zhu, Nonlinear Optics (Fudan University Press, Shanghai, 2005), Chap. 3.
- 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.
- M. V. Hobden, “Phase-matched second-harmonic generation in biaxial crystals,” J. Appl. Phys. 38(11), 4365–4372 (1967). [CrossRef]

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