A REFINEMENT OF THE DETERMINATION METHOD OF THE
LINEAR LOW-PRESSURE UV LAMPS RADIANT FLUX
S. V. Prytkov, S. S. Kapitonov12, A. S. Vinokurov1
1
2
Lodygin Research Institute of Light Source» (Saransk)
National Research Mordovia State University (Saransk)
E-mail: kapitonov_ss@vniiis.su, sergeyvladi88@gmail.com
Annotation
For the measurement of linear low-pressure UV lamps radiant flux the method proposed
by the IUVA, which is based on the Keitz method, has become widely used. For deriving the
equation that connects the irradiance generated by a lamp at a close distance and its radiant
flux, the authors of the method presume that the lamp is the cylinder of equal radiance.
According to our estimates, this assumption leads to the inaccuracy of 3% to 5% with respect to
goniophotometric measurements.
In this investigation, a general formula is derived that connects the irradiance generated
by a linear emitter and its radiant flux. This formula does not impose restrictions on the radiant
intensity curve in the longitudinal plane. The Keitz equation is its particular case. To reduce the
inaccuracy of the IUVA method, the angular distribution of the radiant intensity of the UV lamps
is proposed to be approximated by a cosine polynomial.
In order to find the coefficients of the polynomial, clarify the Keitz formula, as well as to
estimate the inaccuracy of the refined and classical versions of this formula, the series of
goniophotometric measurements of the DB 15, DB 18, DB 30 lamps at various distances was
carried out.
It was found that at a scanning step   5 the first 9 terms of the trigonometric
expansion are sufficient to describe the radiant intensity curve with an accuracy satisfactory for
practical use. It was also shown that the Keitz method needs to be refined only on the basis of
goniophotometric data obtained upon condition r / l  6 where r is the test distance, l is the lamp
length..
It was identified that in the case of a differentiated approach, the approximation of the
low-pressure UV lamps radiant intensity curve by a cosine polynomial makes it possible to
provide an inaccuracy of simplified methods that does not exceed 1% in relation to the
goniophotometric method. It is indicated that in order to find a universal factor applicable for
the entire range of linear low-pressure UV lamps, the development and the analysis of statistical
data is required.
Keywords: linear low-pressure UV lamp, radiant flux, the Keitz equation,
goniophotometric measurements, trigonometric approximation, the distribution curve of radiant
intensity.
Introduction
By now the classical methods of the integrating sphere and the goniophotometer have
found limited application to measure the radiant flux (RF) of the linear low-pressure UV lamps.
This is due to a number of reasons. In the case of an integrating sphere, there are certain
technological difficulties in production of a UV-resistant coating with a high reflectance in the
range of 200-400 nm. In addition, it is necessary to eliminate or take into account its inevitable
luminescence [16]. The goniophotometer does not have such problems, however, its use in the
conditions of industrial enterprises laboratories does not always seem justified due to the fact
that the light distribution of linear UV lamps is axisymmetric and similar to each other. This
circumstance makes it possible to completely restore the light distribution solid and to calculate
the energy RF by measuring the irradiance (IR) in the direction perpendicular to the lamp axis.
So professional community has developed the simplified methods for determining the RF, the
essence of which is reduced to the well-known in lighting engineering problem of IR, formed by
a linear emitter with a given RF. It should be noted that the goniophotometric method is
universal and does not impose restrictions on the design of lamps and their light distribution,
therefore, the accuracy of simplified methods should be evaluated precisely in relation to the
results of goniophotometric measurements. Moreover, these results can be used to refine and
improve simplified methods.
The most well-known method for determining the RF of UV lamps was proposed in 2007
in the article [15]. It uses the luminous line equation presented by H.A.E. Keitz in 1955 in his
book [10]. It should be noted that this problem was set and solved in general terms in the
national lighting engineering science at the end of the 1940s, and it was reflected in the book by
V.V. Meshkov [7]. The release of books by H.A.E. Keitz and V.V. Meshkov coincides with the
period of rapid development and introduction of fluorescent lamps, which led to the development
of new methods for calculating lighting installations.
In 2008, on the basis of this method, a detailed measurement procedure was developed,
composed by the members of the international ultraviolet association IUVA [12]. Figure 1 shows
a measurements scheme according the Keitz method. As one can see, the photometric distance r
(usually r ≥ 2l) is such that the lamp should be considered already as a linear emitter. In this case,
the authors of the article [15] propose the following formula linking the RF and IR of the lamp.
Fig. 1 The scheme of measurements by the Keitz method

2 E 2 LR
2  sin 2
where E is IR, R is the test distance, L is the lamp length.
This equation was derived according the assumption that the element of the lamp length
has the properties of an ideal object such as the cylinder of equal radiance, in which the radiant
intensity in the longitudinal plane changes according to a sine law (if the angle is set aside from
the cylinder axis), i.e. I ( )  I 90 sin  , where I 90 is the radiant intensity in the direction
perpendicular to the cylinder axis. So, the light distribution solid of the cylinder of equal
radiance is a toroid formed by rotating a circle around the tangent aligned with the axis of the
cylinder (Fig. 2).
Fig. 2 The light distribution solid of the cylinder of equal radiance
(1)
The angular distribution of the radiant intensity of real lamps differs from the sinus one (Fig.
3), therefore, in the standard [4], the following expression is proposed to find the RF:
(2)
where  is the coefficient determined during the goniophotometric tests.
It’s necessary to make clear what the coefficient  is. The use of the term “geometrical factor”
for it, as we can see in the standard [4], seems incorrect for us because, by its definition [2], the
geometrical factor is a value independent of the source light distribution. At the same time, the
dependence of
on the angular distribution of the radiant intensity can be described as:
(3)
where I rel ( ) 
I ( )
is the relative angular distribution of the radiant intensity, I ( ) is the
I 90
angular distribution of the radiant intensity, I90 is the radiation strength for   90
Fig.3 The curve of the radiant intensity of the cylinder of equal radiance and a low-pressure UV
lamp
It's clear that I rel (90 )  1 . It is also clear that the values I rel ( ) are non-dimensional, therefore
 takes the dimension of steradians, although in the strict sense  is not a solid angle either.
According to the formula (3) each zonal solid angle is weighted by the corresponding value
I rel ( ) . Further in the article we will use the term relative radiant flux for  .
In the case of the cylinder of equal radiance I rel ( )  sin  we have:
(4)
As one can see,  of the cylinder of equal radiance enters into equation (1).
In general, the value  is included in the equations of all simplified methods [1; 5; 15] and has
a fundamental character, since this is the only term in the equations that is not measured. The
accuracy of the method depends on how much the real lamp  differs from the one, which is
used in the method.
Note that V.V. Meshkov in [7], when deriving the IR equation from the luminous line, in
addition to using the light distribution of the cylinder of equal radiance (Iα = cosα), proposes to
use the following polynomial for their approximation if the light intensity curve of a real lamp
differs from the cosine law:
(5)
where A, B, C are the coefficients that determine the shape of the curve.
Here we can discuss the form of the polynomial, the sufficiency of three terms for an
accurate description of the light intensity curve, but we are convinced that the approach itself
allows us to create a more accurate model of the angular distribution of the radiant intensity of
linear UV lamps, and therefore reduce the error of simplified methods, which in relation to
goniophotometric measurements is on average 5% according to IUVA [12].
Since the publication [15], a number of papers [14] have appeared, refining the formula (1),
and according to them there are other sources of error besides  , for example, the shift of the
receiver relative to the center of the lamp. It also follows from these works that in some cases it
is more convenient to derive the equation using not a horizontal IR, but a value similar to that,
which is called the mean spherical illuminance in the national literature.
Taking the above into account, we decided to derive the equation connecting the IR and the
RF of the UV lamp based on the same initial conditions that were used as the basis for equation
(1), then estimate  based on goniophotometric measurements and, if possible, clarify our
expression keeping the equation simple.
Methods
Analyzing the formula (3) and the relative angular distribution of the radiant intensity
I rel ( ) included in it, we come to the conclusion that:
  I90  
(6)
where  is the RF of the lamp, I90 is the radiant intensity in the direction perpendicular to the
lamp axis,  is the lamp relative radiant flux.
So, I90 is a scale factor, and the value  , as already mentioned, is determined in the course
of goniophotometric measurements or, in the case of the cylinder of equal radiance, is taken
equal to  2 . The task is to determine I90 correctly with the IR measured at a close distance.
Further finding of  is a trivial action.
Deriving the equation, we will assume that each element of the lamp length in the
longitudinal plane has the same curve of the radiant intensity. We will also use the concept of
specific (per length unit) radiant intensity.
In this case, if we place the point E as it shown in fig. 4, then the IR at this point, formed by
the entire lamp, will be described by the following expression:
(7)
If l1  l2  l / 2 then 1   2   and E is equal to:
(8)
where dI is a radiant intensity of a line element dl directed to the point E , r is an
interval between the line element dl and the point E .
The index  at these values indicates that they are the angle functions, but we do integrate
over the lamp length. So, we express the differential of length through the differential of the
angle  .
(9)
Fig. 4 Linear emitter
The radiant intensity dI of the line element dl directed to the point E can be expressed the
next way:
(10)
where I 0 is the specific radiant intensity per length unit dl in the direction perpendicular to
the segment dl .
Then we find the formula for the interval between an elementary segment dl and a point E :
(11)
As a result, substituting (9) in (10), and then the resulting expression and (11) in (8), we get:
(12)
Further, if we take into account that I 0   / (l ) , where  is the RF of a lamp, l is the
lamp length,  is the lamp relative radiant flux, then we can find the following expression
connecting E and  :
(13)
In the case of the cylinder of equal radiance    2 , and I rel ( )  cos( ) , then
expression (13) can be rewritten the next way:
(14)
We should not be deceived by the formula I rel ( )  cos  . The light distribution of the
source remains sinusoidal, i.e. the circle shown in fig. 4, rotates around the axis of the cylinder
forming a toroidal light distribution solid (Fig. 2). The formula I rel ( )  cos  is equal to
I rel ( )  sin  since   90   and cos(90   )  sin( ) .
Calculating the integral in the denominator (14), we get:
(15)
In the case of real lamps, to calculate  and the integral in the denominator (13), we
propose to use the trigonometric approximation of the function I rel ( ) .
Obviously, the function I rel ( ) is 2 -periodic: after rotating through a full angle, we
return to the original direction. If all points  k  k  , where  is a constant angle step, are
distributed over a segment, the sought function can be approximated by a cosine polynomial [9]:
(16)
In the labor [9] it is shown that the series (5) is a special case (16).
Since the light distribution of linear UV lamps is symmetrical about the lamp axis, i.e.
I rel ( )  I rel ( ) , then the experimental data obtained in the diapason [0,  ] can be mirrored to
the area [0,  ] . Then, in order to determine the coefficients of the polynomial (16), one can use
the Euler-Fourier formulas:
ak 
1


 I
rel
( ) cos k d
(2)

Due to the fact that after a certain k, some coefficients become orders of magnitude less
than others and do not significantly affect the curve of the radiant intensity, it is advisable to
exclude small terms from the expansion to simplify the polynom.
The polynom (16) found in this way can be used both to find
(3):
 , if we will substitute it in

  2  T ( )sin  d
(3)
0
and for the integral in the numerator (13).
As a result, the formula for the connection between RF and IR for a linear emitter will be
as follows:


Elr 2  T ( ) sin  d

0
Elr

2 T ( ) cos  d

(4)
2  T ( ) cos  d
0
0
Note the fact that the angle  is substituted in the denominator in the polynomial, and the
angle  is used when finding

(Fig. 4).
Analyzing together Figures 3, 4, as well as expressions (18) and (19), we come to the
following two conclusions.
1. Integration occurs over the entire diapason [0 ,180 ] when  is found. There are areas
in it in which there is a noticeable deviation of the real curve of the radiant intensity from the
sinus one, which on average leads to an inaccuracy of 5% when using the Keitz method
(according to IUVA information) in relation to goniophotometric measurements.
2. On the other hand, the experimental curve does not differ much from the sine curve in


the diapason  [75 ,105 ] . For a standard measurement scheme (R/L≥2), the specified
diapason overlaps the integration area of the formula, which is included in the denominator (19).


Therefore, assuming that in the diapason   [ 15 ,15 ] formula T ( )  cos (19) can be
simplified:

 Elr

2 cos 2 ( ) d

2 Elr
sin 2  2
0
To determine  and evaluate the error of the formula (20), a series of goniophotometric
measurements of the DB18 lamp was made. At the distances of 1m, 1.5 m, 2.5 m, 3.1 m, 3.5 m, 4
m, 5 m, 5.5 m, and 6 m with increments   5 in the diapason  [0 ,180 ] the dependence
E ( ) was determined. To measure IR an ILT 1700 radiometer with a SED 240/W radiometric
(5)
head was used. The lamp worked in conjunction with a semiconductor ballast. Curve of lamp's
run-up time (Fig. 5) and the moment of stabilization of the light parameters was recorded using
our program, in which the ILT 1700 readings were taken every 0.13 seconds. Goniophotometric
measurements began after passing of the IR maximum.
Fig. 5 The graphics of DB18 lamp's run-up
Fig. 6 shows the schematic representation of the measuring equipment. The
measurements were carried out in a dark room on a photometric bench equipped with a
mechanical goniometer Г. The middle of the lamp was aligned with the center of rotation of the
goniometer. The angles were counted along the goniometer limb. The distance r between the
lamp and the receiver was controlled by using the measuring scale of the photometric bench.
Fig. 6 The measuring equipment
To minimize the stray light, we used a screen S2 with the size of 1.1 X 1.1 m with a
rectangular opening 0.12 X 0.03 m, which was installed in front of the detector. The distance
between the screen Э2 and the detector was selected in such a way that the luminous surface of
the lamps to be measured was completely visible from the location of the detector through the
hole in the screen, but the area of the peripheral zone was minimal.
An additional opaque rectangular screen S1 of 0.7 X 0.05 m was used to account for the
diffused light that passed through the hole in the screen E2. The screen E2 was installed between
the lamp and the screen E1. The size of the screen E1 was chosen so that one hand completely
blocked the passage of direct rays from the lamp through the hole in E1, and the other prevented
the passage of scattered light through the same hole as little as possible.
The background radiation was estimated at all indicated distances. The lamp was oriented
perpendicular to the photometric axis ((   90 ), then the technique described in the article [6]
was used, i.e. screen E1 moved between the lamp and screen E2 until the maximum rated
background illumination values.
Next, the angular distribution of the IR with the screen E1 ( EФ ( ) ) and without
( EФЛ ( ) ) was measured alternately. The final E ( ) curve was obtained as a diff
E( )  EФЛ ( )  EФ ( ) .
The Er ( ) determined values for a given distance r were then simultaneously used to
calculate the coefficients of the polynomial (16), the values
 according to the formula (18), and
to calculate the RF according to the formula:

  2 r 2  E ( )sin  d
(6)
0
Next, a  (r / l )
dependency graph was plotted to determine the ratio at which
stabilization was observed. After stabilization, the bogey value

 ср . was found from the values
of the following numbers, which was used for the refined Keitz formula (20). To calculate the IR
by the Keitz formula (15) and by the refined formula (20), we used the IR Er (90 ) from the
same set Er ( ) of measured values.
Goniophotometric measurements were carried out for DB 15 and DB 30 lamps to assess
the possibility of applying to lamps of a different power. The lamps were operated in a set
selection electromagnetic DOI of the corresponding nominal value.
Data obtained during goniophotometric measurements and mathematical calculations
were processed using the python Sage Math language distribution kit [13].
Results
Fig. 6 shows the curves of the radiant intensity in relative units of the equal radiance
cylinder and the DB 18 lamp. T ( ) is a polynom approximating a tabular function I rel ( ) . The
polynom is composed of the first 9 terms of the expansion (16).
Fig. 7 The approximation of the curve of radiant intensity
Table 1 shows

of the DB 18 lamps at different distances r including and excluding
stray light.
Table 1. Relative radiant flux of the DB 18 lamp
r, m
Including stray light
Excluding stray light
1.0
3.083881161784231 π
3.111041037199330 π
1.5
3.033900058017592 π
3.035449496101235 π
2.5
2.995452348096802 π
2.997553844077054 π
3.1
2.991073461237586 π
2.992735533749916 π
3.5
2.990893680555530 π
2.992564478646564 π
4.0
2.992087394842206 π
2.994717008094560 π
5.0
2.993307920225696 π
2.997864109563205 π
5.5
2.995585244061916 π
3.000270088617926 π
2.991045591577655 π
6.0
2.994717008094560 π
Fig. 7 shows the dependence  (r / l ) . The dependence is built on the basis of the data in
Table 1. Inter-cathode distance DB 18: l = 0.49 m.
Table 2 shows the calculations of the EF by the simplified method for γ = π2, γ = 3.084 π
(corresponding r / l  2 ), γ = 2.992 π (arithmetic mean of values following r / l  6 ) and by the
goniophotometric method.
Figure: 8 Dependence of γ on R/L
Table 2. Energy flux of the DB18
r, m
γ = π2, W
γ = 3.08388π, W
γ = 2.99233π, W
gonioph., W
1.0
4.30000
4.22100
4.09570
4.07225
1.5
4.27030
4.19185
4.06741
4.05192
2.5
4.24466
4.16669
4.04300
4.02128
3.1
4.23601
4.15819
4.03475
4.01612
3.5
4.20526
4.12801
4.00546
3.99020
4.0
4.22435
4.14675
4.02365
4.01446
5.0
4.20220
4.12500
4.00254
3.99710
5.5
4.19801
4.12090
3.99856
3.99643
6.0
4.18823
4.11129
3.98924
3.98274
Table 3 shows the relative error of the Keitz method with different values. When
calculating the error, the results of goniophotometric measurements were taken as the actual
value.
Table 3. Relative error of methods for the DB18
r, m
γ = π2, %
γ = 3.08388π, %
γ = 2.99233π, %
1.0
5.59257 %
3.652823
0.575757
1.5
5.38939 %
3.453377
0.382232
2.5
5.55506 %
3.616001
0.540028
3.1
5.47517
3.537579
0.463934
3.5
5.38956
3.453554
0.382389
4.0
5.22831
3.295248
0.228797
5.0
5.13123
3.199958
0.136336
5.5
5.04414
3.114467
0.053382
6.0
5.15946
3.227665
0.163220
Tables 4 and 5 show the results of measurements of the EF by the Keitz method at
different, goniophotometric method, the error of methods for DB15 and DB30 lamps.
Table 4. Energy flux and relative error of methods for DB15 (r = 1.5)
γ = π2
γ = 2.99233π
γ=3π
gonioph.
Flux, W
3.40026
3.23871
3.24701
3.25773
relative error %
4.38
0.58
0.32
—
Table 5. Energy flux and relative error of methods for DB30 (r = 2.5)
γ = π2
γ = 2.99233π
γ=3π
γ = 3.04 π
gonioph.
Flux, W
11.8298
11.26775
11.29663
11.44769
11.48464
relative error %
3.01
1.89
1.64
0.32
—
Discussion
In this work, a general formula (13) was derived that connects the IR formed by the linear
radiator and its RF. This formula does not impose restrictions on the radiant flux curve in the
longitudinal plane. Keitz's equation (1) is a special case of it if we assume that the source is a
cylinder of equal radiance. In the formulation of the problem (see Fig. 4 and formula 7), in
addition to the error caused by the discrepancy between the radiant flux curve of real lamps and
the one adopted in the method, two more sources of error are visible. First, it is the offset of the
detector relative to the center of the lamp. Secondly, this is the error caused by the deviation of
the angular response of the detector from the cosine one. Therefore, when choosing a
photometric distance, one should be guided not only by the IUVA recommendations, but also the
passport data of the radiometer, which indicates the deviation error of the angular characteristic
and the corresponding angle of view.
The light distribution of low-pressure UV lamps is somewhat different from the cylinder
of equal radiance (see Fig. 3); therefore, it was proposed to approximate the relative distribution
function of the radiant flux I rel ( ) , included in (13) with a cosine polynomial (16). Fig. 6 it can
be seen that at the scanning pitch   5 (n = 37), the first 9 terms of expansion (13) are
sufficient to describe the function with sufficient accuracy for practice I rel ( ) .
From table 1 it follows that  , estimated excluding the stray light is always higher. This
is due to the fact that when the lamp is turned as the radiant intensity decreases, the relative
content of stray light in the measured signal increases, after which the curve is somewhat
compressed. The form of the curve changes. When determining  , this is worth paying attention
to.
Fig. 7 shows that at r / l  6 , the coefficient  stabilizes. This means that for a given ratio,
the entire lamp can be considered as a point source, to which the concept of radiant flux is
applicable, and therefore its light distribution in the far field can be described by the angular
distribution of the radiant flux. Strictly speaking, before the specified ratio, the calculated
values  are incorrect.
Goniophotometric measurements and calculations 
were carried out at all the above
distances, because they make it possible to unambiguously record the moment of stabilization of
the photometric body, i.e. transition of the lamp to the status of a point source. At the same time,
when measuring only the axial radiant intensity of low-pressure UV lamps at different distances,
this is not always possible to determine unambiguously.
With regard to goniophotometric measurements, a comment should be made regarding
the maximum radiation reported in the IUVA method. The temperature of the cold point of lowpressure lamps, which determines the vapor pressure of mercury and RF, strongly depends on
the ambient temperature, at a certain value of which a clear maximum of the combustion curve is
observed (see Fig. 5). The IUVA method recommends using the RF value corresponding to this
maximum to calculate the IR. However, in the case of goniophotometric measurements, it is
difficult to record the moment of passage of the RF maximum, even taking into account that they
are carried out for the same plane and are fully automated. They would have to be carried out
instantly. In this work, goniophotometric measurements were carried out after passing the
maximum. That could make certain corrections in the reproducibility of the absolute values of
the RF, but for the formed beam of rays ( r / l  6 ) did not have a noticeable effect on the value 
(see Table 1 and Fig. 8). We also note that having a combustion curve, the results of
goniophotometric measurements can be scaled by multiplying by a factor representing the ratio
of the maximum IR to the IR value corresponding to the stabilized state of the lamp.
Turning to the analysis of the results (see tables 2 and 3), we conclude that when using
the classical version of the Keitz method (    2 ) the error for the DB 18 lamp is just over 5%.
This result is in good agreement with the data given in [11]. If used the updated version of the
method (20), with   2.99233 at r / l  6 (see figure 7) the error does not exceed 1%. If used
  3.08388 at r / l  2 (see figure 7), the error is more than 3%. The results calculated at
  3.08388 demonstrate the fact that it is necessary to refine the Keitz method only on the
basis of goniophotometric data obtained under the condition r / l  6 . On the other hand, when
calculating the RF by the goniophotometric method (the last column in table. 2) we did not put
forward any requirements for the attitude r / l . Moreover, the RF values calculated in this way
were taken as valid when estimating the error of simplified methods. The fact is that if only the
RF is to be determined during goniophotometric measurements, then the use of formula (21) at
almost any distance does not cause difficulties. In this case, the IR is integrated over the surface
of the spherical surface enclosing the source. Nevertheless, in determining the value  , in
addition to the RF in the numerator, which can be found both by integrating the radiant intensety
over the solid angle and over the IR of the spherical surface, there is the radiation force in the
denominator. In this case, the dimensions must already meet the condition r / l  6 .
Analysis of tables 4 and 5 allows us to conclude that the calculated for DB18 also gives a
good result: the error for DB 15 and DB 30 is less than the error of the numerical classical
version of the method. However, in the case of DB 30, when used   2.99233 the difference is
not so noticeable. This means that the angular distribution of the radiant intensity of the DB 30
lamp is closer to sinusoidal than that of DB 15 and DB18. If we create a polynomial specifically
for DB 30 (γ = 3.04 π), the error again does not exceed 1%.
Summing up the results of the study, we can say that the Keitz - Meshkov formula is
quite good and simple for practical measurements, but the coefficient γ = π2 used in it leads to
an error of 3% to 5% in relation to the goniophotometric method. With a differentiated approach,
the approximation of the angular distribution of the radiation intensity of low-pressure UV lamps
by the cosine polynomial (16) allows us to find γ that provides an error of simplified methods not
exceeding 1% with respect to goniophotometric measurements. Finding a convenient from the
point of view of practice, one universal application for the entire range of linear low-pressure
UV lamps is possible after the development and analysis of statistical data. The first results show
that the optimal value for mercury lamps of the DB type is γ = 3π.
Note that the standard [4] specific values of γ are not indicated. It is proposed to find
them as a result of goniophotometric measurements, but the procedure for processing
experimental data (see formula (3)) to obtain γ is not given. Although this is important for
understanding the following: expression (2) from the standard [4] has practical meaning only in
the case of using a predetermined γ, intended for repeated use. Having in the angular distribution
of the radiation intensity of a particular lamp, its RF can be found without γ. In addition,
expression (2) implies that measurements are performed under far-field conditions, although this
is not explicitly specified in [4]. It can be shown that as the photometry distance r increases, the
expression 2l / (2  sin 2 ) in formula (20) converges to r. In this case, we get the formula (2).
In addition, expression (2) implies that the measurements are performed in the far-field
conditions, although this is not explicitly specified in [4]. It can be shown that as the photometry
distance r increases, the expression in formula (20) converges to r. In this case, we get the
formula (2). Thus, the formula (20) is universal and allows us to implement measurements in
both far and near field conditions, which is especially important for low-pressure lamps, given
their overall dimensions. In view of the above, it is obvious that there is a need to clarify the
current standards for measuring tubular and other forms of lamps, taking into account modern
ideas.
In conclusion, we will say that in this paper we used a phenomenological approach to
find the value of γ, i.e. its calculation was based on the results of the goniophotometric
experiment "as is", without describing the factors that determine the radiant intensity curve of the
lamp. At the same time, throughout the article, we emphasized that we are talking about lowpressure UV lamps of the DB type. In the book [3] by A. A. Gershun, a mathematical model of
the radiating volume is given in the form of an infinitely long cylinder, from which it follows
that the radiation intensity curve is a function of the output kz, where k is the absorption
coefficient of the substance enclosed in the volume, and z is the diameter of the cylinder. A
similar model is found in the book [8] by G. N. Rokhlin. These models do not take into account
the properties of the discharge tube glass, but these are details. Most importantly, they show that
for a discharge that is transparent to its own radiation, the radiant intensity curve is uniform, and
for a discharge that is opaque to its own radiation, it is sinus. All intermediate states are
characterized by a curve in the polar system resembling an ellipse. G. N. Rokhlin in the book [8]
on 68 p. says the following: "the nature of the radiation distribution in space gives an idea of the
value of the absorption coefficient." Thus, taking into account the results of goniophotometric
measurements obtained by us, it can be argued that the plasma of low-pressure UV lamps with a
powerful line of 254 nm is opaque to its own discharge. On the other hand, a different picture is
observed for high-pressure lamps. In the book by G. N. Rokhlin, experimental data are given for
some high-pressure mercury lamps, where γ is from 10 to 11.5. the determination of γ for highpressure lamps based on goniophotometric data using the cosine polynomial (16) is the subject
of our future research. Also of interest is the direct measurement of the plasma absorption
coefficient.
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