Soliton Squeezing in Optical Fibers - Semantic Scholar
BARKER MASSACHUSETTS OF TECHNOLOGY INSTITUTE
Soliton Squeezing in Optical Fibers APR 2 420 by
LIBRARIES
Charles Xiao Yu Submitted to the Department of Electrical Engineering and Computer Science in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Electrical Engineering at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY November, 2000 © Massachusetts Institute of Technology, 2000. All Rights Reserved.
A uthor .............................. ....................... Department of Electrical EngineeriAg and Computer Science November 14, 2000 Certified by .....
.................. Hermann A. Haus Institute Professor Thesi§ Supervisor
Accepted by ......... Arthur C. Smith Chairman, Departmental Committee on Graduate Students
2
Soliton Squeezing in Optical Fibers by Charles Xiao Yu Submitted to the Department of Electrical Engineering and Computer Science on November 14, 2000, in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Electrical Engineering
Abstract The use of squeezed light can overcome the standard quantum limit in phase sensitive optical measurements. This thesis is a theoretical and experimental investigation of soliton squeezing at 1.55gm. Theoretically, the effects of the continuum on squeezing have been investigated. Experimentally, modelocked fiber laser sources at 1.55gm have been developed for both schemes and their noise properties have been investigated. The timing jitter of these lasers are found to be quantum limited. Squeezing has been observed with two schemes, both using the optical fiber as the nonlinear medium. As much as 4.4dB of noise reduction has been detected. Thesis Supervisor: Hermann A. Haus Title: Institute Professor
Acknowledgments I want to thank my advisor, Prof. Haus. Prof. Haus matches my idea of a great scholar pretty well, except for the fact of not having white hair. I'm very lucky to have him. I want to thank my co-advisor, Prof. Ippen. Prof. Ippen first served as my undergraduate academic advisor, then as my thesis co-advisor. He always reminds me of something Sam Ulman said about youth, that youth is not a time of life, but a state of mind. "As long as your aerials are up, to catch the wave of optimism, there is hope you may die young at eighty." It's a pity that I won't have his "continued supervision" at Crawford Hill after graduation. I want to acknowledge Prof. Fujimoto. It's nice to see that an Asian American can rise to the top in a still very white world. It's also nice to know that all that work and tuition at MIT will pay off eventually/sometimes. I want to thank Prof. Shapiro. Prof. Shapiro is sharp. He read this thesis and gave lots of advice. I want to thank the people in and around our optics group: Cindy, Donna, Jalal, Krist, Farhan, Pat, Mike, Farzana, John, Jerry, Stu, Will, Shu, Moti, Dan, Juliet, Matt and lots of others whose names escaped me at this moment. It's been great. Finally I'd like to thank my parents, for always giving me lots of freedom and leading me to question the orthodox. I dedicate this thesis to them.
Table of Contents 1 Introduction................................................................................................................11 1.1
M otivations ..................................................................................................
11
1.2
H istorical Background on Soliton Squeezing ...............................................
12
1.3
Thesis Content ..............................................................................................
17
2 Soliton Squeezing in Optical Fibers, Theory ........................................................
19
2.1
Quantized Nonlinear Schr6dinger Equation and its Linearization ..............
20
2.2
Renorm alization of the Soliton Operators ...................................................
24
2.3
The Continuum ............................................................................................
29
2.4
Soliton Squeezing in a Fiber ........................................................................
31
2.5
Continuum Contribution to Squeezing ............................................................
38
2.6
Sum m ary .....................................................................................................
42
3 Squeezing with a short piece of fiber.................................................................... 3.1
3.2
Laser and its N oise......................................................................................
45 45
3.1.1
Introdution..........................................................................................
45
3.1.2
Theoretical Overview ..........................................................................
47
3.1.3
Experim ental Setup and Procedure ...................................................
52
3.1.4
Experim ental Results ..........................................................................
55
3.1.5
Com parison between Theory and Experim ent....................................
66
Cross Phase M odulation(XPM ) Squeezing .................................................
4 Squeezing with a Sagnac loop and l-GHz ps Pulses.............................................
68 75
4.1
GHz Laser Source ........................................................................................
75
4.2
N oise of the GHz laser .................................................................................
79
4.3
The Double Clad Fiber Am plifier...............................................................
86
4.4
The Balanced Detector.................................................................................
88
4.5
Sagnac Loop Squeezing...............................................................................
90
5 Conclusions and Future W ork ..............................................................................
97
5.1
Conclusions.................................................................................................
97
5.2
Future W ork .................................................................................................
98
5
Appendix A Continuum M atrix Elements ...................................................................
101
Appendix B Direct Measurement of Self-Phase Shift due to Fiber Nonlinearity........105 B ibliograp h y ...............................................................................................................
6
113
List of Figures
Figure 1.1: A sample GAWBS spectrum. Inset: Low frequency GAWBS....................14 Figure 1.2: Effect of loss is analogous to going through a beamsplitter.......................16 Figure 2.1: The deformation of uncertainty ellipse. ......................................................
32
Figure 2.2: The squeezing set-up...................................................................................
33
Figure 2.3: The squeezing and anti-squeezing (the minor and major axes of the squeezing ellipse) as functions of 2 ) ............................................................................................
37
Figure 2.4: The root mean square fluctuations as function of the phase angle with respect to local oscillator: 20 = 2, 4, 8 ....................................................................................
37
Figure 2.5: The minimum and maximum fluctuations of the soliton alone as detected by local oscillator of secant hyperbolic shape. Comparison with ideal local oscillator use......40 Figure 2.6: The fluctuations of the soliton alone as detected by local oscillator of secant hyperbolic shape as function of phase angle with respect to local oscillator, (D = 1......40 Figure 2.7: The matrix elements (a) 22 (cont,cont) and (b)
22(sol,cont) as a function of phase
D . N ote the "beats." ...........................................................................................................
43
Figure 2.8: The minimum fluctuations detected by a local oscillator which is orthogonal to continuum, and by a secant hyperbolic local oscillator with and without continuum.......44 Figure 3.1: Stretched pulse laser schem atic...................................................................
46
Figure 3.2: Experim ental setup .....................................................................................
53
Figure 3.3: A typical sampling scope trace(20ps/div)...................................................53 Figure 3.4: (a) Local oscillator of R3265 with RBW=lOHz. (b) Local oscillator of HP 8560E with RBW =1 Hz, the peak is ~15dB above the maximum on the graph. ......
54
Figure 3.5: Hybrid state (a) Optical Spectrum. (b) Harmonic 1. (c) Harmonic 21........57 Figure 3.6: Energy fluctuation at very low frequencies. Note that in this case the noise structure is buried under the local oscillator noise........................................................58 Figure 3.7: Broadband pulse energy noise: (a)without pedestal. (b) with pedestal. RBW =
7
K H z ..............................................................................................................................................
59
Figure 3.8: (a)The optical spectrum, and (b) the autocorrelation of the compressed pulse corresponding to the case of GVD=0.0169ps 2 and highest output power. The dashed line in (a) is a theoretical gaussian fit of the spectrum. ........................................................................................
62
Figure 3.9: Jitter fitting for harmonic 21, measurement time = 0.09 second. Dashed=data, Solid = fitted jitter. ...................................................................................................................................
62
Figure 3.10: (a) Timing jitter due to white noise, jitter=13ppm. (b) Timing jitter as a function of output power. The theoretical curve is the quantum jitter using the following parameters: t0 =5 Ifs, Gain bandwidth =40 nm, D=0.5 * 0.0169 ps 2 , a=0.2, g=0.8, TR=1 8 ns, O= 3.2. Measurement time = 0.0 9 secon d . .................................................................................................................................
63
Figure 3.11: Data taken using the HP 8560E, RBW= 1Hz, Measurement time= 1.92 seconds: (a)Harmonic 1, (b) Harmonic 21. Dashed curve is the experimental data. Solid curve is the sum of harmonic 1 and the theoretically predicted lorentzian-shaped jitter. The signals peaks are 30 dBm above the maximum shown in the graph. ......................................................................................
64
Figure 3.12: (a) The optical spectrum and (b) the autocorrelation of the compressed pulse corresponding to the case of GVD
. 4
(2.54)
1/2= 2 (
(2.55)
The process of squeezing can be treated in a formal way by establishing the correspondence of the solutions (2.51) and (2.52) with the Bogolyubov transformation. We have AA(t) = AAI (t) + i
2 (t) =
(t)AA (0) + v (t)AA (0)
(2.56)
with [t = 1 + i 20 (t) and v = i 20 (t)
input
(2.57)
3 dBY
A~
85/ 15 coupler
coupler
local oscillator
Vacuum
squeezed vacuum
fiber (nonlinear)
-----
pulse transformer
balanced
/ detector
output Figure 2.2: The squeezing set-up. The perturbation (2.56) accompanies the soliton pulse ao (t,x). We are now ready to analyze the generation of squeezed "soliton vacuum" by the set-up illustrated schematically in Fig. 2.2. The transform-limited secant hyperbolic pulse is incident upon one of the ports of the Sagnac loop acting as a nonlinear Mach-Zehnder interferometer. A Sagnac loop is chosen so that index fluctuations of the fiber, slow compared with the transit time, do not lead to imbalance of the interferometer. The single coupler doubles as both input- and output-coupler. The collision of the two pulses at the symmetry point has negligible effect since the Kerr effect is weak and nonlinear phase shifts are accumulated only over fibers
33
that are many meters long. The energy of the secant hyperbolic pulse is adjusted so that the two pulses in the two arms propagate as solitons. The zero-point fluctuations accompanying each of the two pulses are produced by the superposition of the vacuum incident from port (b) and the vacuum accompanying the pulse in input port (a). The fluctuations accompanying each pulse are incoherent with each other. (The situation is analogous to the splitting of thermal power by a beam splitter, each port of which has equal thermal excitations.) The fluctuations are squeezed as indicated by equation (2.56). Upon their return to the coupler, the classical (c-number), excitation exits into port (a) and the squeezed zero-point fluctuations superimpose incoherently into port (b). The classical excitation is used as the local oscillator in a balanced detector after an (optional) reshaping in the pulse transformer. The noise accompanying the local oscillator pulse cancels. We assume that the reshaping produces the local oscillator (L.O.) waveform: ifL (x) = 1/2 {cos
wf1
(x) + sin Ngf
2
(x)} e(i1)
(2.58)
which can be put into the ideal form for the purpose of projecting out a linear combination of AAi (t) and AA2 (t). In a real experiment the L.O. will of course produce gain G. For convenience (2.58) is normalized for G = 1. The squeezed vacuum fluctuations emerging from output port (b) are projected out in the balanced detector resulting in the net charge operator[45] A Q Iq= if dx {fL (x) Aa^0 - fL (x) Aai ts,, =
{cos4 AA1 (t) + sinN AA 2 (t)} = 1/2 {e& AA(t) + elN A At(t)}
=
1/2 {e"
[g AA(O) + v AAt(0)] + elf [p* AAt(O) + v* AA(0)]}
(2.59)
where q is the electrical charge. The mean square fluctuations of the detector current follow from (2.59). Equation (2.59) expresses the normalized difference charge in two ways: (i) as the projection of a vector with components AA, (t), AA 2 (t) onto an axis inclined at an angle V with respect to the (1) axis, and (ii) as the sum of the phase-shifted squeezed input excitations g AA(O) + v AAt (0). The two representations are equivalent, but in particular
34
applications one may be more convenient than the other. We shall determine the degree of squeezing and antisqueezing from representation (i). The mean square fluctuations of the charge are (JA
= Cos2
2
A 21 (t) 1> + sin 212
+ sin (2W) 1/2
If the projection ( A Q2 (t)I)
1/2
(2.60)
Iq is plotted in the (1)-(2) plane as a function of the
orientation angle V, an ellipse is traced out, the locus of the root mean square deviation of the Gaussian distribution of (IAQ 2(t)I) /q 2 . According to (2.52) the component in direction (1) remains unchanged, whereas the component in direction (2) shifts proportionally to []1/2 . The mean square fluctuations along the two axes are
(2.61)
, + 4(D2
The crosscorrelation is 1/2 = 2 D < AA
(2.62)
1>
The probability distribution of the normalized difference charge in the (1)-(2) plane with coordinates
, and 2 is a Gaussian given by
( 1,
2, t) oc exp -1/2 ( 421/0T
(t) + 4 2 2 1a 2 2 (t) + 24, 42/
where
35
12 (t)
)
(2.63)
a 1 1(t) ( 1 2 (t)
(2.64)
U21(t)(522(t)
(AA
2(t))
( AA I(t)AA 2 (t) + AA
I!
2
AA 22))
( A1 (t)AAl2 (t) + AAl1(t)A'2(t)J)
( AA
^2
(0))
2
(t)AA 2 (t)
I
222(t)
_20(t) il + 4(D (t)
and
= / = 0.607. The Fourier transform of the probability
distribution expressed in k-space, the characteristic function, is of the form C(k1 , k2, t) c-< exp - 1/2 (ayi (t) k21 + (22 (t) k2 2+ 2Y12 (t) k, k 2 ) The quadratic form in the exponent of the characteristic
(2.65)
function can be
diagonalized by a reorientation of the axes. A coordinate transformation into new orthogonal coordinates ki' and k2' finds the mutually orthogonal directions along which the fluctuations are uncorrelated. These are the major and minor axes of an ellipse. The transformation is a unitary transformation of the matrix which leaves the eigenvalues of the
matrix (2.64) invariant. The eigenvalues are 2i
± = { (1 +r + 4D 2 )/2±[(( 1 +,q + 42 )/2)
2
/ 1/ 2
}
(.6 (2.66)
The product of the eigenvalues is '2
+X-=i
2
(2.67)
and is constant, independent of the degree of squeezing. The squeezing and antisqueezing is illustrated in Fig. 2.3. With zero phase shift, the fluctuations in the (1) direction are shot noise fluctuations. These are equal to twice the zero point fluctuations of 1/4. In the orthogonal direction, the fluctuations are less, but they are still larger than 1/4. As the
36
nonlinear phase shift increases, the branch that represents shot noise at D = 0 shows monotonically increasing fluctuations, whereas the orthogonal direction decreases and reaches zero asymptotically. Figure 2.4 shows the fluctuations as a function of phase angle V for different degrees of squeezing and antisqueezing. This figure shows that the regime of phase angle within which a large degree of squeezing is observed becomes narrower and 15 10 5 dB 1
2
3
4
5
-5 -10
20D
-0
Figure 2.3: The squeezing and anti-squeezing (the minor and major axes of the squeezing ellipse) as functions of 20.
8 4 20 10 3g
2
dB 0.5
1
1.5
-10 -1
2
2.5
ow
Figure 2.4: The root mean square fluctuations as function of the phaseangle with respect to local oscillator: 2D = 2, 4, 8
37
narrower as the degree of squeezing is increased. The greater the degree of squeezing, the harder it is to find the squeezing angle and stabilize the system at that angle.
2.5 Continuum Contribution to Squeezing The projection with a local oscillator orthogonal to the continuum permitted us to evaluate the perturbation of the soliton without consideration of the continuum. In practice it is inconvenient to reshape the pulse for use in the local oscillator. Any reshaping introduces phase delays that may undergo fluctuations and prevent optimization of the relative phase between LO and the squeezed vacuum. Use of the secant hyperbolic pulse as the local oscillator pulse leads to detection of the continuum contribution. We shall now show that the penalty incurred when a secant hyperbolic LO pulse is used is not severe and, in fact, at some phase shifts the noise is less than for the case of a local oscillator pulse orthogonal to the continuum. The first order perturbation of the pulse is composed of the soliton perturbation and the continuum: Ad = Adsol + Adcont
(2.68)
When the local oscillator pulse is a secant hyperbolic, ifL(x,t) = (1/2) sech (x/4) e(it/2) e(iV) = (1/2)f 1 (x,t) e(1V)
(2.69)
The function f1 (x,t) is the adjoint projection that evaluates AA 1 (t) from the perturbation Aa(x,t). In addition, the local oscillator waveform contains the phase factor exp(iV). We first look at the contribution to the difference current of the soliton part of the squeezed radiation. Equation (2.65) must now be modified to take into account that the quadrature component is projected out with this particular function, namely sin x f 1(x,t), rather than sin AVf
2
(x,t). We find that (2.59) changes to:
AQS0 /q= i f dx{ f*L (x,t) A ^ 0i -fL (x,t) A solI} ={cosipAA 1(t)+2 sinNfAA 2 (t)} The mean square fluctuations due to the soliton part are
38
(2.70)
(t)) soJ/q 2== = D..(T- T') 1 J ii
with "diffusion constants" D
D
= (1 + 0 2 )213
2 T wt T R
(frequency-timing
gh-o
2w
D tt =
PP
2 5/2 = (1+3 2)5
(3.8)
(frequency deviation),
cross-correlation),
and
TR
2 2 3/2to 2g Th 0 (1+1) W
(timing deviation), where 0 is the enhancement factor
TR
due to incomplete inversion of the medium, and wo is the steady state pulse energy. Timing jitter induced by spontaneous emission can be calculated by solving (3.4) using (3.8) as the noise source. The r.m.s value of the quantum limited timing jitter can be obtained by Fourier transforming Equation (3.7). In the limit
T >> rP and large net GVD it has the
form of a random walk as mentioned before and is approximately
= BT
(3.9)
where T is the measurement time and )2
16
g
8L1
+D
g TR 2
2
+D
8g 2
Q 9
2
-
D t +D pt p tt
R TR
When the pulse train impinges on a detector, the power spectrum of the detector cur-
50
rent is [85]:
= A+
Soliton Squeezing in Optical Fibers APR 2 420 by
LIBRARIES
Charles Xiao Yu Submitted to the Department of Electrical Engineering and Computer Science in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Electrical Engineering at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY November, 2000 © Massachusetts Institute of Technology, 2000. All Rights Reserved.
A uthor .............................. ....................... Department of Electrical EngineeriAg and Computer Science November 14, 2000 Certified by .....
.................. Hermann A. Haus Institute Professor Thesi§ Supervisor
Accepted by ......... Arthur C. Smith Chairman, Departmental Committee on Graduate Students
2
Soliton Squeezing in Optical Fibers by Charles Xiao Yu Submitted to the Department of Electrical Engineering and Computer Science on November 14, 2000, in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Electrical Engineering
Abstract The use of squeezed light can overcome the standard quantum limit in phase sensitive optical measurements. This thesis is a theoretical and experimental investigation of soliton squeezing at 1.55gm. Theoretically, the effects of the continuum on squeezing have been investigated. Experimentally, modelocked fiber laser sources at 1.55gm have been developed for both schemes and their noise properties have been investigated. The timing jitter of these lasers are found to be quantum limited. Squeezing has been observed with two schemes, both using the optical fiber as the nonlinear medium. As much as 4.4dB of noise reduction has been detected. Thesis Supervisor: Hermann A. Haus Title: Institute Professor
Acknowledgments I want to thank my advisor, Prof. Haus. Prof. Haus matches my idea of a great scholar pretty well, except for the fact of not having white hair. I'm very lucky to have him. I want to thank my co-advisor, Prof. Ippen. Prof. Ippen first served as my undergraduate academic advisor, then as my thesis co-advisor. He always reminds me of something Sam Ulman said about youth, that youth is not a time of life, but a state of mind. "As long as your aerials are up, to catch the wave of optimism, there is hope you may die young at eighty." It's a pity that I won't have his "continued supervision" at Crawford Hill after graduation. I want to acknowledge Prof. Fujimoto. It's nice to see that an Asian American can rise to the top in a still very white world. It's also nice to know that all that work and tuition at MIT will pay off eventually/sometimes. I want to thank Prof. Shapiro. Prof. Shapiro is sharp. He read this thesis and gave lots of advice. I want to thank the people in and around our optics group: Cindy, Donna, Jalal, Krist, Farhan, Pat, Mike, Farzana, John, Jerry, Stu, Will, Shu, Moti, Dan, Juliet, Matt and lots of others whose names escaped me at this moment. It's been great. Finally I'd like to thank my parents, for always giving me lots of freedom and leading me to question the orthodox. I dedicate this thesis to them.
Table of Contents 1 Introduction................................................................................................................11 1.1
M otivations ..................................................................................................
11
1.2
H istorical Background on Soliton Squeezing ...............................................
12
1.3
Thesis Content ..............................................................................................
17
2 Soliton Squeezing in Optical Fibers, Theory ........................................................
19
2.1
Quantized Nonlinear Schr6dinger Equation and its Linearization ..............
20
2.2
Renorm alization of the Soliton Operators ...................................................
24
2.3
The Continuum ............................................................................................
29
2.4
Soliton Squeezing in a Fiber ........................................................................
31
2.5
Continuum Contribution to Squeezing ............................................................
38
2.6
Sum m ary .....................................................................................................
42
3 Squeezing with a short piece of fiber.................................................................... 3.1
3.2
Laser and its N oise......................................................................................
45 45
3.1.1
Introdution..........................................................................................
45
3.1.2
Theoretical Overview ..........................................................................
47
3.1.3
Experim ental Setup and Procedure ...................................................
52
3.1.4
Experim ental Results ..........................................................................
55
3.1.5
Com parison between Theory and Experim ent....................................
66
Cross Phase M odulation(XPM ) Squeezing .................................................
4 Squeezing with a Sagnac loop and l-GHz ps Pulses.............................................
68 75
4.1
GHz Laser Source ........................................................................................
75
4.2
N oise of the GHz laser .................................................................................
79
4.3
The Double Clad Fiber Am plifier...............................................................
86
4.4
The Balanced Detector.................................................................................
88
4.5
Sagnac Loop Squeezing...............................................................................
90
5 Conclusions and Future W ork ..............................................................................
97
5.1
Conclusions.................................................................................................
97
5.2
Future W ork .................................................................................................
98
5
Appendix A Continuum M atrix Elements ...................................................................
101
Appendix B Direct Measurement of Self-Phase Shift due to Fiber Nonlinearity........105 B ibliograp h y ...............................................................................................................
6
113
List of Figures
Figure 1.1: A sample GAWBS spectrum. Inset: Low frequency GAWBS....................14 Figure 1.2: Effect of loss is analogous to going through a beamsplitter.......................16 Figure 2.1: The deformation of uncertainty ellipse. ......................................................
32
Figure 2.2: The squeezing set-up...................................................................................
33
Figure 2.3: The squeezing and anti-squeezing (the minor and major axes of the squeezing ellipse) as functions of 2 ) ............................................................................................
37
Figure 2.4: The root mean square fluctuations as function of the phase angle with respect to local oscillator: 20 = 2, 4, 8 ....................................................................................
37
Figure 2.5: The minimum and maximum fluctuations of the soliton alone as detected by local oscillator of secant hyperbolic shape. Comparison with ideal local oscillator use......40 Figure 2.6: The fluctuations of the soliton alone as detected by local oscillator of secant hyperbolic shape as function of phase angle with respect to local oscillator, (D = 1......40 Figure 2.7: The matrix elements (a) 22 (cont,cont) and (b)
22(sol,cont) as a function of phase
D . N ote the "beats." ...........................................................................................................
43
Figure 2.8: The minimum fluctuations detected by a local oscillator which is orthogonal to continuum, and by a secant hyperbolic local oscillator with and without continuum.......44 Figure 3.1: Stretched pulse laser schem atic...................................................................
46
Figure 3.2: Experim ental setup .....................................................................................
53
Figure 3.3: A typical sampling scope trace(20ps/div)...................................................53 Figure 3.4: (a) Local oscillator of R3265 with RBW=lOHz. (b) Local oscillator of HP 8560E with RBW =1 Hz, the peak is ~15dB above the maximum on the graph. ......
54
Figure 3.5: Hybrid state (a) Optical Spectrum. (b) Harmonic 1. (c) Harmonic 21........57 Figure 3.6: Energy fluctuation at very low frequencies. Note that in this case the noise structure is buried under the local oscillator noise........................................................58 Figure 3.7: Broadband pulse energy noise: (a)without pedestal. (b) with pedestal. RBW =
7
K H z ..............................................................................................................................................
59
Figure 3.8: (a)The optical spectrum, and (b) the autocorrelation of the compressed pulse corresponding to the case of GVD=0.0169ps 2 and highest output power. The dashed line in (a) is a theoretical gaussian fit of the spectrum. ........................................................................................
62
Figure 3.9: Jitter fitting for harmonic 21, measurement time = 0.09 second. Dashed=data, Solid = fitted jitter. ...................................................................................................................................
62
Figure 3.10: (a) Timing jitter due to white noise, jitter=13ppm. (b) Timing jitter as a function of output power. The theoretical curve is the quantum jitter using the following parameters: t0 =5 Ifs, Gain bandwidth =40 nm, D=0.5 * 0.0169 ps 2 , a=0.2, g=0.8, TR=1 8 ns, O= 3.2. Measurement time = 0.0 9 secon d . .................................................................................................................................
63
Figure 3.11: Data taken using the HP 8560E, RBW= 1Hz, Measurement time= 1.92 seconds: (a)Harmonic 1, (b) Harmonic 21. Dashed curve is the experimental data. Solid curve is the sum of harmonic 1 and the theoretically predicted lorentzian-shaped jitter. The signals peaks are 30 dBm above the maximum shown in the graph. ......................................................................................
64
Figure 3.12: (a) The optical spectrum and (b) the autocorrelation of the compressed pulse corresponding to the case of GVD
. 4
(2.54)
1/2= 2 (
(2.55)
The process of squeezing can be treated in a formal way by establishing the correspondence of the solutions (2.51) and (2.52) with the Bogolyubov transformation. We have AA(t) = AAI (t) + i
2 (t) =
(t)AA (0) + v (t)AA (0)
(2.56)
with [t = 1 + i 20 (t) and v = i 20 (t)
input
(2.57)
3 dBY
A~
85/ 15 coupler
coupler
local oscillator
Vacuum
squeezed vacuum
fiber (nonlinear)
-----
pulse transformer
balanced
/ detector
output Figure 2.2: The squeezing set-up. The perturbation (2.56) accompanies the soliton pulse ao (t,x). We are now ready to analyze the generation of squeezed "soliton vacuum" by the set-up illustrated schematically in Fig. 2.2. The transform-limited secant hyperbolic pulse is incident upon one of the ports of the Sagnac loop acting as a nonlinear Mach-Zehnder interferometer. A Sagnac loop is chosen so that index fluctuations of the fiber, slow compared with the transit time, do not lead to imbalance of the interferometer. The single coupler doubles as both input- and output-coupler. The collision of the two pulses at the symmetry point has negligible effect since the Kerr effect is weak and nonlinear phase shifts are accumulated only over fibers
33
that are many meters long. The energy of the secant hyperbolic pulse is adjusted so that the two pulses in the two arms propagate as solitons. The zero-point fluctuations accompanying each of the two pulses are produced by the superposition of the vacuum incident from port (b) and the vacuum accompanying the pulse in input port (a). The fluctuations accompanying each pulse are incoherent with each other. (The situation is analogous to the splitting of thermal power by a beam splitter, each port of which has equal thermal excitations.) The fluctuations are squeezed as indicated by equation (2.56). Upon their return to the coupler, the classical (c-number), excitation exits into port (a) and the squeezed zero-point fluctuations superimpose incoherently into port (b). The classical excitation is used as the local oscillator in a balanced detector after an (optional) reshaping in the pulse transformer. The noise accompanying the local oscillator pulse cancels. We assume that the reshaping produces the local oscillator (L.O.) waveform: ifL (x) = 1/2 {cos
wf1
(x) + sin Ngf
2
(x)} e(i1)
(2.58)
which can be put into the ideal form for the purpose of projecting out a linear combination of AAi (t) and AA2 (t). In a real experiment the L.O. will of course produce gain G. For convenience (2.58) is normalized for G = 1. The squeezed vacuum fluctuations emerging from output port (b) are projected out in the balanced detector resulting in the net charge operator[45] A Q Iq= if dx {fL (x) Aa^0 - fL (x) Aai ts,, =
{cos4 AA1 (t) + sinN AA 2 (t)} = 1/2 {e& AA(t) + elN A At(t)}
=
1/2 {e"
[g AA(O) + v AAt(0)] + elf [p* AAt(O) + v* AA(0)]}
(2.59)
where q is the electrical charge. The mean square fluctuations of the detector current follow from (2.59). Equation (2.59) expresses the normalized difference charge in two ways: (i) as the projection of a vector with components AA, (t), AA 2 (t) onto an axis inclined at an angle V with respect to the (1) axis, and (ii) as the sum of the phase-shifted squeezed input excitations g AA(O) + v AAt (0). The two representations are equivalent, but in particular
34
applications one may be more convenient than the other. We shall determine the degree of squeezing and antisqueezing from representation (i). The mean square fluctuations of the charge are (JA
= Cos2
2
A 21 (t) 1> + sin 212
+ sin (2W) 1/2
If the projection ( A Q2 (t)I)
1/2
(2.60)
Iq is plotted in the (1)-(2) plane as a function of the
orientation angle V, an ellipse is traced out, the locus of the root mean square deviation of the Gaussian distribution of (IAQ 2(t)I) /q 2 . According to (2.52) the component in direction (1) remains unchanged, whereas the component in direction (2) shifts proportionally to []1/2 . The mean square fluctuations along the two axes are
(2.61)
, + 4(D2
The crosscorrelation is 1/2 = 2 D < AA
(2.62)
1>
The probability distribution of the normalized difference charge in the (1)-(2) plane with coordinates
, and 2 is a Gaussian given by
( 1,
2, t) oc exp -1/2 ( 421/0T
(t) + 4 2 2 1a 2 2 (t) + 24, 42/
where
35
12 (t)
)
(2.63)
a 1 1(t) ( 1 2 (t)
(2.64)
U21(t)(522(t)
(AA
2(t))
( AA I(t)AA 2 (t) + AA
I!
2
AA 22))
( A1 (t)AAl2 (t) + AAl1(t)A'2(t)J)
( AA
^2
(0))
2
(t)AA 2 (t)
I
222(t)
_20(t) il + 4(D (t)
and
= / = 0.607. The Fourier transform of the probability
distribution expressed in k-space, the characteristic function, is of the form C(k1 , k2, t) c-< exp - 1/2 (ayi (t) k21 + (22 (t) k2 2+ 2Y12 (t) k, k 2 ) The quadratic form in the exponent of the characteristic
(2.65)
function can be
diagonalized by a reorientation of the axes. A coordinate transformation into new orthogonal coordinates ki' and k2' finds the mutually orthogonal directions along which the fluctuations are uncorrelated. These are the major and minor axes of an ellipse. The transformation is a unitary transformation of the matrix which leaves the eigenvalues of the
matrix (2.64) invariant. The eigenvalues are 2i
± = { (1 +r + 4D 2 )/2±[(( 1 +,q + 42 )/2)
2
/ 1/ 2
}
(.6 (2.66)
The product of the eigenvalues is '2
+X-=i
2
(2.67)
and is constant, independent of the degree of squeezing. The squeezing and antisqueezing is illustrated in Fig. 2.3. With zero phase shift, the fluctuations in the (1) direction are shot noise fluctuations. These are equal to twice the zero point fluctuations of 1/4. In the orthogonal direction, the fluctuations are less, but they are still larger than 1/4. As the
36
nonlinear phase shift increases, the branch that represents shot noise at D = 0 shows monotonically increasing fluctuations, whereas the orthogonal direction decreases and reaches zero asymptotically. Figure 2.4 shows the fluctuations as a function of phase angle V for different degrees of squeezing and antisqueezing. This figure shows that the regime of phase angle within which a large degree of squeezing is observed becomes narrower and 15 10 5 dB 1
2
3
4
5
-5 -10
20D
-0
Figure 2.3: The squeezing and anti-squeezing (the minor and major axes of the squeezing ellipse) as functions of 20.
8 4 20 10 3g
2
dB 0.5
1
1.5
-10 -1
2
2.5
ow
Figure 2.4: The root mean square fluctuations as function of the phaseangle with respect to local oscillator: 2D = 2, 4, 8
37
narrower as the degree of squeezing is increased. The greater the degree of squeezing, the harder it is to find the squeezing angle and stabilize the system at that angle.
2.5 Continuum Contribution to Squeezing The projection with a local oscillator orthogonal to the continuum permitted us to evaluate the perturbation of the soliton without consideration of the continuum. In practice it is inconvenient to reshape the pulse for use in the local oscillator. Any reshaping introduces phase delays that may undergo fluctuations and prevent optimization of the relative phase between LO and the squeezed vacuum. Use of the secant hyperbolic pulse as the local oscillator pulse leads to detection of the continuum contribution. We shall now show that the penalty incurred when a secant hyperbolic LO pulse is used is not severe and, in fact, at some phase shifts the noise is less than for the case of a local oscillator pulse orthogonal to the continuum. The first order perturbation of the pulse is composed of the soliton perturbation and the continuum: Ad = Adsol + Adcont
(2.68)
When the local oscillator pulse is a secant hyperbolic, ifL(x,t) = (1/2) sech (x/4) e(it/2) e(iV) = (1/2)f 1 (x,t) e(1V)
(2.69)
The function f1 (x,t) is the adjoint projection that evaluates AA 1 (t) from the perturbation Aa(x,t). In addition, the local oscillator waveform contains the phase factor exp(iV). We first look at the contribution to the difference current of the soliton part of the squeezed radiation. Equation (2.65) must now be modified to take into account that the quadrature component is projected out with this particular function, namely sin x f 1(x,t), rather than sin AVf
2
(x,t). We find that (2.59) changes to:
AQS0 /q= i f dx{ f*L (x,t) A ^ 0i -fL (x,t) A solI} ={cosipAA 1(t)+2 sinNfAA 2 (t)} The mean square fluctuations due to the soliton part are
38
(2.70)
(t)) soJ/q 2== = D..(T- T') 1 J ii
with "diffusion constants" D
D
= (1 + 0 2 )213
2 T wt T R
(frequency-timing
gh-o
2w
D tt =
PP
2 5/2 = (1+3 2)5
(3.8)
(frequency deviation),
cross-correlation),
and
TR
2 2 3/2to 2g Th 0 (1+1) W
(timing deviation), where 0 is the enhancement factor
TR
due to incomplete inversion of the medium, and wo is the steady state pulse energy. Timing jitter induced by spontaneous emission can be calculated by solving (3.4) using (3.8) as the noise source. The r.m.s value of the quantum limited timing jitter can be obtained by Fourier transforming Equation (3.7). In the limit
T >> rP and large net GVD it has the
form of a random walk as mentioned before and is approximately
= BT
(3.9)
where T is the measurement time and )2
16
g
8L1
+D
g TR 2
2
+D
8g 2
Q 9
2
-
D t +D pt p tt
R TR
When the pulse train impinges on a detector, the power spectrum of the detector cur-
50
rent is [85]:
= A+
