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Evaluation of noise increase due to pump light crosstalk in quadrature squeezed light generation with a fibre ring reflector
Evalutition of noise increase to pump light crosstalk in quadrature squeezed light generation with a fibre ring reflector N. NISHIZAWA,
due
S. KUME, M. MORI, T. GOTO, A. MIYAUCHI
In generating quadrature squeezed light with a fibre ring reflector, the pump light crosstalk should be minimized to achieve the sub-shot noise levels. The output noise of the homodyne detector, whose signal light is the squeezed light with non-zero amplitude, is analysed theoretically and experimentally for the first time. The requirement for the fibre coupler to generate the squeezed light is derived. KEYWORDS:
squeezed
light, homodyne
detectors,
Introduction squeezed light has been generated with a few non-linear effects-four-wave mixing, degenerate parametric amplification, and the optical Kerr effect’-‘. The squeezing by four-wave mixing and the degenerate parametric amplification were realized in the early day.sm3. But these methods generally depend on phase matching, and the adjustment of the cavity is very difficult. Quadrature
A fibre ring reflector, where the fibre acts as the optical Kerr medium, was proposed by Shirasaki and Haus for the generation of the squeezed vacuum6. This reflector does not depend on the phase matching. So, using this reflector and the pulsed laser, the quadrature squeezed light can be easily generated4,s. The generation model of the squeezed vacuum is shown in Fig. 1. In this method, it is necessary to tune the coupling ratio of the fibre coupler at 1: 1 to get the squeezed vacuum. If the coupling ratio is slightly different from 1: 1, the pump light leaks into the squeezed vacuum. Consequently, the output of the fibre is
0030-3992/94/010049-05
fibre ring reflectors
generally the squeezed amplitude.
light with a non-zero
average
The balanced homodyne detector is used to detect the quadrature squeezed state’. With this method, we can effectively get the variance of the signal light when the intensity of the local oscillator is much higher than that of the signal light. However, for the condition where the signal light has a non-zero average amplitude, it remains an unsettled question how the detector output is given. Squeezed light generation with a fibre ring reflector has been theoretically analysed by Blow et al.‘, and squeezing with unbalanced couplers has also been studied. But no mention has been made of the detection of the squeezed output. In this paper, the detector outputs, whose signal light has a non-zero average amplitude, are analysed both theoretically and experimentally. From the theoretical results, the requirement for the fibre coupler to generate the squeezed light is derived.
NN, SK, MM and TG are at the School of Engineering, Nagoya University, Nagoya-shi, 464-01 Japan. AM is in the Transmission Division, Fujitsu Ltd, Kawasaki-shi, 211 Japan. Received 17 June 1993. Revised 6 September 1993.
Optics & Laser Technology Vol 26 No 1 1994
fibre couplers,
@ 1994
The second section presents the theoretical for two different cases; incoherent crosstalk coherent crosstalk. The requirement for the coupler is also discussed. The third section the experimental method and its results. Butterworth-Heinemann
analysis and fibre describes
Ltd
49
Crosstalk in quadrature squeezed light generation: N. Nishizawa et al.
This squeezed light is shown in Fig. 2. These two light sources input into the balanced homodyne detector. The output variance of the detector is written as
PUMP
(Afi ‘) = ala :[cos{2(8 - 4,)}sinh(2s) +cosh(2s)] + sinh* s + fl/?*
VACUUM Fig. 1 Squeezed light generation in a fibre ring reflector. The pump light and the vacuum are superimposed on the fibre coupler. When the branching ratio of the fibre coupler is 1 : 1, the output from port B is the squeezed vacuum and the pump light returns to port A
Theoretical Incoherent
analysis crosstalk
For the squeezed light generation using degenerate parametric amplification, when the input light is in the coherent state, the output is generally the squeezed light with a non-zero average amplitude. In this case, the variance of the noise of the output light is coherent, but its average amplitude is incoherent to the pump light (local oscillator). In this section, we consider the homodyne detector output noise when the signal light is the squeezed light with a non-zero average amplitude incoherent with the local oscillator. We use the two operators 8, and 8, for the signal light and the local oscillator, respectively. The local oscillator is in the coherent state. The signal light is assumed to be in the squeezed state, and is written as follows9
(2)
where a, is the eigenvalue of 6, and 19is the relative phase between the signal light and the local oscillator. The first and the second terms in the right-hand side qf (2) represent the noise of the squeezed vacuum S([)]O). The third term /I/?* represents the noise increase from the squeezed vacuum due to the displacement D (/? ). Coherent
crosstalk
For the squeezed light generation with a fibre ring reflector, the average amplitude of the squeezed light is attributed to the pump light crosstalk. After the generation process, the pump light is reused as the local oscillator. Therefore, the average amplitude of the squeezed light is coherent with the local oscillator. In this section, we consider the homodyne detector output noise when the squeezed light has a non-zero average amplitude coherent with the local oscillator. We simulate the generation process of the squeezed light with a non-zero average amplitude in a model where the squeezed vacuum is mixed with the coherent light. The calculation model is shown in Fig. 3. Here 6, and 4, are the annihilation operators Coherent
Crosstalk
Model
=mmw
IAT> 6 (B >= ewW4t - P*4)
S(i)=exp(~i*(ci,)*-_tr(ci,t)*) [ = s exp(2$,)
I\
^
rla,+tla,
^
(1)
Balanced
Homodyne
Detector
Fig. 2 Squeezed light with a non-zero amplitude. fi is the average amplitude; I#J,is the angle between the ellipse minor axis and the X axis; &, is the phase of the average amplitude
50
Fig. 3 Generation model of the squeezed light with a non-zero average amplitude coherent with the local oscillator. Branching ratios of BSl and BS2 are t,:r, and t,:r,, respectively. It is assumed that It, I- 1 and 11,I - 0. a^,and I, are the annihilation operators of the coherent light and the squeezed vacuum, respectively
Optics & Laser Technology Vol 26 No 1 1994
Crosstalk in quadrature
squeezed
light generation:
N. Nishizawa
et al.
of the coherent state and the squeezed vacuum, respectively. When the branching ratio of the beam-splitter BSI is t,:r,, the two outputs are written as 1 rl aI + tl a, 1 t,a,+
(3)
^
(4)
II 4
In this model, we assumed that It, I - 1 and Ir, I - 0. In this condition, (3) gives a squeezed state with a small average amplitude. After that, the two lights are input to the balanced homodyne detector where the signal light and the local oscillator are given by (3) and (4), respectively. A phase shift arising from the optical path difference should also be included. The signal light and the local oscillator are superimposed at the second beam-splitter BS2 (t,:r,), and divided into two lights given by ri, = t,(r,ci, + tIci,) + r,(t,ci, + r16,)eie
(5)
ti2 = r,(rlbl + tIci,) + f2(tldl + r,B,)e”
(6)
After subtraction, written as
the output
of the detector
Peak Light Power [W] Fig. 4 Calculated result for the squeezed light generation. The pump source is 1.064pm YAG and the fibre is 50 m long 0.85 pm SM fibre. The abscissa represents the peak power of the pulse pump laser. The ordinate represents the squeezing parameter and the phase difference &-dS
is The second term is the noise increase excess noise of the pump source.
A =fi, -fi, = 4p, p2 cos 8 (cilTd, - 6, Tci,)
due to the
for simplicity. Since BS2 is used for the balanced homodyne detection, p2 = f l/2.
This calculation shows that if the squeezed light has an average amplitude coherent with the local oscillator, a part of the excess noise of the pump source is added to the detector output. The reason for this noise increase is the unbalance between the two output powers of BS2. When the signal light has a non-zero average amplitude coherent with the local oscillator, this amplitude component interferes with the local oscillator. As a result, the two light powers of BS2 become unbalanced, and a part of the excess noise appears even after subtraction.
When pl = 0, the signal light is the squeezed Then the variance of the detector output is
Requirement
- 2ip,(e” - 2r, r f cos B)B,tri, + 2ip,(e-”
- 2r,r : cos O)ti,&,t
where we use real numbers tl r : z ip,,
t,rf
(7)
pl and p2 given by
= ip,
(8)
vacuum.
(Ari *>,, = cllcl 7 [cos{2(B + 4B - 4, + n/2)}sinh(2s) +cosh(2s)]
+ sinh’s
(9)
The phase shift of 71/2 is included in (9), because the phase of the reflected coherent light is shifted by n/2 (see Fig. 3). When pl # 0, the signal light is the squeezed light with a non-zero average amplitude, and the output variance is given as follows
for fibre
coupler
In generating the squeezed light with the fibre ring reflector, the noise increase is mainly caused by the pump light crosstalk which is coherent to the local oscillator. In this section, the requirement for the fibre coupler to generate the squeezed light is discussed based on the calculated results of the previous section. Worst case At first, we assume that the phase difference 8 between the signal light and the local oscillator is equal to 0. In this condition, (11) is written as
(AA *> = (Ari 2)0 + 4p : cos* 8 {(Aii :) + (Ari t)} - 4~ : ~0s 8 (2(ri, )(ri,) +4p: +
(Ali2)-(AA2)0+4p:(Aft:)
+ (ril ) + (fi, ))
cos tkiB((ril>2>((ii,~)2)
4p : cos Oe-ie((cil t)*)((f?,)*)
(10)
When pl is very small and (Ari :) is large, this result is approximated by
(11)
(12)
and the additive noise takes the maximum value. When the second term in (12) is larger than the first, the noise reduction below the shot noise level cannot be observed. So, in order to observe the squeezed light, the condition 4p:(Ati:)
< (Afi*),zSNL
(13)
51
Crosstalk in quadrature squeezed light generation: N. Nishizawa et al.
must be satisfied, where SNL means the shot noise level. This equation shows that the requirement for the crosstalk in the fibre coupler depends on the excess noise of the pump laser. Now, consider an example. We assume that the excess noise of the local oscillator is 20 dB of the shot noise. In this condition, (13) gives p: < 2.5 x 10p3. This means that, in order to observe the squeezed light, the pump light crosstalk in the fibre coupler must be suppressed below - 26 dB. Actual case For the squeezed light generated in the fibre ring reflector, the phase difference & - c$, (see Fig. 2) depends on the squeezing parameter. We have made a brief calculation, and obtained the squeezing parameter and the angle 4B - 4, as functions of the peak pump light power. The calculated results are shown in Fig. 4. When the squeezing parameter is smaller than 1 dB, C&- c$~is about 45”. Then 4B - 4, gradually decreases with the peak pump light power and approaches 0”. In (1 l), the first term in the right-hand side is the variance of the squeezed vacuum. This term takes the minimum value when f3 = 4, - 4B. From Fig. 4, we can see that in the region of the high peak pump light power, the first term in (11) is minimized when 0 N 0. The second term in (11) is the additive noise due to the pump light crosstalk. This term has a maximum when 8 = 0. Therefore, the dependency of the first term of 8 is different from that of the second term in (11). Here, we evaluate the additive noise when the local oscillator phase is adjusted at the minimum point of the variance of the squeezed vacuum. When the squeezing parameter is less than 1 dB, the additive noise is about half of the worst case value in (12) because the local oscillator phase 8 is about -45”. When the squeezing parameter is above 1 dB, however, the additive noise gradually approaches the worst case value because 8 approaches 0”.
ANALYZER
Fig. 5
Experimental configuration
squeezed with the optical Kerr effect along the 50 m optical fibre, and it is not suitable for observing the additive noise. So only the beam component whose polarization is orthogonal to the main component was picked off with a half-wave plate and a polarizer, then used as the signal light. The beam from port A is partially picked off by the beam-splitter BS3 and used as the local oscillator of the homodyne detection. In the BS2, special care is taken to ensure an accurate superposition of the two light beams. For the optical detectors, we use high speed InGaAs-pin photodiodes. Two signal currents are directly subtracted from each other and amplified. The amplifier consists of low-noise OP-amps. The output of the detector is observed with a spectrum analyser. Moving the prism slowly with a micropositioner, the relative phase between the two lights can be changed. The detector output noise at 150 kHz is shown in Fig. 6 as a function of time. The video bandwidth is 10 Hz, the resolution bandwidth is 3 kHz, and the sweep time is 10 s. The upper trace shows the excess noise of the local oscillator. This is obtained when the port B and one of the optical detectors is blocked. The lowest trace shows the shot noise level. This is obtained with the port B blocked. This shot noise level is in good agreement with the theoretical values.
-50 -
From this discussion, we see that the requirement for the fibre coupler should be estimated in the worst case condition given by (12). .
Experiment .
,,.
In order to certify the theoretical results of the previous section, we have performed an experiment to measure the additive noise. The experimental configuration is shown in Fig. 5. We used a mode-locked YAG laser as the pump source. This laser delivers about 100 ps pulses at 1.064 pm at a repetition rate of 82 MHz. A 0.85 pm SM PANDA fibre of 50 m is connected with a polarization maintaining tunable fibre coupler to form a fibre ring reflector. The pump light is coupled to the fibre from port A, then divided into two beams by the fibre coupler BS 1. The branching ratio of the fibre coupler is slightly shifted from 1: 1. In this condition, the output power from port B is of the same order as that from port A. The main polarization beam is
52
I ..;.,
,. --.._. .”
,I
-“‘,...
. ,_.
.
; ;
I
~~,‘l./,
,’
1,::..‘“.;,‘,
I
.,
.. ..- ;, :,_
.‘.
j .,
;.
-1
‘<
:
.,
I...
:
I
0
Fig. 6
I
I
I
I
I
I
5 Time [set]
I
I
II
1
10
Detector output noise at 150 kHz as a function of time
Optics & Laser Technology Vol 26 No 1 1994
Crosstalk in quadrature squeezed light generation: N. Nishizawa et al. The middle trace shows the additive noise measured with the two ports open. This trace corresponds to the worst case value in (11). The relative phase is kept constant, so that the additive noise is at its highest level. The parameter p : in (12) is estimated as p : = 1/15 from the average power of the local oscillator and the signal light. From Fig. 6, the excess noise of the local oscillator is (Aiz F) - 4.2 x SNL. From (12), the additive noise at 150 kHz is estimated as follows. (Afi *) - (Ari *&, = 1.1 x SNL This agrees with the measured 1.2 x SNL in Fig. 6.
(14) additive
noise of
The additive noises at other frequencies are estimated in the same manner. The estimated values are also in good agreement with the measured ones. The additive noise changes with the optical path difference. The measured smallest noise is larger than the shot noise level in our experiment. A part of this noise increase is due to the reflected light from port A. The average power of the reflected light is half of the local oscillator in this measurement. This reflected light can be reduced by using a tilt polished fibre.
Conclusions We have studied the noise increase due to the pump light crosstalk in quadrature squeezed light generation with a fibre ring reflector. The balanced homodyne detector output whose signal light is squeezed light with a non-zero average amplitude is theoretically analysed. As a result, it is shown that when the
average amplitude is incoherent with the local oscillator, the shot noise associated with the average amplitude is added to the detector output. In addition, when the average amplitude is coherent with the local oscillator, a part of the excess noise of the pump light appears. From this analysis, the requirement for the fibre coupler to generate the squeezed light is clearly given. It is shown that when the excess noise of the pump source is large compared with the shot noise level, the crosstalk in the fibre coupler must be sufficiently suppressed. The results of the theoretical analysis are in good agreement with the experimental one. In future, we will use these analytical results to calibrate the measured noise levels. By doing so, accurate squeezing parameters will be obtained.
References Slusher, R. E., Hoberg, L. W., Yurke, B., Mertz, J. C., Valley, J. F. Observation of squeezed states generated by four-wave mixing in an optical cavity, Phys Ra, Lutf, 55 (1985) 2409-2412 Shelby, R. M., Levenson, M. D., Perlmutter, S. H., DeVoe, R. G., Walls, D. F. Broad-band parametric deamplification of quantum noise in an optical fiber, Phys Rev Lett, 57 (1986) 691694 Wu, L. A., Kimble, H. J., Hall, J. L., Wu, H. Generation of squeezed states by parametric down conversion, Whys Rev Lett, 57 (1986) 252&2523 Bergman, K., Haus, H. A. Squeezing in fibers with optical pulses, Opt Left, 16 (1991) 663665 Rosenbluh, M., Shelby, R. M. Squeezed optical solitons, Phys Rev Lett, 66 (1991) 153-156 Shirasaki, M., Haus, H. A. Squeezing of pulses in a nonlinear interferometer, J Opf Sot Am B, 7 (1990) 30-34 Yuen, H. P., Chan, V. W. S. Noise in homodyne and heterodyne detection, Opt Left, 8 (1983) 177-179 Blow, K. J., Loudon, R., Phoenix, S. J. S. Quantum theory of nonlinear loop mirrors, Phys Rev A, 45 (1992) 80648073 Lmdon, R., Knight, P. L. Squeezed light, J Mod Opf, 34 (1987) 7099159
Optics & Laser Technology
Optics & Laser Technology Vol 26 No 1 1994
53
due
S. KUME, M. MORI, T. GOTO, A. MIYAUCHI
In generating quadrature squeezed light with a fibre ring reflector, the pump light crosstalk should be minimized to achieve the sub-shot noise levels. The output noise of the homodyne detector, whose signal light is the squeezed light with non-zero amplitude, is analysed theoretically and experimentally for the first time. The requirement for the fibre coupler to generate the squeezed light is derived. KEYWORDS:
squeezed
light, homodyne
detectors,
Introduction squeezed light has been generated with a few non-linear effects-four-wave mixing, degenerate parametric amplification, and the optical Kerr effect’-‘. The squeezing by four-wave mixing and the degenerate parametric amplification were realized in the early day.sm3. But these methods generally depend on phase matching, and the adjustment of the cavity is very difficult. Quadrature
A fibre ring reflector, where the fibre acts as the optical Kerr medium, was proposed by Shirasaki and Haus for the generation of the squeezed vacuum6. This reflector does not depend on the phase matching. So, using this reflector and the pulsed laser, the quadrature squeezed light can be easily generated4,s. The generation model of the squeezed vacuum is shown in Fig. 1. In this method, it is necessary to tune the coupling ratio of the fibre coupler at 1: 1 to get the squeezed vacuum. If the coupling ratio is slightly different from 1: 1, the pump light leaks into the squeezed vacuum. Consequently, the output of the fibre is
0030-3992/94/010049-05
fibre ring reflectors
generally the squeezed amplitude.
light with a non-zero
average
The balanced homodyne detector is used to detect the quadrature squeezed state’. With this method, we can effectively get the variance of the signal light when the intensity of the local oscillator is much higher than that of the signal light. However, for the condition where the signal light has a non-zero average amplitude, it remains an unsettled question how the detector output is given. Squeezed light generation with a fibre ring reflector has been theoretically analysed by Blow et al.‘, and squeezing with unbalanced couplers has also been studied. But no mention has been made of the detection of the squeezed output. In this paper, the detector outputs, whose signal light has a non-zero average amplitude, are analysed both theoretically and experimentally. From the theoretical results, the requirement for the fibre coupler to generate the squeezed light is derived.
NN, SK, MM and TG are at the School of Engineering, Nagoya University, Nagoya-shi, 464-01 Japan. AM is in the Transmission Division, Fujitsu Ltd, Kawasaki-shi, 211 Japan. Received 17 June 1993. Revised 6 September 1993.
Optics & Laser Technology Vol 26 No 1 1994
fibre couplers,
@ 1994
The second section presents the theoretical for two different cases; incoherent crosstalk coherent crosstalk. The requirement for the coupler is also discussed. The third section the experimental method and its results. Butterworth-Heinemann
analysis and fibre describes
Ltd
49
Crosstalk in quadrature squeezed light generation: N. Nishizawa et al.
This squeezed light is shown in Fig. 2. These two light sources input into the balanced homodyne detector. The output variance of the detector is written as
PUMP
(Afi ‘) = ala :[cos{2(8 - 4,)}sinh(2s) +cosh(2s)] + sinh* s + fl/?*
VACUUM Fig. 1 Squeezed light generation in a fibre ring reflector. The pump light and the vacuum are superimposed on the fibre coupler. When the branching ratio of the fibre coupler is 1 : 1, the output from port B is the squeezed vacuum and the pump light returns to port A
Theoretical Incoherent
analysis crosstalk
For the squeezed light generation using degenerate parametric amplification, when the input light is in the coherent state, the output is generally the squeezed light with a non-zero average amplitude. In this case, the variance of the noise of the output light is coherent, but its average amplitude is incoherent to the pump light (local oscillator). In this section, we consider the homodyne detector output noise when the signal light is the squeezed light with a non-zero average amplitude incoherent with the local oscillator. We use the two operators 8, and 8, for the signal light and the local oscillator, respectively. The local oscillator is in the coherent state. The signal light is assumed to be in the squeezed state, and is written as follows9
(2)
where a, is the eigenvalue of 6, and 19is the relative phase between the signal light and the local oscillator. The first and the second terms in the right-hand side qf (2) represent the noise of the squeezed vacuum S([)]O). The third term /I/?* represents the noise increase from the squeezed vacuum due to the displacement D (/? ). Coherent
crosstalk
For the squeezed light generation with a fibre ring reflector, the average amplitude of the squeezed light is attributed to the pump light crosstalk. After the generation process, the pump light is reused as the local oscillator. Therefore, the average amplitude of the squeezed light is coherent with the local oscillator. In this section, we consider the homodyne detector output noise when the squeezed light has a non-zero average amplitude coherent with the local oscillator. We simulate the generation process of the squeezed light with a non-zero average amplitude in a model where the squeezed vacuum is mixed with the coherent light. The calculation model is shown in Fig. 3. Here 6, and 4, are the annihilation operators Coherent
Crosstalk
Model
=mmw
IAT> 6 (B >= ewW4t - P*4)
S(i)=exp(~i*(ci,)*-_tr(ci,t)*) [ = s exp(2$,)
I\
^
rla,+tla,
^
(1)
Balanced
Homodyne
Detector
Fig. 2 Squeezed light with a non-zero amplitude. fi is the average amplitude; I#J,is the angle between the ellipse minor axis and the X axis; &, is the phase of the average amplitude
50
Fig. 3 Generation model of the squeezed light with a non-zero average amplitude coherent with the local oscillator. Branching ratios of BSl and BS2 are t,:r, and t,:r,, respectively. It is assumed that It, I- 1 and 11,I - 0. a^,and I, are the annihilation operators of the coherent light and the squeezed vacuum, respectively
Optics & Laser Technology Vol 26 No 1 1994
Crosstalk in quadrature
squeezed
light generation:
N. Nishizawa
et al.
of the coherent state and the squeezed vacuum, respectively. When the branching ratio of the beam-splitter BSI is t,:r,, the two outputs are written as 1 rl aI + tl a, 1 t,a,+
(3)
^
(4)
II 4
In this model, we assumed that It, I - 1 and Ir, I - 0. In this condition, (3) gives a squeezed state with a small average amplitude. After that, the two lights are input to the balanced homodyne detector where the signal light and the local oscillator are given by (3) and (4), respectively. A phase shift arising from the optical path difference should also be included. The signal light and the local oscillator are superimposed at the second beam-splitter BS2 (t,:r,), and divided into two lights given by ri, = t,(r,ci, + tIci,) + r,(t,ci, + r16,)eie
(5)
ti2 = r,(rlbl + tIci,) + f2(tldl + r,B,)e”
(6)
After subtraction, written as
the output
of the detector
Peak Light Power [W] Fig. 4 Calculated result for the squeezed light generation. The pump source is 1.064pm YAG and the fibre is 50 m long 0.85 pm SM fibre. The abscissa represents the peak power of the pulse pump laser. The ordinate represents the squeezing parameter and the phase difference &-dS
is The second term is the noise increase excess noise of the pump source.
A =fi, -fi, = 4p, p2 cos 8 (cilTd, - 6, Tci,)
due to the
for simplicity. Since BS2 is used for the balanced homodyne detection, p2 = f l/2.
This calculation shows that if the squeezed light has an average amplitude coherent with the local oscillator, a part of the excess noise of the pump source is added to the detector output. The reason for this noise increase is the unbalance between the two output powers of BS2. When the signal light has a non-zero average amplitude coherent with the local oscillator, this amplitude component interferes with the local oscillator. As a result, the two light powers of BS2 become unbalanced, and a part of the excess noise appears even after subtraction.
When pl = 0, the signal light is the squeezed Then the variance of the detector output is
Requirement
- 2ip,(e” - 2r, r f cos B)B,tri, + 2ip,(e-”
- 2r,r : cos O)ti,&,t
where we use real numbers tl r : z ip,,
t,rf
(7)
pl and p2 given by
= ip,
(8)
vacuum.
(Ari *>,, = cllcl 7 [cos{2(B + 4B - 4, + n/2)}sinh(2s) +cosh(2s)]
+ sinh’s
(9)
The phase shift of 71/2 is included in (9), because the phase of the reflected coherent light is shifted by n/2 (see Fig. 3). When pl # 0, the signal light is the squeezed light with a non-zero average amplitude, and the output variance is given as follows
for fibre
coupler
In generating the squeezed light with the fibre ring reflector, the noise increase is mainly caused by the pump light crosstalk which is coherent to the local oscillator. In this section, the requirement for the fibre coupler to generate the squeezed light is discussed based on the calculated results of the previous section. Worst case At first, we assume that the phase difference 8 between the signal light and the local oscillator is equal to 0. In this condition, (11) is written as
(AA *> = (Ari 2)0 + 4p : cos* 8 {(Aii :) + (Ari t)} - 4~ : ~0s 8 (2(ri, )(ri,) +4p: +
(Ali2)-(AA2)0+4p:(Aft:)
+ (ril ) + (fi, ))
cos tkiB((ril>2>((ii,~)2)
4p : cos Oe-ie((cil t)*)((f?,)*)
(10)
When pl is very small and (Ari :) is large, this result is approximated by
(11)
(12)
and the additive noise takes the maximum value. When the second term in (12) is larger than the first, the noise reduction below the shot noise level cannot be observed. So, in order to observe the squeezed light, the condition 4p:(Ati:)
< (Afi*),zSNL
(13)
51
Crosstalk in quadrature squeezed light generation: N. Nishizawa et al.
must be satisfied, where SNL means the shot noise level. This equation shows that the requirement for the crosstalk in the fibre coupler depends on the excess noise of the pump laser. Now, consider an example. We assume that the excess noise of the local oscillator is 20 dB of the shot noise. In this condition, (13) gives p: < 2.5 x 10p3. This means that, in order to observe the squeezed light, the pump light crosstalk in the fibre coupler must be suppressed below - 26 dB. Actual case For the squeezed light generated in the fibre ring reflector, the phase difference & - c$, (see Fig. 2) depends on the squeezing parameter. We have made a brief calculation, and obtained the squeezing parameter and the angle 4B - 4, as functions of the peak pump light power. The calculated results are shown in Fig. 4. When the squeezing parameter is smaller than 1 dB, C&- c$~is about 45”. Then 4B - 4, gradually decreases with the peak pump light power and approaches 0”. In (1 l), the first term in the right-hand side is the variance of the squeezed vacuum. This term takes the minimum value when f3 = 4, - 4B. From Fig. 4, we can see that in the region of the high peak pump light power, the first term in (11) is minimized when 0 N 0. The second term in (11) is the additive noise due to the pump light crosstalk. This term has a maximum when 8 = 0. Therefore, the dependency of the first term of 8 is different from that of the second term in (11). Here, we evaluate the additive noise when the local oscillator phase is adjusted at the minimum point of the variance of the squeezed vacuum. When the squeezing parameter is less than 1 dB, the additive noise is about half of the worst case value in (12) because the local oscillator phase 8 is about -45”. When the squeezing parameter is above 1 dB, however, the additive noise gradually approaches the worst case value because 8 approaches 0”.
ANALYZER
Fig. 5
Experimental configuration
squeezed with the optical Kerr effect along the 50 m optical fibre, and it is not suitable for observing the additive noise. So only the beam component whose polarization is orthogonal to the main component was picked off with a half-wave plate and a polarizer, then used as the signal light. The beam from port A is partially picked off by the beam-splitter BS3 and used as the local oscillator of the homodyne detection. In the BS2, special care is taken to ensure an accurate superposition of the two light beams. For the optical detectors, we use high speed InGaAs-pin photodiodes. Two signal currents are directly subtracted from each other and amplified. The amplifier consists of low-noise OP-amps. The output of the detector is observed with a spectrum analyser. Moving the prism slowly with a micropositioner, the relative phase between the two lights can be changed. The detector output noise at 150 kHz is shown in Fig. 6 as a function of time. The video bandwidth is 10 Hz, the resolution bandwidth is 3 kHz, and the sweep time is 10 s. The upper trace shows the excess noise of the local oscillator. This is obtained when the port B and one of the optical detectors is blocked. The lowest trace shows the shot noise level. This is obtained with the port B blocked. This shot noise level is in good agreement with the theoretical values.
-50 -
From this discussion, we see that the requirement for the fibre coupler should be estimated in the worst case condition given by (12). .
Experiment .
,,.
In order to certify the theoretical results of the previous section, we have performed an experiment to measure the additive noise. The experimental configuration is shown in Fig. 5. We used a mode-locked YAG laser as the pump source. This laser delivers about 100 ps pulses at 1.064 pm at a repetition rate of 82 MHz. A 0.85 pm SM PANDA fibre of 50 m is connected with a polarization maintaining tunable fibre coupler to form a fibre ring reflector. The pump light is coupled to the fibre from port A, then divided into two beams by the fibre coupler BS 1. The branching ratio of the fibre coupler is slightly shifted from 1: 1. In this condition, the output power from port B is of the same order as that from port A. The main polarization beam is
52
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Detector output noise at 150 kHz as a function of time
Optics & Laser Technology Vol 26 No 1 1994
Crosstalk in quadrature squeezed light generation: N. Nishizawa et al. The middle trace shows the additive noise measured with the two ports open. This trace corresponds to the worst case value in (11). The relative phase is kept constant, so that the additive noise is at its highest level. The parameter p : in (12) is estimated as p : = 1/15 from the average power of the local oscillator and the signal light. From Fig. 6, the excess noise of the local oscillator is (Aiz F) - 4.2 x SNL. From (12), the additive noise at 150 kHz is estimated as follows. (Afi *) - (Ari *&, = 1.1 x SNL This agrees with the measured 1.2 x SNL in Fig. 6.
(14) additive
noise of
The additive noises at other frequencies are estimated in the same manner. The estimated values are also in good agreement with the measured ones. The additive noise changes with the optical path difference. The measured smallest noise is larger than the shot noise level in our experiment. A part of this noise increase is due to the reflected light from port A. The average power of the reflected light is half of the local oscillator in this measurement. This reflected light can be reduced by using a tilt polished fibre.
Conclusions We have studied the noise increase due to the pump light crosstalk in quadrature squeezed light generation with a fibre ring reflector. The balanced homodyne detector output whose signal light is squeezed light with a non-zero average amplitude is theoretically analysed. As a result, it is shown that when the
average amplitude is incoherent with the local oscillator, the shot noise associated with the average amplitude is added to the detector output. In addition, when the average amplitude is coherent with the local oscillator, a part of the excess noise of the pump light appears. From this analysis, the requirement for the fibre coupler to generate the squeezed light is clearly given. It is shown that when the excess noise of the pump source is large compared with the shot noise level, the crosstalk in the fibre coupler must be sufficiently suppressed. The results of the theoretical analysis are in good agreement with the experimental one. In future, we will use these analytical results to calibrate the measured noise levels. By doing so, accurate squeezing parameters will be obtained.
References Slusher, R. E., Hoberg, L. W., Yurke, B., Mertz, J. C., Valley, J. F. Observation of squeezed states generated by four-wave mixing in an optical cavity, Phys Ra, Lutf, 55 (1985) 2409-2412 Shelby, R. M., Levenson, M. D., Perlmutter, S. H., DeVoe, R. G., Walls, D. F. Broad-band parametric deamplification of quantum noise in an optical fiber, Phys Rev Lett, 57 (1986) 691694 Wu, L. A., Kimble, H. J., Hall, J. L., Wu, H. Generation of squeezed states by parametric down conversion, Whys Rev Lett, 57 (1986) 252&2523 Bergman, K., Haus, H. A. Squeezing in fibers with optical pulses, Opt Left, 16 (1991) 663665 Rosenbluh, M., Shelby, R. M. Squeezed optical solitons, Phys Rev Lett, 66 (1991) 153-156 Shirasaki, M., Haus, H. A. Squeezing of pulses in a nonlinear interferometer, J Opf Sot Am B, 7 (1990) 30-34 Yuen, H. P., Chan, V. W. S. Noise in homodyne and heterodyne detection, Opt Left, 8 (1983) 177-179 Blow, K. J., Loudon, R., Phoenix, S. J. S. Quantum theory of nonlinear loop mirrors, Phys Rev A, 45 (1992) 80648073 Lmdon, R., Knight, P. L. Squeezed light, J Mod Opf, 34 (1987) 7099159
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