1. Introduction
 The statement of a maximum laser irradiance to prevent sensor damage:Maximum Permissible Exposure for a Sensor, MPE_{S}
 Statement of a hazard distance corresponding to the MPE_{S}:Nominal Sensor Hazard Distance, NSeHD
 Statement of a laser irradiance that corresponds to a certain dazzle level:Maximum Dazzle Exposure for a Sensor, MDE_{S}
 Statement of a hazard distance corresponding to the MDE_{S}:Nominal Sensor Dazzle Distance, NSeDD
 The possibility to calculate the size of a dazzle spot at the imaging sensor depending on the parameters of laser source, camera lens and imaging sensor.
 Equivalent to laser safety calculations for the human eye, the values of MPE_{S} and MDE_{S} shall be stated for the position of the entrance aperture of the camera lens. In this case, a user can position a power meter at a wellaccessible place to compare calculated exposure values with the laser irradiance.
 Since users, who are not experts in the field, should also be able to be perform such calculations, closedform expressions shall be derived containing only wellknown operations and functions. The equations should be as simple as possible but still sufficiently accurate. In any case, the necessity of performing calculations using a computer should be avoided because of equations that can only be solved numerically.
 As far as possible, the equations shall include only standard parameters that are usually specified by the manufacturer of laser source, camera lens or imaging sensor.
2. Dazzle Scenario
3. Estimation of the Focal Plane Irradiance
3.1. Airy Diffraction Pattern
 As shown in Figure 3, the period of the oscillations of the irradiance profile is in the order of some micrometers (the radius of the first dark ring is $1.22\text{}\mathsf{\lambda}\mathrm{F}$). The pixel size of most imaging sensors is typically larger than 3 µm. Thus, the camera image will show an averaged irradiance pattern.
 As mentioned in the introduction, scattering of light at the optical elements of the camera lens has major influence on the size of the dazzle spot for high laser power. Especially in the wings of the dazzle spot, the scattered component dominates the irradiance distribution.
 Aberrations reduce the contrast of the irradiance oscillations.
 Laser systems may show fluctuations of laser power and have jitter in laser beam pointing, which additionally blurs the Airy diffraction pattern in the camera image.
 In real situations, the laser system and/or the sensor system may move, for example, due to vibrations.
 On long distances between laser and camera sensor, the atmospheric turbulence will cause an additional blur to the laser dazzle spot.
3.2. Diffraction Pattern of a Truncated Gaussian Beam
3.3. Stray Light Irradiance
3.4. Aberrations
 One goal of the laser safety calculations is to estimate the onset of laser damage. By neglecting the aberrations, a higher peak irradiance is estimated, which leads to a lower value of the calculated MPE_{S}. This can be interpreted as a safety factor, equivalent to the MPE for the human eye. The MPE for the human eye is derived from experimentally estimated ED_{50}values for eye damage, and is defined as a value that is usually a factor of 10 below these ED_{50} threshold values.
 Regarding laser dazzle of sensors, the other aim of the laser safety calculation is to estimate the dazzle spot size. In this case, the aberrations have a minor influence, since the size of the dazzle spot at larger dazzle levels is mainly caused by the stray light. Only for very low laser powers, slightly above the onset of laser dazzle, this assumption will cause some error in the dazzle spot size by neglecting aberrations.
 For commercial camera lenses, information on aberrations is typically provided, if at all, in form of diagrams. To transfer this information to values is complex.
 The treatment of aberrations using analytical expressions would increase the complexity of the equations for laser safety calculations a lot.
3.5. Total Focal Plane Irradiance
4. Laser safety Calculations for Sensors
4.1. Maximum Permissible Exposure for a Sensor
4.2. Maximum Dazzle Exposure for a Sensor
4.3. Hazard Distances
4.4. Size of the Dazzle Spot
5. Parameters for Laser Safety Calculations
 the scatter parameters of the camera lens,
 the damage threshold of the imaging sensor and
 the saturation threshold of the imaging sensor.
5.1. Scatter Parameters
5.2. Laser Damage Threshold
5.3. Laser Saturation Threshold
6. Calculation Examples
6.1. Example 1: Calculation of MPE_{S} and NSeHD for a Monochrome CMOS Sensor
 Laser: Laser Quantum Ventus 532 (continuouswave)
 (1)
 Maximum laser output power: ${P}_{0}>500\text{}\mathrm{mW}$
 (2)
 Wavelength: $\lambda =532\text{}\mathrm{nm}$
 (3)
 Beam diameter at the camera lens: (1/e^{2}) ${d}_{86}=3\text{}\mathrm{mm}$
 Camera lens: Qioptiq ApoRodagonN 4.0/80
 (1)
 Focal length: $f=80\text{}\mathrm{mm}$
 (2)
 fnumber: $F=5.6$
 (3)
 No. of optical elements: ${N}_{\mathrm{oe}}\le 8$ (the specification states up to 8 lenses for the lenses of the ApoRodagon series, depending on the focal length)
 (4)
 Focal spot diameter (1/e^{2}): ${d}_{\mathrm{spot}}=25.7\text{}\mathsf{\mu}\mathrm{m}$
 Results for 1 s exposure:
 (1)
 Occurrence of damage observed at a focal peak irradiance of 85 kW/cm^{2}
 (2)
 Estimated LIDT (for 1 s exposure): ${E}_{\mathrm{dam}}=73\frac{\mathrm{kW}}{{\mathrm{cm}}^{2}}$
6.1.1. MPE_{S}
6.1.2. NSeHD
 Divergence (1/e^{2}): $\mathsf{\Phi}=0.55\text{}\mathrm{mrad}$
 Beam diameter at the laser exit port (1/e^{2}): ${d}_{0}=1.25\text{}\mathrm{mm}$
6.2. Example 2: Calculation of Dazzle Spot Size and MDE_{S}
6.2.1. Dazzle Spot Size
 For large dazzle spots (Figure 12c,f), the dazzle spot size cannot be estimated assuming diffraction only. As stated in Section 3.4, the main contribution to the dazzle spot is scattered light. The green circle coincides roughly with the edge of the blue disk.
 When the diffraction and the scatter contribution are nearly equal (Figure 13e), there is some difference of the spot sizes for diffraction and scatter only to the numerically calculated dazzle spot size, but it keeps within limits.
6.2.2. MDE_{S}
6.3. Comparison of MPE_{S} and MDE_{S}
7. Conclusions
Funding
Conflicts of Interest
Appendix A. Comment on Camera Lenses
Approximation 2 used in Equation (30) Includes Two Simplifications
 The size of the beam at the different scattering surfaces of the camera lens is equal to the size of the entering beam.
 The size of the beam at the different scattering surfaces of the camera lens is constant.
Appendix B. Limits of Applicability
Appendix B.1. Applicability of Simplification S1 and Simplification S2
Parameter  Values 

$\lambda $  400 nm, 550 nm, 700 nm, 1000 nm 
$f$  10 mm, 50 mm, 100 mm 
$F$  1.0, 8.0, 22.0 
$s$  −1, −2, −3 
${b}_{0}$  0.01, 0.1, 1.0, 10.0 
$l$  10^{2}, 10^{3}, 10^{4} 
${N}_{\mathrm{ss}}$  2, 10, 20 
Appendix B.2. Applicability of Simplification S3
Appendix C. Spot Size Calculations
${\mathit{E}}_{\mathbf{laser}}\text{}(\mu \mathbf{W}/{\mathbf{cm}}^{2})$  ${\mathit{P}}_{\mathbf{in}}\text{}(\mu \mathbf{W})$  Calculated Dazzle Spot Radius (pixel)  Measured Dazzle Spot Radius (pixel)  $\mathbf{Dazzle}\text{}\mathbf{Level}\text{}\mathit{\u03f5}$  MDE_{S} (µW/cm^{2})  

${\mathsf{\Theta}}_{\mathbf{dazzle},\mathbf{d}}$  ${\mathsf{\Theta}}_{\mathbf{dazzle},\mathbf{s}}.$  ${\mathsf{\Theta}}_{\mathbf{num}}$  
2.94 × 10^{−04}  7.06 × 10^{−04}  2  0  2  2  0.004  1.40 × 10^{−04} 
3.38 × 10^{−03}  8.11 × 10^{−03}  5  0  5  3  0.009  1.33 × 10^{−03} 
4.99 × 10^{−02}  0.120  11  0  12  8  0.02  1.60 × 10^{−02} 
0.151  0.362  16  0  19  17  0.04  0.109 
0.245  0.587  19  7  25  18  0.05  0.122 
0.705  1.69  27  34  41  32  0.08  0.410 
1.77  4.25  37  56  63  46  0.12  0.882 
3.97  9.52  49  82  89  72  0.19  2.41 
7.91  19.0  61  111  118  103  0.27  5.68 
9.96  23.9  66  122  130  115  0.31  7.44 
12.5  30.1  71  135  142  131  0.35  10.3 
15.8  37.9  77  148  156  152  0.40  14.8 
25.0  60.1  90  179  188  204  0.54  31.0 
31.5  75.7  97  197  206  232  0.62  42.8 
39.7  95.2  105  216  225  n.m.  n.m.   
51.1  123  114  239  249  n.m.  n.m.   
60.0  144  120  256  266  n.m.  n.m.   
70.5  169  127  273  283  n.m.  n.m.   
99.6  239  143  314  325  n.m.  n.m.   
125  301  154  344  356  n.m.  n.m.   
158  379  166  377  390  n.m.  n.m.   
379  910  223  536  551  n.m.  n.m.   
488  1170  242  594  609  n.m.  n.m.   
674  1620  270  675  692  n.m.  n.m.   
${\mathit{E}}_{\mathbf{laser}}\text{}(\mu \mathbf{W}/{\mathbf{cm}}^{2})$  ${\mathit{P}}_{\mathbf{in}}\text{}(\mu \mathbf{W})$  Calculated Dazzle Spot Radius (pixel)  Measured Dazzle Spot Radius (pixel)  $\mathbf{Dazzle}\text{}\mathbf{Level}\text{}\mathbf{\u03f5}$  MDE_{S} (µW/cm^{2})  

${\mathsf{\Theta}}_{\mathbf{dazzle},\mathbf{d}}$  ${\mathsf{\Theta}}_{\mathbf{dazzle},\mathbf{s}}.$  ${\mathsf{\Theta}}_{\mathbf{num}}$  
5.45 × 10^{−04}  1.31 × 10^{−03}  3  0  3  3  0.008  4.44 × 10^{−04} 
5.20 × 10^{−03}  1.25 × 10^{−02}  7  0  7  5  0.01  1.96 × 10^{−03} 
5.97 × 10^{−02}  0.143  15  0  15  13  0.03  3.53 × 10^{−02} 
0.161  0.386  21  0  22  20  0.05  0.119 
0.255  0.611  24  0  27  22  0.06  0.149 
0.611  1.47  32  17  40  36  0.09  0.479 
1.47  3.52  43  44  59  53  0.14  1.16 
2.67  6.40  52  62  76  73  0.19  2.40 
5.32  12.8  66  88  102  104  0.28  5.64 
6.40  15.4  70  96  110  113  0.30  6.85 
8.06  19.3  76  106  120  126  0.33  8.96 
10.1  24.3  82  118  132  151  0.40  14.1 
16.1  38.6  95  143  159  196  0.52  27.4 
20.2  48.6  103  158  174  223  0.59  37.91 
24.9  59.7  110  172  189  n.m.  n.m.   
31.3  75.2  119  190  207  n.m.  n.m.   
39.5  94.7  129  209  227  n.m.  n.m.   
46.4  111  136  223  241  n.m.  n.m.   
59.7  143  148  247  267  n.m.  n.m.   
75.2  180  159  272  292  n.m.  n.m.   
94.6  227  172  298  320  n.m.  n.m.   
227  545  230  425  450  n.m.  n.m.   
286  686  249  466  493  n.m.  n.m.   
423  1010  283  546  574  n.m.  n.m.   
${\mathit{E}}_{\mathbf{laser}}\text{}(\mu \mathbf{W}/{\mathbf{cm}}^{2})$  ${\mathit{P}}_{\mathbf{in}}\text{}(\mu \mathbf{W})$  Calculated Dazzle Spot Radius (pixel)  Measured Dazzle Spot Radius (pixel)  $\mathbf{Dazzle}\text{}\mathbf{Level}\text{}\mathbf{\u03f5}$  MDE_{S} (µW/cm^{2})  

${\mathsf{\Theta}}_{\mathbf{dazzle},\mathbf{d}}$  ${\mathsf{\Theta}}_{\mathbf{dazzle},\mathbf{s}}.$  ${\mathsf{\Theta}}_{\mathbf{num}}$  
5.18 × 10^{−04}  7.82 × 10^{−04}  2  0  2  5  0.007  4.42 × 10^{−03} 
4.94 × 10^{−03}  7.47 × 10^{−03}  5  0  5  8  0.01  2.17 × 10^{−02} 
5.67 × 10^{−02}  8.58 × 10^{−02}  11  0  11  14  0.02  9.80 × 10^{−02} 
0.153  0.231  16  0  16  17  0.02  0.188 
0.242  0.366  18  0  19  19  0.03  0.257 
0.581  0.878  25  0  26  26  0.04  0.603 
1.39  2.11  33  0  37  43  0.06  1.97 
2.54  3.83  40  9  48  57  0.08  3.89 
5.06  7.65  50  40  64  75  0.10  7.21 
6.08  9.19  54  47  70  80  0.11  8.45 
7.65  11.6  58  55  77  87  0.12  10.4 
9.64  14.6  63  64  85  94  0.13  12.4 
15.3  23.1  73  82  103  111  0.15  18.4 
19.2  29.1  79  92  113  122  0.17  23.5 
23.7  35.8  84  102  123  130  0.18  27.3 
29.8  45.0  91  114  135  149  0.20  38.6 
37.5  56.7  98  126  148  165  0.23  49.8 
44.1  66.6  104  136  157  181  0.25  63.2 
56.7  85.8  113  152  174  204  0.28  85.0 
71.4  108  122  168  191  236  0.32  124 
89.9  136  132  185  209  276  0.38  185 
216  326  176  266  294  459  0.63  680 
272  411  191  293  322  n.m.  n.m.   
402  607  217  343  375  n.m.  n.m.   
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$$\mathit{\lambda}$$

$$\mathit{F}$$

$$\mathit{\nu}$$

$${\mathit{r}}_{\mathbf{pi}}=\mathit{f}{\mathsf{\Theta}}_{\mathbf{pi}}\text{}(\mu \mathbf{m})$$

$${\mathit{d}}_{\mathit{spot}}/2\text{}(\mu \mathbf{m})$$


445  2.8  0.3  9.7  2.6 
1.0  1.8  1.1  
2.0  1.4  1.0  
11.0  0.3  37.9  10.4  
1.0  7.1  4.3  
2.0  5.5  4.0  
532  2.8  0.3  11.5  3.2 
1.0  2.2  1.3  
2.0  1.7  1.2  
11.0  0.3  45.3  12.4  
1.0  8.5  5.2  
2.0  6.6  4.8  
635  2.8  0.3  13.8  3.8 
1.0  2.6  1.6  
2.0  2.0  1.4  
11.0  0.3  54.1  14.8  
1.0  10.1  6.2  
2.0  7.8  5.7 
Camera Lens.  Focal Length  fNumber  No. of Lenses  Coating  Price (approx.) 

LINOS MeVisC 1.8/50  50.6  1.8  7  unk.  700 € 
Edmund Optics 54690 (Double Gauss)  50  4.0  6  MgF2  500 € 
Edmund Optics 67715  25  1.4  7  BBAR  500 € 
Edmund Optics 86410  100  2.8  7  BBAR  500 € 
SchneiderKreuznach Xenoplan 2.8/50  50.2  2.8  6  unk.  630 € 
Navitar NMV75  75  2.5  5  unk.  185 € 
Navitar NMV100  100  2.8  5  unk.  170 € 
Camera Lens  Scatter Parameters  

$\mathit{s}$  ${\mathit{b}}_{0}\text{}\left({\mathbf{sr}}^{1}\right)$  $\mathit{l}$ (rad)  
LINOS MeVisC 1.8/50  −2.50  1.18  5.29 × 10^{−03} 
Edmund Optics 54690 (Double Gauss)  −2.54  5.83  3.81 × 10^{−03} 
Edmund Optics 67715  −2.14  0.92  7.43 × 10^{−03} 
Edmund Optics 86410  −2.64  10.81  4.09 × 10^{−03} 
SchneiderKreuznach Xenoplan 2.8/50  −2.29  3.36  5.15 × 10^{−03} 
Navitar NMV75  −2.39  0.49  5.46 × 10^{−03} 
Navitar NMV100  −2.45  0.12  5.56 × 10^{−03} 
Mean  −2.42  3.24  5.25 × 10^{−03} 
Median  −2.45  1.18  5.29 × 10^{−03} 
Standard deviation  0.17  3.89  1.18 × 10^{−03} 
Coefficient of variation  0.07  1.20  0.22 
1on1 LIDT (kW/cm^{2})  

Imaging Sensor  Exposure Time (s)  
0.25  1  5  10  
CMOS, monochrome  75 ± 7  73 ± 15  56 ± 4  48 ± 3 
CMOS, color  56.7 ± 1.8  
CCD, monochrome  146 ± 9  118 ± 9  93 ± 19  95 ± 21 
CCD, color  14 ± 2  13 ± 2  11 ± 1  8.1 ± 0.8 
Parameter  Setup 1  Setup 2 

Camera  VRmagic VRmC12/BWPro  Allied Vision Mako G158B 
Imaging sensor  Aptina MT9V024  Sony IMX273 
No. of pixels  754 × 480  1456 × 1088 
Quantum efficiency $\eta $  0.48 (all wavelengths)  0.63 (488, 515, 561 nm) 0.56 (640 nm) 
Pixel size $p$ (µm)  6  3.45 
Fill factor $ff$  $\mathrm{ukn}.=1$  $\mathrm{ukn}.=1$ 
Exposure time ${t}_{\mathrm{exp}}$ (ms)  8.3  8.3 
Maximum pixel value $p{v}_{\mathrm{max}}$  255  255 
Saturation capacity $C$ (e)  6000  10500 
Camera lens  SchneiderKreuznach ApoXenoplan 2.0/352001  Kowa LM25NC3 
Focal length $f$ (mm)  35.1  25 
fnumber $F$  2  1.8 
No. of optical elements ${N}_{\mathrm{oe}}$  7  7 
Laser  Toptica iChrome MLEL  
Wavelength $\lambda $ (nm)  488 / 515 / 561 / 640  
Maximum laser power ${P}_{in}$ (mW)  1.6 / 0.7 / 1.6 / 1.0  0.96 / 0.42 / 0.89 / 0.61 
Beam diameter ${d}_{86}$ (cm)  16.8 / 16.6 / 15.9 / 16.0  
Test chart  Fractal test chart [48]  
Mean pixel value $p{v}_{\mathrm{mean}}$  93  63 
$\mathbf{Focal}\text{}\mathbf{Plane}\text{}\mathbf{Saturation}\text{}\mathbf{Irradiance}\text{}{\mathit{E}}_{\mathbf{sat}}\text{}(\mu \mathbf{W}/{\mathbf{cm}}^{2})$  

Sensor  $\mathbf{Wavelength}\text{}\mathit{\lambda}$  
488 nm  515 nm  561 nm  640 nm  
VRmagic VRmC12/BWPro  1.11  1.03  0.96  0.79 
Allied Vision Mako G158B  5.24  4.81  4.42  4.43 
${\mathit{E}}_{\mathbf{laser}}\text{}(\mu \mathbf{W}/{\mathbf{cm}}^{2})$  ${\mathit{P}}_{\mathbf{in}}\text{}(\mu \mathbf{W})$  Calculated Dazzle Spot Radius (pixel)  Measured Dazzle Spot Radius (pixel)  $\mathbf{Dazzle}\text{}\mathbf{Level}\text{}\mathit{\u03f5}$  MDE_{S} (µW/cm^{2})  

${\mathsf{\Theta}}_{\mathbf{dazzle},\mathbf{d}}$  ${\mathsf{\Theta}}_{\mathbf{dazzle},\mathbf{s}}.$  ${\mathsf{\Theta}}_{\mathbf{num}}$  
2.78 × 10^{−04}  4.20 × 10^{−04}  2  0  2  3  0.004  1.94 × 10^{−03} 
3.19 × 10^{−03}  4.82 × 10^{−03}  4  0  4  13  0.02  0.107 
4.72 × 10^{−02}  7.13 × 10^{−02}  9  0  10  18  0.02  0.263 
0.143  0.215  14  0  14  21  0.03  0.377 
0.231  0.349  16  0  17  23  0.03  0.451 
0.667  1.01  23  0  27  29  0.04  0.771 
1.67  2.53  31  27  42  44  0.06  1.92 
3.75  5.66  40  50  60  64  0.09  4.22 
7.48  11.3  51  72  82  80  0.11  7.12 
9.42  14.2  55  80  90  85  0.12  8.15 
11.9  17.9  59  89  99  91  0.13  9.67 
14.9  22.5  64  99  109  97  0.13  11.3 
23.7  35.7  75  121  132  111  0.15  15.6 
29.8  45.0  81  134  145  121  0.17  19.3 
37.5  56.6  87  148  159  131  0.18  23.0 
48.3  72.9  95  164  176  153  0.21  34.0 
56.7  85.7  100  176  188  169  0.23  43.6 
66.7  101  105  188  200  186  0.26  55.5 
94.2  142  118  216  230  222  0.31  87.1 
119  179  128  238  252  260  0.36  129 
149  225  138  261  276  298  0.41  182 
358  541  185  372  389  475  0.65  596 
461  696  201  412  430  n.m.  n.m.   
637  961  224  469  488  n.m.  n.m.   
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