Laser safety for electro-optical imaging systems: exposure limits and hazard distances

Abstract. Laser safety with regard to the human eye is a well-known topic. Everybody working with laser sources has to follow the long-established occupational safety rules to prevent people from eye damage by accidental irradiation. These rules comprise, for example, the use of laser safety eyewear and the calculation of the maximum permissible exposure (MPE) and its corresponding hazard distance, the nominal ocular hazard distance. At exposure levels below the MPE, glare effects may occur if the laser wavelengths are in the visible spectral range. The physical effects of laser dazzling on the human eye are described by a quite new concept, which defines the maximum dazzle exposure (MDE) and the corresponding nominal ocular dazzle distance (NODD). Triggered by the MDE/NODD concept, we investigated whether similar laser safety calculations could be performed for electro-optical imaging systems. In this publication, we will review our approach for laser safety calculations for such systems. We have succeeded in finding closed-form equations, allowing calculations of exposure limits to prevent electro-optical imaging systems from damage and/or dazzle. Furthermore, we found some interesting effects related to the corresponding hazard distances, which are also discussed.


Introduction
Anyone who works professionally with lasers knows the long-established occupational safety rules to prevent eye damage by accidental laser irradiation.These include, among others, the calculation of the maximum permissible exposure (MPE) and its corresponding hazard distance, the nominal ocular hazard distance (NOHD) 1,2 as well as the choice of proper laser safety eyewear. 3,4Quantities equivalent to MPE and NOHD but related to the reversible effect of laser eye dazzle were recently introduced by Williamson and McLin: [5][6][7] The maximum dazzle exposure (MDE) and nominal ocular dazzle distance (NODD).In the last years, we have attempted to define such quantities for electro-optical imaging systems and to derive equations to calculate those quantities, see Fig. 1.These are: 1. MPE for a sensor (MPE S ): the maximum applicable laser irradiance at the entrance aperture of the camera lens to prevent the image sensor from damage 2. nominal sensor hazard distance (NSeHD): the hazard distance corresponding to the MPE S 3. MDE for a sensor (MDE S ): laser irradiance at the entrance aperture of the camera lens that corresponds to a certain dazzle level (see definition in Sec.4.2) 4. nominal sensor dazzle distance (NSeDD): the hazard distance corresponding to the MDE S .
][10][11][12] This includes the derivation of equations to estimate exposure limits (ELs; Sec. 4) and dazzle spot sizes (Sec.5), approximations to estimate hazard distances (Sec.6), suggestions for values of nonstandard parameters (Sec.7), and the validation of this work by modeling and field trials (Sec.8).We will not go into details here but mainly summarize the results of our work.
A brief note on terminology: We use the term electro-optical imaging system for a combination of a camera lens and a camera.The most important component of the camera for further discussion is the image sensor.Sometimes, we denote an electro-optical imaging system simply as a sensor.

Objectives
When we started our work on this topic some years ago, we initially defined the objectives we wanted to reach.The main goal was to develop a tool for laser safety calculations for electrooptical imaging systems that can be applied also by users, who are not experts in this field, even though this may be associated with some limitation of accuracy.The objectives defined were: Objective 1. Equivalent to laser safety calculations for the human eye, the values of MPE S and MDE S shall be related to the position of the entrance aperture of the camera lens.
Objective 2. All equations to calculate the laser safety quantities shall be given as closedform expressions containing only commonly known operations and functions.
Objective 3.All equations to calculate the laser safety quantities should contain, as far as practical, only standard parameters of the involved devices (laser, camera lens, image sensor/ camera) and the underlying scenario (e.g., distance, atmospheric extinction).
The formulated objectives have the following background: Objective 1 allows the user to position a power meter at the typically easily accessible place in front of the camera lens to compare calculated ELs with the incident laser irradiance.Objective 2 ensures that users who do not have relevant experience in this field can still perform laser safety calculations for sensors.In principle, everybody should be able to perform laser safety calculations using a sheet of paper and a pocket calculator.Therefore, we want to avoid numerical calculations that can only be performed with the help of a computer.Finally, Objective 3 shall enable the user to perform such calculations for a wide range of electro-optical imaging systems, i.e., for different combinations of camera lenses and image sensors, without the necessity to measure unknown parameters beforehand.

Approach
To derive damage-related laser safety quantities for electro-optical imaging systems analogous to those of the human eye (EL MPE and hazard distance NOHD), we had to start from the laserinduced damage threshold (LIDT) of the image sensor, which is located at the focal plane of the camera lens.Such damage thresholds for imaging sensors are usually not known and have to be determined by appropriate measurements, e.g., see the work of Becker et al., 13,14 Théberge et al., 15 Burgess et al., 16 Westgate and James, 17 and Schwarz et al. [18][19][20] In the next step, we had to transfer the image sensor's damage threshold to the corresponding value at the position of the camera lens' entrance aperture to achieve Objective 1.However, this required finding out how the irradiance distribution of the threatening laser beam is related to the irradiance distribution in the focal plane of the camera lens.This issue was the biggest step in our efforts to define the laser safety quantities for sensors since there is not, as with the human eye, only one single kind of camera lens but many different ones.The irradiance distribution at the focal plane depends on the design and quality of the camera lens, comprising its scattering characteristics and image distortions.The scattering properties of camera lenses are usually only very rarely known and must otherwise be determined in time-consuming measurements with a dedicated setup; see, e.g., Ref. 10.
The same considerations are also valid for dazzle-related laser safety quantities.For example, the laser dazzle threshold can be defined as the irradiance, where the pixels of the imaging sensor start to saturate.Such saturation thresholds can be calculated easily from the specifications of the image sensor. 9However, to describe the extent of laser dazzle, i.e., the size of the dazzle spot on the image sensor, we again need the quantitative irradiance distribution at the focal plane to compare it with the saturation threshold.
In summary: the basis for achieving our goal was to set up a theoretical model (based on closed-form equations to achieve Objective 2) that quantitatively describes the irradiance distribution at the focal plane of a camera lens in the case of laser irradiation.For our application, the main focus of the theoretical model was that it is capable of describing the irradiance distribution in an appropriate manner for our laser safety calculations.There was no intent that the theoretical model describes the irradiance distribution perfectly accurate, so it may also be used for other purposes (e.g., stray light analysis in optics design).
Therefore, we concentrated on the estimation of the peak irradiance (which is crucial for laser damage) and the estimation of the large-scale irradiance distribution (which is crucial for the laser dazzle effect).This means that we included diffraction effects in the theoretical model regarding the peak irradiance and scatter/stray light effects for the large-scale irradiance distribution.The importance of including stray light for the calculation of the focal plane irradiance distribution was shown, for example, by Benoist and Schleijpen, 21 who modeled the size of laser dazzle spots for charge-coupled device (CCD) cameras.
However, we did not include aberrations in our theoretical model since aberrations have typically only an influence on the spatial irradiance distribution near the center of the laser spot at the image sensor as long as the imaging system is focused.The inclusion of aberrations is only required for an accurate description of very small laser dazzle spots in the order of some pixels in diameter, which is rather irrelevant for practical use.Regarding laser dazzle, our interest laid on the description of dazzle spots that affect a considerable amount of the sensor's field of view (FOV; >10%).Also, specific optical effects, such as diffraction spikes, have not been considered since they have an influence on the sensor image only in very limited areas and thus have little impact on the overall perceptibility of the scene.

Scenario
For our laser safety calculations, we assumed the scenario shown in Fig. 2. A laser emits a Gaussian beam characterized by the output power, P 0 ; the laser wavelength, λ; the output beam diameter, d 0 ; and the full angle divergence, Φ.The laser radiation illuminates an electro-optical imaging system consisting of a camera lens and a camera with an image sensor.The laser beam diameter at the camera lens is denoted by d 86 , which refers (in the case of Gaussian beams) to the two opposing positions at the radial irradiance profile, where the irradiance dropped to 1∕e 2 of the peak irradiance.
The camera lens is described by the focal length f and the diameter of the entrance pupil d ap .We would like to point out that the location and size of the entrance pupil and the rear principle plane as well as the beam paths in Fig. 2 are drawn for illustrative purposes only.Further parameters of the camera lens are the f-number F ¼ f∕d ap , the number of optical elements N oe , the transmittance T, and three scatter parameters s, b 0 (or alternatively b), and l, which will be described in more detail in Sec.3.1.
The ratio of the beam diameter d 86 to the diameter of the camera lens' entrance pupil d ap is called the truncation factor ν and, as we will see later, has a determining influence on the distribution of the laser light in the focal plane of the camera lens E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 1 ; 1 1 4 ; 5 6 5 ν ¼ d 86 ∕d ap ; (1) where the laser beam diameter d 86 in a distance z to the laser source can be calculated by ; t e m p : i n t r a l i n k -; e 0 0 2 ; 1 1 4 ; 5 2 9 Using the truncation factor ν, the fraction P in of the laser power P 0 that enters the lens can be calculated by E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 3 ; 1 1 4 ; 4 7 2 The image sensor is characterized by several parameters: the number of pixel columns and rows N col , N row , pixel size p, total quantum efficiency η, saturation capacity C, and the integration time t exp .We assume that the sensor system is well focused, i.e., the imaging sensor is placed at (or very near to) the focal plane of the camera lens.The onset of damage and dazzle is described by irradiance values E dam (LIDT) and E sat (saturation irradiance).
The attenuation of laser power by the atmosphere may be included by substituting P 0 with P 0 • expð−μzÞ, where μ is the atmospheric extinction coefficient.

Parameters
Table 1 summarizes all parameters that we use for our laser safety calculations.
According to Objective 3, our equations shall be based only on standard parameters that are typically specified by the manufacturers of lasers, camera lenses, or image sensors.However, besides such standard parameters, some of the aforementioned parameters are usually not specified or known: 1. the LIDT of the image sensor E dam 2. the saturation irradiance of the image sensor E sat and 3. the scatter parameters of the camera lens: s, b 0 , l.
In Sec. 7, we state values/equations for these nonstandard parameters that may be used, if measured values for a specific electro-optical imaging system are not available.

Estimation of the Focal Plane Irradiance Distribution
In our theoretical model, the incident power P in contributes to the focal plane irradiance distribution E fp by two components: (a) the scatter/stray light component E s and (b) the diffraction component E d : E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 4 ; 1 1 4 ; 1 1 5 where η d describes the fraction of the incident laser power that is diffracted and r is the radial coordinate.In Eq. ( 4), as well as in all subsequent equations, the dependency on the radial coordinate r (in the focal plane) can be replaced by the dependency on the viewing angle Θ using the relationship E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 5 ; 1 1 7 ; 1 5 4

Stray Light Component
To estimate the contribution of the stray light E s to the focal plane irradiance distribution E fp , we relied on the work of Peterson, 22 who published an analytical approach for this task using the three-parameter Harvey scatter model as a bidirectional scattering distribution function (BSDF).This type of BSDF describes the angular distribution of stray light caused by scatter from smooth surfaces of optical elements using three parameters, s, b 0 , and l.Other sources of stray light such as multiple reflections or scatter from the lens housing are not considered.For a detailed explanation of the BSDF and the meaning of the scatter parameters, we refer the reader to other publications, e.g., Ref. 23.
For our theoretical model, we applied some simplifications (see Ref. 9 for more details) to the work of Peterson to keep equations manageable for typical camera lenses with five or more optical elements.As a result, we calculate the stray light component by E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 6 ; 1 1 4 ; 6 4 0 where v Ã is defined by ; t e m p : i n t r a l i n k -; e 0 0 7 ; 1 1 4 ; 5 9 1 N ss is the number of scattering surfaces of the camera lens, which we assume is twice the number of optical elements N oe .The modified truncation factor ν Ã considers that the beam diameter within the camera lens cannot increase, if a laser beam, which is already much larger than the entrance aperture, would expand further, e.g., with increasing distance to the laser source.
The ratio of scattered power to the incident power (for a single scattering surface) is called the total integrated scatter (TIS) and can be calculated by 23 Please note: Eq. ( 8) uses an alternative scatter parameter b that has its origins in the two-parameter Harvey scatter model (see, e.g., Ref. 21).This scatter parameter b is linked to scatter parameter b 0 by the following equation: The quantity TIS is used to calculate the fraction η d of the incident power that is diffracted; see Eq. ( 4) The scatter parameters are wavelength dependent; the wavelength dependency is described by the equation 24 Reference 24 stated that these wavelength scaling laws are valid only "over a limited range of λ," but limits are not given.Wein, 25 for example, stated that the "wavelength scaling approximately predicts the scattering in the visible but not in the far infrared."

Diffraction Component
For the diffraction component E d , we assumed in our theoretical model a Gaussian beam and used for our calculations the work of Urey 26 regarding the diffraction pattern of a truncated Gaussian beam.This diffraction pattern can be imagined to look like something between the Airy diffraction pattern and a pure Gaussian distribution.The shape depends on the value of the truncation factor ν and consists of a central lobe that can be approximated by a Gaussian distribution and diffraction rings of lower power similar to the Airy diffraction pattern.
The central lobe is approximated by ; t e m p : i n t r a l i n k -; e 0 1 2 ; 1 1 4 ; 1 0 9 where the peak irradiance of the diffraction pattern is given by E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 1 3 ; 1 1 7 ; 7 2 4 and the spatial extent of the central lobe is calculated by 26 E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 1 4 ; 1 1 7 ; 6 7 5 Here, K is a spot size constant, which also depends on the truncation factor ν, see Ref. 26 for details E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 1 5 ; 1 1 7 ; 6 2 6 For a typical camera lens designed for the visible spectral range, the diffraction spot size d spot is usually in the order of some micrometers.Since the pixel size of common CCD or complementary metal-oxide-semiconductor (CMOS) image sensors is of the same order, the central lobe is usually not resolved by an image sensor.Thus, for our laser safety calculations, we relied on the peak irradiance E 0 ðνÞ only.
Outside the central lobe, the wings of the diffraction pattern are described by the mean of the diffraction ring irradiance, which is given by E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 1 6 ; 1 1 7 ; 4 9 1 Since Eq. ( 16) describes the mean of the diffraction ring irradiance, there is no oscillating term in the formula.Here, the diffraction ring irradiance E dr is a monotonically decreasing function with a 1∕r 3 dependence.

Focal Plane Irradiance Distribution
Some examples of radial irradiance profiles as calculated with Eq. ( 4) are plotted in Fig. 3.The three graphs show the normalized irradiance as a function of the radial coordinate r for different values of the truncation factor ν.For the calculations, the parameters listed in Table 2 were used.

Exposure Limits
After setting up the theoretical model for the focal plane irradiance distribution, the EL MPE S and MDE S could be derived.For the derivation of these two quantities, we did not employ the exact equation for E fp ðrÞ but used only the respective essential components.This means that we used the peak irradiance E 0 ðrÞ only for the derivation of the MPE S since the onset of laser damage depends on the peak irradiance.By contrast, the derivation of the MDE S is based on the irradiance of the diffraction rings E dr ðrÞ and the stray light irradiance E s since the spatial extent of the dazzle spot essentially depends only on these components; see Fig. 3.Only for the treatment of tiny dazzle spots, the irradiance of the central lobe E cl ðrÞ would be needed.As mentioned before, we considered this case as not relevant for practical use and focused on larger dazzle spots that comprise considerable amounts of the sensors FOV (>10%).

MPE for a Sensor
The MPE S can be calculated by equalizing the focal plane peak irradiance E 0 ðνÞ of Eq. ( 13) and the image sensor's damage threshold E dam .This leads to E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 1 7 ; 1 1 7 ; 1 5 4 Resolving for the maximum permissible laser power P max and calculating the corresponding laser peak irradiance by dividing From Eq. ( 18), we see that the MPE S depends on the truncation factor, which means that the MPE S also depends on the distance of the laser to the imaging system.This has a crucial impact on the calculation of the corresponding hazard distance, as we will explain in more detail in Sec. 6.
The lowest value of the MPE S (worst case) occurs for ν → ∞ E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 1 9 ; 1 1 7 ; 6 2 5 Equation ( 19) is applicable when the laser source is far away from the electro-optical imaging system and the laser beam overfills the aperture of the camera lens.In this case, no distance dependence exists.

MDE for a Sensor
For the human eye, the MDE is not just a single value but is stated for specific dazzle levels. 6,7he dazzle levels range from very low to low, medium, and high, which corresponds to angular dazzle fields of 2, 10, 20, and 40 deg, respectively.For electro-optical imaging systems, an equivalent definition of such default values for the dazzle field is not useful since the system's FOV changes with the focal length of the camera lens.
For an imaging system, we defined the dazzle level as the fraction ϵ of the system's FOV that is dazzled.This means, e.g., that for an incident irradiance of MDE S ðϵ ¼ 0.1Þ MDE S ðϵ ¼ 0.5Þ, and MDE S ðϵ ¼ 1.0Þ, a tenth of the FOV, half of the FOV, and the full FOV are dazzled, respectively.Figure 4 illustrates that approach.The fraction ϵ shall be understood as the diameter of the dazzle spot divided by the size of the longer side of the image sensor.
Using this definition, the angular radius of the dazzle spot is E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 2 0 ; 1 1 7 ; 3 9 5 where the FOV of the imaging system can be calculated by E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 2 1 ; 1 1 7 ; 3 4 9 We can find the MDE S for a specific dazzle level ϵ by equating the focal plane irradiance distribution E fp ðΘ ϵ Þ and the image sensor's saturation threshold E sat .As previously addressed, we approximate the focal plane irradiance distribution E fp ðΘ ϵ Þ for this case through the diffraction ring irradiance E dr ðrÞ and the stray light component E s ðrÞ.Thus, we obtain the following equation: E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 2 2 ; 1 1 7 ; 2 5 1 P max TλF Equivalent to the MPE S , the searched quantity MDE S can be derived by resolving for the maximum permissible laser power P max and calculating the corresponding laser irradiance by dividing P max by π ; t e m p : i n t r a l i n k -; e 0 2 4 ; 1 1 4 ; 6 6 5 As in the case of the MPE S , the equation simplifies for the case of ν → ∞, which gives the minimum value of the MDE S E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 2 5 ; 1 1 4 ; 5 8 8 MDE S;min ðϵÞ ¼ lim Equations ( 24) and ( 25) should not be used for vanishing values of ϵ, such as ϵ ¼ 0 or ϵ very close to zero, see Ref. 9 for more details.The value MDE S ðϵ ¼ 0Þ would correspond to the onset of laser dazzle.This onset can be estimated using Eq. ( 18) or ( 19) but replacing the focal plane damage threshold E dam by the saturation threshold E sat E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 2 6 ; 1 1 4 ; 4 9 4 E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 2 7 ; 1 1 4 ; 4 3 2

Dazzle Spot Size
The size of a dazzle spot Θ ds can be calculated by solving Eq. ( 22) for Θ E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 2 8 ; 1 1 4 ; 3 6 6 Unfortunately, there is no closed-form expression for this solution.This equation can be solved numerically using a computer.However, an approximate analytic solution can be given by solving the equations for the diffracted irradiance and the scattered irradiance individually and choosing the maximum value as the dazzle radius.
The equation to solve for the irradiance of the diffraction rings E dr is E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 2 9 ; 1 1 4 ; 2 8 1 and the equation to solve for the scattered irradiance E s is E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 3 0 ; 1 1 4 ; 2 3 1 Solving Eqs. ( 29) and (30) for Θ ds;d and, Θ ds;s respectively, we obtain E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 3 1 ; 1 1 4 ; 1 8 0 E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 3 2 ; 1 1 4 ; 1 2 4 and finally E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 3 3 ; 1 1 7 ; 7 3 6 Θ ds ≈ maxðΘ ds;d ; Θ ds;s Þ: (33)

Hazard Distances
Unfortunately, the calculation of hazard distances for electro-optical imaging systems is different compared with those for the human eye.As we can see from Eqs. ( 18) and ( 24), both the MPE S and the MDE S depend on the truncation factor ν.This implicates a dependence of these ELs on the distance between the laser source and imaging system since the beam diameter changes with distance and thus the truncation factor does.The conventional equations to calculate the human eye-related NOHD and NODD (neglecting atmospheric extinction) are E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 3 4 ; 1 1 7 ; 6 1 2 E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 3 5 ; 1 1 7 ; 5 5 8 Please note that the beam diameter d 0 and divergence Φ in Eqs.(34) and ( 35) have to refer to the opposite positions of the laser beam profile at which the irradiance has dropped to 1∕e of the peak irradiance.This is a common practice in laser safety, whereas, in this publication, we typically refer to the 1∕e 2 points unless otherwise stated.
Nevertheless, these equations cannot be applied in our case, even if the correct quantities are used since Eqs.(34) and (35) were derived for distance-independent ELs.The distance-dependent values of MPE S and MDE S are plotted in Fig. 5 for a generic electro-optical imaging system and laser source; the parameters assumed for this calculation are listed in Table 2.
For the graph in Fig. 5(a), the independent parameter is the distance z of the electro-optical imaging system to the laser source, whereas, for the graph in Fig. 5(b), we used the truncation factor p ∕d ap as the independent parameter.While the MPE S is a single curve (plotted in red), the MDE S is represented by a light blue band, which indicates the range of MDE S values for various dazzle levels ϵ ∈ ½0.1; 1.0.The MDE S curve for ϵ ¼ 0.5 is highlighted by a blue line; the upper and lower border of the blue band correspond to ϵ ¼ 1.0 and ϵ ¼ 0.1, respectively.In addition, the laser peak irradiance at the front face of the camera lens is shown as a green curve.Please note that the parameter E dam in Table 2 is related to the focal plane, whereas the ordinate of the plot is related to the front face of the camera lens.
In Fig. 5, we can recognize how the MPE S and MDE S vary with distance.Following the curves starting from large distance values, the ELs are quite constant, which correspond to the minimum values as given by Eqs. ( 19) and (25).For closer distances from ∼40 to ∼4 m, the ELs increase strongly (for the given example) with decreasing distance.Finally, for distances below ∼1 m, the ELs stay constant again.
A very important result concerns the hazard distance regarding the MPE S curve: Looking at the green curve, which indicates the laser peak irradiance, we can see that this curve intersects the MPE S curve twice.As long as the irradiance curve is below the MPE S curve, the imaging system is safe from damage.In the distance range where the irradiance curve is above the MPE S curve, the imaging system is not safe from damage.By the example of Fig. 5, we can recognize two intersection points ν hd , which cut the MPE S curve into three sections (see vertical dashed red lines): In the first section, the image sensor is safe from damage within the distance from the laser to the first intersection point.Between the two intersection points, the image sensor is not safe from damage, and in section three, beyond the second intersection point, the image sensor is again safe from damage.This means that the laser hazard zone for imaging systems may have a distance dependent upper as well as a lower limit.This is in contrast to laser hazard distances valid for the human eye.
We acknowledge that we intentionally chose the laser parameters in such a way that this effect is eye-catching in the example of Fig. 5.If we choose a sufficiently higher value of laser power P 0 , the irradiance curve would be shifted so much upward that the irradiance curve would intersect the MPE S curve only once.In our example, this is the case for the MDE S curves in Fig. 5.For each dazzle level ϵ (0.1, 0.5, or 1.0), the irradiance curve will intersect a specific MDE S curve only once.For another choice of system parameters (mainly the scatter parameter s) that may be different.

Derivation of Hazard Distances
The hazard distances NSeHD and NSeDD are defined by the fact that the laser irradiance at the system's camera lens E laser is equal to the MPE S and MDE S , respectively.Using EL as a synonym for either the MPE S or the MDE S , this means that E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 3 6 ; 1 1 4 ; 4 2 4 with E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 3 7 ; 1 1 4 ; 3 7 7 where ν hd is the truncation factor in the hazard distance z hd .Resolving Eq. (36) for z hd would result in the hazard distance NSeHD/NSeDD.Unfortunately, there is no solution in closed form, when we apply Eqs. ( 18) and ( 24) for the MPE S and the MDE S in Eq. (36), respectively.We recognized that the truncation factor ν is the determining parameter for the MPE S and MDE S .Thus, we chose in the first step the approach not to determine z hd but the corresponding truncation factor ν hd .This led to somewhat easier equations for the derivation of the NSeHD and the NSeDD, although these equations were still not resolvable analytically but only numerically.
Then, in a second step, the terms of the equations to be solved, which depended on the truncation factor ν, were substituted by approximations that allowed solutions for ν hd in closed form.We refer the reader to Ref. 8 for detailed explanations and will present here the results only.
Finally, having an approximation for ν hd , the corresponding hazard distance z hd (NSeHD or NSeDD) can then be calculated by E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 3 8 ; 1 1 4 ; 1 6 3

Nominal Sensor Hazard Distance
For the NSeHD, we found approximate estimates both for the upper and lower values of ν hd 8 E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 3 9 ; 1 1 7 ; 7 1 2 which can be used to calculate the upper and lower values of the laser hazard zone by E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 4 0 ; 1 1 7 ; 6 2 1 These approximations are well usable to determine hazard distances for sensor damage.In Ref. 8, approximate values of NSeHD calculated using Eqs.( 39) and (40) (NSeHD appr ) were compared with values computed by numerically solving Eq. (36) (NSeHD num ).The calculations were performed for a total of 5734 different parameter sets.The relative error of the approximate NSeHD values E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 4 1 ; 1 1 7 ; 5 2 3 δNSeHD ¼ ðNSeHD appr − NSeHD num Þ∕NSeHD num ; (41) was between [0, +0.5] in more than 90% of cases.The results are graphically displayed in Fig. 6: Fig. 6(a) as a histogram of the δNSeHD values (bin width 0.5) and Fig. 6(b) a plot of the relative error δNSeHD as a function of the numerically calculated value of the truncation factor ν hd .
In both graphs, red and blue color correspond to the upper and lower values of the hazard distance, respectively.We found an overestimation of values rather than an underestimation, which is good in terms of laser safety.However, it should be noted that these investigations were based on a limited database and the results may not be generally valid.

Nominal Sensor Dazzle Distance
The search for a closed-form expression to calculate the NSeDD was quite demanding.This is due to the fact that the equation that has to be solved to calculate the NSeDD is quite complex.As for the NSeHD, there may be an upper and lower limit of the laser hazard zone.Furthermore, for a specific dazzle level E, there may be areas within the laser hazard zone where the dazzle level ϵ is lower than that specific value: E < ϵ.This means that the size of the dazzle spot in the camera image may decrease within the laser hazard zone (for the specific dazzle level E) as the imaging system approaches the laser source.For more details, see Ref. 8. Due to this complexity, we decided to state only an approximation for the upper limit of the laser hazard zone in case of laser dazzle.Even for this limited case, we could not find a generally applicable approximation in a closed form.Nevertheless, we can give approximations for the NSeDD for two special cases: (a) For extended laser beams, i.e., the laser beam diameter is much larger than the aperture of the camera lens (ν ≥ ffiffi ffi 2 p ).(b) For beam diameters that are smaller than the lens aperture (ν < ffiffi ffi 2 p ) in combination with larger dazzle levels.However, to anticipate, the accuracy of these approximations is not as good as in the case of the NSeHD.
The NSeDD in the case of extended laser beams (ν ≥ ffiffi ffi 2 p ) can be approximately calculated by E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 4 2 ; 1 1 4 ; 6 2 8 For laser beams with a diameter smaller than the aperture of the camera lens (ν < ffiffi ffi 2 p ), the NSeDD can be approximated by E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 4 3 ; 1 1 4 ; 5 6 6 For Eq. ( 43), the following constraints have to be fulfilled: ; t e m p : i n t r a l i n k -; e 0 4 4 ; 1 1 4 ; 5 1 5 Both constraints imply that the dazzle level ϵ should be sufficiently large, particularly that the influence of the scatter component outweighs that of the diffraction component regarding the focal plane irradiance distribution.
Finally, the NSeDD can be calculated using Eq. ( 42) or ( 43) by E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 4 6 ; 1 1 4 ; 3 9 1 Equivalently to the NSeHD, we also investigated the accuracy of these approximations.The NSeDD was first estimated using the approximation of Eqs.(42) and (43).Since both equations give an approximation for the upper limit of the laser hazard zone, we chose the maximum of both values for further processing (NSeDD appr ).Second, the NSeDD was computed by numerically solving Eq. (36) (NSeDD num ).Subsequently, the relative error of NSeDD appr was calculated as before for the NSeHD E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 4 7 ; 1 1 4 ; 2 7 3 These calculations were performed for 2,292,924 different parameter sets.In Fig. 7, the results are plotted for the NSeDD.In these graphs, the red color corresponds to the approximation for ν ≥ ffiffi ffi 2 p calculated using Eq.(42).The blue color corresponds to the approximation for ν < ffiffi ffi 2 p calculated using Eq.(43).In the case of the NSeDD, we see a similar distribution in the histogram but not as good as for the NSeHD.Larger relative errors occur for the approximation for ν ≥ ffiffi ffi 2 p .We can recognize in the graph on the right-hand side that these cases mainly occur for values of the truncation factor ν hd ≈ ffiffi ffi 2 p but not exclusively.Furthermore, we can see that the relative error is negative for a considerable amount of the calculations.A negative relative error corresponds to an underestimation of the hazard distance.We conclude that Eqs. ( 42) and (43) have to be used with care for the estimation of hazard distances for sensor dazzle.
7 Nonstandard Parameters for Laser Safety Calculations 7.1 Damage Thresholds of Image Sensors Information on LIDTs of image sensors for continuous wave (cw) laser radiation in the visible spectral range is rare.Publications related to measured cw LIDT of CCD and CMOS cameras are, for example, Becker et al., 13,14 Théberge et al., 15 Burgess et al., 16 Westgate and James 17 and Schwarz et al. [18][19][20] Here, we refer to a publication by Schwarz et al. 18 and summarize these threshold values in Table 3. Please note that Schwarz et al. measured these values for specific image sensors (CCD sensor: Sony ICX098, CMOS sensor: Aptina MT9V024).Laser damage thresholds for other types of image sensors may vary, but the order of magnitude (10 − 100 kW∕cm 2 ) should be similar.

Saturation Thresholds of Image Sensors
The saturation threshold of an image sensor may be calculated using its technical specifications.At a specific irradiance E sat;pixel (i.e., the saturation irradiance), the number of photoelectrons μ e generated by the incident photons within the camera's exposure time t exp will equal the saturation capacity C of a pixel.Using this relation, the saturation irradiance of a pixel of size p can then be estimated by E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 4 8 ; 1 1 7 ; 2 9 5  Putting just the pixel saturation irradiance E sat;pixel from Eq. ( 48) into the MDE S equations [Eqs.( 24)-( 27)] would imply that the image sensor is illuminated by the dazzle laser only.In a real situation, the imaging system typically observes a scene, which means that the charge generated by a pixel is determined by the ambient light of the scene and the laser light.To meet this condition, we assume that the operator or the camera's automatic exposure control will set the exposure time to a level, such that the image sensor's mean pixel signal equals roughly half of the maximum pixel signal.Thus, applying a factor of 0.5 to Eq. ( 48), we obtain a first estimate for the saturation irradiance E sat E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 4 9 ; 1 1 4 ; 6 3 9 If the saturation level sl of a camera image is known in more detail, e.g., when using a specific test chart in a laboratory environment, the saturation threshold may be estimated more precisely by E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 5 0 ; 1 1 4 ; 5 6 3 where a saturation level sl ¼ 0 means not saturated and sl ¼ 1 means completely saturated.

Scatter Parameters
In our simple theoretical model in Sec. 3, the scatter component E s contributing to the focal plane irradiance distribution E fp depends on the scatter parameters s∕bb 0 and l of the camera lens used.
Unfortunately, these scatter parameters are usually not known for commercial off-the-shelf (COTS) camera lenses.Thus, we performed a series of measurements where we recorded images of the focal plane irradiance distribution for a selection of seven typical camera lenses with focal lengths ranging from 25 to 100 mm.We then derived radial irradiance profiles from the image data and subsequently fitted our Eq.( 4) to these irradiance profiles, where we used the quantities s∕bb 0 and l as fit parameters.The outcome of this work was an individual set of scatter parameters for each camera lens under test; see Table 4. Since these sets describe the corresponding camera lens as a whole (and not the scattering properties of the single scattering surfaces), we denote them as integrated scatter parameters.Details can be found in Ref. 10.We noticed that the integrated scatter parameters for the different camera lenses are of the same order of magnitude.Therefore, we additionally derived a generic set of integrated scatter parameters based on a statistical analysis of the individual sets.This generic set may be applied together with our equations to predict the laser dazzle of an electro-optical imaging system when a camera lens with unknown scatter parameters is used: E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 5 1 ; 1 1 4 ; 2 9 7 S ¼ −1.86; B ¼ 0.36 sr −1 ; B 0 ¼ 6.92 sr −1 ; L ¼ 2.04 mrad: (51) underestimation occurs.Nevertheless, we got an adequate agreement between our simplified theoretical model and the output of the corresponding FRED simulation.

Validation Using Data from Outdoor Measurements
During a field trial, an electro-optical imaging system at a distance of 660 m was dazzled on a slant path by cw laser radiation. 29Camera images were acquired for various laser wavelengths and camera exposure times.In the subsequent data analysis, dazzle spot sizes were assessed using overexposed pixel counting (OPC) and then compared to the theoretical predictions of the dazzle spot size according to Eq. (33).
As dazzler, we used the Toptica iChrome MLE multiwavelength laser device.This laser source comprises four different lasers (wavelengths 488, 515, 561, and 640 nm) with output powers ranging from 40 to 100 mW.The data of the laser device are listed in Table 5.The actually used laser output power P 0 was measured during the field trial, whereas the output diameter d 0 and full angle divergence Φ were measured in the laboratory using an M 2 measurement system (Thorlabs M2MS-BC106VIS/M).The laser beam diameter d 86 at the imaging system was calculated from these data using Eq. ( 2).For the atmospheric attenuation of the laser radiation, we assumed an extinction coefficient of μ ¼ 0.2 km −1 since the weather conditions were good.
The imaging system consisted of a Schneider-Kreuznach Xenoplan 2.8/50 COTS camera lens 30 mounted to an Allied Vision Mako G-223B NIR camera. 31The specifications of both devices are also listed in Table 5.The transmittance T of the camera lens was not known.Thus, we assumed a broadband antireflection coating with a reflection below <0.5% per surface, resulting in T ¼ ð1 − 0.005Þ 2•N oe ¼ 0.94.Furthermore, we intentionally did not measure the scatter parameters of this camera lens but simply used the generic set of integrated scatter parameters as stated in Eq. (51) to investigate their applicability.The data of the camera were taken from the datasheet, see Ref. 31.From the corresponding graph of the data sheet, we estimated a quantum efficiency of η ≈ 0.7 for the laser wavelengths used.In summary, our set of measured, specified, and estimated values represents a quite suitable mix for testing the usability of our approach.
In Fig. 9, some examples of camera images are shown for the laser wavelengths of 515 and 640 nm.The images were taken using three different exposure times of 100 μs, 1 ms, and 10 ms.
The data was processed according to the following way: 1. Calculation of a mean image: For each parameter setting (combination of laser wavelength and camera exposure time), a mean image from 20 consecutive frames of the video stream was calculated.2. OPC: The number of overexposed pixels in each mean image was counted.Using this number, the nominal size of the dazzle spot could be estimated for each mean image.We denote this quantity as nominal size since the method assumes a circular dazzle spot, which of course is not exactly true, particularly for measurements affected by the atmosphere.3. Estimation of the dazzle spot size using a saturation level of sl ¼ 0.5: The dazzle spot size according to our theoretical model was calculated using Eq.(33) by applying the parameters of Table 5 and estimating the saturation threshold according to Eq. (49), i.e., the saturation level of the whole image was assumed to be sl ¼ 0.5.4. Estimation of the dazzle spot size using an image-specific saturation level (ISSL): The dazzle spot size according to our theoretical model was calculated using Eq. ( 33) by applying the parameters in Table 5 and estimating the saturation threshold according to Eq. (50).In this case, the saturation level sl was determined from the mean images.For this, the mean value of strips of pixels at the left and right border of the mean images (width every 200 pixels) was divided by the maximum possible pixel value to estimate roughly the saturation level sl.
The three calculated dazzle spot sizes are drawn into the camera images of Fig. 9 as colored circles.Red circles indicate the dazzle spot size as calculated by the OPC method.The dazzle spot sizes according to our theoretical model are drawn as green and blue circles for a saturation level of 0.5 and the image-specific saturation level, respectively.
Looking at Fig. 9, we can see that our approach for laser safety calculations can indeed give a reasonable estimate for the dazzle spot sizes.In the case of the underexposed images (t exp ¼ 100 ms and t exp ¼ 1 ms), the use of ISSL leads to a very good agreement between the OPC result and the calculated dazzle spot size using the theoretical model.Using the saturation level sl ¼ 0.5 slightly overestimates the dazzle effect but not to an extraordinary degree.
In the case of the slightly overexposed images (t exp ¼ 10 ms), the theoretical prediction of the dazzle spot size for saturation level 0.5 fits quite well to the dazzled area of the camera image, whereas there is an overestimation using the image-specific saturation level.At first glance, one might say that the calculated dazzle spot size using the ISSL fits quite well with the dazzle spot size computed using the OPC method.However, one has to keep in mind that the camera image was slightly overexposed, which means that the OPC method also counts pixels overexposed by ambient light and not only by laser light.Thus, the dazzle spot size computed with the OPC method overestimates the laser effect.
The results for all laser wavelengths used at the field trial and for all values of camera exposure time are presented in the graphs of Fig. 10.For each laser wavelength, there is a graph that contains the results of the calculations according to the theoretical model and the OPC evaluation applied to the image data.The theoretical results comprise both the use of saturation level sl ¼ 0.5 and the ISSL.The graphs also confirm the above statements for the other laser wavelengths of 488 and 561 nm.

Fig. 1
Fig. 1 ELs for electro-optical imaging systems (MPE S and MDE S ) and corresponding hazard distances (NSeHD and NSeDD).Image reproduced with permission from Ref. 8.

Fig. 2
Fig. 2 Schematic view of a dazzle scenario.RPP, rear principal plane.Please note that the location and size of the apertures and pupils are drawn for illustrative purposes only.Image reproduced with permission from Ref. 8.

Fig. 5
Fig. 5 Example of MPE S (red line) and MDE S (light blue band corresponding to various values of dazzle level ϵ) for an arbitrarily chosen scenario as the function of the distance between the imaging system and laser source.In addition, the peak irradiance of the assumed laser source at the position of the lens is shown (green line).The parameters used for the calculations are listed in Table 2. Independent parameter: (a) distance z and (b) truncation factor ν. Graphs reproduced with permission from Ref. 8.

Fig. 6
Fig. 6 Relative error δNSeHD according to Eq. (41) regarding the approximate NSeHD calculations.The red and blue data points correspond to the upper and lower values of the hazard distance, respectively.(a) Histogram showing the frequency of the δNSeHD values.(b) δNSeHD as a function of the numerically estimated truncation factor ν hd .Image reproduced with permission from Ref. 8.

Fig. 7
Fig. 7 Relative error δNSeDD according to Eq. (47) regarding the approximate NSeDD calculations.The red and blue data points correspond to the approximations for large and low values of the truncation factor (ν ≥ ffiffiffi 2 p and ν < ffiffiffi 2 p ), respectively.(a) Histogram showing the frequency of the δNSeDD values.(b) δNSeDD as a function of the numerically estimated truncation factor ν hd .Image reproduced with permission from Ref. 8.

Fig. 9
Fig.9Dazzling of an electro-optical imaging system by laser radiation of the wavelength 515 and 640 nm.The images were acquired during an outdoor field trial on a slant path.Dazzle spot sizes computed by OPC are indicated by red circles.Furthermore, dazzle spot sizes as calculated by Eq. (33) using saturation levels of 0.5 (green circles) or image-specific saturation levels (blue circles) are shown.

Fig. 10
Fig. 10 Comparison of theoretically and experimentally determined dazzle spot sizes for different laser wavelengths.Abbreviations: ISSL, image-specific saturation level; sl, saturation level; OPC, overexposed pixel counting.

Table 1
Parameters used for the laser safety calculations.

Table 2
Parameters used for the example calculations of Figs.3 and 5.
E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 1 8 ; 1 1 7 ; 7 3 6

Table 4
Integrated scatter parameters for various commercial off-the-shelf camera lenses.Data reproduced with permission from Ref. 10.

Table 5
Device parameters used for the evaluation of the measurement data gained during the field trial.