55 

 This work is licensed under a Creative Commons Attribution 4.0 International License. 

 

  

 

 

Calculation of Modes Properties for Single-Mode and Multimode Fibers at 

633 nm 

 

 

 
 

 

 

Abstract   

The need for optical fibers has emerged for its ability to transmit information with less 

attenuation and over long distances. In this work, four optical fibers with core radii from 1 µm to 

4.75 µm in steps of 1.25 µm and a numerical aperture of 0.17 were studied and their modes 

properties have been calculated at a wavelength of 633 nm by using RP Fiber Calculator (free 

version 2022). Also, the effect of increasing the core radius on these properties has been studied. 

Multimode fibers can be obtained when the radius of the fiber core is large compared to the 

operating wavelength of the fiber which is less than the cutoff wavelength of the mode. Otherwise, 

a single-mode fiber is obtained. It has been concluded that all the calculated properties increase 

with increasing core radius. More than half of the power is contained in the core. Intensity profiles 

of all modes were illustrated.  

 

Keywords: Optical Fibers, Single-Mode Fiber, Multimode Fibers, Step Index Fibers, RP Fiber 

Calculator. 

 

1. Introduction 

Optical fibers are very fine fibers of glass [1]. A step index fiber (SIF) consists of a central glass 

core (having refractive index 𝑛1) surrounded by a cladding layer of slightly lower refractive index 
𝑛2 [2]. Figure 1 illustrates the two major types of SIF. In a single-mode fiber (SMF), only one 
mode can propagate; while a multimode fiber (MMF),  allows many modes to propagate [1]. The 

main difference between the SMF and MMF is the core size [2]. The bandwidth of SMF is so large 

compared to that of the MMF that SMF is used in all long-haul communications [3]. Because of 

their larger core cross-sectional area, MMFs are more suitable than SMFs for power transmission 

applications [4]. 

 

Doi: 10.30526/35.4.2851 

 

Article history: Received 28 April 2022, Accepted 14 June 2022, Published in October 2022. 

 

 

 

Ibn Al-Haitham Journal for Pure and Applied Sciences 

Journal homepage: http://jih.uobaghdad.edu.iq/index.php/j/index 

 

Wasan M. Hmood 

General Directorate of Education in Baghdad, Second 

Rusafa, Baghdad, Iraq. 

wasn.mahdi1204a@ihcoedu.uobaghdad.edu.iq 

 

 

Aqeel R. Salih 
Department of Physics, College of Education for Pure 

Science (Ibn-AL-Haitham), University of Baghdad, 

Baghdad, Iraq. 
aqeel.r.s@ihcoedu.uobaghdad.edu.iq 

 

 

 

https://creativecommons.org/licenses/by/4.0/
mailto:wasn.mahdi1204a@ihcoedu.uobaghdad.edu.iq
mailto:aqeel.r.s@ihcoedu.uobaghdad.edu.iq
mailto:aqeel.r.s@ihcoedu.uobaghdad.edu.iq


IHJPAS. 53 (4)2022 
 

56 

 
Figure 1. Schematic diagram showing a SM SIF and MM SIF, where 𝑟 is the radial position [5]. 

 

Before 1970, optical fibers have high losses (about 1000 dB/km). Their use for communication 

purposes was considered impractical [6]. Kao and Hockham [7] suggested that glass fibers could 

be a good transmission medium if impurities could be removed. In 1970, SMFs with a loss of about 

17 dB/km at a wavelength near 633 nm were produced, making fiber-optic communications 

practical [8]. This is recognized as the birth of optical fiber communication. Since then, the 

progress in this field has been phenomenal [9].     

In 2020, Salih [10] used RP Fiber Calculator to design SM SIFs at the wavelengths of       1310 

nm and 1550  nm. In the same year, Ibrahim and Salih [11, 12] used this calculator for studying 

modes properties for SIFs at 850  nm and 1300 nm. In 2021, Salih [13] designed a MM SIF at 

1300 nm. In the same year, Shnain and Salih [14–16] designed SIFs and calculated their guided 

modes properties at 1550 nm. Also, Salih [17] calculated properties of the fundamental mode for 

SM SIFs at 1550 nm. 

In this work, the modes properties for SM and MM SIFs at 633 nm have been calculated with 

free fiber optics software RP Fiber Calculator (version 2022). Also, the effect of increasing the 

core radius on these properties has been studied. 

 

2.  Theoretical Background 

If the angle of incidence is larger than the critical angle (𝜃c), the incident light undergoes total 
internal reflection. The critical angle is determined by [18]:  

𝜃c
= sin−1(𝑛2/𝑛1)                                                                                                                                        (1) 

The numerical aperture (NA) is defined by: 

NA

= √𝑛1
2 − 𝑛2

2                                                                                                                                         (2) 

The normalized frequency (V) governs the number of modes (M) and their propagation 

constants [19]. It is given by [2]:   

V
= 𝑘0aNA                                                                                                                                                      (3) 

where (𝑘0 = 2 π λ0),⁄  λ0 is the wavelength of light and a is the core radius. In practice, a< 2 µ m 
for a SMF in the visible region.  

The effective refractive index (𝑛eff) is related to the propagation constant (𝛽) by: 

𝑛eff
= 𝛽 𝑘0⁄                                                                                                                                                     (4) 
It varies approximately between 𝑛1 and 𝑛2.  

The effective area of the fundamental mode is [6]: 

𝐴eff
= 𝜋𝜔0

2                                                                                                                                                    (5) 



IHJPAS. 53 (4)2022 
 

57 

where the spot size (𝜔0) of the fundamental mode is given by: 

(𝜔0 a)⁄ ≈ 0.65 + 1.619V
−3/2

+ 2.879V−6                                                                                            (6) 
The percentage power in core of the fundamental mode is: 

P in core = [1 − exp (−
2a2

𝜔0
2

)]

× 100%                                                                                                                      (7) 
The mode cutoff wavelength is given by [9]: 

𝜆co

=
2π

Vco
aNA                                                                                                                                                (8) 

where Vco is the cutoff frequency of the linearly polarized (LP𝑙,𝑚) mode below which it cannot 
exist [5] as shown in Table 1, where 𝑙 = 0, 1, 2, … and 𝑚 = 1, 2, 3, … 

 
Table 1. Cutoff frequencies of the LP𝑙,𝑚 modes in a SIF [9]. 

 LP𝑙,𝑚 modes Vco 

LP0,1 0 

LP1,1 2.4048 

LP2,1 3.8317 

LP3,1 5.1356 

LP4,1 6.3802 

LP5,1 7.5883 

LP0,2 3.8317 

LP1,2 5.5201 

LP2,2 7.0156 

LP0,3 7.0156 

     

A SIF with 0<V<2.4048 supports only the fundamental (LP0,1) mode. Similarly, for 

2.4048<V<3.8317, only LP0,1 and LP1,1 modes; for 3.8317<V<5.1356, only LP0,1 , LP1,1 , LP2,1 

and LP0,2 modes will exist, and so forth [9]. Figure 2 shows the intensity profiles of the modes 

existing at V=8.  



IHJPAS. 53 (4)2022 
 

58 

  

 Figure 2. Observed intensity profiles of all LP𝑙,𝑚 modes at V=8 [20]. 

3.  Results and Discussion  

In this work, RP Fiber Calculator (free version 2022) (Figure 3) is used to calculate the modes 

properties of SIFs. Inputs in this calculator are: a = (1 − 4.75) µm,  𝑛1 = 1.45,  𝑛2 = 1.44 and 
λ0 = 633 nm. Outputs are: NA= 0.17, V and the modes properties. The NA of the fiber can be 
calculated either from this calculator or from Equation (2). The critical angle is 83.3º which can be 

determined from Equation (1).  

 



IHJPAS. 53 (4)2022 
 

59 

 
Figure 3.  RP Fiber Calculator (index profile tab). 

 

Table 2 shows normalized frequencies and fiber modes number. If V<2.4048, there may be 

only one mode. On the other hand, if V>2.4048, there is more than one mode. 

 
Table 2. Normalized frequencies of the fibers and their modes number. 

M V a 

(µm) From RP Fiber Calculator From Equation (3) 

1 1.6874 1 

2 3.7967 2.25 

6 5.9060 3.5 

10 8.0153 4.75 

 

Five properties of modes have been calculated. These are propagation constant, effective 

refractive index, effective area, percentage power in core and cutoff wavelength. Table 3 shows 

propagation constant and effective refractive index of the LP0,1 mode of a SMF calculated from 

RP Fiber Calculator. This fiber has a core radius which is close to the wavelength of light.  

 
Table 3. Propagation constant and effective refractive index of LP0,1 mode (a=1 µm).  

𝑛eff 𝛽 (μm
−1) Mode 

1.443026 14.3235 LP0,1 

 

MMFs can be obtained when the radius of the fiber core is large compared to the wavelength. 

Tables 4 to 6 show propagation constants and effective refractive indices of all modes obtained 

from RP Fiber Calculator. It can be noted that the first mode has the highest propagation constant 

and effective refractive index which are related by Equation (4). These two properties increase 

with increasing core radius. 

   
Table 4. Propagation constants and effective refractive indices of LP𝑙,𝑚 modes (a=2.25 µm). 

𝑛eff 𝛽 (μm
−1) LP𝑙,𝑚 modes 

1.447528 14.3682 LP0,1 

1.443935 14.3326 LP1,1 



IHJPAS. 53 (4)2022 
 

60 

 

Table 5. Propagation constants and effective refractive indices of LP𝑙,𝑚 modes (a=3.5 µm). 

𝑛eff 𝛽 (μm
−1) LP𝑙,𝑚 modes 

1.448791 14.3808 LP0,1 

1.446957 14.3626 LP1,1 

1.444599 14.3391 LP2,1 

1.441805 14.3114 LP3,1 

1.443849 14.3317 LP0,2 

1.440591 14.2994 LP1,2 

 

Table 6. Propagation constants and effective refractive indices of LP𝑙,𝑚 modes (a=4.75 µm). 

𝑛eff 𝛽 (μm
−1) LP𝑙,𝑚 modes 

1.449288 14.3857 LP0,1 

1.448198 14.3749 LP1,1 

1.446777 14.3608 LP2,1 

1.445053 14.3437 LP3,1 

1.443048 14.3238 LP4,1 

1.440793 14.3014 LP5,1 

1.446295 14.3560 LP0,2 

1.444091 14.3341 LP1,2 

1.441663 14.3100 LP2,2 

1.441297 14.3064 LP0,3 

 

Table 7 shows effective area and percentage power in core of the LP0,1 mode of a SMF obtained 

from RP Fiber Calculator.  

 
Table 7. Effective area and percentage power in core of LP0,1 mode (a=1 µm).  

P in core (%) 𝐴eff (μm
2) Mode 

63.1 6.4 LP0,1 

 

Tables 8 to 10 show effective areas and percentage powers in core of all modes of MMFs 

calculated from RP Fiber Calculator. These two properties increase with increasing core radius. It 

can be noted that the first mode has the highest percentage power.  

 
Table 8. Effective areas and percentage powers in core of LP𝑙,𝑚 modes (a=2.25 µm). 

P in core (%) 𝐴eff (μm
2) LP𝑙,𝑚 modes 

94.3 12.8 LP0,1 

82.8 13.4 LP1,1 

 

 
Table 9. Effective areas and percentage powers in core of LP𝑙,𝑚 modes (a=3.5 µm). 

P in core (%) 𝐴eff (μm
2) LP𝑙,𝑚 modes 

98.2 25.4 LP0,1 

95.1 23.7 LP1,1 

90.1 25.4 LP2,1 

82.0 27.5 LP3,1 

87.1 22.5 LP0,2 

63.4 33.3 LP1,2 

 

Table 10. Effective areas and percentage powers in core of LP𝑙,𝑚 modes (a=4.75 µm). 

P in core (%) 𝐴eff (μm
2) LP𝑙,𝑚 modes 

99.2 42.8 LP0,1 

97.9 39.1 LP1,1 

96.1 40.4 LP2,1 



IHJPAS. 53 (4)2022 
 

61 

93.5 40.3 LP3,1 

89.9 40.4 LP4,1 

84.5 41.5 LP5,1 

95.2 34.4 LP0,2 

90.8 33.9 LP1,2 

82.4 41.7 LP2,2 

77.9 40.9 LP0,3 

 

Table 11 shows effective areas and percentage powers in core of the first mode obtained from 

Equations (5) and (7). A comparison with those in Tables 7 to 10 shows a small difference which 

is due to that the spot size is calculated approximately from Equation (6). 

 
Table 11. Effective areas and percentage powers in core of LP0,1 mode. 

P in core (%) 𝐴eff (μm
2) a 

(µm) From Equation (7) From Equation (5) 

58.2 7.19 1 

92.9 12.0 2.25 

96.8 22.4 3.5 

97.9 36.9 4.75 

 

Tables 12 to 14 show the cutoff wavelengths of LP𝑙,𝑚 modes of the MMFs. It can be noted that 
the first mode has no cutoff. Cutoff wavelengths which obtained from RP Fiber Calculator are in 

good agreement with those calculated from Equation (8). Cutoff wavelengths are more than 633 

nm. This is due to that cutoff frequencies of the modes are less than the normalized frequency of 

the fiber. Cutoff wavelengths of the modes increase with increasing core radius. 

 
Table 12. Cutoff wavelengths (nm) of LP𝑙,𝑚 modes (a=2.25 µm). 

From Equation (8) From RP Fiber Calculator LP𝑙,𝑚 modes 

  LP0,1 

999.38 995.22 LP1,1 

 

 

Table 13. Cutoff wavelengths (nm) of LP𝑙,𝑚 modes (a=3.5 µm). 

From Equation (8) From RP Fiber Calculator LP𝑙,𝑚 modes 

  LP0,1 

1554.60 1548.12 LP1,1 

975.68 971.63 LP2,1 

727.96 724.95 LP3,1 

975.68 971.60 LP0,2 

677.25 674.44 LP1,2 

 

Table 14. Cutoff wavelengths (nm) of LP𝑙,𝑚 modes (a=4.75 µm). 

From Equation (8) From RP Fiber Calculator LP𝑙,𝑚 modes 

  LP0,1 

2109.81 2101.02 LP1,1 

1324.13 1318.65 LP2,1 

987.94 983.86 LP3,1 

795.22 791.96 LP4,1 

668.62 665.88 LP5,1 

1324.13 1318.60 LP0,2 

919.13 915.31 LP1,2 

723.20 720.20 LP2,2 

723.20 720.19 LP0,3 

 



IHJPAS. 53 (4)2022 
 

62 

Figure 4 shows intensity profile of the LP0,1 mode of a SMF. This mode has a single spot 

profile. 

 

 
Figure 4. Intensity profile of LP0,1 mode (a=1 µm). 

   

Figures 5 to 7 show modes profiles of MMFs. Modes with 𝑙 = 0 (LP0,1, LP0,2 and LP0,3) have 
a single spot in their profiles. Other modes have (2𝑙) spots. The LP𝑙,𝑚 modes and their number in  
Figure 7 are equal to those in Figure 2 which obtained from experimental observation.  

 

 

 
 

 

 
 

Figure 5. Intensity profiles of LP0,1 and LP1,1 modes (a=2.25 µm). 

 
 

 

 

 

 
 

 

 

 

 

 
 



IHJPAS. 53 (4)2022 
 

63 

 
 

 

 

 
 

 

Figure 6. Intensity profiles of LP𝑙,𝑚 modes (a=3.5 µm). 

 

 
 

 
 

 

 
 

 

 
 

 

 
 

 

 

 

 

 
 

 

 
 

 
 

 

 



IHJPAS. 53 (4)2022 
 

64 

 
 

 

 
 

Figure 7. Intensity profiles of LP𝑙,𝑚 modes (a=4.75 µm). 

 

4.  Conclusions 

It has been concluded that the modes number and their properties are affected by the fiber core 

radius. If the core radius is close to the operating wavelength, a SMF is obtained. As the core radius 

becomes much larger than the operating wavelength, MMFs are obtained. Cutoff wavelengths of 

the modes are more than the operating wavelength of the fiber. All the modes properties increase 

with increasing core radius. 

 

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