# Safety Analysis for Laser-based Optical Wireless Communications: A Tutorial

Mohammad Dehghani Soltani, Elham Sarbazi, Nikolaos Bamiedakis, Priyanka
de Souza, Hossein Kazemi, Jaafar M. H. Elmirghani, Ian H. White, Richard V.
Penty, Harald Haas and Majid Safari M. D. Soltani and M. Safari are with the
Institute for Digital Communications (IDCOM), School of Engineering, the
University of Edinburgh, Edinburgh, UK. e-mails: {msoltan,
<EMAIL_ADDRESS>Sarbazi, H. Kazemi and H. Haas are with the LiFi
Research and Development Centre, Department of Electronic & Electrical
Engineering, University of Strathclyde, Glasgow, UK. e-mails: {e.sarbazi,
h.kazemi<EMAIL_ADDRESS>M. H. Elmirghani is with the School
of Electronic and Electrical Engineering, University of Leeds, Leeds, UK.
e-mails<EMAIL_ADDRESS>H. White is with the University of
Bath, Bath, UK. e-mail<EMAIL_ADDRESS>Bamiedakis and R. Penty are
with the Centre for Photonic Systems, Electrical Engineering Division,
Department of Engineering, University of Cambridge, Cambridge CB3 0FA, U.K.
<EMAIL_ADDRESS>author: M. D. Soltani,
<EMAIL_ADDRESS>
###### Abstract

Light amplification by stimulated emission of radiation (laser) sources have
many advantages for use in high data rate optical wireless communications. In
particular, the low cost and high-bandwidth properties of laser sources such
as vertical-cavity surface-emitting lasers (VCSELs) make them attractive for
future indoor optical wireless communications. In order to be integrated into
future indoor networks, such lasers should conform to eye safety regulations
determined by the international electrotechnical commission (IEC) standards
for laser safety. In this paper, we provide a detailed study of beam
propagation to evaluate the received power of various laser sources, based on
which as well as the maximum permissible exposure (MPE) defined by the IEC
60825-1:2014 standard, we establish a comprehensive framework for eye safety
analyses. This framework allows us to calculate the maximum allowable transmit
power, which is crucial in the design of a reliable and safe laser-based
wireless communication system. Initially, we consider a single-mode Gaussian
beam and calculate the maximum permissible transmit power. Subsequently, we
generalize this approach for higher-mode beams. It is shown that the
$M$-squared-based approach for analysis of multimode lasers ensures the IEC
eye safety limits, however, in some scenarios, it can be too conservative
compared to the precise beam decomposition method. Laser safety analyses with
consideration of optical elements such as lens and diffuser, as well as for
VCSEL array have been also presented. Skin safety, as another significant
factor of laser safety, has also been investigated in this paper. We have
studied the impacts of various parameters such as wavelength, exposure
duration and the divergence angle of laser sources on the safety analysis by
presenting insightful results.

###### Index Terms:

Eye safety, skin safety, maximum permissible exposure (MPE), laser source,
VCSEL, Gaussian beam, Hermite-Gaussian, Laguerre-Gaussian and optical wireless
communication.

##  I Introduction

The number of mobile users is increasing rapidly, with expected numbers to
exceed $5.7$ billion, that is anticipated to generate $79$% of the global data
traffic [1]. It is forecast that smartphones, laptops, tablets and wireless
sensors will struggle to get their required share of the radio frequency (RF)
spectrum without a significant capacity increase [2]. In the 5th generation
($5$G) and beyond of cellular networks, $1000$-fold capacity growth is
expected with respect to the 4th generation ($4$G) and long-term evolution
(LTE) wireless networks [3]. Therefore, the RF spectrum is becoming congested
and wireless devices have to cope with increased co-channel interference. As a
result, the data throughput of the devices is severely affected, and the
established connection may be poor.

Optical wireless communication (OWC) enables wireless connectivity harnessing
a huge spectrum, which is available in infrared, visible or ultraviolet bands
[4]. Moreover, OWC is license free thus leading to a cost-effective service
[5]. It is a solution to alleviate RF spectrum congestion for both indoor and
outdoor scenarios [6, 7]. OWC enables the creation of smaller communication
cells in the network, known as attocells [8] which is a vital key to unlock
the path to exponential capacity growth for indoor scenarios [9]. Indoor OWC
can offload heavy traffic loads from congested RF wireless networks, therefore
making the room available for extreme low-capacity streams such as in the
Internet of Things (IoT). It can ensure higher security compared to the RF
counterpart in the physical layer because optical signals do not penetrate
walls or opaque objects [10]. Outdoor applications of OWC, which are widely
referred to as free-space optical (FSO) communications, include satellite-to-
satellite communications, inter-building connections as well as satellite-to-
plane communications [11]. Underwater OWC is another application which has
recently gained much attention since it is able to provide a much higher
transmission bandwidth, and, as a result, much higher data rate in comparison
to acoustic and RF counterparts [12, 13]. Vehicle-to-vehicle (V2V)
communication via the optical spectrum is another recent area of focus, where
experimental results have confirmed its reliability even under heavy-fog
conditions [14]. All the aforementioned applications of OWC and its capability
to offer low-cost, high-speed, secure and reliable communication links have
made OWC an indispensable piece of future generations of communications.

Compared to light emitting diodes (LEDs), light amplification by stimulated
emission of radiation (laser) diodes provide a larger modulation bandwidth
that can achieve multi-Gb/s data rates, making them appealing to be used for
Tb/s indoor networks [15, 16, 17] required for the next generations of
wireless networks. A data rate of $4~{}\times~{}12.5$ Gbps has been
successfully demonstrated by experiments in [18] via optical fiber over the
entire beam footprint of about 1 m. Among different types of laser diodes,
vertical cavity surface emitting lasers (VCSELs) are one of the strongest
candidates to fulfil this role due to several outstanding features such as
[19]: high-speed modulation (bandwidths over $28$ GHz) [20, 21, 22], high
power conversion efficiency and low cost. Furthermore, high aggregate bit-
rates exceeding $100$ Gbit/s have been confirmed by means of VCSEL arrays
through experiments in many studies [23, 24, 25, 16, 17]. These attributes
make VCSELs noteworthy for many applications, particularly for high-speed
indoor networks [26]. However, one of the major challenges of utilizing laser
sources as transmitters is to meet eye safety regulations [27]. In the
following section, we review the relevant standards, research papers and
handbooks on laser safety.

IEC 60825 Part 1 [27] Part 2 [28] Part 3 [29] Part 4 [30] Part 5 [31] Part 8
[32] Part 12 [33] Part 13 [34] Part 14 [35] Part 17 [36] General laser safety
in the wavelength range $180$ nm to $1$ mm Guidance for the safe operation and
maintenance of optical fiber Guidance on the design, set-up and conduct of
laser displays which use high power lasers Specifies the requirements for
laser guards Ensures that each new or modified design complies with the
requirements of IEC 60825-1:2014 Guidance for operators on the safe use of
lasers classified as class 3B or class 4 Safety of point-to-point or point-to-
multipoint free space optical data transmission (180 nm to 1 mm) Guidance on
safe radiometric measurements and information about calculating AELs and MPEs
Guidance on best practice in the safe use of laser products that conform to
IEC 60825-1:2014 Safety measures against passive optical components and
optical cables used in high power OFCS Figure 1: IEC 60825 standard series on
the safety of laser products along with their main focus. ANSIANSI Z136.1
[48]ANSI Z136.2 [49]ANSI Z136.3 [50]ANSI Z136.4 [51]ANSI Z136.5 [52]ANSI
Z136.6 [53]ANSI Z136.7 [54]ANSI Z136.8 [55]ANSI Z136.9 [56] General safe use
of lasers for industry, military, research and development Safe use of optical
communication systems utilizing laser diode and LED sources Safe use of high
power lasers (class 3B and class 4) in health care Guidance on safe
measurements of the classification and hazard evaluation Safe use of lasers in
educational institutions Safe use of lasers outdoors, e.g., construction,
laser lightshows and military Guidance on the test methods and protocols used
to provide eye protective equipment Guidance on the safe use of lasers in
research, development, or testing Policies to ensure laser safety in both
public and private manufacturing environments Figure 2: Classification of ANSI
standards on laser safety including their main focus.

###  I-A Literature Review of Laser Safety

With the increased use of lasers in OWC for high-speed links, the safety of
lasers111We note that safety of optical fiber is out of the focus of this
paper. The IEC has released a standard for safety of optical fiber
communication systems (OFCS) (part 2 of IEC 60825) [28]. has emerged as a
topic of interest. There are several standards on the safety of lasers. IEC
60825 consists of 10 parts222EN 60825 is European requirements documents which
follow the IEC versions and BS EN 60825 is the UK national requirements
documents [37]. The USA has always had its own regulation on lasers which is
known as 21 CFR 1040.10 [38]. This is a USA government regulation rather than
a standard, which has not been updated for about 30 years and is consequently
very outdated compared to the IEC / EN standard., under the general title
safety of laser products. Fig. 1 illustrates the principle target of each
part. The most relevant parts to OWC design with lasers are part 1 [27] and 12
[33], where general laser safety and the safety of free space optical
communications are specified respectively. In addition, the American national
standards institute (ANSI) has published a 9 part ANSI Z-136 series of
standards. This series, along with an outline of the primary focus of their
work, are presented in Fig. 2. Parts 1 and 2 are the most relevant with
regards to the design of safe optical wireless links, where the safe use of
lasers and the safety of fixed terrestrial point-to-point free-space links are
discussed, respectively. The international commission on non-ionizing
radiation protection (ICNIRP) has specified the maximum level of exposure to
coherent laser sources for wavelengths between $180$ nm to $1$ mm [39]333This
includes the revision of ICNIRP on maximum levels of exposure laser radiation
for wavelengths between 400 nm and 1.4 $\mu$m, which was released in October
2000.. Table I summarizes the most relevant standards on eye safety for laser
products. It is worth mentioning that ICNIRP have determined exposure limits
for the whole electromagnetic spectrum including static magnetic fields (SMF)
[40], static electric fields (SEF) [41], low frequencies (LF) from $1$ Hz to
$100$ kHz [42], RF electromagnetic fields (EMF) from $100$ kHz to $300$ GHz
[43], Infrared from $780$ nm to $1$ mm [44, 39, 45], visible spectrum from
$380$ nm to $780$ nm [44, 39] and ultraviolet (UV) from $100$ nm to $400$ nm
[46, 47]. Fig. 3 illustrates the ICNIRP classification on limits of exposure
for the electromagnetic spectrum.

TABLE I: Related standards on eye safety for laser products. Year | Standard | Remark
---|---|---
2014 | IEC 60825-1 [27] | General laser safety
2019 | IEC 60825-12 [33] | Free space optical communications
2014 | ANSI Z-136.1 [48] | Safe use of lasers
2012 | ANSI Z-136.2 [49] |  | Safe use of optical communications systems utilizing laser
---
diodes or LEDs, including end-to-end optical fiber based
links, fixed terrestrial point-to-point free-space links, or a
combination of both.
2013 | ICNIRP [39] |  | Maximum levels of exposure to laser radiation for wavelengths
---
between 180 nm and 1000 $\mu$m.
ICNIRP SMF (0 Hz) SEF (0 Hz)
[40, 41] LF
(1 Hz-100 kHz)
[42] RF EMF
(100 kHz- 300 GHz)
[43] Infrared
(780 nm- 1 mm)
[44, 39, 45] Visible
(380- 780 mm)
[44, 39] UV
(100- 400 mm)
[46, 47] Figure 3: Classification of ICNIRP guidelines on limits of exposure.

Laser safety has been discussed in plenty of research papers with different
applications such as industry, medicine, military, education and so on since
the late 1960s [57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70]. The
main contributions and achievements of these studies are outlined in Table
LABEL:table:relevant_work. There are many research papers on laser safety,
however, we have included relatively more recent papers in Table
LABEL:table:relevant_work, in which important factors such as MPE, exposure
duration and subtense angle have been evaluated. There are also several
handbooks on general techniques for the safe use of lasers in different
environments such as medicine, industry, education, outdoors and so on. Some
of the well-know books on laser safety are presented in Table III. They mainly
focus on the safety analysis for ideal Gaussian beams, the radiation hazards,
protective techniques and laser classifications.

TABLE II: Selected papers on laser safety. Year | Paper | Main contribution/achievement
---|---|---
1968 | Makous and Gould [57] |  | Development of a theoretical estimate of retinal irradiance
---
for threshold damage
1978 | Avdeev _et al._ [58] |  | Determination of MPEs for the wavelength of 1.54 $\mu$m
---
and pulse durations of $40$ ns and $1$ ms to be $0.16$ J/cm2
and $0.3$ J/cm2, respectively.
1979 | Dmitriev _et al._ [59] |  | Possibility of nonlinear effect and damage on the retina
---
for ns pulsed lasers.
1985 | Birngruber _et al._ [60] |  | A thermal damage model for retinal lesions in accordance
---
with experimental measurements in the regime of $1$ ms to
$300$ ms.
2001 | Galbiati _et al._ [61] |  | Determination of the location and size of the apparent
---
source for laser beams.
2011 | Schulmeister _et al._ [62] |  | The dependency of the maximum angular subtense on
---
exposure durations.
2013 | Gustafsson [63] | Atmospheric influence on the NOHD has been evaluated.
2015 | Heussner _et al._ [64] |  | A prediction model for ocular damage is demonstrated,
---
which is validated for long-term exposure.
2018 | Ramos _et al._ [65] |  | Experimental measurements of nano seconds laser pulses
---
to the mammalian eye are presented.
2019 | Heussner _et al._ [66] |  | A software based on the photochemical, thermomechanical
---
and photomechanical damages is developed to evaluate the
eye safety of laser products.
2019 | Jiao _et al._ [67] |  | A series of experiments were conducted to determine the
---
retinal damage thresholds induced by a 420-750 nm
supercontinuum source and a CW 532 nm laser, where no
significant difference could be found between them.
2019 | Jiao _et al._ [68] |  | The retinal damage thresholds are determined for two
---
groups of chinchilla grey rabbits with the ocular axial
lengths of 15.97 and 17.25 mm, respectively, using a
1319 nm CW laser.
2019 | Sliney [69] |  | Laser products that intentionally radiate towards the face
---
necessitate higher reliability and protection to be assured
of always remaining below the Class 1, however, the
current limits are considered sound.
2021 | Jean _et al._ [70] |  | Computer models are developed to predict injury thresholds
---
in the wavelength range from $1050$ nm to $10.6$ $\mu$m.
TABLE III: Handbooks on laser safety. Year | Book | Remark
---|---|---
1985 | Winburn [71] |  | This book includes characteristics of lasers; eye
---
components; skin damage thresholds; classification
of lasers by ANSI Z136.1; selecting laser-protective
eyewear; hazards associated with lasers.
1997 | Henderson [72] |  | A guide for the safe use of laser products, which embraces
---
hazards of laser radiation, laser classification, approaches
for hazard control and biological effect of laser radiation.
2003 | Henderson _et al._ [73] |  | A handbook that covers laser radiation hazards,
---
laser classification and protective measures and safety.
2013 | Sliney _et al._ [74] |  | Topics such as the effect of laser radiation on the eye and
---
skin, laser hazard classification and laser safety for outdoor
applications are covered.
2014 | Barat [75] |  | A handbook that determines general protective approaches
---
for the safe use of lasers.

###  I-B Scope and structure of the Tutorial

Most of the literature discussing laser eye safety has involved calculations
using single-mode Gaussian beams, and the literature suggests that laser
classification for non-single-mode-Gaussian beams should be evaluated
experimentally. Furthermore, there is a lack of a solid analytical framework
for eye safety where various beam propagation models, as well as optics such
as lenses and diffusers, are considered. Furthermore, to the best of the
authors’ knowledge, there is no such a wide study with the main focus of
utilizing laser sources as transmitters for indoor OWC where long time
exposure is a possibility.

In this study, we provide in-depth assessments of beam propagation for ideal
single-mode Gaussian and multimode beams. We also investigate the beam
propagation for arrays and beam shaping optics including lenses and diffusers.
Beam propagation investigations have been supported by demonstrating extensive
results. We have also developed a generalized framework which recommends the
maximum power operation of non-single-mode Gaussian beams, based upon known
laser parameters, rather than purely experimental evaluation. This allows us
to calculate the theoretical maximum power at which we can operate our lasers
and feed into our link budgets. The calculated maximum permissible transmit
power of the laser source ensures both eye and skin safety, which is derived
for various cases including single mode Gaussian/non-Gaussian beams and
multimode beams. The multimode results are based on experimental measurements
reported in the literature. The eye safety of multimode lasers is investigated
using different approaches based on analytical methods and the calculation of
$M$-squared method. A comparison between these techniques is carried out in
this paper. It is confirmed that the $M$-squared-based approach is a
dependable method to determine the maximum transmit power, however, in some
scenarios, it provides a significantly lower transmit power level compared to
the precise beam decomposition method. For those scenarios, the power budget
can be improved by evaluating the maximum transmit power based on the
presented precise beam decomposition method. We have also presented unique
algorithms to obtain the maximum transmit power of a laser source with optical
components such as lenses and diffusers. We further note that algorithms
proposed in this study consider not only the parameters of a laser including
the wavelength and beam waist but also the exposure duration and the most
hazardous position. We have also presented perceptive results of eye safety by
evaluating various parameters and delivering in-depth discussions. To the best
of the authors’ knowledge, there are no papers that provide such wide
mathematical analyses for eye safety supported by a broad range of results on
laser sources of interest in modern OWC systems.

The remainder of this paper is organized as follows. Laser safety terminology
is described in Section II. Laser beam propagation models, including single
transverse mode, higher-mode, using an optical element such as a lens or a
diffuser and for an array of VCSELs are detailed in Section III. Eye safety
analyses along with insightful results for each of the described beam
propagation model are presented in Section IV. This section has been wrapped
up with skin safety. Finally, conclusions are drawn in Section V. The acronyms
used in the paper are listed in Table IV.

TABLE IV: List of acronyms Acronym | Description
---|---
AEL | Accessible emission limit
ANSI | American national standards institute
CW | Continuous wave
EL | Exposure limit
EMF | Electromagnetic field
FSO | Free-space optical
FWHM | Full width half maximum power
ICNIRP | International commission on non-ionizing radiation protection
IEC | International electrotechnical commission
IoT | Internet of things
ISO | International organization for standardization
LASER | Light amplification by stimulated emission of radiation
LF | Low frequency
LED | Light emitting diodes
LTE | Long-term evolution
MHP | Most hazardous position
MPE | Maximum permissible exposure
MRP | Most restrictive position
NOHD | Nominal ocular hazard distance
OFCS | Optical fibre communication system
OWC | optical wireless communication
PMN | Principal mode number
RF | Radio frequency
SEF | Static electric field
SMF | Static magnetic field
VCSEL | Vertical-cavity surface-emitting laser
V2V | Vehicle-to-vehicle
UV | Ultraviolet

##  II Laser Safety Terminology

There are various standards specifying safe levels of laser radiation [37, 27,
39, 48]. We will briefly review the measures and conditions specified in these
standards for the safe use of lasers. In this section, we discuss various
laser radiation hazards and in order to clarify our later discussions, we
define the nomenclatures that will be used throughout this article.

###  II-A Spectral Regions and Laser Radiation Hazards

Laser radiation can cause biological damages through temperature effects due
to the absorbed energy and also through some photochemical reactions. Damage
is mostly due to temperature effects with the critical organs considered being
the eyes and skin. These biological effects vary depending upon the wavelength
of the laser radiation, the exposure duration and the tissue being exposed. In
this article, we denote the wavelength by $\lambda$ and mainly focus on
visible and infrared wavelength regimes444It is worth mentioning that the
algorithms presented in this paper can be readily extended to other
wavelengths as well., since the commercially available laser sources operate
in these spectral regions. In the following, we briefly discuss the primary
hazards in different spectral regions.

####  II-A1 Effects of visible and near infrared radiation

The primary effect on the eye in this spectral region is damage to the retina.
The ability of the cornea and the eye lens to focus these wavelengths directly
onto a small spot on the retina makes these wavelengths significantly more
hazardous than the longer infrared wavelengths, despite the equivalent
received optical irradiances on the surface of the eye. Injury to the skin in
the visible and near infrared wavelength range ($400-1400$ nm) results from
temperature rises above $45^{\circ}$C [39].

####  II-A2 Effects of mid and far infrared radiation

For wavelengths greater than $1400$ nm, the laser radiation primarily causes
damage to the cornea and the skin, as the radiation at these wavelengths is
absorbed by water, and the vitreous humor of the eye protects the retina from
damage [39].

###  II-B Exposure Duration

The exposure duration is defined as the time span of a pulse, a pulse train or
a continuous laser radiation on the human body555For a single pulse, this is
the duration between the half-peak power point of the leading edge and the
corresponding point on the trailing edge. For a train of pulses (or
subsections of a train of pulses), this is the duration between the first
half-peak power point of the leading pulse and the last half-peak power point
of the trailing pulse. [37]. Laser safety standards specify exposure durations
between 100 fs and 30 ks (about 8 hours) [37, 27, 39, 48]. The applicable
exposure duration itself is dependent on the wavelength and the nature of the
exposure, i.e. intended or accidental exposure, intra-beam exposure, or
exposure to diffused radiation. As an example, for accidental intra-beam
viewing of a continuous visible light laser radiation by a person with normal
health condition, the exposure time is limited to $0.25$ s which is equal to
the aversion response time [62].

It should also be noted that the longer the exposure duration is, the more
damage to the eye and skin it can cause. In this paper, the exposure duration
is denoted by $t_{\rm{ex}}$ and we derive the maximum transmit power that
ensures eye safety regulations for the highest exposure duration.

Figure 4: Geometric representation of the angular subtense, $\alpha$, of the
apparent source.

###  II-C Angular Subtense of the Apparent Source

The apparent source is defined as the real or virtual object that forms the
smallest retinal image by eye accommodation. Human eye accommodation refers to
the act of physiologically adjusting the focal length of the eye lens to keep
an object in focus on the retina as its distance from the eye varies.

The angular subtense of the apparent source is the angle subtended by the
apparent source as viewed from a point in space666The angular subtense of the
source should not be confused with the divergence angle of the beam.. It
relates to the ability of the eye to focus the laser’s power onto a small spot
on the retina. (see Fig. 4). The angular subtense, denoted by $\alpha$, is
calculated as:

$\alpha=2\tan^{-1}{\left(\dfrac{D_{\mathrm{s}}}{2z}\right)}$ (1)

where $D_{\mathrm{s}}$ is the source size (assuming that the source is
circular) and $z$ is the distance of the apparent source from the considered
position in space. For non-circular sources, the effective angular subtense is
the arithmetic mean of two angular subtense values calculated for the maximum
and minimum dimensions of the apparent source [39].

If $\alpha<\alpha_{\rm{min}}=1.5$ mrad, the source is considered as a point
source, otherwise, it is regarded as an extended source [37, 73]. According to
the IEC $60825-1$ standard [27], the extended sources are further classified
into two types depending on their angular subtense; intermediate sources and
large sources. For intermediate sources the condition
$\alpha_{\rm{min}}<\alpha<\alpha_{\rm{max}}$ holds, while for large sources
$\alpha\geq\alpha_{\rm{max}}$, where $\alpha_{\rm{max}}=100$ mrad. In
practice, by using optical elements, a point source can be turned into an
extended source.

###  II-D Maximum Permissible Exposure

The maximum permissible exposure (MPE), also known as the exposure limit (EL),
is the maximum irradiance level to which the eye or skin can be exposed
without consequential injury immediately or after a long time. The MPE is
reported in units of J$/$m2 or W$/$m2 and depends on the laser wavelength, the
exposure time and the type of tissue at risk777In the context of eye safety,
MPE also depends on the image size on the retina where the image size is
specified in terms of the angular subtense of the apparent source.. The MPE
values are set by the ICNIRP [39] and in collaboration with the IEC [27] and
the ANSI [48] standards. Since the injury levels for eye and skin tissues are
different, these standards have specified different sets of MPEs for ocular
and skin exposure hazards.

The MPE not only depends on the wavelength and the exposure duration, but also
on whether a source is viewed as a point source or an extended one, determined
by the subtense angle. Table 8 summarizes the eye-safe MPE expressions for
point and extended sources in different wavelength regimes and for their
corresponding applicable exposure duration. To obtain these MPEs, a limiting
aperture diameter of $7$ mm has been considered. It is defined in terms of the
diameter of a circular area over which the average irradiance or radiant is
considered [27]. Note that for the range of wavelengths considered in this
table, eye safety is more restricting than skin safety, hence, the provided
MPEs suffice for the laser radiation safety analysis.

TABLE V: MPE calculation for eye safety analysis888For $10^{-3}\leq
t_{\rm{ex}}<0.25$, we have $\alpha_{\rm max}=200t_{\rm{ex}}^{0.5}$ mrad while
for $t_{\rm{ex}}>0.25$, we have $\alpha_{\rm max}=100$ mrad. [37]. Spectral
regions and exposure durations
---
$\lambda$ [nm] | $700\ \leq\lambda<1050$ | $1050\ \leq\lambda<1150$ | $1150\ \leq\lambda<1400$
$t_{\rm{ex}}$ [s] |  $10^{-3}$ to 10 | 10 to $3\times 10^{4}$ |  $10^{-3}$ to 10 | 10 to $3\times 10^{4}$
Expressions for MPE [W$/$m2]
Point source | $18t_{\rm{ex}}^{0.75}{\rm{C}}_{4}/t_{\rm{ex}}$ | 10${\rm{C}}_{4}{\rm{C}}_{7}$ | $90t_{\rm{ex}}^{0.75}/t_{\rm{ex}}$ | $10C_{4}C_{7}$
Extended source |  [.5]0 $\begin{aligned} &18t_{\rm{ex}}^{0.75}{\rm{C}}_{4}{\rm{C}}_{6}/t_{\rm{ex}},\ {\rm if}\ t_{\rm{ex}}\leq T_{2}\\\ &18{\rm{C}}_{4}{\rm{C}}_{6}T_{2}^{-0.25},\quad\;{\rm if}\ t_{\rm{ex}}>T_{2}\end{aligned}$ | $\begin{aligned} &90t_{\rm{ex}}^{0.75}{\rm{C}}_{6}{\rm{C}}_{7}/t_{\rm{ex}},\ {\rm if}\ t_{\rm{ex}}\leq T_{2}\\\ &90{\rm{C}}_{6}{\rm{C}}_{7}T_{2}^{-0.25},\quad\;{\rm if}\ t_{\rm{ex}}>T_{2}\end{aligned}$
Required coefficients and parameters for calculating MPE
${\rm{C}}_{4}$ | $10^{0.002(\lambda-700)}$ | 5
${\rm{C}}_{6}$ |  [.5]0 ${\rm C}_{6}=\begin{cases}1,&\alpha\leq\alpha_{\rm min}\\\\[5.0pt] \dfrac{\alpha}{\alpha_{\rm min}},&\alpha_{\rm min}<\alpha\leq\alpha_{\rm max}\\\\[10.0pt] \dfrac{\alpha_{\rm max}}{\alpha_{\rm min}},&\alpha>\alpha_{\rm max}\end{cases}$
${\rm{C}}_{7}$ |  [.5]0 $C_{7}=\begin{cases}1,&\lambda<1150\\\ 10^{0.018(\lambda-1150)},&1150\ \leq\lambda<1200\\\ 8,&1200\ \leq\lambda<1400\end{cases}$
$T_{2}$ [s] |  [.5]0 $T_{2}\\!=\\!\begin{cases}10,&\alpha\leq\alpha_{\rm min}\\\ 10\times 10^{\frac{(\alpha-\alpha_{\rm min})}{98.5}},&\alpha_{\rm min}<\alpha\leq\alpha_{\rm max}\\\ 100,&\alpha>\alpha_{\rm max}\end{cases}$

###  II-E Most Hazardous Position

For a given laser beam, the most hazardous position, MHP, denoted by $z_{\rm
haz}$, is defined as the location at which the ratio of the exposure level to
the MPE value is at its maximum. It can be expressed mathematically as [76]:

$z_{\rm haz}=\ \operatorname*{arg\,max}_{z}\ \frac{I(z)}{\rm MPE},$ (2)

where $I(z)$ is the exposure level (i.e., irradiance in W$/$m2). We note that
MPE depends on the subtense angle which is a function of $z$ (see (1)). It is
clear from (2) that the MHP for different laser beam profiles can be
different. Therefore, it is necessary to carefully specify MHP based on the
characteristics of the laser source, e.g., whether the beam is single-mode or
multimode, and whether the source is a single element or an array.

The MHP also depends upon the type of laser hazard; i.e, the tissue being
exposed (retinal, corneal or skin). For corneal and skin hazards, the MHP is
where the laser beam has the smallest area and hence, the highest irradiance.
For retinal hazards, the MHP is not necessarily where the irradiance is the
highest, but where the cornea and the eye lens focus the highest optical power
onto the smallest retinal spot. It is this irradiance that can be most
damaging to the retina. Therefore, the MHP in this case depends on the angular
subtense of the apparent source. This means that various points in space with
different distances from the eye should be evaluated in order to specify the
MHP.

According to [37, 27, 39, 48], the closest evaluation position which needs to
be considered in a retinal hazard MPE analysis is $10$ cm from the apparent
source. This is based on the assumption of a near point of $10$ cm, i.e. $10$
cm is the closest position at which a healthy human eye is able to image the
apparent source onto the retina. Closer distances result in higher optical
powers entering the eye, but forming blurred, less hazardous images.

###  II-F Laser Classification

Lasers are often categorized, based on their operating wavelength and transmit
power, into four main classes, with a few subclasses [73]. Old classification
includes, Class 1, Class 2, Class 3A, Class 3B and Class 4. The revised
version was demonstrated in 2001 under the IEC and EN classification [77],
which specifies the following classes:

  * •

Class 1: This class is eye-safe under all operating conditions, even for long
term intentional viewing.

  * •

Class 1M: This class is safe for an unaided eye, but might be hazardous when
viewed with optical instruments such as binoculars.

  * •

Class 2: These are visible lasers. The class is safe for unintentional
exposure durations less than $0.25$ s.

  * •

Class 2M: These are visible lasers. This class is safe for unintentional
exposure durations less than $0.25$ s, but may be hazardous when viewed with
collecting optics such as binoculars.

  * •

Class 3R: Unintentional or accidental exposure to direct, reflected beam or
radiation in this class is considered as low risk, but potentially hazardous.

  * •

Class 3B999Terms ‘M’, ‘R’ are derived from “magnifying optical viewing
instruments”, “reduced or relaxed requirements”, respectively; and ‘B’ is
based on the old classification.: Radiation in this class is very likely to be
hazardous. For a continuous wave laser the maximum output power into the eye
must not exceed $500$ mW. The radiation can be hazardous to both the eye and
skin. However, viewing of the diffuse reflection is safe.

  * •

Class 4: This is the highest class of laser radiation and it is severely
dangerous. Radiation in this class is very hazardous, and viewing of the
diffuse reflections may be dangerous as well.

These classes are summarized in Table VI. It is noted that higher wavelengths
can accommodate higher power and still be a class 1 product.

TABLE VI: Laser classifications. Class | Risk or injury | Exposure duration | Viewing condition
---|---|---|---
Class 1 | eye-safe | all durations | also with optics
Class 1M | eye-safe | all duration | naked eye only
Class 2 | eye-safe | less than $0.25$ s | also with optics
Class 2M | eye-safe | less than $0.25$ s | naked eye only
Class 3R | low risk | less than $0.25$ s | also with optics
Class 3B |  | diffuse reflections are safe
---
low risk for skin injury
all duration | also with optics
Class 4 |  | severely eye-hazardous
---
diffuse reflections may be hazardous
all duration | also with optics

##  III Laser Beam Propagation

In this section, we review single mode and multimode laser beams, and beam
transformation by optical elements such as lenses and diffusers. The
understanding of various laser beam profiles and their characteristics is
essential for performing laser safety analyses in the next section.

###  III-A Single Transverse Mode Beam

Figure 5: Gaussian beam geometry.

An ideal single-mode Gaussian beam has an intensity distribution given by the
following equation [78]:

$I(r,z)=\frac{2P_{\mathrm{t}}}{\pi
W^{2}(z)}{\exp{\left(-\dfrac{2r^{2}}{W^{2}(z)}\right)}},$ (3)

where $P_{\mathrm{t}}$ is the optical power carried by the beam, and $r$ and
$z$ are the radial and axial positions, respectively. For ideal Gaussian beams
at a distance of $z$, the spot radius101010Spot radius or spot size is the
radius at which the intensity values fall to $1/e^{2}$ of their axial values,
in the transverse plane at position $z$ along the beam axis. and the wavefront
radius of the curvature are respectively expressed as below [78]:

$\displaystyle W(z)=w_{0}\sqrt{1+\left(\frac{z}{z_{0}}\right)^{2}},$ (4a)
$\displaystyle R(z)=z\left(1+\left(\frac{z_{0}}{z}\right)^{2}\right),$ (4b)

where $z_{0}$ is the Rayleigh range and is given by:

$z_{0}=\frac{\pi w_{0}^{2}}{\lambda}$ (5)

and $\lambda$ is the wavelength of the laser source. The complex $q$-parameter
of the Gaussian beam is defined as $q(z)=z+jz_{0}$ where $j=\sqrt{-1}$. The
reciprocal of $q(z)$ contains the radius of the curvature and the spot radius
in its real and imaginary parts, respectively [78]:

$\frac{1}{q(z)}=\frac{1}{R(z)}-j\frac{\lambda}{\pi W^{2}(z)}.$ (6)

The beam divergence angle111111In this paper, the term “beam divergence angle”
specifies the half-angle divergence as shown in Fig. 5. In some studies, beam
divergence angle stands for full-angle divergence., as shown in Fig. 5, is
defined as:

$\theta=\lim_{z\to\infty}\tan^{-1}\left(\frac{W(z)}{z}\right).$ (7)

In the paraxial case (i.e, very small $\theta$), the beam divergence (in
radians) is approximated as:

$\theta\approx\frac{\lambda}{\pi w_{0}}.$ (8)

TABLE VII: Commonly used measures for the beam spot radius. Measure | Power ratio | Spot radius | Divergence angle
---|---|---|---
$1/e^{2}$ | $86\%$ | $W(z)$ | $\theta$
$1/e$ | $63\%$ | $\frac{\sqrt{2}}{2}W(z)$ | $\frac{\sqrt{2}}{2}\theta$
$1/2$ | $50\%$ | $\frac{\sqrt{2\ln{2}}}{2}W(z)$ | $\frac{\sqrt{2\ln{2}}}{2}\theta$

The received power at a distance of $z$ through an aperture of radius $r_{0}$,
can be obtained as:

$P_{\rm{r}}=\int_{0}^{r_{0}}I(r,z)2\pi{r{\rm{d}}}r=P_{\mathrm{t}}\left(1-\exp{\left(-\dfrac{2r_{0}^{2}}{W^{2}(z)}\right)}\right).$
(9)

Therefore, a circle of $r_{0}=W(z)$ contains $1-e^{-2}$ or $86\%$ of the
Gaussian beam’s total power. We refer to this as the “$1/e^{2}$” or “$86\%$
measure” and use it throughout this paper. Apart from this measure, in other
studies $1/e$ and $1/2$ are also used. These commonly used measures are
compared in Table VII in terms of the spot radius and the beam divergence
angle. The third row in Table VII represents the case where the beam spot
radius is measured based on the full width at half maximum (FWHM) criterion.
In fact, FWHM specifies the beam diameter for which the optical power
contained within the beam spot is $1/2$ of the total power.

###  III-B Multimode Laser Beams

The Gaussian beam is not the only solution of the paraxial Helmholtz equation
[78]. Of practical interest are solutions which have non-Gaussian intensity
distributions. In general, non-Gaussian irradiance profiles can be expressed
as the incoherent superposition of transverse modes. The total optical
intensity of the beam can be represented as:

$I=\sum_{l=0}^{L_{\rm max}}\sum_{m=0}^{M_{\rm max}}C_{l,m}|U_{l,m}|^{2},$ (10)

where $U_{l,m}$ is the complex amplitude of the mode $(l,m)$, hence,
$|U_{l,m}|^{2}$ is the optical intensity of the mode, and $C_{l,m}$ is the
mode power coefficient.

Two well known orthogonal sets of transverse modes are Hermite Gaussian modes,
which are used mainly in Cartesian coordinate systems, and Laguerre Gaussian
modes, which are more suitable for circularly or cylinderically symmetric
beams. In the following, these two polynomials are discussed in more detail.

####  III-B1 Hermite-Gaussian Beams

A Hermite-Gaussian beam of order $(l,m)$ is often represented as ${\rm
HG}_{l,m}$, where $l$ and $m$ refer to the mode numbers in transverse $x$ and
$y$ directions in the Cartesian coordinate system, respectively. The complex
field amplitude of ${\rm HG}_{l,m}$ propagated at distance $z$ is represented
by [78]:

$\displaystyle U_{l,m}(x,y,z)$
$\displaystyle=A_{l,m}\left[\frac{w_{0}}{W(z)}\right]\mathbb{G}_{l}\left(\frac{\sqrt{2}x}{W(z)}\right)\mathbb{G}_{m}\left(\frac{\sqrt{2}y}{W(z)}\right)$
(11) $\displaystyle\times\exp{\left(-jkz-
jk\frac{x^{2}+y^{2}}{2R(z)}+j(l+m+1)\zeta(z)\right)},$

where $\zeta(z)=\tan^{-1}(z/z_{0})$ and $k=2\pi/\lambda$ is the wave number
(in radian per meter) for the free-space wavelength of $\lambda$. The mode
normalisation coefficient $A_{l,m}$ for Hermite-Gaussian beams is defined as
below:

$\displaystyle A_{l,m}=\sqrt{\dfrac{2^{-l-m+1}}{\pi l!m!}}$ (12)

In (11), $\mathbb{G}_{l}(u)$ is known as the Hermite-Gaussian function of
order $l$ and is given as:

$\mathbb{G}_{l}(u)=\mathbb{H}_{l}(u)\exp{\left(\frac{-u^{2}}{2}\right)},\ \ \
l=0,1,2,...$ (13)

The Hermite polynomials, $\mathbb{H}_{l}(u)$, are defined as:

$\mathbb{H}_{l}(u)=l!\sum_{i=0}^{\left\lfloor\tfrac{l}{2}\right\rfloor}\frac{(-1)^{i}}{i!(l-2m)!}(2u)^{l-2i}.$
(14)

The irradiance of the mode $(l,m)$ at $x,y,z$ is represented by [78]:

$\displaystyle|U_{l,m}\left(x,y,z\right)|^{2}=A_{l,m}^{2}\left[\frac{w_{0}}{W(z)}\right]^{2}\\!\\!\mathbb{H}_{l}^{2}\left(\frac{x\sqrt{2}}{W_{x}\left(z\right)}\right)\\!\mathbb{H}_{m}^{2}\left(\frac{y\sqrt{2}}{W_{y}\left(z\right)}\right)\exp{\left(-\frac{2x^{2}}{W_{x}^{2}\left(z\right)}-\frac{2y^{2}}{W_{y}^{2}\left(z\right)}\right)}.$
(15)

(a) $\rm{PMN}=1$

(b) $\rm{PMN}=2$

(c) $\rm{PMN}=2$

(d) $\rm{PMN}=3$

(e) $\rm{PMN}=5$

(f) $\rm{PMN}=5$

Figure 6: Example irradiance profiles (normalised to peak) of Hermite-Gaussian
transverse modes ($\lambda=0.85~{}\mu$m and $w_{0}=5~{}\mu$m).

The modes can be grouped into degenerate (i.e. very similar) modes based on
their principal mode number (PMN) defined as ${\rm{PMN}}=l+m+1$ for Hermite-
Gaussian beams. Fig. 6 presents some examples of the Hermite-Gaussian
irradiance profiles for different PMNs. As can be seen in Fig. 6a, the
Hermite-Gaussian beam of order $(0,0)$, namely ${\rm HG}_{0,0}$, is the ideal
Gaussian beam.

A multimode laser beam can be decomposed into the principal transverse modes
as introduced above. The number of modes that a multimode laser supports
depends on the device characteristics and operating conditions, while the mode
partitioning in the device constitutes a complex dynamic system [79, 80, 81,
82]. Based on different values of the mode coefficient, $C_{l,m}$, and the
mode combinations diverse beam profiles can be generated. Fig. 7 presents an
example of three different mode combinations and their corresponding output
power profiles. The mode coefficients of each combination are illustrated in
Fig. 7a and the total power profile of these combinations are depicted in
Figs. 7b-7d. The purple dots indicate the point of maximum irradiance in each
case, which is substantial for laser safety analysis.

(a) Combinations

(b) HG-Comb 1

(c) HG-Comb 2

(d) HG-Comb 3

Figure 7: Hermite-Gaussian mode combination: (a) normalized mode coefficients
for three different combinations, (b),(c), and (d) power distribution for each
combination.

####  III-B2 Laguerre-Gaussian Beams

An alternative set of solutions to the paraxial Helmholtz equation are
Laguerre-Gaussian beams. Circularly symmetric beam profiles (e.g., lasers with
cylindrical cavities) are often best represented using the Laguerre-Gaussian
modal decomposition. Let ${\rm LG}_{l,m}$ denote the Laguerre-Gaussian beam of
order $(l,m)$ in the cylindrical coordinate system $(r,\phi,z)$. The integers
$l\in\mathbb{Z}^{+}$ and $m\in\mathbb{Z}^{+}$ are azimuthal and radial
indices, respectively. The complex amplitude of ${\rm LG}_{l,m}$ is expressed
as [78]:

$\displaystyle U_{l,m}(r,\phi,z)$
$\displaystyle=A_{l,m}\left[\frac{w_{0}}{W(z)}\right]\left(\frac{\sqrt{2}r}{W(z)}\right)^{l}\mathbb{L}_{m}^{l}\left(\frac{2r^{2}}{W^{2}(z)}\right)\exp\left(-\frac{r^{2}}{W^{2}(z)}\right)$
(16) $\displaystyle\times\exp{\left(-jkz-jk\frac{r^{2}}{2R(z)}\mp
jl\phi+j(l+2m+1)\zeta(z)\right)},$

where $\mathbb{L}_{m}^{l}(\cdot)$ denotes generalized Laguerre polynomials
expressed as [78]:

$\mathbb{L}_{m}^{l}(x)=(m+l)!\sum_{i=0}^{m}\frac{(-x)^{i}}{i!(m-i)!(l+i)!},$
(17)

The ideal Gaussian beam corresponds the lowest-order Laguerre-Gaussian beam
$\rm LG_{0,0}$. In (16), $A_{l,m}$ is the normalized mode coefficient which is
used to ensure that the total power of each mode is equal to $1$ and is given
by:

$A_{l,m}=\sqrt{\frac{2m!}{\pi(m+l)!}}.$ (18)

The irradiance of each Laguerre-Gaussian mode, can be expressed as [78]:

$\displaystyle\left|U_{l,m}(r,\phi,z)\right|^{2}=\Big{|}A_{l,m}\left[\frac{w_{0}}{W(z)}\right]\left(\frac{\sqrt{2}r}{W(z)}\right)^{l}\mathbb{L}_{m}^{l}\left(\frac{2r^{2}}{W^{2}(z)}\right)\exp{\left(-\frac{r^{2}}{W^{2}(z)}\right)}\Big{|}^{2}.$
(19)

For Laguerre-Gaussian beams, PMN is defined as ${\rm{PMN}}=l+2m+1$. Fig. 8
shows some examples of irradiance profiles of low order Laguerre-Gaussian
beams.

(a) $\rm{PMN}=1$

(b) $\rm{PMN}=4$

(c) $\rm{PMN}=4$

Figure 8: Example irradiance profiles (normalised to peak) of VCSEL Laguerre-
Gaussian transverse modes ($\lambda=0.85~{}\mu$m and $w_{0}=5~{}\mu$m).

####  III-B3 $M$-Squared Factor

In practice, rather than decomposing the profile of the actual laser beam to
Hermite or Laguerre Gaussian modes, the propagation of the laser beam is
described approximately by an ideal Gaussian beam but with a modified beam
radius $W_{\rm R}(z)$ given by:

$W_{\rm R}(z)=w_{0\rm R}\sqrt{1+\left(\frac{z\lambda M^{2}}{\pi w_{0\rm
R}}\right)^{2}},$ (20)

where the degree of deviation of the practical laser beam from the ideal
Gaussian beam is quantified by the quality factor $M^{2}$, often known as the
$M$-squared factor or the beam propagation factor [83, 84] and defined as [85,
86]:

$M^{2}=\frac{w_{0\rm R}\theta_{\rm R}}{w_{0}\theta}=\frac{\pi w_{0\rm
R}\theta_{\rm R}}{\lambda},$ (21)

In (21), $w_{0\rm R}$ and $\theta_{\rm R}$ are the beam waist and far-field
divergence angle of the practical beam, respectively, and $w_{0}$ and $\theta$
are those of the benchmark ideal Gaussian beam, respectively. Note that for an
ideal Gaussian beam, we have $w_{0}\theta=\lambda/\pi$.

The $M$-squared factor reflects how well a collimated beam can be focused to a
small spot, or how well a divergent beam can be collimated. For an ideal
Gaussian beam, $M^{2}$ is equal to $1$. Lower values of $M^{2}$ represent a
better beam quality. Practical laser beams have factors greater than one. The
$M^{2}$ factor of VCSELs can be as high as $20$ [83]. This is due to their
transverse multimode emission which results from their large active diameters.
In [87], an experimental methodology for measuring the beam propagation factor
$M^{2}$ has been developed based on the technique described in the ISO 11146
series of standards [88, 89, 90].

####  III-B4 VCSEL Beam Decomposition Using Experimental Measurements

Far field measurements of VCSEL devices allow the estimation of mode power
coefficients based on the ideal models described in Sections III-B1 and
III-B2. Such measurements consist of recording the far field beam profile at a
given distance from the laser source. Although the exact mode power
coefficients of multimode VCSELs depend on the laser dynamics and the
operating conditions, a common approach is to use a set of pre-defined mode
combinations for system design and/or the performance analysis. As an example,
in [91], a set of Laguerre-Gaussian mode combinations is presented. Table
III-B4 provides the mode power coefficients of these combinations and Fig.
III-B4 illustrates the normalized radial power profile assuming $\lambda=850$
nm, $w_{0}=5~{}\mu$m, and $z=0$. In this figure, the red dashed line indicates
the radial position where the power falls into the $1/{e}^{2}$ of the maximum
value. For a single-mode Gaussian beam, this position is known as the spot
radius (see Section III-A). This definition of the spot radius can also be
used for non-Gaussian modes and multimode beams. The spot radius can be found
analytically or numerically from the mode power profiles. Once the spot radius
is known, the divergence angle can be calculated according to (7). Let
$\theta_{l,m}$ denote the divergence angle of LGl,m. Table IX presents the
spot radii and divergence angles of the first six Laguerre-Gaussian transverse
modes. The approximation values in this table correspond to the small value of
${\lambda}/{(\pi w_{0})}$.

In Table III-B4, the ratio $\theta_{\rm R}/\theta$ is provided for each
combination, where $\theta_{\rm R}$ represents the divergence angle of each
combination and $\theta$ is that of LG0,0 (assuming that $\lambda/(\pi w_{0})$
is small). The $M^{2}$ factor can be found according to this ratio and
therefore, it can help simplify the eye safety analysis. This approximation
method has been used in Section IV-B and it is compared to the exact
analytical technique.

TABLE VIII: Mode power coefficients for various combinations of the first six
Laguerre-Gaussian transverse modes [91].

| Mode combinations
---|---
| LG-
---
Comb 1
| LG-
---
Comb 2
| LG-
---
Comb 3
| LG-
---
Comb 4
| LG-
---
Comb 5
| LG-
---
Comb 6
| LG-
---
Comb 7
Modes | 1 | $1$ | $0.5$ | $0$ | $0$ | $0.125$ | $0$ | $0.06$
2 | $0$ | $0.5$ | $0.5$ | $0$ | $0.75$ | $0.125$ | $0.44$
3 | $0$ | $0$ | $0.25$ | $0.25$ | $0.0625$ | $0.375$ | $0.22$
4 | $0$ | $0$ | $0$ | $0.25$ | $0$ | $0.0625$ | $0.03$
5 | $0$ | $0$ | $0.25$ | $0.25$ | $0.0625$ | $0.375$ | $0.22$
6 | $0$ | $0$ | $0$ | $0.25$ | $0$ | $0.0625$ | $0.03$
| $\theta_{\rm R}/{\theta}=M^{2}({w_{0}}/w_{0\rm R})$
$1$ | $1.3248$ | $1.7346$ | $1.9451$ | $1.5432$ | $1.7511$ | $1.7237$

Figure 9: Normalised radial power for various mode combinations given in Table
III-B4 for $\lambda=850$ nm, $w_{0}=5~{}\mu$m, and $z=0$.

TABLE IX: Spot radii and divergence angles of the first six Laguerre-Gaussian transverse modes. Mode | $(l,m)$ | PMN | Spot radius | Divergence angle, $\theta_{l,m}$
---|---|---|---|---
$1$ | $(0,0)$ | $1$ | $W(z)$ | $\theta_{0,0}=\theta_{0}\approx\dfrac{\lambda}{\pi w_{0}}$
$2$ | $(1,0)$ | $2$ |  $1.500W(z)$. | $\theta_{1,0}\approx 1.5\theta_{0}$
$3$ | $(2,0)$ | $3$ | $1.774W(z)$ | $\theta_{2,0}\approx 1.774\theta_{0}$
$4$ | $(0,1)$ | $3$ | $1.502W(z)$ | $\theta_{0,1}\approx 1.502\theta_{0}$
$5$ | $(3,0)$ | $4$ | $1.999W(z)$ | $\theta_{3,0}\approx 1.999\theta_{0}$
$6$ | $(1,1)$ | $4$ | $2.008W(z)$ | $\theta_{1,1}\approx 2.008\theta_{0}$

###  III-C Beam Propagation in Lens Systems

A Gaussian beam remains Gaussian after passing through a lens (if there are no
aberrations). However, the parameters of the output Gaussian beam are
different from those of the input beam. The ray transfer method is a simple
method which relates the input and output beam parameters of any optical
system using ABCD matrices [92]. This method is schematically shown in Fig.
10a. A $2\times 2$ matrix associates the position and angle of paraxial rays
at the input and output of the optical system by means of simple linear
algebraic equations. This method is described briefly in the following
paragraphs.

(a)

(b)

Figure 10: (a) Transformation of a Gaussian beam by an arbitrary paraxial
system described by an $ABCD$ matrix. Here, $R_{1}$ and $R_{2}$ are the radii
of curvature of the input and output wavefronts, respectively. (b) Gaussian
beam propagation after a thin lens.

Let us assume that the input and output beam waists corresponds to the object
and image, respectively. The object and/or the image may be real or virtual,
depending on the type of lens and the location of the beam waist with respect
to lens. Let $q$ and $w_{0}$ be the parameters of the input Gaussian beam, and
$q_{2}$ and $w_{2}$ be the corresponding parameters of the output Gaussian
beam. The relation between the complex beam parameters of the input and the
output, $q$ and $q_{2}$, are given as follows:

$q_{2}=\frac{Aq+B}{Cq+D},$ (22)

where the matrix elements are expressed as [93]:

$\displaystyle A=1+\frac{\rho(n-1)}{n\upsilon_{1}},$ (23a) $\displaystyle
B=\frac{\rho}{n},$ (23b) $\displaystyle
C=(n-1)\left(\frac{1}{\upsilon_{1}}-\frac{1}{\upsilon_{2}}\right)-\frac{\rho(n-1)^{2}}{n\upsilon_{1}\upsilon_{2}},$
(23c) $\displaystyle D=1-\frac{\rho(n-1)}{n\upsilon_{2}}.$ (23d)

Here, $n$ is the refractive index of the lens and $\rho$ shows the lens
thickness. The parameters $\upsilon_{1}$ and $\upsilon_{2}$ are the radii of
the input and output surfaces of the lens, respectively.

As an example, consider a thin lens of focal length $f$ and let $d_{1}$ and
$d_{2}$ denote the distances of the input and output beam waists from the
lens, respectively, as illustrated in Fig. 10b. The parameters $d_{2}$ and
$w_{2}$ are given as [78]:

$d_{2}=\frac{\dfrac{1}{f}-(1-\dfrac{d_{1}}{f})d_{1}(\dfrac{\lambda}{\pi
w_{0}^{2}})^{2}}{\dfrac{1}{f^{2}}+(1-\dfrac{d_{1}}{f})^{2}(\dfrac{\lambda}{\pi
w_{0}^{2}})^{2}},$ (24) $w_{2}=\frac{\lambda f}{\pi
w_{0}\sqrt{1+(1-\dfrac{d_{1}}{f})^{2}(\dfrac{\lambda f}{\pi
w_{0}^{2}})^{2}}}.$ (25)

The beam divergence after the lens is also given by:

$\theta_{2}=\frac{\theta}{\kappa},$ (26)

where $\kappa=w_{2}/w_{0}$ is the magnification factor of the lens.

###  III-D Beam Propagation for Array of VCSELs

An $N\times N$ two dimensional ($2$D) VCSEL array is illustrated in Fig. 11a.
The total output power of a VCSEL array is simply the sum of the output powers
of all single VCSEL emitters. Practically, the emission per VCSEL might have
to be somewhat decreased if the marginal constraints of the cooling system are
reached. It should be noted that the total dissipated power of a VCSEL array
and the thermal power density can be considerable (often several hundreds of
Watt per cm2). The cooling can be made simpler by increasing the spacing
between the emitters, however, this does decrease the beam quality. This
spacing may be referred to as pitch distance in some studies and is denoted by
$d_{\sigma}$ in Fig. 11a.

(a)

(b)

Figure 11: (a) An array of $N\times N$ VCSELs with the pitch of $d_{\sigma}$.
(b) Collimation of the output of a VCSEL array.

Assuming that all VCSELs in the array are identical, the received power
collected by an aperture with an area of $A_{\rm a}$ which is located at
$(x_{\rm a},y_{\rm a})$ is given by:

$P_{\rm r}=\sum_{i=1}^{N^{2}}\iint\limits_{A_{\rm a}}\frac{2P_{\rm t}}{\pi
W(z)^{2}}\exp{\left(-2\frac{(x_{i}-x_{\rm a})^{2}+(y_{i}-y_{\rm
a})^{2}}{W(z)^{2}}\right)}{\rm d}x{\rm d}y,$ (27)

where $(x_{i},y_{i})$ is the location of $i$th VCSEL of the array. In the far-
field region, i.e., $z\gg N\times d_{\sigma}$, (27) can be approximated as:

$P_{\rm r}\approx N^{2}\iint\limits_{A_{\rm a}}\frac{2P_{\rm t}}{\pi
W(z)^{2}}\exp{\left(-2\frac{(x-x_{\rm a})^{2}+(y-y_{\rm
a})^{2}}{W(z)^{2}}\right)}{\rm d}x{\rm d}y\approx N^{2}P_{\rm r,single},$ (28)

where $P_{\rm r,single}$ is the received power of a single VCSEL at the
considered position.

It should be noted that, in some applications, a lens might be used in front
of the VCSEL arrays to separate beam spots at the receiver plane (see Figs.
3b, 3d and 3f of [16]). In such cases, the total received power at the
photodetector can be approximated by the received power of a single VCSEL. On
the other hand, some applications necessitate the reduction of beam divergence
by the use of a microlens array, where a separate lens is in front of each
VCSEL element. A set of collimated beamlets are shown in Fig. 11b. In such a
case, the total received power depends on the area of receiver [17].

(a)

(b) $z=0.1$ cm

(c) $z=1$ cm

(d) $z=10$ cm

Figure 12: Power profiles of a $3\times 3$ multimode VCSEL array at different
distances for $w_{0}=5~{}\mu$m and $\lambda=850$ nm. The VCSELs are assumed to
transmit the same output power. The mode power coefficients of the multimode
VCSEL for each VCSEL is shown in the first sub-figure.

In Fig. 12, a $3\times 3$ multimode VCSEL array is considered. For these
results, the mode power coefficients of the different VCSELs are varied, while
their output power is assumed to be the same. The beam waist of each VCSEL and
the pitch distance are assumed to be $w_{0}=5\ \mu$m and $d_{\sigma}=250\
\mu$m, respectively. The total output power at different $z$-distances
($z=0.1,1,10$ cm) and the point of maximum irradiance are found and indicated
with a purple dot in the plots.

Figure 13: System model of a diffuser.

###  III-E Beam Propagation through Diffusers

A diffuser can be used to alter the intensity pattern of a laser beam. In this
paper, we will consider two types of diffusers: the Lambertian diffuser, and
the uniform diffuser[94, 95]. The system model of a diffuser is shown in Fig.
13. The input beam of a diffuser should be a collimated beam, therefore, a
lens is required before the diffuser. The laser source is located at the focal
point of the lens and a collimated beam is generated. The distance between the
lens and the diffuser is denoted by $z_{\rm d}$. The diameter and the FWHM
angle121212The FWHM angle is often reported in the datasheets for practical
diffusers. of the diffuser are denoted by $D$ and $\Theta_{\rm d}$,
respectively. The received power at a distance of $z$ from the diffuser can be
obtained as:

$P_{\rm r}=\iint\limits_{\Omega_{\rm p}}I(\theta,\phi)\sin(\theta){\rm
d}\theta{\rm d}\phi,$ (29)

where $I(\theta,\phi)$ is the intensity in W/Sr (i.e. Watt per
steradian)131313The steradian (Sr) is the unit of a solid angle which has its
vertex at the center of a sphere. The solid angle cuts off an area of the
surface of the sphere equal to that of a square with side lengths equal to the
radius of the sphere., $\theta$ and $\phi$ are the polar and azimuthal angles,
respectively, in the spherical coordinate system. Note that the integral is
taken over the solid angle, $\Omega_{\rm p}$, defined by the area of relevant
aperture at a given distance from the source. In the following, we study the
Lambertian and uniform diffusers.

####  III-E1 Lambertian Pattern Engineered Diffusers

The Lambertian diffusers are available with various diffusing angles (from
$0.5^{\circ}$ to $50^{\circ}$ at FWHM angle) to control the diffuse area of
illumination and increase the transmission efficiency. It is important to note
that diffusing angles are given for a collimated input beam and angular
divergence will vary for different incidence angles. The intensity of a
Lambertian diffuser depends only upon the azimuthal angle, as it is symmetric
around the polar angle$,\phi$, that is $I(\theta,\phi)$ reduces to
$I(\theta)$. The radiation pattern of a generalized Lambertian diffuser of
order $m$ is given by [96, 97]:

$I(\theta)=\frac{m+1}{2\pi}\cos^{m}(\theta)P_{\rm t},$ (30)

where $P_{\rm t}$ is the transmit optical power. Substituting (30) into (29)
and calculating the integral, the received power is obtained as:

$P_{\rm r}=P_{\rm t}\int_{0}^{2\pi}\int_{0}^{\psi_{\rm
c}}\frac{m+1}{2\pi}\cos^{m}(\theta)\sin(\theta){\rm d}\theta{\rm
d}\phi=(1-\cos^{m+1}(\psi_{\rm c}))P_{\rm t},$ (31)

where, $\psi_{\rm c}=\tan^{-1}(\frac{r_{\rm{a}}}{z})$ where $z$ is the viewing
distance and $r_{\rm{a}}$ is the radius of the aperture (see Fig. 13).

(a) $\Theta_{\rm d}=20^{\circ}$

(b) $\Theta_{\rm d}=50^{\circ}$

Figure 14: Normalized intensity versus scatter angle of uniform diffusers with
$\lambda=785$ nm [95]. The normalization is based on the total intensity.

####  III-E2 Uniform Pattern Engineered Diffusers

Uniform diffusers are designed to create non-Gaussian intensity distributions
typically in circular, square or hexagonal shapes. The FWHM, $\Theta_{\rm d}$,
of practical diffusers ranges from several milliradian to $1.4$ rad
($80^{\circ}$). As an example, the normalized irradiance profile of two
diffusers with of $\Theta_{\rm d}=20^{\circ}$ and $\Theta_{\rm d}=50^{\circ}$
are shown in Fig. 14 and compared with an ideal diffuser [95]. As it can be
seen, the normalized intensity, particularly for square pattern diffusers, is
almost the same for all scatter angles. Hence, we can assume that the
irradiance profile is independent of the scatter angle. Another point is that
the maximum transmit power based on the assumption of an ideal diffuser is the
worst case, and therefore, the eye safety constraint on the transmit power for
practical diffusers (with the irradiance profile shown in Fig. 14) would be
relaxed to some degree.

The solid angle of a cone whose cross-section subtends the angle $\Theta_{\rm
d}$ is $\Omega=2\pi(1-\cos(\Theta_{\rm d}/2))$. Therefore, for a circularly
uniform diffuser with a FWHM of $\Theta_{\rm d}$, the irradiance is given as:

$I=\frac{P_{\rm t}}{2\pi(1-\cos(\Theta_{\rm d}/2))}.$ (32)

It can be seen that the irradiance presented in the above equation is
independent of the polar and azimuthal angles141414It can be readily justified
that $\int_{0}^{2\pi}\int_{0}^{\Theta_{\rm d}/2}I\sin(\theta){\rm d}\theta{\rm
d}\phi=P_{\rm t}$.. Substituting the above equation in (29) and calculating
the integral over $\phi$ and $\theta$ from $0$ to $2\pi$ and from $0$ to
$\psi_{\rm c}$, respectively, the received power in the pupil of the eye is
obtained as:

$P_{\rm r}=\frac{P_{\rm t}}{{(1-\cos(\Theta_{\rm d}/2))}}(1-\cos(\psi_{\rm
c})).$ (33)

In the next Section, we present eye safety evaluations for the discussed beam
propagation models.

##  IV Eye Safety Analysis

The hazards from laser sources can best be classified into four categories:
optical radiation hazards to the eye and skin; chemical hazards; electrical
hazards; and casualty hazards151515Chemical hazards are essentially due to
vaporization of material, metal oxides and other fumes. Electrical hazards
refer to electrical shock and possibly resulting in electrocution. Casualty
hazards might be due to several reasons like causing an accidental fire by
means of lasers.. Eye hazards are by far the most significant [74]. Hence, we
mainly focus on the analysis of eye safety under different laser beam
propagation conditions (single mode, multimode, with lens, array of VCSELs and
with diffusers) and skin safety will be discussed at the end of this section.

###  IV-A Safety Analysis for Single Mode laser

For an ideal single-mode (Gaussian) beam, the maximum received power is at the
center of the beam. Therefore, the maximum transmit power to ensure eye safety
is calculated based on the received power into the eye pupil (the pupil can be
as large as 6–7 mm when it is wide open, which translates into the maximum
physical aperture [98]) located in the center of the beam at the most
hazardous distance from the laser source (see Fig. 4).

Substantially, the maximum transmit power can be obtained based on the MPE
defined by ANSI standard [48]. In order to fulfill eye safety regulations, the
following condition should be met:

$\frac{P_{\rm r}}{\pi r_{\rm p}^{2}}<{\rm MPE},$ (34)

where $P_{\rm r}$ is the fraction of the power ($86\%$161616This metric here
is more conservative in comparison to the $63\%$ by [27] and $72\%$ adopted in
[73]) into the eye at the MHP, $z_{\rm haz}$ and $r_{\rm p}$ is the cornea
radius. Based on (9), the maximum transmit is given as:

$P_{\rm t,max}=\frac{1}{\eta}{\pi r_{\rm p}^{2}}\times{\rm MPE},$ (35)

where $\eta=\left(1-\exp{\left(-\dfrac{2r_{\rm p}^{2}}{W^{2}(z_{\rm
haz})}\right)}\right)$.

The subtense angle is required in order to specify the MPE. The subtence angle
can be determined based on the distance between the eye and the source, and
the beam waist of the laser. Let denote the distance for which the $86\%$ of
the power passes into a $7$ mm pupil by $d_{86}$. If $d_{86}$ is less than
$0.1$ m, then the MHP is set to $0.1$ m. For this case, we have
$\alpha=2\tan^{-1}({w_{0}}/{z_{\rm{haz}}})$. Otherwise,
$\alpha=2\tan^{-1}({w_{0}}/{d_{86}})$. Next, the MPE can be calculated
according to the operating wavelength, exposure duration and the calculated
subtense angle from Table 8. The approach to obtain the maximum transmit power
that satisfies eye safety regulations is summarized in algorithm 1. It should
be mentioned that this algorithm covers the analysis for both point and
extended sources. There is only one consideration in reading the MPE from the
Table 8. The MPE can be distinguished based on the subtense angle for point
and extended sources.

Algorithm 1 Determining maximum transmit power of a single mode laser source.

0: Denote the wavelength of the laser source by $\lambda$; the beam waist by
$w_{0}$; the radius of the pupil by $r_{\rm p}$; exposure duration by
$t_{\rm{ex}}$.

0: Maximum transmit power, $P_{\rm{t,max}}$ of the laser source.

1: Compute $d_{86}=\dfrac{\pi w_{0}}{\lambda}\sqrt{\dfrac{-2r_{\rm
p}^{2}}{\ln{(1-\varepsilon_{86})}}}$, where $\varepsilon_{86}=0.86$

2: if $2\theta>92$ mrad AND $d_{86}<0.1$ then

3: Set $z_{\rm{haz}}=0.1$

4: Compute
$\alpha=2\tan^{-1}\left(\dfrac{w_{0}}{z_{\rm{haz}}}\right)=2\tan^{-1}(10w_{0})$

5: else

6: Compute $\alpha=2\tan^{-1}\left(\dfrac{w_{0}}{d_{86}}\right)$

7: if $\alpha<\alpha_{\rm min}$ then

8: Set $z_{\rm{haz}}=\dfrac{w_{0}}{0.0021}$

9: else

10: Set $z_{\rm{haz}}=\dfrac{9.2\pi w_{0}}{2\lambda}$

11: end if

12: end if

13: Compute $\eta=1-\exp{\left(\dfrac{-2r_{\rm p}^{2}(\pi
w_{0}^{2})^{2}}{(\lambda z_{\rm{haz}})^{2}}\right)}$

14: Calculated the MPE based on the given wavelength, exposure duration,
$t_{\rm{ex}}$, and the calculated $\alpha$ from Table 8.

15: Calculate $P_{\rm{t,max}}=\dfrac{1}{\eta}({\rm MPE})\pi r_{\rm p}^{2}$
using the calculated MPE

16: return $P_{\rm{t,max}}$ as the maximum transmit power of the laser source

Fig. 15a shows the maximum transmit power versus exposure duration for a point
source. Two wavelengths of $\lambda=850$ nm and $\lambda=950$ nm are
considered and it is assumed that $w_{0}=5\ \mu$m. As shown in the results,
the maximum transmit power increases as the wavelength of the laser source
increases, generating an increment of $1.14$ mW between operating wavelengths
of $850$ nm and $950$ nm for exposure durations higher than $10$ s. As shown
in these simulations, $P_{\rm t,max}$ highly depends on exposure durations of
less than $10$ s while it is constant for exposure durations of higher than
$10$ s. Fig. 15b illustrates $P_{\rm t,max}$ versus beam divergence angle,
$\theta$, of a laser source with $w_{0}=5\ \mu$m assuming an exposure duration
greater than $10$ s. As shown in this set of results, the maximum transmit
power is constant up to the divergence angle of $2.6^{\circ}$ and then after
that, the maximum transmit power increases as the beam divergence increases.
This is due to the reduction of the amount of power passing into the eye as
the beam divergence increases, which leads to a more relaxed condition on the
transmit power limitation.

Fig. 16 presents the results of $P_{\rm t,max}$ versus the wavelength of the
laser source assuming an exposure duration greater than $10$ s. Results are
provided for three practical ranges of the beam waist. It can be seen from
these simulation results that as the operating wavelength of a laser source
increases, the maximum transmit power increases as well. In particular, the
maximum transmit power increases exponentially beyond wavelength $1050$ nm and
up to the wavelength $1200$ nm, where the eye safety constraint is much more
relaxed in comparison to lower wavelengths. For wavelengths higher than $1200$
nm, $P_{\rm t,max}$ increases linearly as the wavelength increases. The other
observation is related to the relation between the beam waist and the $P_{\rm
t,max}$. It can be inferred from this set of results that the $P_{\rm t,max}$
reduces as the beam waist increases. This is due to the fact that the laser
beam is less divergent for a larger beam waist.

(a)

(b)

Figure 15: $P_{\rm{t,max}}$ versus (a) exposure duration and (b) divergence
angle, $\theta$. Figure 16: $P_{\rm t,max}$ versus $\lambda$ assuming an
exposure duration greater than $10$ s.

###  IV-B Safety Analysis for Multimode Lasers

Eye safety analysis of multimode laser sources is normally a complicated task.
An accurate approach is based on the beam profile of such laser, which can be
obtained through experimental measurements. Once, the beam profile is
measured, the maximum power that passes through an aperture of diameter $7$ mm
can be specified. Then, based on this calculated irradiance, the maximum
transmit power to ensure eye safety can be determined. Evaluating the received
power passes through an aperture of radius $r_{\mathrm{p}}$ centered at
($x_{0}$, $y_{0}$) is a case of calculating the appropriate integral of the
intensity. Assuming that the multimode laser beam is decomposed into a set of
Hermite-Gaussian modes, the power that passes through the considered aperture
can be specified as:

$\displaystyle P_{\mathrm{r}}\left(x,y,z\right)$ $\displaystyle=P_{\rm
t}\sum_{l=0}^{L_{\rm max}}\sum_{m=0}^{M_{\rm
max}}C_{l,m}\int_{x_{0}-r_{\mathrm{p}}}^{x_{0}+r_{\mathrm{p}}}\int_{y_{0}-\sqrt{r_{\mathrm{p}}^{2}-\left(x-x_{0}\right)^{2}}}^{y_{0}+\sqrt{r_{\mathrm{p}}^{2}-\left(x-x_{0}\right)^{2}}}|U_{l,m}\left(x,y,z\right)|^{2}\
{\rm d}x\ {\rm d}y.$ (36)

For the purposes of the eye safety analysis, this double integral should be
calculated at the point at which the maximum power is gathered by the
aperture. We assume that the integral evaluated around the position of peak
irradiance ($x_{0}$, $y_{0}$) will provide the maximum (or close to the
maximum) received power. The coordinates of the maximum point can be found by
solving the following set of equations at $z=z_{\rm haz}$:

$\displaystyle\frac{\partial}{\partial x}{\left(\sum_{l=0}^{L_{\rm
max}}\sum_{m=0}^{M_{\rm max}}C_{l,m}|U_{l,m}\left(x,y,z\right)|^{2}\right)}=0$
(37) $\displaystyle\frac{\partial}{\partial y}\left(\sum_{l=0}^{L_{\rm
max}}\sum_{m=0}^{M_{\rm max}}C_{l,m}|U_{l,m}\left(x,y,z\right)|^{2}\right)=0.$

Setting $r_{x}=\frac{x\sqrt{2}}{W_{x}\left(z\right)}$ and
$r_{y}=\frac{y\sqrt{2}}{W_{y}\left(z\right)}$, the coordinates of the peak
irradiance will be the roots of the polynomials defined by the following
equations:

$\displaystyle\sum_{l=0}^{L_{\rm max}}\sum_{m=0}^{M_{\rm
max}}C_{l,m}A_{l,m}\mathbb{H}_{m}^{2}\left(r_{y}\right)\mathbb{H}_{l}\left(r_{x}\right)\left[2l\mathbb{H}_{l-1}\left(r_{x}\right)-r_{x}\mathbb{H}_{l}\left(r_{x}\right)\right]=0$
(38a) $\displaystyle\sum_{l=0}^{L_{\rm max}}\sum_{m=0}^{M_{\rm
max}}C_{l,m}A_{l,m}\mathbb{H}_{l}^{2}\left(r_{x}\right)\mathbb{H}_{m}\left(r_{y}\right)\left[2m\mathbb{H}_{m-1}\left(r_{y}\right)-r_{y}\mathbb{H}_{m}\left(r_{y}\right)\right]=0.$
(38b)

These roots can be calculated numerically if the mode weights (i.e.,
$C_{l,m}$) are known, and accordingly the received power can be evaluated
around the peak intensities to calculate the maximum permissible transmit
power.

As an example, let’s consider the results shown in Fig. 7. The coordinates of
the maximum point are calculated based on (38) and depicted by a purple dot in
the subsets of Fig. 7. The purple circle corresponds to an aperture of
diameter $7$ mm. The incident power fraction passed through this circle
located $10$ cm away from the laser source is calculated analytically for mode
combinations $\\#1-3$ to be respectively $0.14$, $0.144$ and $0.149$ of the
total emitted power. The beam waist is assumed to be $5\ \mu$m. Since
$\alpha=2\tan^{-1}\left(\frac{w_{0}}{z_{\rm haz}}\right)=0.2$ mrad less than
$\alpha_{\rm min}$, then, the laser is treated as a point source. The MPE for
such a laser operating at $850$ nm for the exposure duration of higher than 10
s, is $20$ W/m2. Hence, the maximum transmit powers are $5.5$ mW, $5.3$ mW and
$5.2$ mW, respectively.

For a Laguerre-Gaussian beam, the power passes through an aperture of radius
$r_{\mathrm{p}}$ centered at ($r_{0}$,$\theta_{0}$) is given by:

$P_{\mathrm{r}}\left(r,z\right)=P_{\rm t}\sum_{l=0}^{L_{\rm
max}}\sum_{m=0}^{M_{\rm
max}}C_{l,m}{A_{l,m}\int_{\theta_{1}}^{\theta_{2}}\int_{r_{1}}^{r_{2}}{\left|U_{l,m}\left(r,z\right)\right|^{2}r\
{\rm d}r\ {\rm d}\theta}},$ (39)

where $\theta_{1}$ and $\theta_{2}$ represent the maximum and minimum polar
angles that span the aperture. Their values depend on the relative size of the
aperture $r_{\rm p}$ with respect to its position $(r_{0},\theta_{0})$. The
two cases and the equations to obtain $\theta_{1}$, $\theta_{2}$,
$r_{1}\left(\theta\right)$ and $r_{2}\left(\theta\right)$ are provided below.
If $r_{\rm p}\leq r_{0}$,

$\displaystyle r_{1}\left(\theta\right)$
$\displaystyle=r_{0}\cos{\left(\theta-\theta_{0}\right)}-\sqrt{r_{\mathrm{p}}^{2}-r_{0}^{2}\sin^{2}{\left(\theta-\theta_{0}\right)}}$
$\displaystyle\theta_{1}$
$\displaystyle=\theta_{0}+\sin^{-1}{\left(-\frac{r_{\mathrm{p}}}{r_{0}}\right)}$
$\displaystyle r_{2}\left(\theta\right)$
$\displaystyle=r_{0}\cos{\left(\theta-\theta_{0}\right)}+\sqrt{r_{\mathrm{p}}^{2}-r_{0}^{2}\sin^{2}{\left(\theta-\theta_{0}\right)}}$
$\displaystyle\theta_{2}$
$\displaystyle=\theta_{0}+\sin^{-1}{\left(\frac{r_{\mathrm{p}}}{r_{0}}\right)}.$

If $r_{\rm p}>r_{0}$,

$\displaystyle r_{1}\left(\theta\right)$ $\displaystyle=0$
$\displaystyle\theta_{1}$ $\displaystyle=0$ $\displaystyle
r_{2}\left(\theta\right)$
$\displaystyle=r_{0}\cos{\left(\theta-\theta_{0}\right)}+\sqrt{r_{\mathrm{p}}^{2}-r_{0}^{2}\sin^{2}{\left(\theta-\theta_{0}\right)}}$
$\displaystyle\theta_{2}$ $\displaystyle=2\pi.$

Similarly, to fulfil the laser safety requirement, this double integral should
be determined at the point at which the maximum power is gathered by the
aperture. Again, we assume that the double integral evaluated around the
position of peak irradiance will provide the maximum (or close to the maximum)
received power. We can find the position of the peak irradiance by solving the
following equation:

$\frac{\partial}{\partial r}\left(\sum_{l=0}^{L_{\rm max}}\sum_{m=0}^{M_{\rm
max}}C_{l,m}A_{l,m}|U_{l,m}\left(r,z\right)|^{2}\right)=0.$ (40)

The roots of this solution can be obtained numerically and are the roots of
the polynomial defined by the following equation:

$\sum_{l=0}^{L_{\rm max}}{\sum_{m=0}^{M_{\rm
max}}2{C_{l,m}}^{2}A_{l,m}\left(\frac{2}{W^{2}\left(z\right)}\right)^{l}\mathbb{L}_{m}^{l}\left(\frac{2r^{2}}{W^{2}\left(z\right)}\right)}\\\
\times\left[\mathbb{L}_{m}^{l}\left(\frac{2r^{2}}{W^{2}\left(z\right)}\right)\left\\{lr^{2l-1}-\frac{{2r}^{2l+1}}{W^{2}\left(z\right)}\right\\}-\frac{4r^{2l+1}}{W^{2}\left(z\right)}\mathbb{L}_{m-1}^{l+1}\left(\frac{2r^{2}}{W^{2}\left(z\right)}\right)\right]=0.$
(41)

Once this peak intensity position is found, then, we can specify the power
collected by an aperture with a radius of $r_{\rm p}$ by means of (39), and
comparing this power to the value permitted by the exposure limit defined by
the standard.

Algorithm 2 Determining maximum transmit power of a multi mode laser source.

0: Denote the wavelength of the laser source by $\lambda$;the beam divergence
of the actual beam by $\theta_{\rm R}$;the radius of the pupil by $r_{\rm
p}$;exposure duration by $t_{\rm{ex}}$;

0: Maximum transmit power, $P_{\rm{t,max}}$ of the multimode laser source

1: Calculate $\theta_{\rm em}$ and $w_{0,\rm em}$ of the embedded Gaussian
beam:

2: $\theta_{\rm em}=\theta_{\rm R}$

3: $w_{0,\rm em}=\lambda/(\pi\theta_{\rm em})$

4: Use $\theta_{\rm em}$ and $w_{0,\rm em}$ in the equations of algorithm 1
instead of $\theta$ and $w_{0}$, respectively (start from step 1 and continue
until step 16)

5: return $P_{\rm{t,max}}$ for the multimode laser

The method described above is analytically complex, however, it can provide an
accurate value for the maximum transmit power. To deal with such intricacy
analyses of the multimode beams, the $M$-squared approximation is suggested as
an alternative approach in the literature [83, 87, 99, 100, 101]. By fitting
the multimode beam to the so-called embedded Gaussian beam based on the beam
divergence angle, a straightforward framework to analyse eye safety for such
practical multimode lasers can then be established. The embedded Gaussian beam
does not necessarily have any physical reality, i.e., it does not necessarily
indicate the mode $(0,0)$ of the actual beam. Once the parameters of the
embedded Gaussian beam are obtained (the beam waist and divergence angle),
these values can be used in algorithm 1 to identify the maximum transmit power
for such non-Gaussian beams. The approach to obtain $P_{\rm{t,max}}$ for a
multimode laser is summarized in algorithm 2. Next, we aim to assess the
validity of this approach by comparing its results with the analytical beam
decomposition ones.

(a)

(b)

Figure 17: $P_{\rm{t,max}}$ versus (a) exposure duration and (b) wavelength
for different modes of a Laguerre-Gaussian laser. The beam waist is assumed to
be $w_{0}=5\ \mu$m.

Fig. 17a shows results of $P_{\rm t,max}$ for a Laguerre-Gaussian laser versus
the exposure duration assuming $\lambda=850$ nm and $w_{0}=5\ \mu$m. The
simulation results are presented for different mode combinations of Laguerre-
Gaussian beams given in Table III-B4. In these simulations, LG-Comb 6 and LG-
Comb 7 are not chosen since their $P_{\rm t,max}$ are very close to the LG-
Comb 3. The results are presented and compared for both $M$-squared and beam
decomposition-based methods. The beam decomposition approach is based on the
calculation of the integral given in (39) at the maximum point of the beam.
For the purpose of short representation, it is denoted as “Decomp.” in the
legend of Fig. 17a. The outcomes indicate that for LG-Comb $2$ only,
$M$-squared and beam decomposition methods perform similarly. For LG-Comb $3$,
$4$ and $5$, the difference between the $M$-squared and beam decomposition
techniques are $25\%$, $30\%$ and $23\%$, respectively, where $M$-squared is
more conservative. In fact, a pure Gaussian beam has more pronounced peak
leading to higher relative irradiance than the multimode beams. The values of
$P_{\rm t,max}$ calculated by the beam decomposition method are in agreement
with the IEC standard for eye safety, which means that the values of $P_{\rm
t,max}$ obtained by the $M$-squared-based ensure the IEC regulations as well.
However, one can use the precise beam decomposition method to relax the power
budget limitations to some degree. It can be also observed from these results
that the fourth mode combination ensures the highest transmit power in
comparison to other mode combinations. This is due to the fact that LG-Comb
$4$ provides a higher beam divergence at the MHP, therefore, a lower amount of
power passes through the eye pupil.

Fig. 17b represents the simulation results of $P_{\rm{t,max}}$ versus the
wavelength of the laser for the four mode combinations using the $M$-squared
based approach. For these results, the exposure duration is set to $10$ s. It
is noted that the results given in Table III-B4 are for the wavelength of
$850$ nm. A similar ratio of $\theta_{\rm R}/\theta$ is assumed in the
simulations of Fig. 17b for other wavelengths. From these results, the same
conclusion, i.e., the LG-Comb $4$ relaxes the transmit power limitation, can
be drawn. The other observation is about the gap between $P_{\rm{t,max}}$ of
various combinations, which increases for higher wavelengths. This is due to
the dependency of the MPE on wavelength, which is larger for higher
wavelengths.

Figure 18: Beam transformation after a thin lens and illustration of the MHP.

###  IV-C Laser Safety Analysis for Lens systems

In cases where an optical lens is used at the output of a laser source, the
eye safety calculations remain the same, but the beam parameters need to be
transformed. Assuming the thin lens approximation, the lens will result in the
transformation of the output beam according to the $ABCD$ formulation
described in Section III-C. The beam waist and its location after the thin
lens are given in (24) and (25), respectively. In order to determine the MHP,
there is one consideration when dealing with such lenses, which is about the
location of the image of the beam waist. The location of the image from the
lens is denoted by $d_{2}$ in Fig. 18 and, depending on the relative location
of the laser source from the lens and the focal length, it can be either
positive or negative. Hence, if the location of the image is after the lens
($d_{2}>0$), the MHP would be considered $z_{\rm haz}$ m away from the
location of $w_{2}$ (or $d_{2}+z_{\rm haz}$ m away from the lens). Otherwise,
the MHP is considered $z_{\rm haz}$ m away from the lens. In this case, the
size of the apparent source for the calculation of the subtense angle is the
one at the lens location, which is $W(d_{1})$. The procedure to calculate
$P_{\rm{t,max}}$ for a laser with a lens placed in front of it, is described
in algorithm 3.

Algorithm 3 Determining maximum transmit power of a laser source with a lens.

0: Denote the wavelength of the laser source by $\lambda$;the beam waist by
$w_{0}$;the radius of the pupil by $r_{\rm p}$;exposure duration by
$t_{\rm{ex}}$;the distance between laser and lens by $d_{1}$;the focal length
of the lens by $f$.

0: Maximum transmit power, $P_{\rm{t,max}}$

1: Calculate $d_{2}$, $w_{2}$ and $\theta_{2}$ based on (24), (25) and (26),
respectively

2: if $d_{2}>0$ then

3: Use $w_{2}$ and $\theta_{2}$ as the new beam waist and beam divergence
respectively in the equations of algorithm 1

4: Calculate $P_{\rm{t,max}}$

5: else

6: Find the spot radius at the lens location, $W(d_{1})$

7: Calculate $\alpha$ based on $W(d_{1})$

8: Use $w_{2}$ and $\theta_{2}$, $\alpha$ as the new beam waist, beam
divergence and subtense angle respectively in the equations of algorithm 1

9: Calculate $P_{\rm{t,max}}$

10: end if

11: return $P_{\rm{t,max}}$ for the laser with a lens

(a)

(b)

Figure 19: Eye safety analysis for a leaser with a lens: (a) $P_{\rm{t,max}}$
versus $t_{\rm ex}$ for different locations of the laser from the lens, (b)
$P_{\rm{t,max}}$ versus $\lambda$ for different focal lengths.

Fig. 19 depicts the eye safety results for a laser with a thin lens placed in
front of it. The beam waist of the laser is assumed to be $w_{0}=5\ \mu$m in
these simulations. Fig. 19a represents the results of $P_{\rm{t,max}}$ versus
the exposure duration assuming the laser source operates at $\lambda=850$ nm.
A lens with a focal length of $4$ cm is considered. Four locations for $d_{1}$
are studied. When the laser is located 8 cm away from the lens, the beam waist
after the lens is $w_{2}=5\ \mu$m, therefore, the magnification factor is
$\kappa=w_{2}/w_{0}=1$. Consequently, the results of $P_{\rm{t,max}}$ is the
same as the one when there is no lens in front of the laser source (see Fig.
15a). It can be seen that the most limiting case is related to $d_{1}=4$ cm
where the laser is placed at the focal length of the lens. For this case, a
collimated beam will be constructed after the lens, as a result, most of the
power passes through the eye pupil. If the laser is located between the lens
and the focal point, $P_{\rm{t,max}}$ can be increased since the subtense
angle in this case is quite large, therefore the parameter
$C_{6}=\alpha/\alpha_{\rm min}$ is large and the MPE value is higher. As the
laser becomes closer to the lens, the size of the apparent source $W(d_{1})$
becomes smaller, and therefore, $\alpha$ and $C_{6}$ are smaller, which
correspond to lower values for MPE and $P_{\rm{t,max}}$. Fig. 19b illustrates
the results of $P_{\rm{t,max}}$ versus the spectral region from 700 nm to 1400
nm for three different focal lengths. For these three curves, it is assumed
that $d_{1}=8$ cm. Compared to the case without a lens shown in Fig. 16,
$P_{\rm{t,max}}$ increases as $\lambda$ increases. However, the case of $f=6$
cm imposes a more limiting condition on the transmit power. When $f=4$ cm, the
results are the same as the case without a lens since $w_{2}=w_{0}$. Finally,
for the case of $f=3$ cm, $P_{\rm{t,max}}$ can be increased more since the
beam divergence of the transformed beam is higher and a lower amount of power
is collected at the eye pupil.

Figure 20: $P_{\rm{t,max}}$ versus focal length, $f$, and the distance between
lens and laser, $d_{1}$.

Fig. 20 shows the impact of geometric parameters such as the focal length of
the lens as well as the distance between the laser source and lens on
$P_{\rm{t,max}}$. The exposure duration is set to $100$ s for these
simulations. Let’s first focus on the curves of $P_{\rm{t,max}}$ versus $f$.
We set $d_{1}=8$ cm. The results are shown for two wavelengths of $850$ nm and
950 nm. Two different behaviors can be seen in these simulations. When the
focal lengths decreases by up to 8 cm, $P_{\rm{t,max}}$ becomes less; while
for $f>8$ cm, $P_{\rm{t,max}}$ increases. The minimum value for
$P_{\rm{t,max}}$ is related to the focal length of 8 cm where the beam after
the lens is collimated and the collected power into the eye is maximum. The
inset of Fig. 20 illustrated the effect of $d_{1}$ on $P_{\rm{t,max}}$. For
this curve, we set $\lambda=850$ nm and $f=8$ cm. The peak value corresponds
to $d_{1}=7$ cm, where the laser source is still between the focal point and
the lens, and the apparent size, $W(d_{1})$ is near to its maximum value.

###  IV-D Laser Safety Analysis for Array of VCSELs

The beam propagation of a VCSEL array is studied in Section III-D. The general
approach to analyze the eye safety for sources that consist of several sources
such as arrays or bundles of fibers is based on combinational methodology,
where in addition to the evaluation of the source as a whole, each individual
source needs to be evaluated separately, as well as all combinations of
sources. Each individual source or combination is associated with a certain
value of $\alpha$, and for each individual source or combination, a particular
transmit power level will be measured within the pupil diameter of $r_{\rm
p}=7$ mm. The pairs of received power values and MPE (depending on $\alpha$)
are subsequently compared, and the condition $P_{\rm r}<{\rm MPE}\times\pi
r_{\rm p}^{2}$ should be guaranteed for the individual sources and all
combinations. This condition imposes the limitation that the maximum transmit
power for each individual source of an array is always less than the maximum
transmit power of a single source. For an $N\times N$ array with the pitch
distance of $d_{\sigma}$, the subtense angle for each combination can be
obtained by:

$\alpha^{(i)}=2\tan^{-1}\left(\frac{(i-1)d_{\sigma}+2w_{0}}{2z_{\rm
haz}}\right),$ (42)

for $i\in\\{1,2,\cdots,N\\}$ and $w_{0}$ is the beam waist of each individual
source. The subtence angle for the combinational approach should be limited to
a minimum value of $1.5$ mrad and to a maximum value of $100$ mrad. Therefore,
if the pitch distance is small enough that the subtense angle of an array is
smaller than $1.5$ mrad, the whole array is treated as one source. The
condition on pitch distance to fulfil this requirement is
$d_{\sigma}<(0.00015-2w_{0})/(N-1)$. On the other hand, when spaces between
all sources of the array are in such a way that the subtense angle is larger
than $100$ mrad, sources are treated as completely independent and not
evaluated in a combination (as the corresponding images on the retina can be
considered thermally independent) [73]. The constraint
$d_{\sigma}>(0.01-2w_{0})/(N-1)$ ensures that individual sources can be
evaluated independently.

Figure 21: Eye safety analysis for a VCSEL array with $\lambda=850$ nm and
$w_{0}=5\ \mu$m.

Fig. 21 illustrates the results of the maximum transmit power per laser source
versus the exposure duration assuming that the laser source operates at
$\lambda=850$ nm with a beam waist of $5\ \mu$m. The results are given for
different array sizes and for two pitch distances of $d_{\sigma}=0$ and
$d_{\sigma}=250\ \mu$m. Obviously, as the size of the array increases, the
maximum transmit power for each individual source decreases. The reason for
this is that the output power of each individual VCSEL will be summed up at
the eye pupil, particularly when the distance between the array and eye is
large enough in comparison to the size of the array. As the pitch distance
increases, the subtence angle also increases, which means that the source is
categorized as an extended one, for which the MPE value and $P_{\rm t,max}$
increase as well. We can see from these results that an increase of $7$ times
in the maximum transmit power of a $5\times 5$ VCSEL array is possible when
the pitch distance increases from $0$ to $250\ \mu$m. Another important note
regarding these results is that the maximum transmit power for each individual
source of an array is less than the maximum transmit power of a single source
(see results shown in Fig. 15a). It is noted that the values of $P_{\rm
t,max}$ for $10<t_{\rm ex}<3\times 10^{4}$ s is the same as the values of
$P_{\rm t,max}$ for $t_{\rm ex}=10$, which are not represented in this figure.

###  IV-E Laser Safety Analysis Considering Diffusers

####  IV-E1 Lambertian Pattern Diffusers

For a diffuser in front of a laser source, the condition described in (34)
should be fulfilled to guarantee the eye safety regulation. Hence, the maximum
transmit power can be obtained by substituting (31) into (34), which should be
evaluated at $z=z_{\rm{haz}}$ and for an aperture radius equals to the radius
of pupil, i.e., $r_{\rm a}=r_{\rm p}$. Therefore, we have:

$P_{\rm{t,max}}=\frac{({\rm{MPE}})\pi
r_{\rm{p}}^{2}}{1-\cos^{m+1}(\psi_{\rm{haz}})},$ (43)

where $\psi_{\rm{haz}}=\tan^{-1}(\frac{r_{\rm{p}}}{z_{\rm{haz}}})$ calculated
at the hazardous distance, $z_{\rm{haz}}$. To find out the $\rm MPE$, we
require to calculate the subtense angle. Let’s denote $\alpha_{\rm
haz}=2\tan^{-1}(\frac{D}{2z_{\rm{haz}}})$, if $\alpha_{\rm
haz}>\alpha_{\rm{min}}$, which corresponds to the regions close to the
diffuser, the MPE of an extended source should be adopted to be used in (43).
The values of MPE are provided in Table 8. Let’s denote $z_{\rm
ps}=\frac{D}{2\tan(\frac{\alpha_{\rm{min}}}{2})}$ as the location after which
the source is viewed as a point source. If $z_{\rm{haz}}<z_{\rm ps}$ (or
equivalently $\alpha_{\rm haz}<\alpha_{\rm{min}}$) the MPE of a point source
should be used in (43).

Practical diffusers work more efficiently with collimated beams [102]. In
order to provide a collimated beam, a lens is required. By placing the point
source at the focal point of the lens, a collimated beam will be generated
after the lens. This beam passes through the diffuser and diverges according
to the divergence angle of the diffuser. The geometry of a collimated beam
using a lens is shown in Fig. 13. In general, and depending on the focal
length of the lens, denoted as $f$, the diameter of the spot on the diffuser
might be less than the diameter of the diffuser. In this case, we have:

$\alpha_{\rm{haz}}=2\tan^{-1}\left(\frac{W(f)}{z_{\rm{haz}}}\right),$ (44)

where based on (4a), $W(f)=w_{0}\sqrt{1+\left(\dfrac{f}{z_{0}}\right)^{2}}$.

Practical lasers have a unique beam divergence angle, $\theta$, up to several
degrees. If we want to ensure the condition $2W(f)=D$, $f$ should be:

$f=\frac{D}{2\tan(\theta)},$ (45)

which means that for a diffuser with $D=2.5$ cm and a laser with beam
divergence angle of $1^{\circ}$, the focal length should be $70$ cm.
Therefore, we usually have $W(f)<D$ for practical lenses. On the other side,
$W(f)$ cannot be smaller than the minimum beam requirement of a diffuser,
which is specified in a datasheet. In fact, the diameter of the beam that hits
a diffuser should be higher than the minimum beam requirement. This value for
practical diffusers changes from $1$ mm to $5$ mm.

####  IV-E2 Uniform Pattern Engineered Diffusers

For uniform diffusers with a specific beam divergence angle of $\Theta_{\rm
d}$, the geometry of the beam is shown in Fig. 13. Using (33) and (34), the
maximum transmit power that meets the eye safety requirement is expressed as:

$P_{\rm t,max}={\rm MPE}\times\pi r_{\rm p}^{2}\times\frac{1-\cos(\Theta_{\rm
d}/2)}{1-\cos(\psi^{{}^{\prime}}_{\rm haz})},$ (46)

where
$\cos(\psi^{{}^{\prime}}_{\rm{haz}})=\frac{z^{{}^{\prime}}_{\rm{haz}}}{\sqrt{{z^{{}^{\prime}}}_{\rm{haz}}^{2}+r_{\rm
p}^{2}}}$, where
$z^{{}^{\prime}}_{\rm{haz}}=z_{\rm{haz}}+\frac{W(f)}{\tan(\Theta_{\rm d}/2)}$.

Algorithm 4 Determining maximum transmit power of a laser source with a
diffuser

0: Denote the diameter of the diffuser by $D$; the radius of the pupil by
$r_{\rm p}$; the most hazardous distance by $z_{\rm{haz}}$; the focal length
of the lens by $f$ and the divergence angle at FWHM of a diffuser by
$\Theta_{\rm d}$.

0: Maximum transmit power, $P_{\rm{t,max}}$ of the laser source

1: Compute $\alpha$ at $z=z_{\rm{haz}}$, as
$\alpha_{\rm{haz}}=2\tan^{-1}\left(\frac{W(f)}{z_{\rm{haz}}}\right)$

2: Calculate the associated MPE to the $\alpha_{\rm{haz}}$ from Table 8

3: if Diffuser is Lambertian one then

4: Obtain $\psi_{\rm{haz}}=\tan^{-1}(\frac{r_{\rm{p}}}{z_{\rm{haz}}})$

5: Calculate $P_{\rm{t,max}}=\frac{({\rm{MPE}})\pi
r_{\rm{p}}^{2}}{1-\cos^{m+1}(\psi_{\rm{haz}})}$

6: else if Diffuser is uniform one then

7: Obtain
$\cos(\psi^{{}^{\prime}}_{\rm{haz}})=\frac{z^{{}^{\prime}}_{\rm{haz}}}{\sqrt{{z^{{}^{\prime}}}_{\rm{haz}}^{2}+r_{\rm
p}^{2}}}$, where
$z^{{}^{\prime}}_{\rm{haz}}=z_{\rm{haz}}+\frac{W(f)}{\tan(\Theta_{\rm d}/2)}$

8: Calculate $P_{\rm t,max}={\rm MPE}\times\pi r_{\rm
p}^{2}\times\frac{1-\cos(\Theta_{\rm d}/2)}{1-\cos(\psi^{{}^{\prime}}_{\rm
haz})}$

9: end if

10: return $P_{\rm{t,max}}$ as the maximum transmit power of the laser source

The approach for determining the maximum transmit power for a laser source
with a diffuser has been summarized in algorithm 4. Next, we provide some
results of the maximum transmit power. The enclosed extended source is assumed
to be constructed with a diffuser which is placed in front of a point source.
Furthermore, no limitations have been considered on the minimum beam
requirement of a diffuser in the following simulations (this assumption indeed
imposes more constraint on the maximum transmit power, particularly for
smaller beam divergence angles).

(a) $m=1$

(b) $m=5$

Figure 22: $P_{\rm{t,max}}$ versus $\theta$ for a Lambertian diffuser of order
$m$ and with different focal lengths of the lens, $f$.

Fig. 22 represents the maximum transmit power, $P_{\mathrm{t,max}}$, versus
the beam divergence angle, $\theta$, with the consideration of eye safety for
Lambertian diffusers of orders $m=1$ and $m=5$. Different lenses with focal
lengths of $1$ mm, $5$ mm and $10$ mm are assumed to collimate the beam. The
wavelength of the laser source is assumed to be $850$ nm. As it can be
inferred from these results, smaller focal lengths impose more constraints on
the maximum transmit power. Comparing the sets of results shown in Fig. 22a
and Fig. 22b, one can observe that as the Lambertian order of the difusser
increases, the maximum transmit power increases. This is because higher
Lambertian orders focus the beam more and, as a result, a higher amount of
power is collected at the eye. Another note regarding these results is that,
even though the maximum transmit power limit for $f=10$ is theoretically
several Watts, this would be limited by the practical maximum transmit power
of the laser source reported in the datasheets.

(a) $\Theta_{\rm d}=20^{\circ}$

(b) $\Theta_{\rm d}=50^{\circ}$

Figure 23: $P_{\rm{t,max}}$ versus $\theta$ for a uniform pattern diffuser
with FWHM divergence angle of $\Theta_{\rm d}$ and for different focal lengths
of the lens, $f$.

Fig. 23 represents the maximum transmit power, $P_{\mathrm{t,max}}$, versus
the beam divergence angle, $\theta$, with eye safety considerations for a
uniform diffuser with FWHM divergence angles of $20^{\circ}$ and $50^{\circ}$.
The wavelength of the laser source is assumed to be $850$ nm. The results are
obtained based on (46). Again, different lenses with focal lengths of $1$ mm,
$5$ mm and $10$ mm are assumed to collimate the beam. As it can be observed
from these curves, a similar conclusion can be drawn as the one for a
Lambertian diffuser, i.e., smaller focal lengths limit the maximum transmit
power more. It can be seen that the limitations on the maximum transmit power
for the diffuser with the FWHM divergence of $50^{\circ}$ have been been more
relaxed, compared to that of the diffuser with the FWHM divergence of
$20^{\circ}$.

###  IV-F Skin Safety

Injuring the skin is far less likely than injuring the eye unless one is
working with very high-powered laser sources and protects only the eye. Skin
injuries are typically not possible with low power and medium power lasers.
Skin safety analysis depends not only upon the area of the skin absorbing the
laser radiation, but also how long the skin is exposed to the laser and also
the wavelength of the laser source. The MPE levels required to cause injury to
the skin are quite high, at least several Watts-per-square-centimeter (W/cm2 )
[74].

TABLE X: MPE calculation for skin safety analysis [37]. $\lambda$ [nm] | $1400\ \leq\lambda<1500$ | $1500\ \leq\lambda<10^{5}$
---|---|---
$t_{\rm{ex}}$ [s] |  $10^{-3}$ to 0.35 |  0.35 to 10 |  10 to $3\times 10^{4}$ |  $10^{-3}$ to 0.35 |  0.35 to 10 |  10 to $3\times 10^{4}$
MPE [W$/$m2] | $A_{\rm s}\leq 0.01$ | $5600t_{\rm{ex}}^{-0.75}$ | $1000$ | $\dfrac{10^{4}}{t_{\rm{ex}}}$ | $1000$
| $0.01<A_{\rm s}\leq 0.1$ | $\dfrac{56t_{\rm{ex}}^{-0.75}}{A_{\rm s}}$ | $\dfrac{10}{A_{\rm s}}$ | $\dfrac{10^{2}}{t_{\rm{ex}}A_{\rm s}}$ | $\dfrac{10}{A_{\rm s}}$
| $A_{\rm s}>0.1$ | $560t_{\rm{ex}}^{-0.75}$ | 100 | $\dfrac{10^{3}}{t_{\rm{ex}}}$ | $100$
Limiting aperture diameter (mm) for skin | $3.5$ | $3.5$ | $3.5$ | $3.5$ | $3.5$ | $3.5$
Limiting aperture diameter (mm) for eye skin | 1 | $1.5t_{\rm{ex}}^{3/8}$ | $3.5$ | 1 | $1.5t_{\rm{ex}}^{3/8}$ | $3.5$

For skin safety, the MHP is where the intensity of the beam is the highest,
and in most cases this will be at the position closest to the laser beam
waist.

For most beams in the $700-1400$ nm wavelength range, the transmit power will
ideally be limited by retinal power limits. In this case, for the purposes of
skin safety, it is only necessary to limit access to the beam closest to the
waist, so that the beam irradiance is not high enough to cause damage.
Shielding provides sufficient distance to the beam waist and it helps to avoid
direct contact with the laser. This allows a further increase in transmit
power.

In the 1400 nm – 105 nm spectral region, and for exposure durations between 10
and $3\times 10^{4}$ seconds, the average irradiance measured through a 3.5 mm
diameter aperture should not exceed 1000 W$/$m2, unless the exposed area of
skin is greater than 0.1 m2. For values when the exposed area of skin is
greater than 0.1 m2, the maximum irradiance is 100 W$/$m2, and for skin areas
between 0.01 m2 and 0.1 m2, the maximum permissible exposure is
$10/A_{\mathrm{s}}$ [W$/$m2], where $A_{\mathrm{s}}$ is the area of exposed
skin. In addition, for beams with a 1/e2 diameter of less than 1 mm, the
actual radiant exposure (not averaged over the 3.5 mm aperture) should be
calculated [37]. The MPE values for the spectral region of 1400 nm to 105 nm
for exposure durations higher than $10^{-3}$ s are presented in Table X. It
can be seen from this table, the MPE depends on the limiting aperture
diameter. For skin safety considerations, the limiting aperture diameter
depends on the exposure duration, which are given in the table. In the
following, we will provide the skin safety analysis for single transverse mode
and multimode beams.

####  IV-F1 Single Transverse Mode

In order to calculate the skin-hazard for an ideal Gaussian beam, it is clear
that the maximum transmit power should be evaluated at the point at which the
accessible beam diameter is smallest, and in the case of an ideal Gaussian
laser beam, this will be at the beam waist location. If shielding is used,
then the skin hazrdous distance will shift to the location where the shield is
placed. The skin-hazard-limited transmit power for an ideal Gaussian beam with
the shielding distance of $d_{\rm sh}$ is given as:

$P_{\mathrm{t,max}}=\frac{\pi r_{\rm a}^{2}\times{\rm
MPE}}{1-\exp{\left(-\frac{2r_{\rm a}^{2}}{W^{2}\left(d_{\rm
sh}\right)}\right)}},$ (47)

where $r_{\rm a}$ is the radius of the limiting aperture given in Table X. The
area of exposed skin at $z=d_{\rm sh}$ is $A_{\rm s}=\pi W^{2}(d_{\rm sh})$.
Let’s assume $t_{\rm ex}>10$ s, then, $r_{\rm a}=1.75$ mm and the maximum
transmit power to ensure skin safety requirements for different conditions of
$W(d_{\rm sh})$ are presented in Table XI.

TABLE XI: Maximum permissible transmit power for skin safety. $W(d_{\rm sh})$ [m] | MPE [W$\cdot$ m-2] |  $P_{\rm t,max}$ [W]
---|---|---
$W(d_{\rm sh})<1.75\times 10^{-3}$ | 1000 | 1000$\times{\pi W^{2}(d_{\rm sh})}$
$1.75\times 10^{-3}\leq W(d_{\rm sh})<\sqrt{\dfrac{0.01}{\pi}}$ | 1000 | $\dfrac{1000\pi r_{\rm a}^{2}}{1-\exp\left(-\dfrac{2r_{\rm a}^{2}}{W^{2}(d_{\rm sh})}\right)}$
$\sqrt{\dfrac{0.01}{\pi}}\leq W(d_{\rm sh})<\sqrt{\dfrac{0.1}{\pi}}$ | $\dfrac{10}{\pi W^{2}(d_{\rm sh})}$ | $\dfrac{10r_{\rm a}^{2}}{W^{2}(d_{\rm sh})\left(1-\exp\left(-\dfrac{2r_{\rm a}^{2}}{W^{2}(d_{\rm sh})}\right)\right)}$
$W(d_{\rm sh})\geq\sqrt{\dfrac{0.1}{\pi}}$ | 100 | $\dfrac{100\pi r_{\rm a}^{2}}{1-\exp\left(-\dfrac{2r_{\rm a}^{2}}{W^{2}(d_{\rm sh})}\right)}$
Figure 24: Skin safety results for $\lambda=1550$ nm.

The skin safety results for $\lambda=1550$ nm are illustrated in Fig. 24. As
discussed in Section II-A, skin safety becomes notable for higher wavelengths
($>1400$ nm). The wavelength $\lambda=1550$ nm is chosen as an example as
there are commercial APDs working at this wavelength. As it can be observed
from these results, the maximum transmit power increases as the beam
divergence increases. Furthermore, as the shielding distance decreases, the
maximum transmit power decreases. For high beam divergence angles, $P_{\rm
t,max}$ can be upto few Watts, however, for laser sources with very small beam
divergence angles, $P_{\rm t,max}$ is considerably small. Shielding is a
practical method that can help to increase $P_{\rm t,max}$ for those lasers.

####  IV-F2 Multimode Laser Sources

For multimode beams, the maximum transmit power will be dependent upon the
maximum power gathered through an aperture of 3.5 mm diameter [37], calculated
at the accessible position where the beam has the smallest area. This can be
calculated as per the formula presented in (36) and (39) for Hermite-Gaussian
and Laguerre-Gaussian beams, respectively.

##  V Summary and Concluding Remarks

This tutorial provided technical information necessary for the safety analysis
of laser sources under different conditions. Initially, we studied beam
propagation for ideal Gaussian and multimode laser beams, including two well-
known solutions of the paraxial Helmholtz equation, i.e., Hermite-Gaussian and
Laguerre-Gaussian modes. We investigated beam shaping by using optical
elements such as lenses and two types of practical diffusers. We also examined
the beam propagation for VCSEL arrays, where multimode VCSELs with various
mode power coefficients were assumed.

The main objective of this study is to derive the maximum permissible transmit
power that fulfills both eye and skin safety requirements for laser products
according to the IEC standard regulations. Analytical derivations were
developed for various beam propagation models and optics. Specifically, first,
we considered the ideal Gaussian beam, for which an algorithm was provided to
obtain the maximum transmit power. Then, we looked into higher-mode beams by
comparing safety results of the mode decomposition and $M$-squared approaches.
It was demonstrated that the $M$-squared method leads to a more conservative
maximum transmit power with respect to the precise beam decomposition method
in some scenarios, however, for the purpose of power budget improvement, one
can use the intricate beam decomposition approach. We also studied the effect
of using a lens in front of a laser source for various focal lengths and
distances between the lens and laser, showing that the maximum transmit power
is minimal when the laser source is placed at the focal point of the lens.

Thereupon, we analyzed VCSEL arrays by considering practical aspects such as
pitch distance (i.e., the center-to-center distance between adjacent VCSEL
elements along rows or columns). The results suggest that by increasing the
size of arrays, the transmit power per VCSEL reduces, and this effect is more
pronounced for small values of pitch distance. We showed that an increase of
$7$ times in the maximum transmit power of a $5\times 5$ VCSEL array is
possible when the pitch distance increases from $0$ to $250\ \mu$m. For lasers
with diffusers, we showed that the transmit power limitation can always be
relaxed to some degree, especially when using diffusers with a higher FWHM
divergence angle. Finally, we investigated skin safety as another important
factor in laser safety analysis, which in particular plays a significant role
for higher wavelengths ($\geq 1400$ nm). The discussions in the paper were
supported by extensive simulation results to characterize the impacts of
various parameters of laser sources such as wavelength and beam divergence
angle, as well as transmitter optics and different exposure durations. This
paper provides guidelines for the design of high-speed laser-based OWC systems
in consideration of the safe transmit power for both indoor and outdoor
applications.

##  VI Acknowledgement

The authors acknowledge financial support from the EPSRC under program grant
EP/S016570/1 ‘Terabit Bidirectional Multi-User Optical Wireless System (TOWS)
for 6G LiFi’. Prof. Harald Haas also acknowledges financial support from the
Engineering and Physical Sciences Research Council (EPSRC) under the
Established Career Fellowship grant EP/R007101/1 as well as the Wolfson
Foundation and the Royal Society. The authors also acknowledge helpful
technical discussions provided by Dr. Ravinder Singh.

## References

  * [1] “Cisco visual networking index: Forecast and trends, 2017-2022,” Cisco, Tech. Rep., Mar. 2019.
  * [2] M. Cave _et al._ , “Review of Radio Spectrum Management,” _An independent review for Department of Trade and Industry and HM Treasury (www. spectrumreview. radio. gov. uk)_ , 2002.
  * [3] Q. C. Li, H. Niu, A. T. Papathanassiou, and G. Wu, “5G Network Capacity: Key Elements and Technologies,” _IEEE Vehicular Technology Magazine_ , vol. 9, no. 1, pp. 71–78, 2014.
  * [4] M. A. Khalighi and M. Uysal, “survey on free space optical communication: A communication theory perspective,” _IEEE Communications Surveys Tutorials_.
  * [5] M. Uysal and H. Nouri, “Optical Wireless Communications — An Emerging Technology,” in _2014 16th International Conference on Transparent Optical Networks (ICTON)_ , 2014, pp. 1–7.
  * [6] H. Haas, J. Elmirghani, and I. White, “Optical wireless communication,” _Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences_ , vol. 378, no. 2169, p. 20200051, 2020. [Online]. Available: https://royalsocietypublishing.org/doi/abs/10.1098/rsta.2020.0051
  * [7] T. Koonen, “Indoor Optical Wireless Systems: Technology, Trends, and Applications,” _IEEE J. Light. Technol._ , vol. 36, no. 8, pp. 1459–1467, Apr. 2018.
  * [8] H. Haas, L. Yin, Y. Wang, and C. Chen, “What is LiFi?” _J. Lightw. Technol._ , vol. 34, no. 6, pp. 1533–1544, 2016.
  * [9] H. Elgala, R. Mesleh, and H. Haas, “Indoor Optical Wireless Communication: Potential and State-of-the-Art,” _IEEE Commun. Mag._ , vol. 49, no. 9, pp. 56–62, Sep. 2011.
  * [10] F. R. Gfeller and U. Bapst, “Wireless in-house data communication via diffuse infrared radiation,” _Proceedings of the IEEE_ , vol. 67, no. 11, pp. 1474–1486, 1979.
  * [11] H. Kaushal and G. Kaddoum, “Optical Communication in Space: Challenges and Mitigation Techniques,” _IEEE Communications Surveys Tutorials_ , vol. 19, no. 1, pp. 57–96, 2017.
  * [12] ——, “Underwater Optical Wireless Communication,” _IEEE Access_ , vol. 4, pp. 1518–1547, 2016.
  * [13] Z. Zeng, S. Fu, H. Zhang, Y. Dong, and J. Cheng, “A Survey of Underwater Optical Wireless Communications,” _IEEE Communications Surveys Tutorials_ , vol. 19, no. 1, pp. 204–238, 2017.
  * [14] Y. H. Kim, W. A. Cahyadi, and Y. H. Chung, “Experimental Demonstration of VLC-Based Vehicle-to-Vehicle Communications Under Fog Conditions,” _IEEE Photonics Journal_ , vol. 7, no. 6, pp. 1–9, 2015.
  * [15] K. Wang, Z. Yuan, E. Wong, K. Alameh, H. Li, K. Sithamparanathan, and E. Skafidas, “Experimental Demonstration of Indoor Infrared Optical Wireless Communications With a Silicon Photonic Integrated Circuit,” _Journal of Lightwave Technology_ , vol. 37, no. 2, pp. 619–626, 2019.
  * [16] E. Sarbazi, H. Kazemi, M. D. Soltani, M. Safari, and H. Haas, “A Tb/s Indoor Optical Wireless Access System Using VCSEL Arrays,” in _2020 IEEE 31st Annual International Symposium on Personal, Indoor and Mobile Radio Communications_ , 2020, pp. 1–6.
  * [17] H. Kazemi, E. Sarbazi, M. D. Soltani, M. Safari, and H. Haas, “A Tb/s Indoor Optical Wireless Backhaul System Using VCSEL Arrays,” in _2020 IEEE 31st Annual International Symposium on Personal, Indoor and Mobile Radio Communications_ , 2020, pp. 1–6.
  * [18] K. Wang, A. Nirmalathas, C. Lim, and E. Skafidas, “4$\,\times\,$ 12.5 Gb/s WDM Optical Wireless Communication System for Indoor Applications,” _Journal of Lightwave Technology_ , vol. 29, no. 13, pp. 1988–1996, 2011\.
  * [19] K. Iga, “Surface-Emitting Laser-Its Birth and Generation of New Optoelectronics Field,” _IEEE Journal of Selected Topics in Quantum Electronics_ , vol. 6, no. 6, pp. 1201–1215, 2000.
  * [20] P. Westbergh, R. Safaisini, E. Haglund, J. S. Gustavsson, A. Larsson, and A. Joel, “High-Speed 850-nm VCSELs with 28 GHz Modulation Bandwidth for Short Reach Communication,” in _Vertical-Cavity Surface-Emitting Lasers XVII_ , K. D. Choquette and J. K. Guenter, Eds., vol. 8639, International Society for Optics and Photonics. SPIE, 2013, pp. 251 – 256.
  * [21] P. Westbergh, R. Safaisini, E. Haglund, J. S. Gustavsson, A. Larsson, and A. Joel, “High-Speed Oxide Confined 850-nm VCSELs Operating Error-Free at 47 Gbit/s at Room Temperature and 40 Gbit/s at 85∘ C,” in _2013 Conference on Lasers Electro-Optics Europe International Quantum Electronics Conference CLEO EUROPE/IQEC_ , 2013, pp. 1–1.
  * [22] E. Haglund, P. Westbergh, J. S. Gustavsson, E. P. Haglund, A. Larsson, M. Geen, and A. Joel, “30 GHz Bandwidth 850 nm VCSEL with sub-100 fJ/bit Energy Dissipation at 25–50 Gbit/s,” _Electronics Letters_ , vol. 51, no. 14, pp. 1096–1098, 2015.
  * [23] X. Xiao, N. K. Fontaine, H. Chen, B. Huang, D. T. Neilson, K. W. Kim, J. H. Sinsky, R. R. Ryf, G. Raybon, P. Winzer, A. Daly, C. Neumeyr, M. Ortsiefer, and S. J. B. Yoo, “High-Speed IQ Modulator Based on Injection-Locked VCSEL Array,” in _2017 Conference on Lasers and Electro-Optics (CLEO)_ , 2017, pp. 1–2.
  * [24] C. Xie, S. Spiga, P. Dong, P. Winzer, M. Bergmann, B. Kögel, C. Neumeyr, and M. Amann, “400-Gb/s PDM-4PAM WDM System Using a Monolithic 2×4 VCSEL Array and Coherent Detection,” _Journal of Lightwave Technology_ , vol. 33, no. 3, pp. 670–677, 2015.
  * [25] P. Westbergh, J. S. Gustavsson, and A. Larsson, “240 Gbit/s VCSEL Array for Multicore Fiber Interconnects,” in _2014 IEEE Photonics Conference_ , 2014, pp. 91–92.
  * [26] A. Liu, P. Wolf, J. A. Lott, and D. Bimberg, “Vertical-Cavity Surface-Emitting Lasers for Data Communication and Sensing,” _Photon. Res._ , vol. 7, no. 2, pp. 121–136, Feb 2019.
  * [27] _Safety of Laser Products - Part 1: Equipment Classification, Requirements and User’s Guide_ , International Electrotechnical Commission (IEC) 60825-1:2014 Std., Aug. 2014.
  * [28] _Safety of Laser Products - Part 2: Safety of Optical Fiber Communication Systems (OFCS)_ , International Electrotechnical Commission (IEC) 60825-2:2004 Std., June 2004.
  * [29] _Safety of Laser Products - Part 3: Guidance for Laser Displays and Shows_ , International Electrotechnical Commission (IEC) 60825-3:2008 Std., Mar. 2008.
  * [30] _Safety of Laser Products - Part 4: Laser Guards_ , International Electrotechnical Commission (IEC) 60825-4:2006 Std., Aug. 2006.
  * [31] _Safety of Laser Products - Part 5: Manufacturer’s Checklist for IEC 60825-1_ , International Electrotechnical Commission (IEC) 60825-5:2019 Std., Nov. 2019.
  * [32] _Safety of Laser Products - Part 8: Guidelines for the Safe Use of Laser Beams on Humans_ , International Electrotechnical Commission (IEC) 60825-8:2006 Std., Dec. 2006.
  * [33] _Safety of laser products - Part 12: Safety of Free Space Optical Communication Systems Used for Transmission of Information_ , International Electrotechnical Commission (IEC) 60825-12:2019 Std., Feb. 2019.
  * [34] _Safety of Laser Products - Part 13: Measurements for Classification of Laser Products_ , International Electrotechnical Commission (IEC) 60825-13:2011 Std., Oct. 2011.
  * [35] _Safety of Laser Products - Part 14: A User’s Guide_ , International Electrotechnical Commission (IEC) 60825-14:2004 Std., Feb. 2004.
  * [36] _Safety of Laser Products - Part 17: Safety Aspects for Use of Passive Optical Components and Optical Cables in High Power Optical Fiber Communication Systems_ , International Electrotechnical Commission (IEC) 60825-17:2015 Std., Oct. 2015.
  * [37] _Safety of laser products - Part 1: Equipment classification, requirements and user’s guide_ , British Standards Institution (BSI) Std. BS EN 60825-1:2014 Std., Aug. 2014.
  * [38] Code of Federal Regulations Title 21: Sec. 1040.10 Laser products. [Online]. Available: https://www.govinfo.gov/content/pkg/CFR-2000-title21-vol8/pdf/CFR-2000-title21-vol8-sec1040-10.pdf. Regulatory Information.
  * [39] International Commission on Non-Ionizing Radiation Protection, “Guidelines on limits of exposure to laser radiation of wavelengths between 180 nm and 1,000 $\mu$m,” _Health Physics_ , vol. 105, no. 3, pp. 271–295, 2013.
  * [40] ——, “ICNIRP Guidelines on Limits of Exposure to Static Magnetic Fields,” _Health Physics_ , vol. 96, no. 4, pp. 504–514, 2009.
  * [41] J. Bernhardt, R. Matthes, A. McKinlay, P. Vecchia, and B. Veyret, “Exposure to Static and Low Frequency Electromagnetic Fields, Biological Effects and Health Consequences (0-100kHz)-Review of the Scientific Evidence and Health Consequences,” _Munich: International Commission on Non-Ionizing Radiation Protection_ , 2003.
  * [42] International Commission on Non-Ionizing Radiation Protection, “ICNIRP Guidelines for Limiting Exposure to Time‐Varying Electric and Magnetic Fields (1Hz – 100 kHz),” _Health Physics_ , vol. 99, no. 6, pp. 818–836, 2010.
  * [43] ——, “ICNIRP Guidelines for Limiting Exposure to Electromagnetic Fields (100 kHz to 300 GHz),” _Health Physics_ , vol. 118, no. 5, pp. 483–524, 2020.
  * [44] ——, “ICNIRP Guidelines on Limits of Exposure to Incoherent Visible and Infrared Radiation,” _Health Physics_ , vol. 105, no. 1, pp. 74–96, 2013\.
  * [45] ——, “ICNIRP Statements on Far Infrared Exposure,” _Health Physics_ , vol. 91, no. 6, pp. 630–645, 2006.
  * [46] ——, “ICNIRP Guidelines on Limits of Exposure to Ultraviolet Radiation of Wavelengths Between 180 nm and 400 nm (Incoherent Optical Radiation),” _Health Physics_ , vol. 87, no. 2, pp. 171–186, 2004.
  * [47] ——, “ICNIRP Guidelines on Protection of Workers against Ultraviolet Radiation,” _Health Physics_ , vol. 99, no. 1, pp. 66–87, 2010.
  * [48] _American National Standard for Safe Use of Lasers_ , American National Standards Institute (ANSI) Z136.1-2014 Std., 2014.
  * [49] _American National Standard for Safe Use of Optical Fiber Communication Systems Utilizing Laser Diode and LED Sources_ , American National Standards Institute (ANSI) Z136.2-2012 Std., Dec. 2012.
  * [50] _American National Standard for Safe Use of Lasers in Health Care_ , American National Standards Institute (ANSI) Z136.3-2018 Std., May 2018.
  * [51] _Recommended Practice for Laser Safety Measurements for Hazard Evaluation_ , American National Standards Institute (ANSI) Z136.4-2021 Std., 2021.
  * [52] _American National Standard for Safe Use Of Lasers In Educational Institutions_ , American National Standards Institute (ANSI) Z136.5-2020 Std., May 2020.
  * [53] _American National Standard for Safe Use of Lasers Outdoors_ , American National Standards Institute (ANSI) Z136.6-2015 Std., Oct. 2015.
  * [54] _American National Standard for Testing and Labeling of Laser Protective Equipment_ , American National Standards Institute (ANSI) Z136.7-2020 Std., July 2020.
  * [55] _American National Standard for Safe Use of Lasers in Research, Development, or Testing_ , American National Standards Institute (ANSI) Z136.8-2012 Std., Apr. 2012.
  * [56] _American National Standard for Safe Use of Lasers in Manufacturing Environments_ , American National Standards Institute (ANSI) Z136.9-2013 Std., Aug. 2013.
  * [57] W. L. Makous and J. D. Gould, “Effects of Lasers on the Human Eye,” _IBM Journal of Research and Development_ , vol. 12, no. 3, pp. 257–271, 1968\.
  * [58] A. Ps, Y. D. Berezin, Y. P. Gudakovskiĭ, V. R. Muratov, A. G. Murzin, and V. Fromzel, “Experimental Determination of Maximum Permissible Exposure to Laser Radiation of $1.54$ $\mu$ wavelength,” _Soviet Journal of Quantum Electronics_ , vol. 8, pp. 137–139, 1978.
  * [59] V. G. Dmitriev, V. Emel’yanov, M. Kashintsev, V. V. Kulikov, A. Solov’ev, M. Stel’makh, and O. B. Cherednichenko, “Nonlinear Perception of Infrared Radiation in the 800–1355 nm Range with Human Eye,” _Soviet Journal of Quantum Electronics_ , vol. 9, no. 4, p. 475, 1979.
  * [60] B. R, H. F, and G. VP, “Theoretical Investigations of Laser Thermal Retinal Injury,” _Health Physics_ , vol. 48, no. 6, pp. 781–796, Jun. 1985.
  * [61] E. Galbiati, “Evaluation of the apparent source in laser safety,” _Journal of Laser Applications_ , vol. 13, no. 4, pp. 141–149, 2001.
  * [62] K. Schulmeister, B. E. Stuck, D. J. Lund, and D. H. Sliney, “Review of Thresholds and Recommendations for Revised Exposure Limits for Laser and Optical Radiation for Thermally Induced Retinal Injury,” _Health Physics_ , vol. 100, no. 2, pp. 210–220, 2011.
  * [63] K. O. S. Gustafsson, “Questions about Using of Atmospheric Attenuation Calculating the Nominal Ocular Hazard Distance,” in _Electro-Optical Remote Sensing, Photonic Technologies, and Applications VII; and Military Applications in Hyperspectral Imaging and High Spatial Resolution Sensing_ , G. W. Kamerman, O. K. Steinvall, G. J. Bishop, and J. D. Gonglewski, Eds., vol. 8897, International Society for Optics and Photonics. SPIE, 2013, pp. 207 – 214.
  * [64] N. Heussner, M. Vagos, M. S. Spitzer, and W. Stork, “A prediction model for ocular damage - experimental validation,” _Journal of Thermal Biology_ , vol. 52, pp. 38 – 44, 2015.
  * [65] S. Ramos, M. Reh, G. Zeck, and N. Heussner, “Evaluation of Short Pulse Laser Damage to the Retinal Pigment Epithelium Layer: A Key Point for the Assessment of Devices Using the Nanosecond Regime,” in _Biophotonics: Photonic Solutions for Better Health Care VI_ , vol. 10685. International Society for Optics and Photonics, 2018, p. 106850H.
  * [66] N. Heussner, S. Ramos, M. Lücking, C. Schwarz, and A. Frederiksen, “Eye Safety Evaluation of Laser Systems Based on Damage Calculations,” in _International Laser Safety Conference_ , vol. 2019, no. 1. Laser Institute of America, 2019, p. 204.
  * [67] L. Jiao, Y. Fan, J. Wang, and Z. Yang, “Comparative Study of Retinal Injuries Induced by a 420–750 nm Supercontinuum Source and a 532 nm Laser,” _Biomed. Opt. Express_ , vol. 10, no. 6, pp. 3018–3029, Jun 2019.
  * [68] L. Jiao, J. Wang, J. Yang, Y. Fan, and Z. Yang, “Retinal Damage Thresholds from 100-Millisecond Laser Radiation Exposure at 1319 nm: a Comparative Study for Rabbits with Different Ocular Axial Lengths,” _Biomed. Opt. Express_ , vol. 10, no. 4, pp. 1882–1890, Apr 2019.
  * [69] D. H. Sliney, “Revisiting Laser Exposure Limits for Intended Viewing,” in _International Laser Safety Conference_ , vol. 2019, no. 1. Laser Institute of America, 2019, p. 103.
  * [70] M. Jean, K. Schulmeister, D. J. Lund, and B. E. Stuck, “Laser-Induced Corneal Injury: Validation of a Computer Model to Predict Thresholds,” _Biomed. Opt. Express_ , vol. 12, no. 1, pp. 336–353, Jan 2021.
  * [71] D. Winburn, “Practical Laser Safety,” 1985.
  * [72] A. Henderson, _Guide to Laser Safety_. Springer Science & Business Media, 1997, vol. 1.
  * [73] R. Henderson and K. Schulmeister, _Laser safety_. CRC Press, 2003.
  * [74] D. H. Sliney and J. Mellerio, _Safety with Lasers and Other Optical Sources: A Comprehensive Handbook_. Springer Science & Business Media, 2013.
  * [75] K. Barat, _Laser Safety: Tools and Training_. CRC Press, 2014.
  * [76] K. Schulmeister, “Classification of Extended Source Products According to IEC 60825-1,” in _Proceeding of the International Laser Safety Conference_ , 2015, pp. 271–280.
  * [77] _Safety of Laser Products - Part 1: Equipment Classification, Requirements and User’s Guide_ , International Electrotechnical Commission (IEC) 60825-1:2001 Std., Aug. 2001.
  * [78] B. E. Saleh and M. C. Teich, _Fundamentals of Photonics_ , 3rd ed. John Wiley & Sons, Inc., 2019, vol. Part I: Optics.
  * [79] K. H. Hahn, M. R. Tan, Y. M. Houng, and S. Y. Wang, “Large area multitransverse-mode vcsels for modal noise reduction in multimode fibre systems,” _Electronics Letters_ , vol. 29, no. 16, pp. 1482–1483, 1993.
  * [80] J. Heinrich, E. Zeeb, and K. J. Ebeling, “Butt-coupling efficiency of vcsels into multimode fibers,” _IEEE Photonics Technology Letters_ , vol. 9, no. 12, pp. 1555–1557, 1997.
  * [81] A. Valle, “Selection and modulation of high-order transverse modes in vertical-cavity surface-emitting lasers,” _IEEE Journal of Quantum Electronics_ , vol. 34, no. 10, pp. 1924–1932, 1998.
  * [82] Guoyu Yu and D. R. Selviah, “Modeling of optical coupling to multimode polymer waveguides: axial and lateral misalignment tolerance,” in _The 17th Annual Meeting of the IEEELasers and Electro-Optics Society, 2004. LEOS 2004._ , vol. 2, 2004, pp. 981–982 Vol.2.
  * [83] I. Kardosh and F. Rinaldi, “Beam Properties and $M^{2}$ Measurements of High-power VCSELs,” _NIVERSITÄT ULM_ , p. 17, 2004.
  * [84] A. E. Siegman and S. W. Townsend, “Output beam propagation and beam quality from a multimode stable-cavity laser,” _IEEE Journal of Quantum Electronics_ , vol. 29, no. 4, pp. 1212–1217, 1993.
  * [85] A. E. Siegman, “How to (Maybe) Measure Laser Beam Quality,” in _DPSS (Diode Pumped Solid State) Lasers: Applications and Issues_. Optical Society of America, 1998, pp. 184–199.
  * [86] “Gaussian Beam Optics,” CVI Melles Griot, Tech. Rep. [Online]. Available: http://experimentationlab.berkeley.edu/sites/default/files/MOT/Gaussian-Beam-Optics.pdf
  * [87] S. R. G. Hall and S. D. Knox, “Traceable Measurements for Beam Propagation Ratio $M^{2}$,” _Journal of Physics: Conference Series_ , vol. 85, p. 012014, Oct. 2007.
  * [88] _Lasers and Laser-Related Equipment - Test Methods for Laser Beam Widths, Divergence Angles and Beam Propagation Ratios - Part 1: Stigmatic and Simple Astigmatic Beams_ , International Organization for Standardization (ISO) 11146-1:2005 Std., Jan. 2005.
  * [89] _Lasers and Laser-Related Equipment - Test Methods for Laser Beam Widths, Divergence Angles and Beam Propagation Ratios - Part 2: General Astigmatic Beams_ , International Organization for Standardization (ISO) 11146-2:2005 Std., Feb. 2005.
  * [90] _Lasers and Laser-Related Equipment - Test Methods for Laser Beam Widths, Divergence Angles and Beam Propagation Ratios - Part 3: Intrinsic and Geometrical Laser Beam Classification, Propagation and Details of Test Methods_ , International Organization for Standardization (ISO) 11146-3:2004 Std., Feb. 2004.
  * [91] P. Pepeljugoski, M. J. Hackert, J. S. Abbott, S. E. Swanson, S. E. Golowich, A. J. Ritger, P. Kolesar, Y. C. Chen, and P. Pleunis, “Development of System Specification for Laser-Optimized 50-µm Multimode Fiber for Multigigabit Short-Wavelength LANs,” _J. Lightwave Technol._ , vol. 21, no. 5, p. 1256, May 2003. [Online]. Available: http://jlt.osa.org/abstract.cfm?URI=jlt-21-5-1256
  * [92] H. Kogelnik and T. Li, “Laser beams and resonators,” _Applied optics_ , vol. 5, no. 10, pp. 1550–1567, 1966.
  * [93] E. Hecht, “Optics,” _Addison Wesley_ , pp. 384–441, 2002.
  * [94] Thorlabs, “Lambertian Pattern Engineered Diffuser,” [Revised Oct. 2008]. [Online]. Available: https://www.thorlabs.com/catalogpages/784.pdf
  * [95] ——, “Engineered Diffusers.” [Online]. Available: https://www.thorlabs.com/newgrouppage9.cfm?objectgroup_id=1660
  * [96] J. M. Kahn and J. R. Barry, “Wireless Infrared Communications,” _Proceedings of the IEEE_ , vol. 85, no. 2, pp. 265–298, 1997.
  * [97] S. Dimitrov, R. Mesleh, H. Haas, M. Cappitelli, M. Olbert, and E. Bassow, “On the SIR of a Cellular Infrared Optical Wireless System for an Aircraft,” _IEEE Journal on Selected Areas in Communications_ , vol. 27, no. 9, pp. 1623–1638, 2009.
  * [98] D. A. Atchison and G. Smith, _Optics of the Human Eye_. Butterworth-Heinemann, 2000.
  * [99] A. E. Siegman, “Defining, Measuring, and Optimizing Laser Beam Quality,” in _Laser Resonators and Coherent Optics: Modeling, Technology, and Applications_ , vol. 1868. International Society for Optics and Photonics, 1993, pp. 2–12.
  * [100] ——, “How to (Maybe) Measure Laser Beam Quality,” in _Diode Pumped Solid State Lasers: Applications and Issues_. Optical Society of America, 1998, p. MQ1.
  * [101] _Laser and Laser-Related Equipment–Test Methods for Laser Widths, Divergence Angle and Beam Propagation Factor_ , ISO 11146 Std., 2003.
  * [102] G. Stefansson _et al._ , “Toward Space-like Photometric Precision from the Ground with Beam-shaping Diffusers,” _The Astrophysical Journal_ , vol. 848, no. 1, p. 9, oct 2017.