## 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

The reliable, accurate and precise measurement for timing jitter between the signal after transmitting through an optical link and the original reference is necessary in many areas such as frequency transfer and timing distribution [1–6]. In the application of frequency transfer and timing distribution through fiber link, a local reference is transmitted to the remote end by the fiber link. However, the performance of the transmitted frequency/timing signal at the remote end is degraded by the fiber-induced phase noise (timing jitter). In order to achieve a reliable and precise frequency signal at the remote end, an active compensation loop is used to cancel the phase noise. In this way, the timing jitter between the transmitted signal and the reference must be measured for the action of the loop.

In the case of the frequency comb as a reference [7], 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 [3–6]. 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]. 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, 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 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]. 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.

In the application, the reference pulse is generated by a mode-locked laser which is stabilized to a frequency reference, and the transmitted pulse is the signal after propagating through the fiber link. In practice, the timing jitter cannot be measured directly between the transmitted pulse at the remote end and the reference at the local, so a partial reflector is used at the remote end to return part of the transmitted signal to the local through a round trip of the fiber link and the jitter measurement can be performed between this round-trip signal and the reference. In this case, the reference’s coherence time must be longer than the round-trip time of the pulse propagation through the fiber link in order to achieve an accurate jitter measurement.

## 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]. 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]. 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 [9,11]. 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

**(**

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

*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.

Considering the case of no absorption in the crystal, if the phase is totally matched (Δ*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]*

_{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.

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

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

*t*′ =

*t*-

*z*/

*υ*

_{g}_{2}, Eq. (2)c) becomes

*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}.

By substituting the variation *t*′ for the original variation *t* at the crystal end *z = L*, we can express Eq. (5) as

*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 as

*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

*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

*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

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

and the intensities of the hyperbolic secant pulse can be expressed as where*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

_{P}*S*, can be written as

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

*α*-

_{s}*α*)

_{f}*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 *S _{diff}* 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
shows the final output signal *S _{diff}* varying with the delay time

*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.

In the linear region, the expression of the sensitivity *K _{S}* can be obtained approximately as following through calculating the derivative of Eq. (14)a) at

*τ*= 0

*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}Figure 3
shows *K _{S}* and Δ

*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}The parameter *ξ/T _{P}* 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/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}It is worth noticing that all the above curves and results are given in the case of *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}As an example, we choose BBO crystal for second harmonic generation at the fundamental wavelength of 800nm. In this case, *υ _{g}*

_{1}

*and*

_{s}*υ*

_{g}_{1}

*are 1.780 × 10*

_{f}^{8}m/s and 1.843 × 10

^{8}m/s respectively [10], from which we can obtain that the value of

*α*-

_{s}*α*is about 192fs/mm. With the pulse half-width

_{f}*T*of a mode-locked laser about 100fs, the crystal length

_{P}*L*needs to be about 2mm or longer in order to arrive at a maximal dynamic range of 384fs. The value of the sensitivity

*K*is not only related to the crystal and the two fundamental pulses, but also to the performance of photodetectors. In general, the sensitivity can achieve the level of ~10mV/fs.

_{S}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.

In the application of jitter measuring, the level of timing jitter can be characterized by the root-mean-square (rms) timing jitter, Δ*T*
_{rms}, which is specified over a frequency bandwidth range from *f*
_{1} (lower frequency of the bandwidth) to *f*
_{2} (upper frequency of the bandwidth). We define the power spectral density (PSD) of the timing jitter, *S _{T}*(

*f*), as the mean squared timing fluctuation at Fourier frequency

*f*in a 1Hz bandwidth, then the rms timing jitter Δ

*T*

_{rms}in the bandwidth (

*f*

_{1},

*f*

_{2}) can be expressed as

*S*(

_{T}*f*) has the unit of s

^{2}/Hz and the rms timing jitter Δ

*T*

_{rms}has the unit of s. The timing jitter PSD is related to the PSD of the phase fluctuation,

*S*(

_{φ}*f*), which represents the mean-squared phase fluctuation at Fourier frequency

*f*in a 1Hz bandwidth. The relation between the two

*S*(

_{T}*f*) and the

*S*(

_{φ}*f*) can be described bywhere

*f*is the pulse repetition frequency.

_{rep}The timing resolution of this rms timing jitter in the measuring process can reflect the measuring precision. The timing resolution of conventional RF method for timing jitter measurement is limited by the use of microwave components and there is drift timing error because of the thermal drifts of these microwave components. These limitations make it hard to measure the timing jitter better than the resolution of 100fs [8]. The proposed method uses optical crystals instead of microwave components such as mixers to avoid these limitations so as to obtain more precise jitter measurement of two pulses in the same wavelength, and this method is not affected by the temperature drifts. The timing resolution of the proposed method can achieve a level of better than fs. In addition, the RF method can be affected by amplitude noise of the pulse introduced by fiber link transmission, while the proposed method can be robust against the amplitude noise. However, we should note that the dynamic range is limited by the pulse width in this method. Furthermore, comparing this proposed method with the previous optical method in Ref. 8, 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 as we have mentioned in section 1.

## 5. Conclusion

In summary, a new measuring method for timing jitter of the transmitted pulse relative to the reference pulse has been proposed. The method is based on type II phase-matched SHG. The polarizations of the measuring pulses are exchanged in two nonlinear crystals and the final output can reflect the timing jitter between the two pulses. The non-stationary nonlinear wave-coupled equations have been used to obtain the theoretical result. The effect of the important parameter *ξ*/*T _{P}* on the measuring sensitivity and dynamic range is analyzed. For a given crystal length

*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

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