FM CW incoherent by MikeJenny

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									        Research on Doppler Frequency in Incoherent FM/CW Laser
                                Detection
                                           LIU Kai
  National Defense Key Laboratory of Mechatronics Engineering and Control,Beijing Institute of
                             Technology,Beijing 100081,China


                                                              ABSTRACT

The principle of transmitted and received laser in incoherent FM/CW laser detection is different from the one in coherent
FM/CW laser detection. The methods for distance solution in both detections are similar. Incoherent FM/CW laser
detection uses subcarrier to modulate the intensity of laser, and the photodetector detects the intensity of received signal.
The amplified photocurrent is mixed with local oscillator signal, and the intermediate frequency (IF) signal contains the
information of distance from sensor to target. The Doppler frequency for this detection is related with the relative radial
velocity between sensor and target. The optical frequency is directly modulated with electro-optic device in coherent
FM/CW laser detection and the received laser signal is photomixed with transmitted laser signal. The Doppler frequency
in the detection relates to the optical frequency. In distance-measuring lidar, the Doppler frequency affects the solution.
The Doppler frequency in incoherent FM/CW laser detection is unrelated with optical frequency, and it is much less than
the one in coherent FM/CW laser detection, correspondingly. The error in incoherent FM/CW laser detection is smaller.
As a result, the incoherent FM/CW laser detection is more suitable for the use of distance-measuring lidar.
Keywords: Incoherent FM/CW laser detection, Doppler frequency, Lidar

                                                           1. INTRODUCTION

1.1 Principle of coherent FM/CW laser detection
The transmitted frequency in coherent FM/CW laser detection is cyclic variation, and optical frequency is modulated by
the voltage of modulation signal. The modulation can be intracavity modulation or outer cavity modulation. The
intracavity laser modulation is achieved by cutting photoelectric crystal with modulation signal. The refracting index is
linearly changed with the voltage of modulation signal, and the optical frequency is correspondingly changed with the
refracting index. Thus, the optical frequency is modulated by the signal. The outer cavity modulation is achieved by the
acousto-optic modulator which is consist of acousto-optic medium and energy transducer. The energy transducer
transforms the voltage signal to ultrasonic waveform going through the acousto-optic medium to diffract the optical
carrier. The frequency of diffracted light is the summation of optical carrier’s frequency and ultrasonic frequency. As a
result, the diffracted light’s frequency is linearly changed with the signal’s voltage. Whichever modulation is, the output
optical frequency is linearly controlled by the signal.
One portion of the modulated optical signal is transmitted to the atmosphere, and the reflected laser is photomixed with
the other portion of the modulated optical signal. The difference frequency of photomixing is related to the distance and
velocity between sensor and target. Difference frequency can be used to resolve the distance and velocity between sensor
and target. Figure 1 shows the schematic diagram of coherent FM/CW laser detection.


                                 Modulation                    Photoelectric crystal or
                                  signal                      acousto-optic modulator

                                                                                                        Target


                                                Signal
                                                                                  Coherent processing
                                              processing




                              Figure 1: Schematic diagram of the coherent FM/CW laser detection
1.2 Principle of incoherent FM/CW laser detection
In the transmitting system of incoherent FM/CW laser detection, the modulation signal modulates the carrier into
subcarrier. A portion of the subcarrier is used to modulate the optical intensity of laser, therefore, the amplitude of laser is
linearly changed with the voltage of subcarrier and the optical frequency is constant. Another portion of the subcarrier is
mixed with the reflected signal as local oscillation.
The photoreceivers pick up the reflected optical signal from the target viaing the lenses which focus the light on the
active region of the photodiodes. The photodiodes convert the light into low photocurrent which is proportional to the
light power. Therefore, the output photocurrent is the envelope of the reflected optical signal. This kind of detection is
also named as envelope detection or direct detection. The current transforms into voltage waveform with a
transimpedance amplifier, and voltage signal is mixed with the subcarrier. The output of the mixer is fed into a low pass
filter (LPF) whose output is named intermediate frequency (IF). IF signal is the difference frequency between subcarrier
and envelope of received signal. IF signal contents the information of distance and velocity between sensor and the target.
Figure 2 shows the schematic diagram of incoherent FM/CW laser detection.

                                                                       Carrier




                                                                                      Subcarrier




                                    Modulation                         Modula                         Optical
                                     signal                             tion                         modulation

                                                                                                                                           Target

                                      Signal                                                                       Envelope
                                                                                     Mixing
                                    processing                                                                     detection




                              Figure 2: Schematic diagram of the incoherent FM/CW laser detection

                                                                2. SIGNAL ANALYSIS

2.1 Subcarrier in incoherent FM/CW laser detection
Using periodic sawtooth signal as the modulation signal, the instantaneous frequency and instantaneous phase can be
written as:
                                                                                     F
                                                                  ft n (t )  f 0        (t  nT )
                                                                                      T
                                                                              f 0   (t  nT )
                                                                              f 0   tm             nT  t  (n  1)T
                                                       t
                                      tn (t )  2  ft (t )dt
                                                       0

                                                       n kT                  t            
                                                  2    ft k 1 (t )dt   ft n (t )dt 
                                                       k 1 ( k 1)T
                                                                            nT            
                                                                                           
                                                            n     kT                                          t
                                                  2 {                  f 0   [t  (k  1)T )]dt            f 0   (t  nT )dt}
                                                           k 1 ( k 1)T                                      nT

                                                                        1        n(n  1)
                                                  2 [( f 0  n T )t   t 2           T 2 ]                               nT  t  (n  1)T
                                                                        2           2
The subcarrier is described as:
                                                                           It (t )  It 0  Itm cos t (t )
where I t 0 is the direct current (DC) in subcarrier, I tm is the amplitude of the alternating current (AC) in subcarrier, f0
is the frequency of the carrier, F is the difference between the start and stop frequencies, T is the modulation cycle,
     F
        is the slope in sawtooth modulation.
     T
2.2 Transmitted optical signal
The transmitted optical intensity is proportional with the current of the subcarrier in incoherent FM/CW laser detection.
The output instantaneous intensity changes with the driving instantaneous current when using laser diode (LD).
Supporting output of the LD as TEM 00 single longitudinal mode laser, the transmitted optical signal can be written as:
                                                                                       P (t )  [P0  P cos t (t )]cos wt
                                                                                        t         t    tm                l

where Pt 0 is the direct component of the transmitted optical signal intensity, Ptm is the amplitude of alternating
component in the transmitted optical signal intensity, t (t ) is the envelope phase which is the instantaneous phase of the
subcarrier, wl  2 fl is the light circular frequency. The transmitted optical signal is a kind of amplitude modulation
(AM) with frequency modulation (FM) subcarrier, so this kind of modulation is also named as FM-AM modulation. The
transmitted optical signal is shown in figure 3.




                                               Figure 3: Transmitted optical signal in incoherent FM/CW laser detection

2.3 Received optical signal
The received optical signal can be treated as the phase-delay signal if distance from sensor to target is far enough to
                                                                                                                                        t

consider the target as a point. The distance from sensor to the target yields: R(t )  R0   v(t ' )dt ' , round-trip time delay is:
                                                                                                           the
                                                                                                                                        0
                            t
                           2 v(t )dt
                                 '      '

           2R(t )
 (t )            0     0
                                            , the received optical signal is written as:
            C                   C
                                                                                Pr (t )  [ Pr 0  Prm cos r (t )]cos wl (t   )
                                                                                        [ Pr 0  Prm cos t (t   )]cos wl (t   )
where Pr 0 is the direct component of the received optical signal intensity, Ptm is the amplitude of alternating
component in received optical signal intensity. They yield the expression below:
                                                                                                     Pr 0 Prm
                                                                                                             k
                                                                                                     Pt 0 Ptm
where k is a reduction factor considering the transmitting system attenuation, the atmospheric transmission
attenuation, the target reflection factor and the receiving system attenuation. The factor characters the power reduction in
incoherent FM/CW laser detection.

2.4 Mathematical analysis for IF signal
The IF signal can be produced by multiplication of alternating component in transmitted optical signal and alternating
component in received optical signal. The IF signal is produced by:
                                                 Ptm cos t (t ) cos wl t  Prm cos r (t )e cos wl (t   )
                                                  Ptm Prm [cos t (t )  cos r (t )]  [cos( wl t )  cos wl (t   )]
                                                            1                        1                           1         1
                                                  Ptm Prm { cos[t (t )  r (t )]  cos[t (t )  r (t )]}  [ cos wl  cos wl (2t   )]
                                                            2                        2                           2         2
where     (t )  t (t )  r (t ) is the envelope phase difference between transmitted and received signals, wl is the optical
phase difference between transmitted and received signals. The expressions of envelope phase difference are different in
different observational time regions:
                                                                                  nT
                                                                                2  v(t ' )dt '
When observational time region yields nT   0                                   0
                                                                                                   t  (n  1)T   , the expression is:
                                                                                       C
                                                              (t )  t (t )  r (t )
                                                                        t (t )  t (t   )
                                                                                               1       n(n  1)
                                                                        2 [( f 0  n T )t   t 2            T 2 ]
                                                                                               2          2
                                                                                                    1               n(n  1)
                                                                       2 [( f 0  n T )(t   )   (t   ) 2           T 2 ]
                                                                                                    2                   2
                                                                                                    1
                                                                        2 [  (t  nT )  f 0   2 )]
                                                                                                    2
                                                                                              1 2
                                                                        2 (  tm  f 0   )
                                                                                              2
                                                                              1 d                   d      d          d
The envelope difference frequency is                                    fi                 tm       f0              , the                   optical difference frequency is
                                                                             2 dt                    dt       dt         dt
          1 dwl      d
 fi*             fl          .
         2 dt        dt
                                                                                                      nT
                                                                                                   2  v(t ' )dt '
When observational time region yields nT  t  nT   0                                              0
                                                                                                                     ,the envelope difference frequency is written as :
                                                                                                           C
          1 d                   d      d      d
 fi              F     tm     f0                      .
         2 dt                    dt      dt      dt

2.5 IF signal in different motion states
The motion between sensor and the                                               target           is        uniform       rectilinear             motion        which              is   described            as
                            2R(t ) 2( R0  v0t )
                                                  , d   0 . The envelope difference
                                                            2v                                                                                                           2v
R(t )  R0  v0t ,  (t )                                                                                                         frequency is                fi    0 [ f0   (tm   )]             or
                              C             C         dt     C                                                                                                            C
                  2v0                                                                     2v
fi  F           [ f 0   (tm   )] .The optical difference frequency is fi*   fl 0 .
                   C                                                                       C
The motion between sensor and the target is uniformly accelerated rectilinear motion which is described
                          1              d   2(v  at )
as: R(t )  R0  v0t  at 2 ,                0               .The envelope difference frequency is:
                          2              dt      C
              2(v0  at )                                                    2(v0  at )                                                                                                      2(v0  at )
 fi                   [ f 0   (tm   )]   or fi  F                          [ f0   (tm   )] . The        optical difference frequency is fi*   fl                                         .
                  C                                                              C                                                                                                                C
The motion between sensor and the target is variable accelerated rectilinear motion which is described as:
                t 
                                         d
                                                      t                                                                                                                       t
                                               2                                                                                                                           2
                                                                                                                                                                           C
R(t )  R0   [  a(  )d  ]d     ,         a (t ' )dt '    . The envelope difference frequency is:                                                     f i          a(t ' )dt ' [ f 0   (tm   )]
                0 0
                                         dt    C0                                                                                                                            0
                           t                                                                                                                 t
                        2                                                                                                                 2
                        C                                                                                                                C
or   f i  F           a (t ' )dt ' [ f 0   (tm   )] . The   optical difference frequency is                        fi*   fl       a(t ' )dt ' .
                          0                                                                                                                 0




                                                                        3. DOPPLER FREQUENCY

3.1 Photoelectric detection and mixing
The output current of the photodiode is proportional to the incident optical power focused onto the active region of
photodiode. The output photocurrent yields the expression when using avalanche photo diode (APD):
                                                                                                           e
                                                                                           iAPD  M         Pr (t )
                                                                                                       hv
                                                                                                   Mir (t )
                                                                                                   iAPD 0  iAPDm cos r (t )
The photocurrent is fed into a signal pretreatment circuit including a block circuit and a transconductance amplifier. The
pretreatment circuit converts the photocurrent into an amplified voltage waveform. The amplitude of the signal matches
with the subcarrier signal which is one of the inputs of the mixer. The amplified alternating voltage signal yields:
                                                                                         urm  G pre MiAPDm cos r (t )
                                                                                                  U rm cos r (t )
The amplified alternating voltage signal is fed into the mixer as the radio frequency (RF) and the alternating portion of
the subcarrier is fed into the mixer as the local oscillator (LO). Output of the mixer is IF signal between the RF and LO
though a LPF. The IF signal is written as:
                                                               1
                                                           um  k1U tmU rm cos[t (t )  r (t )]
                                                               2
where k1 is the mixing efficiency. The IF instantaneous phase is the envelope phase difference between transmitted
signal and received optical signals.

3.2 Doppler frequency in IF signal
The Doppler frequency of IF signal is different in different motion states. The  or F   is difference frequency
caused by the distance from sensor to target. The expression of Doppler frequency is decided by the observational time
region. Doppler frequency in IF signal in uniform rectilinear motion yields:
                                                     2v0                                  2v0 nT
                                                     C [ f 0   (tm   )]
                                                    
                                                                                      0 
                                                                                             C
                                                                                                    tm  T
                                               fi  
                                                     2v0 [ f   (t   )]                        2v nT
                                                                                     0  tm   0  0
                                                     C 0
                                                    
                                                                    m
                                                                                                      C
Doppler frequency of IF signal in uniform rectilinear motion yields:
                                                                                                  1
                                          2(v0  at )                                  2(v0 nT  an 2T 2 )
                                                      [ f 0   (tm   )]      0              2          tm  T
                                               C                                                 C
                                    fi  
                                                                                                          1
                                                                                                 2(v0 nT  an 2T 2 )
                                          2(v0  at ) [ f   (t   )]          0  tm   0            2
                                         
                                              C
                                                          0        m
                                                                                                          C


                                                            4. EMULATION

Emulating Doppler frequency to estimate the error in distance-meauring lidar with MATLAB. Doppler frequency relates
to the distance and velocity between sensor and the target. The distance relates to the motion states and time. We emulate
the Doppler frequency VS. time in uniform rectilinear motion and uniformly accelerated rectilinear motion and Doppler
frequency VS. time and velocity in these motions mentioned above.




                                                  a                                             b
                                 Figure 4: (a) is Doppler frequency in uniform rectilinear motion;
                               (b) is Doppler frequency in uniformly accelerated rectilinear motion




                      Figure 5: (a) is Doppler frequency VS. time and velocity in uniform rectilinear motion;
                    (b) is Doppler frequency VS. time and velocity in uniformly accelerated rectilinear motion
                                                                                                  nT
                                                                                                 2  v(t ' )dt '
There are some dots meaning the observational time region nT  t  nT   0                       0
                                                                                                                   is quite short in Figure 4 and
                                                                                                       C
Figure 5.The slope is variable in Figure 4 and Figure 5 which means the Doppler frequency relates to the velocity. The
Doppler frequency doesn’t change too much in different velocity in Figure 5. We can estimate the maximum Doppler
frequencies in two of the motions are about 3 to 4 kilohertz at the speed of 1.3 Mach (about 450m/s), and the IF signal is
about 250kHz if the distance is about 2 kilometers. Ignoring Doppler frequency, the relation between distance and IF
signal yields R  TC fi . Considering Doppler frequency in IF signal, the relation between distance and IF signal yields
                      2F
     TC                                                        R   '
                                                                       R      fd
R'      ( fi   f d ) . The   error rate can be written as                        , we can estimate the maximum error rate is        3% .
     2F                                                               R'    fi f d


                                                           5. DISCUSSIONS

Doppler frequency in incoherent FM/CW laser detection relates to the distance and velocity between sensor and the
target, the frequency of the carrier f0 and difference between the start and stop frequencies F . The Doppler
frequency is a kind of error signal in distance-measuring lidar which gets the distance information only. The emulations
show Doppler frequencies in the limiting case. The Doppler frequency in incoherent FM/CW laser detection which is
irrelevant with the optical frequency is not as immense as the one in coherent FM/CW laser detection, and the error rate
is correspondingly smaller. Therefore, the incoherent FM/CW laser detection is more suitable for distance-meauring
lidar.

                                                            REFERENCES

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