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A novel noise model based on balanced detection for an ultrafast line-scan imaging system
Journal Pre-proof A novel noise model based on balanced detection for an ultrafast line-scan imaging system Kaimin Wang, Shanshan Jiang, Bo Dai, Yu Huang, Wei Li, Meiyong Xu, Xiangjun Xin, Dawei Zhang
PII: DOI: Reference:
S0030-4018(19)30791-6 https://doi.org/10.1016/j.optcom.2019.124508 OPTICS 124508
To appear in:
Optics Communications
Received date : 5 June 2019 Revised date : 28 August 2019 Accepted date : 30 August 2019 Please cite this article as: K. Wang, S. Jiang, B. Dai et al., A novel noise model based on balanced detection for an ultrafast line-scan imaging system, Optics Communications (2019), doi: https://doi.org/10.1016/j.optcom.2019.124508. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier B.V.
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A novel noise model based on balanced detection for an ultrafast line-scan imaging system
a
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Xiangjun Xinb, Dawei Zhanga,*
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Kaimin Wanga, Shanshan Jianga, Bo Daia, Yu Huanga, Wei Lia, Meiyong Xub,
School of Optical-Electrical and Computer Engineering, University of Shanghai for Science and Technology, No.516 Jungong Road, 200093 Shanghai, China School of Electronic Engineering, Beijing University of Posts and
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b
Telecommunications, No.10 Xitucheng Road, 100876 Beijing, China
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Abstract—— A novel noise model is proposed for a balanced-detection ultrafast imaging system to comprehensively analyze the noise in the system. The whole noise model is obtained by solving for erbium-doped fiber amplifier (EDFA) noise, intersymbol interference (ISI) noise and low-pass filter (LPF) noise successively with stochastic process theory. Then, the nonmonotonic relationship between the cut-off frequency and the signal-to-noise ratio(SNR) for certain experimental parameters are derived quantitatively and qualitatively: the SNR first increases and then decreases
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when the cut-off frequency increases, while the unique extreme point is located at 6.3 GHz. In addition, imaging results are obtained accordingly. The model and the results can provide a reference for system design and parameter selection. Keywords: noise model; ultrafast imaging;optical signal processing * Corresponding author. Tel.: +86 137 646 94608; E-mail address: [email protected]
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I. Introduction Ultrafast imaging plays an important role in the study of dynamical phenomena and is widely applied in scientific and industrial fields, such as shock waves[1],
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chemical dynamics in living cells[2-4], neural activity[5-6], laser surgery[7-9], microfluidics[10-12], and intelligent manufacturing[13]. Currently, charged-coupled
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device (CCD) or complementary metal oxide semiconductor (CMOS) image sensors are employed in traditional imaging approaches. However, conventional CCD and CMOS image sensors with a frame rate of kHz are too slow to catch rapid dynamical
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events. Meanwhile, if the frame rate increases to MHz or THz, the cost and difficulty in synchronization will constitute a bottleneck and affect the application prospects[14-
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16]. Consequently, serial time-encoded amplified microscopy (STEAM), a novel ultrafast imaging system has been proposed and studied in recent years[17-19]. However, the noise model in STEAM still needs to be improved, so as to precisely analyze the imaging performance.
STEAM was proposed by a research team engaged in optical signal coding and decoding technology at the University of California in 2009[20]. STEAM can achieve a frame rate of 37 MHz, which is approximately 104 times higher than the current technical imaging rate. The imaging system is shown in Fig. 1-a, and wide-spectrum
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pulses are applied as the source beam. By means of a spatial light modulator (grating), the source beam is then mapped from the spectral domain to the 1D spatial domain and then is incident on a 2D object line by line (scanning). In each line, since the absorption varies spatially, the intensity of the reflected beam accordingly varies spatially. When * Corresponding author. Tel.: +86 137 646 94608; E-mail address: [email protected]
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the optical signal emerges from the optical circulator, it is demapped from the 1D spatial domain to the spectral domain. The intensity of the optical signal varies in the spectrum and can then be unfolded in the time domain through a dispersion compensation fiber
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(DCF). Then, the optical signal is amplified by an erbium-doped fiber amplifier (EDFA) and converted to an electronic signal by a balanced photodetector (BPD). The BPD
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performs differential detection, which highlights the edge pixel and cancels out the interior pixels so that it is easier to make the edge of the object more distinguishable
thus achieving ultrafast imaging.
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from the background[21]. Finally, the 2D image can be restored by a digital processor,
The newly invented STEAM has many problems that need to be solved, among
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them image distortion is a crucial and harmful problem. Except for the signal intensity, the factor that most affects the imaging definition is the noise produced during amplification, photodetection and filtering. To make the image distortion more comprehensive and adequate, a novel noise model that contains noise from the EDFA, BPD and low-pass filter (LPF) is proposed. Using this model, the relationship between the system noise and the imaging quality can be solved qualitatively and quantitatively. Therefore, a reference can be provided to help adjust the parameters in an imaging system (e.g., the gain of the EDFA and the cut-off frequency of the LPF) appropriately
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and achieve the required imaging resolution in an application. This noise model can be applied in the STEAM to reduce the image distortions which will improve the accuracy of cell identification and classification. As a result, ultrafast imaging technique with this model can be effective in the detection of the rare cells and the diseases. * Corresponding author. Tel.: +86 137 646 94608; E-mail address: [email protected]
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II. Theory : II.A. Modeling procedures : We propose a comprehensive noise model according to the structure of the
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balanced-detection ultrafast imaging system, as shown in Fig. 1-a. The femtosecond laser source produces femtosecond pulses with a high pulse repetition rate (100 MHz)
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and a narrow pulse width (100 fs). Then, the pulses pass through a diffraction grating, and the spectrum of the pulse sequence spreads out to different spatial positions of the object under test. The reflectivity of the object varies spatially, so the spectrum of the
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reflected modulated light is modulated. Next, along the direction of the optical circulator, the modulated light signal is first amplified by the EDFA and then
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demodulated into an electrical signal by the BPD. According to ultrafast imaging processing, a comprehensive noise model can be built through the following procedures, as shown in Fig. 1-b:
1. Obtain the electric field of the EDFA noise Eamp (t ) . 2. Obtain the current of 2 parts of the EDFA noise: the signal-spontaneous beat noise (1) is sp (t ) , spontaneous-spontaneous beat noise (2) ispsp (t ) , and the BPD receiver noise (3) in (t ) caused by the BPD.
3. Obtain the multiplicative intersymbol interference (ISI) noise (4) iISI (t ) caused
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by the BPD and the LPF band-limited noise (5) iLPF (t ) of noise (1), (2), and (3) caused by the LPF. Then obtain the whole system noise: the sum of noise (4) and noise (5).
* Corresponding author. Tel.: +86 137 646 94608; E-mail address: [email protected]
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(a)
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(b)
Fig. 1-a Structure of the balanced-detection ultrafast imaging system Fig. 1-b The corresponding comprehensive noise model.
II.B. Model of the EDFA noise (1) and (2) and the BPD receiver noise (3) The EDFA introduces additive amplified spontaneous emission (ASE) noise to the
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input signal. Let the electric field of the signal and noise be Esig (t ) and Eamp (t ) , respectively. The total electric field from the EDFA is defined by expression (1) [22]: Eafter _ EDFA (t ) E forward _ EDFA (t ) Eamp (t ) Esig (t ) Eamp (t )
= GPin a (t ) cos( ws t )+
B / 2
k ( B / 2
2nsp (G 1)h cos(( ws 2 k )t k )
)
* Corresponding author. Tel.: +86 137 646 94608; E-mail address: [email protected]
(1)
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Eamp (t )
is the sum of the M B / 2
components, which correspond to M
frequencies ws / 2 k ranging from B / 2 to B / 2 . B is the bandwidth of the ASE noise and can be obtained by a conversion of the wavelength range of 16 nm
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around a central wavelength of 1550 nm. For simplicity, is a certain small frequency width to make M B / 2 a large integer. The M components have the
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same average intensity but vary in a random phase k . k obeys a uniform distribution in the interval [2 , 2 ] . nsp is the inversion parameter of the EDFA and is typically between 1~4, and G is the amplification gain. Then, the amplified signal
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is received by the BPD. The signal is divided into an upper arm part and a lower arm part. The corresponding electric field is defined as (2) and (3): (2)
Edown (t ) 0.5Eafter _ EDFA (t )
(3)
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Eup (t ) 0.5Eafter _ EDFA (t T )
Both parts are converted to photocurrents generated by two photodetectors. Then, the output of the BPD I BPD (t ) is the difference of the photocurrents I up (t ) and
I down (t ) from the upper and lower arms. Accordingly, the power of the ASE noise can be obtained by solving the power density function of I BPD (t ) . I down (t ) is first obtained, and then I up (t ) is obtained similarly: I down (t ) e / h | Edown (t ) |2 +in (t )
e 2 2 e B / 2 a (t ) GPin nsp (G 1)h cos(ws t ) cos((ws 2 k )t k ) h h k ( B / 2 ) 0
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0.5GPin
nsp (G 1) e
B0 /2
k ( B0 / 2 )
0
cos(( ws 2 k )t k )
B0 / 2
j ( B0 / 2 )
(4) cos(( ws 2 j )t j ) in (t )
is (t ) is sp (t ) isp sp (t ) in (t )
* Corresponding author. Tel.: +86 137 646 94608; E-mail address: [email protected]
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In photodetection, the noise current consists of two parts: the current of the signalspontaneous beat noise (1) is sp (t ) and the spontaneous-spontaneous beat noise (2) isp sp (t ) .
The corresponding power is obtained by solving the power density function
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(PDF). In addition, the current of the additive white Gaussian receiver noise (AWGN) (3) in (t ) is also considered. is the quantum efficiency of the photodetector, e is
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the electron charge, h is the Planck constant and is the central frequency of the optical signal.
The noise power (1), (2) and (3) can be obtained by integrating the PDF, while the
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PDF can be obtained by taking a Fourier transform of the corresponding autocorrelation function (AF), under the condition that the current of the noise is a stationary stochastic process (SSP). In particular, the noise can be an SSP if the corresponding AF is related
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to the time difference but not the time t , namely, Rs sp (t , t ) Rs sp ( ) . First, the current of noise (1) is sp (t ) can be proved to be an SSP by analyzing the corresponding AF Rs sp (t , t ) , as defined in (5) ~ (8): Rs sp (t , t ) E[is sp (t )is sp (t )] (0.5
B /2 B / 2 4e GPin nsp (G 1)h )2 E[cos(2 kt k ) cos(2 j (t ) j )] h k ( B /2 ) j ( B /2 )
(5)
Since k and j are independent, when k j , E[cos(2 kt k ) cos(2 j (t ) j )] 0
(6)
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When k = j , the AF is related to but not t , as shown in (7) and (8):
E[cos(2 kt k ) cos(2 j (t ) j )] E[cos(2 kt k ) cos(2 k (t ) k )]
1 1 1 E[cos(2 k ) cos(2 k (t ) 2 k 2 k )] E[cos(2 k )] 0 cos(2 k ) 2 2 2
* Corresponding author. Tel.: +86 137 646 94608; E-mail address: [email protected]
(7)
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Rs sp (t , t ) Rs sp ( )
B / 2 1 4 e (0.5 GPin nsp nsp (G 1)h ) 2 cos(2 k ) 2 h k ( B / 2 )
B / 2 e =2( ) 2 GPin nsp (G 1)h cos(2 k ) h k ( B / 2 )
(8)
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According to the central limit theorem, when B / ( ) , is sp (t ) approximately obeys a Gaussian distribution, with a mean of zero and a variance (equal to power) as
1 2
s2sp Rs sp (0) (0.5
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follows:
B /2 4 e e GPin nsp (G 1)h ) 2 cos(2 k 0) 2( ) 2 GPin nsp (G 1)h B h h k ( B /2 )
(9)
The corresponding PDF is a constant with a rectangular window function in the optical bandwidth [B / 2, B / 2] :
2(e)2 Pin nsp (G 1)G h
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Nssp Rssp (0) / B
(10)
N s sp , f [ B / 2, B / 2] Ps sp ( f ) else 0,
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The corresponding power is
Ps sp N s sp B
(11)
2( e) 2 Pin nsp (G 1)GB h
(12)
Similarly, the current of noise (2) isp sp (t ) can also be proven to be an SSP. The corresponding AF Rsp sp (t , t ) can be expressed by the product of 2 components Rsp 1 ( ) and Rsp 2 ( ) . Obviously, Psp 2 ( f ) and Psp 2 ( f ) are also rectangular
windows. The triangular PDF of noise (2) Psp sp ( f ) can be solved by a convolution of the above 2 rectangular windows, as defined in (13) ~ (16): Rsp sp (t , t ) ( n sp (G 1) e) 2
j ( B0 / 2 )
j ( B0 / 2
cos(( ws 2 j )t j )
)
cos(( ws 2 j )(t ) j )
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k )
B0 / 2
B0 / 2
B0 / 2
k ( B0 / 2 )
B0 / 2
k ( B0 / 2 )
cos(( ws 2 k )t
(13)
cos(( ws 2 k )(t ) k )
( nsp (G 1) e) 2 Rsp 1 (t , t ) Rsp 2 (t , t )
Rsp1 (t , t ) Rsp1 ( )
1 B0 /2 1 B0 /2 cos(2 j ) , Rsp2 (t, t ) Rsp2 ( )= cos(2 k ) (14) 2 j ( B0 /2 ) 2 k ( B0 /2 )
* Corresponding author. Tel.: +86 137 646 94608; E-mail address: [email protected]
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1 , f [ B / 2, B / 2] Psp 1 ( f ) Psp 2 ( f ) 2 0, else
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(0.5nsp (G 1) e)2 (1 f / B), f [0, B] Psp sp ( f ) (nsp (G 1) e)2 Psp 1 ( f )* Psp 2 ( f ) (0.5nsp (G 1) e)2 (1 f / B), f [ B, 0] else 0, 2 Thus, the corresponding power is Pspsp 0.25(nsp (G 1)e) B
(15)
(16)
(17)
PT and the shot noise power Pshot [23]: Pn PT Pshot
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The power of noise (3) in (t ) can be solved by obtaining the thermal noise power
4kBT 2eGPin B RL
(18)
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kB is the Boltzmann constant, RL is the load resistance of the receiver, T is the absolute temperature and B is the baseband noise bandwidth of the receiver front end.
e . e is the electronic charge. It is obvious that h
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is the receiver responsivity,
the thermal noise is a constant when kB , RL and T are chosen, while the shot noise is AWGN. The whole power of noise (3) can be obtained by a sum of the constant part and the AWGN part. The latter will be described by the PDF-bandwidth curve as shown at the end of this section.
Then, the influence of the BPD is analyzed. The current of the upper arm in the balanced detector I up (t ) is expressed in (19):
(19)
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Iup (t ) e / h | Eup (t ) |2 +in (t ) e / h | Edown (t T ) |2 +in (t )
Except for AWGN noise (3) in (t ) , I up (t ) also consists of the signal is (t T ) and
the current of the signal-spontaneous beat noise (1) is sp (t T ) and spontaneousspontaneous beat noise (2) isp sp (t T ) . The noise is independently and identically distributed between the upper arm and lower arm is independently; thus, the powers are * Corresponding author. Tel.: +86 137 646 94608; E-mail address: [email protected]
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identical.
The
baseband
symbol
components
a (t ) 2 |t nT At2
and
a (t T ) 2 |t nT At2T are taken into the formula, and the output current of the balanced
I in (t ) |t nT I up (t ) |t nT I down (t ) |t nT
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detector is:
GPin e 2 ( At T At2 )+is sp (nT ) is sp (nT T ) 2h
(20)
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+isp sp (nT ) isp sp (nT T )+in (nT ) in (nT )
Finally, we achieve the output signal-to-noise ratio (SNR) of the balanced receiver under the EDFA noise model
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GPin e 2 ( At T At2 ))2 2 h SNR (
2( Pn Ps sp Psp sp )
GPin e 2 ( At T At2 )) 2 2h
=
8e 2 Pin nsp (G 1)GB ( nsp (G 1) e) 2 B h
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(
16k B Th / RL +8ehGPsi B
(21)
The parameters are set as: nsp 2 ， h 6.626176*1034 ， e 1.602 *10 19 ， k B 1.380662 *10 23 ， 1.931*1014 , G 30 dB ，RL 50 ，Pin 5mW ，T 300 K ， 0.9 A / W , and B 1.998*1012 Hz .
According to formulas (11), (16) and (18), the PDFs of noise (1), (2) and (3) can
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be seen in Figs. 2(a), (b) and (c), respectively.
* Corresponding author. Tel.: +86 137 646 94608; E-mail address: [email protected]
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Fig. 2 (a) PDF of noise (1) is sp (t ) . (b) PDF of noise (2) isp sp (t ) . (c) PDF of noise (3) in (t ) .
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II.C. Model of the ISI noise (4) and the LPF band-limited noise (5): To improve the research on the output SNR of the balanced receiver under the
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EDFA noise model, we consider the effect of the ISI noise and LPF noise. The output of the photodetector (PD) includes a large current due to the femtosecond pulse in each balanced photodetection period: M 1
iout (t ) iin (t ) * h(t )=[ ak g ( kTrep )+2is sp (t ) 2isp sp (t ) 2in (t )]* h(t ) k 0
M 1
=[ ak g ( kTrep )]* h(t )+[2is sp (t ) 2isp sp (t ) 2in (t )]* h(t ) k 0
M 1
[ ak g ( kTrep )h(t )]d iLPF (t )
(22)
k 0
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where iout (t ) is the output that is achieved by a convolution of iin (t ) and the unit impulse response h (t ) . Trep is the impulse period (reciprocal of the repetition rate f rep ) of the femtosecond laser. The first term contains the ISI noise (4) and the useful signal, while the second term represents the LPF band-limited noise (5) iLPF (t ) , which is * Corresponding author. Tel.: +86 137 646 94608; E-mail address: [email protected]
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obtained by integrating the PDFs of noise (1), (2) and (3) in formula (21) in the interval [ f c , f c ] : k BT 4( e) 2 Pin nsp (G 1)Gf c 0.5( nsp (G 1) e) 2 f c (2 B f c ) / B (23) +4eGPsi f c RL h
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PLPF 8
Accordingly, the PDFs of the LPF band-limited noise (5) iLPF (t ) and the
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components (LPF band-limited noise (1) , (2) and (3)) can be seen in Figs. 3(a), (b),
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and (c), respectively.
Fig. 3 PDF of the LPF band-limited noise (5) iLPF (t ) and the components. Now, the ISI noise (4) is analyzed. For simplicity, the typical sampling function
h(t ) 2 fc
sin(2 fct ) in the unit impulse response function is chosen with a cut-off 2 fct
frequency of f c 1 / Trep . Since the duty cycle of the femtosecond laser source is 5 extremely low ( Tpulse / Trep 100 fs /10ns 10 ) in every pulse period, the first term of M 1
a k 0
k
( kTrep )
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the input signal iin (t ) is approximately the impulse sequence
regardless of g ( kTrep ) . There are 4 mainstream functions (hyperbolic secant, Gaussian, Lorentzian and asymmetric hyperbolic secant functions) to approximate g ( kTrep ) [24], and the corresponding curves are shown in Fig. 4 as a reference.
* Corresponding author. Tel.: +86 137 646 94608; E-mail address: [email protected]
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Fig. 4 Waveform of a femtosecond pulse.
According to the integral properties of the impulse function (t ) , the simplified output signal can be achieved:
M 1
M 1
k 0
k 0
iout (t ) [ ak ( kTrep )h(t )]d inLPF = ak h(t kTrep ) inLPF (t )
(24)
Then, the ISI noise (4) of the simplified output signal is analyzed. At each sampling time kTrep , (l 0,1, 2,..., M 1) , the sampling signal is obtained: k
ik iout (kTrep ) ak l h(lTrep ) inLPF (kTrep )
(25)
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l 0
Since the cut-off frequency is f c 1/ Trep , Trep 1/ f c , the values of the ISI noise
components ak l h(lTrep ) (l≥2) are too small (far less) to be considered. The repetition rate of the femtosecond pulse f rep 1/ Trep is approximately 100 MHz, and the size
* Corresponding author. Tel.: +86 137 646 94608; E-mail address: [email protected]
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relation can be shown in (26) and (27). Let f c 1/ Trep be approximately 1 GHz, and the ISI noise components ak l h(2Trep ) and ak l h(3Trep ) be 0.0083 and 0.0054, respectively.
h(3Trep ) h(0)
max{[sin c(2 f c / f rep 1),sin c(2 f c / f rep )]}
(26)
max{[sin c(3 f c / f rep 1),sin c(3 f c / f rep )]}
(27)
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h(0)
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h(2Trep )
As a result, only the relatively serious interference
ak 1 h(Trep )
and
ak 1 h( Trep ) (l≥1) from neighboring symbols at time (k 1)Trep , (k 1)Trep should
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be considered. On the other hand, the amount of LPF band-limited noise (5) is too small compared with Pin 5mW as shown in Fig. 3. Thus, the sampling signal can be obtained :
(28)
iISI ak 1 h(Trep ) ak 1 h(Trep ), is ak h(0) inLPF (kTrep ) ak h(0)
(29)
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ik iout (kTrep ) ak h(0) ak 1 h(Trep ) ak 1 h(Trep ) inLPF (kTrep ) is iISI
where the first term ak h(0) is the useful signal, and the second and third terms ak 1 h(Trep ) and ak 1 h( Trep ) are the ISI noise (4) iISI . iISI can be regarded as
multiplicative noise and the multiplicative factor ISI iISI / is is the ratio of is and iISI . Then, the relationship between ISI and f c is obtained in Fig. 5. Although there
is an extremum in each impulse period Trep , ISI first increases and then decreases.
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In general, the extremum of ISI will decrease when f c increases. According to [25], the intensity of the optical pulses a k 1 , ak , and a k 1 are all constants, and the power of the ISI noise (4) at the sampling time can be expressed by (30): 2 PISI ik2 is2 (is iISI )2 is2 2* is * iISI iISI (2* ISI ISI2 ) Psi
* Corresponding author. Tel.: +86 137 646 94608; E-mail address: [email protected]
(31)
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Fig. 5 Relationship between the multiplicative factor ISI and f c / f rep .
Fig. 5 shows that when f c increases, the ISI noise (4) will decrease in general, but the LPF band-limited noise (5) iLPF (t ) will increase due to a wider filter bandwidth.
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Finally, the noise of the whole system Ptotal can be obtained by a sum of noise (4) and noise (5) according to Section II.A.
* Corresponding author. Tel.: +86 137 646 94608; E-mail address: [email protected]
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III. Results and analysis Let the SNR be the power ratio of the signal and the whole system noise, SNRisi be the power ratio of the signal and the ISI noise (4), and SNRadditional be the power
of
ratio of the signal and the LPF band-limited noise (5). To demonstrate the relationship between the f c and the SNR, we use 3 kinds of images to simulate the effect of the
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ultrafast line-scan imaging system image quality for different values of f c from 100 MHz to 100 GHz, and the corresponding results are shown in Figs. 6-8. According to the whole noise model in our previous paper, differential detection-
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based STEAM [28] and Fig. 1, different degrees of imaging can be obtained by varying f c of the LPF, as shown in Fig. 6 and Fig. 7. First, we simulate a Gaussian beam and
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university logo –USST with a frequency of 100 MHz and the reconstructed images are seriously distorted. Then we find that the reconstructed images become clearer when f c is 6.3 GHz. However, the qualities of the reconstructed images decreases, and the
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influence of the noise becomes obvious when f c increases to 100 GHz.
* Corresponding author. Tel.: +86 137 646 94608; E-mail address: [email protected]
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Fig. 6 Imaging results of a Gaussian beam with (a) Ptotal =0.0241 W at f c =100 MHz, (b) Ptotal = 5.1709*10-4 W at f c =6.3 GHz, (c) Ptotal = 0.0042 W at f c =100 GHz, and (d)
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Ptotal =0 (a reference system with no noise).
* Corresponding author. Tel.: +86 137 646 94608; E-mail address: [email protected]
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Fig. 7 Imaging results of a university logo with (a) Ptotal =0.0241 W at f c =100 MHz, (b) Ptotal = 5.1709*10-4 W at f c =6.3 GHz, (c) Ptotal = 0.0042 W at f c =100 GHz, and (d)
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Ptotal =0 (a reference system with no noise).
From Fig. 8, the relationship between the SNR and fc is analyzed quantitatively: the maximum SNR is approximately 19.85 dB at an
f c of
6.3 GHz. SNRisi
increases rapidly in the beginning and then grows steady when f c is above 20 GHz. SNRadditional decreases when f c increases throughout the whole process. According to Fig. 8, we can observe that the reconstructed image quality will be improved greatly when fc is below 6.3 GHz, and the best image quality can be achieved at 6.3 GHz. In
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this case, a distinction can hardly be found between the reconstructed image and original image. When f c is larger than 6.3 GHz, the quality of the image decreases slowly, and the images are blurry compared with the original pictures. In addition, the
* Corresponding author. Tel.: +86 137 646 94608; E-mail address: [email protected]
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system shows a robust effect in resconstructing 2D codes, which can be recognized
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under all conditions.
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Fig. 8 Relationship between the SNR and cut-off frequency f c .
Fig. 9 Imaging results of 2D codes with (a) Ptotal =0.0241 W at f c =100 MHz, (b) Ptotal
* Corresponding author. Tel.: +86 137 646 94608; E-mail address: [email protected]
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= 5.1709*10-4 W at f c =6.3 GHz, (c) Ptotal = 0.0042 W at fc =100 GHz, and (d) Ptotal =0 (a reference system with no noise).
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IV. Conclusion
In conclusion, a novel noise model based on a balanced-detection ultrafast line-scan
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imaging system is obtained with 5 noise parts considering the effect of the EDFA, BPD and LPF. Due to the opposite trends of noise (4) and noise (5), the relationship between the cut-off frequency f c and the SNR is nonmonotonic, while the unique extreme
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value of the SNR is approximately 19.85 dB at an fc of 6.3 GHz for certain evaluated parameters of nsp , h , e , k B , , G , RL , Pin , T , and B . According to the proposed
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model and results, the above parameters of the ultrafast imaging system can be optimized to improve the SNR performance. This noise model can be applied to the ultrafast imaging systems such as ultrafast automated image cytometry cancer detection[26], data compression for time-stretch imaging based on differential detection and run-length encoding[21] and ultrafast cell edge detection by line-scan time-stretch microscopy[27]. The noise model can help the ultrafast imaging system achieve the target recognition, the cell identification and the cancer diagnosis with high accuracy
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by finding the max SNRs in their imaging progress.
V. Acknowledgment The financial supports from NSFC (No.61805144) and Shanghai Sailing Program (No.17YF1429400) are gratefully acknowledged. * Corresponding author. Tel.: +86 137 646 94608; E-mail address: [email protected]
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References [1] K. Nakagawa, Sequentially timed all-optical mapping photography for observation of ultrafast phenomena, Opto-Electronics and Communications Conference. 2015. [2] M. Tanter and M. Fink, Ultrafast imaging in biomedical ultrasound, IEEE Trans. Ultrason.
of
Ferroelectr. Freq. Control. 61(1) (2014) 102-119.
[3] C. Papadacci, M. Pernot, M. Couade, M. Fink, and M. Tanter, High-contrast ultrafast imaging of the heart, IEEE Trans. Ultrason. Ferroelectr. Freq. Control. 61(2) (2014) 288-
pro
301.
[4] David J. Flannigan, Brett Barwick, and Ahmed H. Zewail.Biological imaging with 4D ultrafast electron microscopy.Proceedings of the National Academy of Sciences of the United States of America. 107(22) (2010) 9933-9937.
re-
[5] D.J. Flannigan, B. Barwick, and A.H. Zewail, Biological imaging with 4D ultrafast electron microscopy, Pro. Natl. Acad. Sci. U.S.A. 107(22) (2010) 9933-9937. [6] Q.T.K. Lai, K.C. M. Lee, A.H.L. Tang, K.K.Y. Wong, H.K.H. So, and K.K. Tsia, High-
urn al P
throughput time-stretch imaging flow cytometry for multi-class classification of phytoplankton, Opt. Express. 24(25) (2016) 28170-28184. [7] C.L. Hoy, O. Ferhanoglu, M. Yildirim, K.H. Kim, S.S. Karajanagi, K.M.C. Chan, J.B. Kobler, S.M. Zeitels, and A. Ben-Yakar, Clinical ultrafast laser surgery: recent advances and future directions, IEEE J. Sel. Top. Quantum Electron. 20(2) (2014) 7100814. [8] O. Ferhanoglu, M. Yildirim, K. Subramanian, and A. Ben-Yakar, A 5-mm piezo-scanning fiber device for high speed ultrafast laser microsurgery, Biomed. Opt. Express. 5(7) (2014) 2023-2036.
[9] M. Han, L. Zickler, G. Giese, M. Walter, F.H. Loesel, J.F. Bille, Second-harmonic imaging of cornea after intrastromal femtosecond laser ablation, J. Biomed. Opt. 9(4) (2004) 760–
Jo
766.
[10] T.T.W. Wong, A.K.S. Lau, K.K.Y. Ho, M.Y. H. Tang, J.D.F. Robles, X.M. Wei, A.C.S. Chan, A.H.L. Tang, E.Y. Lam, K.K.Y. Wong, G.C.F. Chan, H.C. Shum and K.K. Tsia, Asymmetric-detection time-stretch optical microscopy (ATOM) for ultrafast highcontrast cellular imaging in flow, Sci Rep. (2014)4:3656.
[11] A.K.S. Lau, H.C. Shum, K.K.Y. Wong and K.K. Tsia, Optofluidic time-stretch imaging– * Corresponding author. Tel.: +86 137 646 94608; E-mail address: [email protected]
Journal Pre-proof
an emerging tool for high-throughput imaging flow cytometry, Lab Chip. 16(10) (2016) 1743-1756. [12] K. Godaa, A. Ayazia, D.R. Gossettb, J. Sadasivama, C.K. Lonappana, E. Sollierb, A.M. Farda, S.C. Hurb, J.Adama, C. Murrayc, C. Wanga, N. Brackbilla, D.D. Carlob, and B.
of
Jalalia, High-throughput single-microparticle imaging flow analyzer, Proc. Natl. Acad. Sci. U. S. A. 109(29) (2012) 11630–11635.
[13] T. Pezeril, C. Klieber, V. Shalagatskyi, G. Vaudel, V. Temnov, O.G. Schmidt, and D.
pro
Makarov, Femtosecond imaging of nonlinear acoustics in gold, Opt. Express. 22(4) (2014) 4590-4598.
[14] A. Barty, S. Boutet, M.J. Bogan, S. Hau-Riege, S. Marchesini, K. Sokolowski-Tinten, N. Stojanovic, R. Tobey, H. Ehrke, A. Cavalleri, S. Düsterer, M. Frank, S. Bajt, B.W. Woods, M.M. Seibert, J. Hajdu, R. Treusch and H.N. Chapman, Ultrafast single-shot diffraction
re-
imaging of nanoscale dynamics, Nat. Photonics. 2(7) (2008) 415-419.
[15] A. Feist, K.E. Echternkamp, J. Schauss, S.V. Yalunin, S. Schäfer and C. Ropers, Quantum coherent optical phase modulation in an ultrafast transmission electron microscope,
urn al P
Nature. 521(7551) (2015) 200–203.
[16] H. Park, Z. Hao, X. Wang, S. Nie, R. Clinite, and J. Cao, Synchronization of femtosecond laser and electron pulses with subpicosecond precision, Rev. Sci. Instrum. 76(8) (2005) 083905.
[17] K.K. Tsia, K. Goda, D. Capewell and B. Jalali, Performance of serial time-encoded amplified microscope, Opt. Express. 18(10) (2010) 10016-10028. [18] T.T.W. Wong, A.K.S. Lau, K.K.Y. Wong, and K.K. Tsia, Optical time-stretch confocal microscopy at 1 mu m, Opt. Lett. 37(16) (2012) 3330-3332. [19] A.K.S. Lau, T.T.W. Wong, K.K.Y. Ho, M.T.H. Tang, A.C.S. Chan, X.M Wei, E.Y. Lam, H. C. Shum, K.K.Y. Wong and K.K. Tsiaa, Interferometric time-stretch microscopy for
Jo
ultrafast quantitative cellular and tissue imaging at 1 μm, J. Biomed Opt. 19(7) (2014) 076001.
[20] K.K. Tsia, K. Goda, D. Capewell, and B. Jalali, Performance of serial time-encoded amplified microscope, Opt. Express. 18(10) (2010) 10016-10028.
[21] B. Dai, S.CH. Ying, Zh.S. Gao, K.M. Wang, D.W. Zhang, S.L Zhuang, and X. Wang, Data compression for time-stretch imaging based on differential detection and run-length * Corresponding author. Tel.: +86 137 646 94608; E-mail address: [email protected]
Journal Pre-proof
encoding, J. Lightwave Technol. 35(23) (2017) 5098-5104. [22] P.C. Becker, N.A. Olsson, and J.R. Simpson, Eebium-Doped Fiber Amplifiers Fundamantals and Technology, Academic Press.1999. [23] R.G. Smith and S.D. Personick, Semiconductor devices for optical communications, Topics in Applied Physics. 39 (1982) 89-160.
of
[24] Z.G. Zhang, Femtosecond laser technique.Beijing: Science Press.2017.
[25] Q. Guo, H.W. Chen, H.W. Chen, S.G. Yang, and S.Z. Xie, Analysis of detection bandwidth limitations in time-stretch-based single-pixel imaging systems, Appl. Optics. 56(5) (2017)
pro
1327-1332.
[26] K. Goda, D.D. Carlo, and B. Jalali, Ultrafast automated image cytometry for cancer detection, 35th Annual International Conference of the IEEE EMBS. (2013) 129-132. [27] B. Dai, L. He, L.L. Zheng, Y.F. Fu, K.M. Wang and X. Wang, Ultrafast cell edge detection
Jo
urn al P
re-
by line-scan time-stretch microscopy, Bio. Photonic. 12(1) (2019) 1-7.
* Corresponding author. Tel.: +86 137 646 94608; E-mail address: [email protected]
PII: DOI: Reference:
S0030-4018(19)30791-6 https://doi.org/10.1016/j.optcom.2019.124508 OPTICS 124508
To appear in:
Optics Communications
Received date : 5 June 2019 Revised date : 28 August 2019 Accepted date : 30 August 2019 Please cite this article as: K. Wang, S. Jiang, B. Dai et al., A novel noise model based on balanced detection for an ultrafast line-scan imaging system, Optics Communications (2019), doi: https://doi.org/10.1016/j.optcom.2019.124508. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier B.V.
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A novel noise model based on balanced detection for an ultrafast line-scan imaging system
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Xiangjun Xinb, Dawei Zhanga,*
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Kaimin Wanga, Shanshan Jianga, Bo Daia, Yu Huanga, Wei Lia, Meiyong Xub,
School of Optical-Electrical and Computer Engineering, University of Shanghai for Science and Technology, No.516 Jungong Road, 200093 Shanghai, China School of Electronic Engineering, Beijing University of Posts and
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b
Telecommunications, No.10 Xitucheng Road, 100876 Beijing, China
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Abstract—— A novel noise model is proposed for a balanced-detection ultrafast imaging system to comprehensively analyze the noise in the system. The whole noise model is obtained by solving for erbium-doped fiber amplifier (EDFA) noise, intersymbol interference (ISI) noise and low-pass filter (LPF) noise successively with stochastic process theory. Then, the nonmonotonic relationship between the cut-off frequency and the signal-to-noise ratio(SNR) for certain experimental parameters are derived quantitatively and qualitatively: the SNR first increases and then decreases
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when the cut-off frequency increases, while the unique extreme point is located at 6.3 GHz. In addition, imaging results are obtained accordingly. The model and the results can provide a reference for system design and parameter selection. Keywords: noise model; ultrafast imaging;optical signal processing * Corresponding author. Tel.: +86 137 646 94608; E-mail address: [email protected]
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I. Introduction Ultrafast imaging plays an important role in the study of dynamical phenomena and is widely applied in scientific and industrial fields, such as shock waves[1],
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chemical dynamics in living cells[2-4], neural activity[5-6], laser surgery[7-9], microfluidics[10-12], and intelligent manufacturing[13]. Currently, charged-coupled
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device (CCD) or complementary metal oxide semiconductor (CMOS) image sensors are employed in traditional imaging approaches. However, conventional CCD and CMOS image sensors with a frame rate of kHz are too slow to catch rapid dynamical
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events. Meanwhile, if the frame rate increases to MHz or THz, the cost and difficulty in synchronization will constitute a bottleneck and affect the application prospects[14-
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16]. Consequently, serial time-encoded amplified microscopy (STEAM), a novel ultrafast imaging system has been proposed and studied in recent years[17-19]. However, the noise model in STEAM still needs to be improved, so as to precisely analyze the imaging performance.
STEAM was proposed by a research team engaged in optical signal coding and decoding technology at the University of California in 2009[20]. STEAM can achieve a frame rate of 37 MHz, which is approximately 104 times higher than the current technical imaging rate. The imaging system is shown in Fig. 1-a, and wide-spectrum
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pulses are applied as the source beam. By means of a spatial light modulator (grating), the source beam is then mapped from the spectral domain to the 1D spatial domain and then is incident on a 2D object line by line (scanning). In each line, since the absorption varies spatially, the intensity of the reflected beam accordingly varies spatially. When * Corresponding author. Tel.: +86 137 646 94608; E-mail address: [email protected]
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the optical signal emerges from the optical circulator, it is demapped from the 1D spatial domain to the spectral domain. The intensity of the optical signal varies in the spectrum and can then be unfolded in the time domain through a dispersion compensation fiber
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(DCF). Then, the optical signal is amplified by an erbium-doped fiber amplifier (EDFA) and converted to an electronic signal by a balanced photodetector (BPD). The BPD
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performs differential detection, which highlights the edge pixel and cancels out the interior pixels so that it is easier to make the edge of the object more distinguishable
thus achieving ultrafast imaging.
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from the background[21]. Finally, the 2D image can be restored by a digital processor,
The newly invented STEAM has many problems that need to be solved, among
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them image distortion is a crucial and harmful problem. Except for the signal intensity, the factor that most affects the imaging definition is the noise produced during amplification, photodetection and filtering. To make the image distortion more comprehensive and adequate, a novel noise model that contains noise from the EDFA, BPD and low-pass filter (LPF) is proposed. Using this model, the relationship between the system noise and the imaging quality can be solved qualitatively and quantitatively. Therefore, a reference can be provided to help adjust the parameters in an imaging system (e.g., the gain of the EDFA and the cut-off frequency of the LPF) appropriately
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and achieve the required imaging resolution in an application. This noise model can be applied in the STEAM to reduce the image distortions which will improve the accuracy of cell identification and classification. As a result, ultrafast imaging technique with this model can be effective in the detection of the rare cells and the diseases. * Corresponding author. Tel.: +86 137 646 94608; E-mail address: [email protected]
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II. Theory : II.A. Modeling procedures : We propose a comprehensive noise model according to the structure of the
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balanced-detection ultrafast imaging system, as shown in Fig. 1-a. The femtosecond laser source produces femtosecond pulses with a high pulse repetition rate (100 MHz)
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and a narrow pulse width (100 fs). Then, the pulses pass through a diffraction grating, and the spectrum of the pulse sequence spreads out to different spatial positions of the object under test. The reflectivity of the object varies spatially, so the spectrum of the
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reflected modulated light is modulated. Next, along the direction of the optical circulator, the modulated light signal is first amplified by the EDFA and then
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demodulated into an electrical signal by the BPD. According to ultrafast imaging processing, a comprehensive noise model can be built through the following procedures, as shown in Fig. 1-b:
1. Obtain the electric field of the EDFA noise Eamp (t ) . 2. Obtain the current of 2 parts of the EDFA noise: the signal-spontaneous beat noise (1) is sp (t ) , spontaneous-spontaneous beat noise (2) ispsp (t ) , and the BPD receiver noise (3) in (t ) caused by the BPD.
3. Obtain the multiplicative intersymbol interference (ISI) noise (4) iISI (t ) caused
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by the BPD and the LPF band-limited noise (5) iLPF (t ) of noise (1), (2), and (3) caused by the LPF. Then obtain the whole system noise: the sum of noise (4) and noise (5).
* Corresponding author. Tel.: +86 137 646 94608; E-mail address: [email protected]
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(a)
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(b)
Fig. 1-a Structure of the balanced-detection ultrafast imaging system Fig. 1-b The corresponding comprehensive noise model.
II.B. Model of the EDFA noise (1) and (2) and the BPD receiver noise (3) The EDFA introduces additive amplified spontaneous emission (ASE) noise to the
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input signal. Let the electric field of the signal and noise be Esig (t ) and Eamp (t ) , respectively. The total electric field from the EDFA is defined by expression (1) [22]: Eafter _ EDFA (t ) E forward _ EDFA (t ) Eamp (t ) Esig (t ) Eamp (t )
= GPin a (t ) cos( ws t )+
B / 2
k ( B / 2
2nsp (G 1)h cos(( ws 2 k )t k )
)
* Corresponding author. Tel.: +86 137 646 94608; E-mail address: [email protected]
(1)
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Eamp (t )
is the sum of the M B / 2
components, which correspond to M
frequencies ws / 2 k ranging from B / 2 to B / 2 . B is the bandwidth of the ASE noise and can be obtained by a conversion of the wavelength range of 16 nm
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around a central wavelength of 1550 nm. For simplicity, is a certain small frequency width to make M B / 2 a large integer. The M components have the
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same average intensity but vary in a random phase k . k obeys a uniform distribution in the interval [2 , 2 ] . nsp is the inversion parameter of the EDFA and is typically between 1~4, and G is the amplification gain. Then, the amplified signal
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is received by the BPD. The signal is divided into an upper arm part and a lower arm part. The corresponding electric field is defined as (2) and (3): (2)
Edown (t ) 0.5Eafter _ EDFA (t )
(3)
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Eup (t ) 0.5Eafter _ EDFA (t T )
Both parts are converted to photocurrents generated by two photodetectors. Then, the output of the BPD I BPD (t ) is the difference of the photocurrents I up (t ) and
I down (t ) from the upper and lower arms. Accordingly, the power of the ASE noise can be obtained by solving the power density function of I BPD (t ) . I down (t ) is first obtained, and then I up (t ) is obtained similarly: I down (t ) e / h | Edown (t ) |2 +in (t )
e 2 2 e B / 2 a (t ) GPin nsp (G 1)h cos(ws t ) cos((ws 2 k )t k ) h h k ( B / 2 ) 0
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0.5GPin
nsp (G 1) e
B0 /2
k ( B0 / 2 )
0
cos(( ws 2 k )t k )
B0 / 2
j ( B0 / 2 )
(4) cos(( ws 2 j )t j ) in (t )
is (t ) is sp (t ) isp sp (t ) in (t )
* Corresponding author. Tel.: +86 137 646 94608; E-mail address: [email protected]
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In photodetection, the noise current consists of two parts: the current of the signalspontaneous beat noise (1) is sp (t ) and the spontaneous-spontaneous beat noise (2) isp sp (t ) .
The corresponding power is obtained by solving the power density function
of
(PDF). In addition, the current of the additive white Gaussian receiver noise (AWGN) (3) in (t ) is also considered. is the quantum efficiency of the photodetector, e is
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the electron charge, h is the Planck constant and is the central frequency of the optical signal.
The noise power (1), (2) and (3) can be obtained by integrating the PDF, while the
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PDF can be obtained by taking a Fourier transform of the corresponding autocorrelation function (AF), under the condition that the current of the noise is a stationary stochastic process (SSP). In particular, the noise can be an SSP if the corresponding AF is related
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to the time difference but not the time t , namely, Rs sp (t , t ) Rs sp ( ) . First, the current of noise (1) is sp (t ) can be proved to be an SSP by analyzing the corresponding AF Rs sp (t , t ) , as defined in (5) ~ (8): Rs sp (t , t ) E[is sp (t )is sp (t )] (0.5
B /2 B / 2 4e GPin nsp (G 1)h )2 E[cos(2 kt k ) cos(2 j (t ) j )] h k ( B /2 ) j ( B /2 )
(5)
Since k and j are independent, when k j , E[cos(2 kt k ) cos(2 j (t ) j )] 0
(6)
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When k = j , the AF is related to but not t , as shown in (7) and (8):
E[cos(2 kt k ) cos(2 j (t ) j )] E[cos(2 kt k ) cos(2 k (t ) k )]
1 1 1 E[cos(2 k ) cos(2 k (t ) 2 k 2 k )] E[cos(2 k )] 0 cos(2 k ) 2 2 2
* Corresponding author. Tel.: +86 137 646 94608; E-mail address: [email protected]
(7)
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Rs sp (t , t ) Rs sp ( )
B / 2 1 4 e (0.5 GPin nsp nsp (G 1)h ) 2 cos(2 k ) 2 h k ( B / 2 )
B / 2 e =2( ) 2 GPin nsp (G 1)h cos(2 k ) h k ( B / 2 )
(8)
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According to the central limit theorem, when B / ( ) , is sp (t ) approximately obeys a Gaussian distribution, with a mean of zero and a variance (equal to power) as
1 2
s2sp Rs sp (0) (0.5
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follows:
B /2 4 e e GPin nsp (G 1)h ) 2 cos(2 k 0) 2( ) 2 GPin nsp (G 1)h B h h k ( B /2 )
(9)
The corresponding PDF is a constant with a rectangular window function in the optical bandwidth [B / 2, B / 2] :
2(e)2 Pin nsp (G 1)G h
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Nssp Rssp (0) / B
(10)
N s sp , f [ B / 2, B / 2] Ps sp ( f ) else 0,
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The corresponding power is
Ps sp N s sp B
(11)
2( e) 2 Pin nsp (G 1)GB h
(12)
Similarly, the current of noise (2) isp sp (t ) can also be proven to be an SSP. The corresponding AF Rsp sp (t , t ) can be expressed by the product of 2 components Rsp 1 ( ) and Rsp 2 ( ) . Obviously, Psp 2 ( f ) and Psp 2 ( f ) are also rectangular
windows. The triangular PDF of noise (2) Psp sp ( f ) can be solved by a convolution of the above 2 rectangular windows, as defined in (13) ~ (16): Rsp sp (t , t ) ( n sp (G 1) e) 2
j ( B0 / 2 )
j ( B0 / 2
cos(( ws 2 j )t j )
)
cos(( ws 2 j )(t ) j )
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k )
B0 / 2
B0 / 2
B0 / 2
k ( B0 / 2 )
B0 / 2
k ( B0 / 2 )
cos(( ws 2 k )t
(13)
cos(( ws 2 k )(t ) k )
( nsp (G 1) e) 2 Rsp 1 (t , t ) Rsp 2 (t , t )
Rsp1 (t , t ) Rsp1 ( )
1 B0 /2 1 B0 /2 cos(2 j ) , Rsp2 (t, t ) Rsp2 ( )= cos(2 k ) (14) 2 j ( B0 /2 ) 2 k ( B0 /2 )
* Corresponding author. Tel.: +86 137 646 94608; E-mail address: [email protected]
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1 , f [ B / 2, B / 2] Psp 1 ( f ) Psp 2 ( f ) 2 0, else
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(0.5nsp (G 1) e)2 (1 f / B), f [0, B] Psp sp ( f ) (nsp (G 1) e)2 Psp 1 ( f )* Psp 2 ( f ) (0.5nsp (G 1) e)2 (1 f / B), f [ B, 0] else 0, 2 Thus, the corresponding power is Pspsp 0.25(nsp (G 1)e) B
(15)
(16)
(17)
PT and the shot noise power Pshot [23]: Pn PT Pshot
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The power of noise (3) in (t ) can be solved by obtaining the thermal noise power
4kBT 2eGPin B RL
(18)
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kB is the Boltzmann constant, RL is the load resistance of the receiver, T is the absolute temperature and B is the baseband noise bandwidth of the receiver front end.
e . e is the electronic charge. It is obvious that h
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is the receiver responsivity,
the thermal noise is a constant when kB , RL and T are chosen, while the shot noise is AWGN. The whole power of noise (3) can be obtained by a sum of the constant part and the AWGN part. The latter will be described by the PDF-bandwidth curve as shown at the end of this section.
Then, the influence of the BPD is analyzed. The current of the upper arm in the balanced detector I up (t ) is expressed in (19):
(19)
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Iup (t ) e / h | Eup (t ) |2 +in (t ) e / h | Edown (t T ) |2 +in (t )
Except for AWGN noise (3) in (t ) , I up (t ) also consists of the signal is (t T ) and
the current of the signal-spontaneous beat noise (1) is sp (t T ) and spontaneousspontaneous beat noise (2) isp sp (t T ) . The noise is independently and identically distributed between the upper arm and lower arm is independently; thus, the powers are * Corresponding author. Tel.: +86 137 646 94608; E-mail address: [email protected]
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identical.
The
baseband
symbol
components
a (t ) 2 |t nT At2
and
a (t T ) 2 |t nT At2T are taken into the formula, and the output current of the balanced
I in (t ) |t nT I up (t ) |t nT I down (t ) |t nT
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detector is:
GPin e 2 ( At T At2 )+is sp (nT ) is sp (nT T ) 2h
(20)
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+isp sp (nT ) isp sp (nT T )+in (nT ) in (nT )
Finally, we achieve the output signal-to-noise ratio (SNR) of the balanced receiver under the EDFA noise model
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GPin e 2 ( At T At2 ))2 2 h SNR (
2( Pn Ps sp Psp sp )
GPin e 2 ( At T At2 )) 2 2h
=
8e 2 Pin nsp (G 1)GB ( nsp (G 1) e) 2 B h
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(
16k B Th / RL +8ehGPsi B
(21)
The parameters are set as: nsp 2 ， h 6.626176*1034 ， e 1.602 *10 19 ， k B 1.380662 *10 23 ， 1.931*1014 , G 30 dB ，RL 50 ，Pin 5mW ，T 300 K ， 0.9 A / W , and B 1.998*1012 Hz .
According to formulas (11), (16) and (18), the PDFs of noise (1), (2) and (3) can
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be seen in Figs. 2(a), (b) and (c), respectively.
* Corresponding author. Tel.: +86 137 646 94608; E-mail address: [email protected]
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Fig. 2 (a) PDF of noise (1) is sp (t ) . (b) PDF of noise (2) isp sp (t ) . (c) PDF of noise (3) in (t ) .
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II.C. Model of the ISI noise (4) and the LPF band-limited noise (5): To improve the research on the output SNR of the balanced receiver under the
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EDFA noise model, we consider the effect of the ISI noise and LPF noise. The output of the photodetector (PD) includes a large current due to the femtosecond pulse in each balanced photodetection period: M 1
iout (t ) iin (t ) * h(t )=[ ak g ( kTrep )+2is sp (t ) 2isp sp (t ) 2in (t )]* h(t ) k 0
M 1
=[ ak g ( kTrep )]* h(t )+[2is sp (t ) 2isp sp (t ) 2in (t )]* h(t ) k 0
M 1
[ ak g ( kTrep )h(t )]d iLPF (t )
(22)
k 0
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where iout (t ) is the output that is achieved by a convolution of iin (t ) and the unit impulse response h (t ) . Trep is the impulse period (reciprocal of the repetition rate f rep ) of the femtosecond laser. The first term contains the ISI noise (4) and the useful signal, while the second term represents the LPF band-limited noise (5) iLPF (t ) , which is * Corresponding author. Tel.: +86 137 646 94608; E-mail address: [email protected]
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obtained by integrating the PDFs of noise (1), (2) and (3) in formula (21) in the interval [ f c , f c ] : k BT 4( e) 2 Pin nsp (G 1)Gf c 0.5( nsp (G 1) e) 2 f c (2 B f c ) / B (23) +4eGPsi f c RL h
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PLPF 8
Accordingly, the PDFs of the LPF band-limited noise (5) iLPF (t ) and the
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components (LPF band-limited noise (1) , (2) and (3)) can be seen in Figs. 3(a), (b),
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and (c), respectively.
Fig. 3 PDF of the LPF band-limited noise (5) iLPF (t ) and the components. Now, the ISI noise (4) is analyzed. For simplicity, the typical sampling function
h(t ) 2 fc
sin(2 fct ) in the unit impulse response function is chosen with a cut-off 2 fct
frequency of f c 1 / Trep . Since the duty cycle of the femtosecond laser source is 5 extremely low ( Tpulse / Trep 100 fs /10ns 10 ) in every pulse period, the first term of M 1
a k 0
k
( kTrep )
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the input signal iin (t ) is approximately the impulse sequence
regardless of g ( kTrep ) . There are 4 mainstream functions (hyperbolic secant, Gaussian, Lorentzian and asymmetric hyperbolic secant functions) to approximate g ( kTrep ) [24], and the corresponding curves are shown in Fig. 4 as a reference.
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Fig. 4 Waveform of a femtosecond pulse.
According to the integral properties of the impulse function (t ) , the simplified output signal can be achieved:
M 1
M 1
k 0
k 0
iout (t ) [ ak ( kTrep )h(t )]d inLPF = ak h(t kTrep ) inLPF (t )
(24)
Then, the ISI noise (4) of the simplified output signal is analyzed. At each sampling time kTrep , (l 0,1, 2,..., M 1) , the sampling signal is obtained: k
ik iout (kTrep ) ak l h(lTrep ) inLPF (kTrep )
(25)
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l 0
Since the cut-off frequency is f c 1/ Trep , Trep 1/ f c , the values of the ISI noise
components ak l h(lTrep ) (l≥2) are too small (far less) to be considered. The repetition rate of the femtosecond pulse f rep 1/ Trep is approximately 100 MHz, and the size
* Corresponding author. Tel.: +86 137 646 94608; E-mail address: [email protected]
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relation can be shown in (26) and (27). Let f c 1/ Trep be approximately 1 GHz, and the ISI noise components ak l h(2Trep ) and ak l h(3Trep ) be 0.0083 and 0.0054, respectively.
h(3Trep ) h(0)
max{[sin c(2 f c / f rep 1),sin c(2 f c / f rep )]}
(26)
max{[sin c(3 f c / f rep 1),sin c(3 f c / f rep )]}
(27)
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h(2Trep )
As a result, only the relatively serious interference
ak 1 h(Trep )
and
ak 1 h( Trep ) (l≥1) from neighboring symbols at time (k 1)Trep , (k 1)Trep should
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be considered. On the other hand, the amount of LPF band-limited noise (5) is too small compared with Pin 5mW as shown in Fig. 3. Thus, the sampling signal can be obtained :
(28)
iISI ak 1 h(Trep ) ak 1 h(Trep ), is ak h(0) inLPF (kTrep ) ak h(0)
(29)
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ik iout (kTrep ) ak h(0) ak 1 h(Trep ) ak 1 h(Trep ) inLPF (kTrep ) is iISI
where the first term ak h(0) is the useful signal, and the second and third terms ak 1 h(Trep ) and ak 1 h( Trep ) are the ISI noise (4) iISI . iISI can be regarded as
multiplicative noise and the multiplicative factor ISI iISI / is is the ratio of is and iISI . Then, the relationship between ISI and f c is obtained in Fig. 5. Although there
is an extremum in each impulse period Trep , ISI first increases and then decreases.
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In general, the extremum of ISI will decrease when f c increases. According to [25], the intensity of the optical pulses a k 1 , ak , and a k 1 are all constants, and the power of the ISI noise (4) at the sampling time can be expressed by (30): 2 PISI ik2 is2 (is iISI )2 is2 2* is * iISI iISI (2* ISI ISI2 ) Psi
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(31)
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Fig. 5 Relationship between the multiplicative factor ISI and f c / f rep .
Fig. 5 shows that when f c increases, the ISI noise (4) will decrease in general, but the LPF band-limited noise (5) iLPF (t ) will increase due to a wider filter bandwidth.
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Finally, the noise of the whole system Ptotal can be obtained by a sum of noise (4) and noise (5) according to Section II.A.
* Corresponding author. Tel.: +86 137 646 94608; E-mail address: [email protected]
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III. Results and analysis Let the SNR be the power ratio of the signal and the whole system noise, SNRisi be the power ratio of the signal and the ISI noise (4), and SNRadditional be the power
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ratio of the signal and the LPF band-limited noise (5). To demonstrate the relationship between the f c and the SNR, we use 3 kinds of images to simulate the effect of the
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ultrafast line-scan imaging system image quality for different values of f c from 100 MHz to 100 GHz, and the corresponding results are shown in Figs. 6-8. According to the whole noise model in our previous paper, differential detection-
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based STEAM [28] and Fig. 1, different degrees of imaging can be obtained by varying f c of the LPF, as shown in Fig. 6 and Fig. 7. First, we simulate a Gaussian beam and
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university logo –USST with a frequency of 100 MHz and the reconstructed images are seriously distorted. Then we find that the reconstructed images become clearer when f c is 6.3 GHz. However, the qualities of the reconstructed images decreases, and the
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influence of the noise becomes obvious when f c increases to 100 GHz.
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Fig. 6 Imaging results of a Gaussian beam with (a) Ptotal =0.0241 W at f c =100 MHz, (b) Ptotal = 5.1709*10-4 W at f c =6.3 GHz, (c) Ptotal = 0.0042 W at f c =100 GHz, and (d)
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Ptotal =0 (a reference system with no noise).
* Corresponding author. Tel.: +86 137 646 94608; E-mail address: [email protected]
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Fig. 7 Imaging results of a university logo with (a) Ptotal =0.0241 W at f c =100 MHz, (b) Ptotal = 5.1709*10-4 W at f c =6.3 GHz, (c) Ptotal = 0.0042 W at f c =100 GHz, and (d)
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Ptotal =0 (a reference system with no noise).
From Fig. 8, the relationship between the SNR and fc is analyzed quantitatively: the maximum SNR is approximately 19.85 dB at an
f c of
6.3 GHz. SNRisi
increases rapidly in the beginning and then grows steady when f c is above 20 GHz. SNRadditional decreases when f c increases throughout the whole process. According to Fig. 8, we can observe that the reconstructed image quality will be improved greatly when fc is below 6.3 GHz, and the best image quality can be achieved at 6.3 GHz. In
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this case, a distinction can hardly be found between the reconstructed image and original image. When f c is larger than 6.3 GHz, the quality of the image decreases slowly, and the images are blurry compared with the original pictures. In addition, the
* Corresponding author. Tel.: +86 137 646 94608; E-mail address: [email protected]
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system shows a robust effect in resconstructing 2D codes, which can be recognized
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under all conditions.
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Fig. 8 Relationship between the SNR and cut-off frequency f c .
Fig. 9 Imaging results of 2D codes with (a) Ptotal =0.0241 W at f c =100 MHz, (b) Ptotal
* Corresponding author. Tel.: +86 137 646 94608; E-mail address: [email protected]
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= 5.1709*10-4 W at f c =6.3 GHz, (c) Ptotal = 0.0042 W at fc =100 GHz, and (d) Ptotal =0 (a reference system with no noise).
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IV. Conclusion
In conclusion, a novel noise model based on a balanced-detection ultrafast line-scan
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imaging system is obtained with 5 noise parts considering the effect of the EDFA, BPD and LPF. Due to the opposite trends of noise (4) and noise (5), the relationship between the cut-off frequency f c and the SNR is nonmonotonic, while the unique extreme
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value of the SNR is approximately 19.85 dB at an fc of 6.3 GHz for certain evaluated parameters of nsp , h , e , k B , , G , RL , Pin , T , and B . According to the proposed
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model and results, the above parameters of the ultrafast imaging system can be optimized to improve the SNR performance. This noise model can be applied to the ultrafast imaging systems such as ultrafast automated image cytometry cancer detection[26], data compression for time-stretch imaging based on differential detection and run-length encoding[21] and ultrafast cell edge detection by line-scan time-stretch microscopy[27]. The noise model can help the ultrafast imaging system achieve the target recognition, the cell identification and the cancer diagnosis with high accuracy
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by finding the max SNRs in their imaging progress.
V. Acknowledgment The financial supports from NSFC (No.61805144) and Shanghai Sailing Program (No.17YF1429400) are gratefully acknowledged. * Corresponding author. Tel.: +86 137 646 94608; E-mail address: [email protected]
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References [1] K. Nakagawa, Sequentially timed all-optical mapping photography for observation of ultrafast phenomena, Opto-Electronics and Communications Conference. 2015. [2] M. Tanter and M. Fink, Ultrafast imaging in biomedical ultrasound, IEEE Trans. Ultrason.
of
Ferroelectr. Freq. Control. 61(1) (2014) 102-119.
[3] C. Papadacci, M. Pernot, M. Couade, M. Fink, and M. Tanter, High-contrast ultrafast imaging of the heart, IEEE Trans. Ultrason. Ferroelectr. Freq. Control. 61(2) (2014) 288-
pro
301.
[4] David J. Flannigan, Brett Barwick, and Ahmed H. Zewail.Biological imaging with 4D ultrafast electron microscopy.Proceedings of the National Academy of Sciences of the United States of America. 107(22) (2010) 9933-9937.
re-
[5] D.J. Flannigan, B. Barwick, and A.H. Zewail, Biological imaging with 4D ultrafast electron microscopy, Pro. Natl. Acad. Sci. U.S.A. 107(22) (2010) 9933-9937. [6] Q.T.K. Lai, K.C. M. Lee, A.H.L. Tang, K.K.Y. Wong, H.K.H. So, and K.K. Tsia, High-
urn al P
throughput time-stretch imaging flow cytometry for multi-class classification of phytoplankton, Opt. Express. 24(25) (2016) 28170-28184. [7] C.L. Hoy, O. Ferhanoglu, M. Yildirim, K.H. Kim, S.S. Karajanagi, K.M.C. Chan, J.B. Kobler, S.M. Zeitels, and A. Ben-Yakar, Clinical ultrafast laser surgery: recent advances and future directions, IEEE J. Sel. Top. Quantum Electron. 20(2) (2014) 7100814. [8] O. Ferhanoglu, M. Yildirim, K. Subramanian, and A. Ben-Yakar, A 5-mm piezo-scanning fiber device for high speed ultrafast laser microsurgery, Biomed. Opt. Express. 5(7) (2014) 2023-2036.
[9] M. Han, L. Zickler, G. Giese, M. Walter, F.H. Loesel, J.F. Bille, Second-harmonic imaging of cornea after intrastromal femtosecond laser ablation, J. Biomed. Opt. 9(4) (2004) 760–
Jo
766.
[10] T.T.W. Wong, A.K.S. Lau, K.K.Y. Ho, M.Y. H. Tang, J.D.F. Robles, X.M. Wei, A.C.S. Chan, A.H.L. Tang, E.Y. Lam, K.K.Y. Wong, G.C.F. Chan, H.C. Shum and K.K. Tsia, Asymmetric-detection time-stretch optical microscopy (ATOM) for ultrafast highcontrast cellular imaging in flow, Sci Rep. (2014)4:3656.
[11] A.K.S. Lau, H.C. Shum, K.K.Y. Wong and K.K. Tsia, Optofluidic time-stretch imaging– * Corresponding author. Tel.: +86 137 646 94608; E-mail address: [email protected]
Journal Pre-proof
an emerging tool for high-throughput imaging flow cytometry, Lab Chip. 16(10) (2016) 1743-1756. [12] K. Godaa, A. Ayazia, D.R. Gossettb, J. Sadasivama, C.K. Lonappana, E. Sollierb, A.M. Farda, S.C. Hurb, J.Adama, C. Murrayc, C. Wanga, N. Brackbilla, D.D. Carlob, and B.
of
Jalalia, High-throughput single-microparticle imaging flow analyzer, Proc. Natl. Acad. Sci. U. S. A. 109(29) (2012) 11630–11635.
[13] T. Pezeril, C. Klieber, V. Shalagatskyi, G. Vaudel, V. Temnov, O.G. Schmidt, and D.
pro
Makarov, Femtosecond imaging of nonlinear acoustics in gold, Opt. Express. 22(4) (2014) 4590-4598.
[14] A. Barty, S. Boutet, M.J. Bogan, S. Hau-Riege, S. Marchesini, K. Sokolowski-Tinten, N. Stojanovic, R. Tobey, H. Ehrke, A. Cavalleri, S. Düsterer, M. Frank, S. Bajt, B.W. Woods, M.M. Seibert, J. Hajdu, R. Treusch and H.N. Chapman, Ultrafast single-shot diffraction
re-
imaging of nanoscale dynamics, Nat. Photonics. 2(7) (2008) 415-419.
[15] A. Feist, K.E. Echternkamp, J. Schauss, S.V. Yalunin, S. Schäfer and C. Ropers, Quantum coherent optical phase modulation in an ultrafast transmission electron microscope,
urn al P
Nature. 521(7551) (2015) 200–203.
[16] H. Park, Z. Hao, X. Wang, S. Nie, R. Clinite, and J. Cao, Synchronization of femtosecond laser and electron pulses with subpicosecond precision, Rev. Sci. Instrum. 76(8) (2005) 083905.
[17] K.K. Tsia, K. Goda, D. Capewell and B. Jalali, Performance of serial time-encoded amplified microscope, Opt. Express. 18(10) (2010) 10016-10028. [18] T.T.W. Wong, A.K.S. Lau, K.K.Y. Wong, and K.K. Tsia, Optical time-stretch confocal microscopy at 1 mu m, Opt. Lett. 37(16) (2012) 3330-3332. [19] A.K.S. Lau, T.T.W. Wong, K.K.Y. Ho, M.T.H. Tang, A.C.S. Chan, X.M Wei, E.Y. Lam, H. C. Shum, K.K.Y. Wong and K.K. Tsiaa, Interferometric time-stretch microscopy for
Jo
ultrafast quantitative cellular and tissue imaging at 1 μm, J. Biomed Opt. 19(7) (2014) 076001.
[20] K.K. Tsia, K. Goda, D. Capewell, and B. Jalali, Performance of serial time-encoded amplified microscope, Opt. Express. 18(10) (2010) 10016-10028.
[21] B. Dai, S.CH. Ying, Zh.S. Gao, K.M. Wang, D.W. Zhang, S.L Zhuang, and X. Wang, Data compression for time-stretch imaging based on differential detection and run-length * Corresponding author. Tel.: +86 137 646 94608; E-mail address: [email protected]
Journal Pre-proof
encoding, J. Lightwave Technol. 35(23) (2017) 5098-5104. [22] P.C. Becker, N.A. Olsson, and J.R. Simpson, Eebium-Doped Fiber Amplifiers Fundamantals and Technology, Academic Press.1999. [23] R.G. Smith and S.D. Personick, Semiconductor devices for optical communications, Topics in Applied Physics. 39 (1982) 89-160.
of
[24] Z.G. Zhang, Femtosecond laser technique.Beijing: Science Press.2017.
[25] Q. Guo, H.W. Chen, H.W. Chen, S.G. Yang, and S.Z. Xie, Analysis of detection bandwidth limitations in time-stretch-based single-pixel imaging systems, Appl. Optics. 56(5) (2017)
pro
1327-1332.
[26] K. Goda, D.D. Carlo, and B. Jalali, Ultrafast automated image cytometry for cancer detection, 35th Annual International Conference of the IEEE EMBS. (2013) 129-132. [27] B. Dai, L. He, L.L. Zheng, Y.F. Fu, K.M. Wang and X. Wang, Ultrafast cell edge detection
Jo
urn al P
re-
by line-scan time-stretch microscopy, Bio. Photonic. 12(1) (2019) 1-7.
* Corresponding author. Tel.: +86 137 646 94608; E-mail address: [email protected]