# ITU 100 GHz Frequency Grid Math

Quote of the Day

A hero is someone who has given his or her life to something bigger than oneself.

— Joseph Campbell

## Introduction

Figure 1: Typical Optical Fiber Cable.

A physicist in my group and I were having a discussion about how the wavelengths (i.e. colors) for lasers are specified by an international standard and I thought this discussion would provide a nice example of a differential approximation. The widespread deployment of fiber optic cable (see Figure 1, Wikipedia) is a game changer for networking and may be our most important new infrastructure -- remember that high-speed wireless depends on cell towers interconnected with fiber optic cables.

## Analysis

Fiber optic cable is an incredible media for transmitting light. We are greatly increasing the amount of information that we can transfer over fiber by adding additional wavelengths of light onto the fiber. Our discussion this morning centered on the Dense Wavelength Division Multiplexing (DWDM) wavelengths specified in ITU-T G.694.1. This standard specifies laser wavelengths in terms of a frequency grid. Adjacent wavelengths in the grid are separated in frequency by 100 GHz. Each wavelength is referred to as a channel.

I normally think of light in terms of wavelength and not frequency. The wavelength and frequency of light are related by Equation 1.

 Eq. 1 $\displaystyle \lambda =\frac{c}{\nu }$

where

• λ is wavelength of the light channel.
• ν is is the frequency of the light channel.
• c is speed of light in a vacuum.

I have seen engineers use Equation 2 to approximate the wavelength difference between two wavelengths separated by a defined frequency difference.

 Eq. 2 $\displaystyle d\lambda =d\left( \frac{c}{\nu } \right)=-\frac{c}{{{\nu }^{2}}}\cdot d\nu =-\frac{{{\lambda }^{2}}}{c}\cdot d\nu$

where

• is is the frequency difference between adjacent wavelengths.

As you can see from Equation 2, holding the frequency difference between adjacent λ's constant means that will vary for each wavelength. I have seen engineers incorrectly assume that the is constant for all wavelengths -- not true -- only the frequency difference is fixed.

Using Mathcad, we can easily compute the wavelengths associated with each frequency (Figure 2).

Figure 2: Computing the ITU Frequency Grid Using Mathcad.

I have included that actual grid specification in Appendix A. It is identical to what I generated in Mathcad.

## Conclusion

Just a quick post to illustrate a quick use of differentials.

# Table 1: ITU 100 GHz Frequency Grid.

 Channel Frequency (GHz) Wavelength (nm) 1 190,100 1577.03 2 190,200 1576.2 3 190,300 1575.37 4 190,400 1574.54 5 190,500 1573.71 6 190,600 1572.89 7 190,700 1572.06 8 190,800 1571.24 9 190,900 1570.42 10 191,000 1569.59 11 191,100 1568.77 12 191,200 1567.95 13 191,300 1567.13 14 191,400 1566.31 15 191,500 1565.5 16 191,600 1564.68 17 191,700 1563.86 18 191,800 1563.05 19 191,900 1562.23 20 192,000 1561.42 21 192,100 1560.61 22 192,200 1559.79 23 192,300 1558.98 24 192,400 1558.17 25 192,500 1557.36 26 192,600 1556.55 27 192,700 1555.75 28 192,800 1554.94 29 192,900 1554.13 30 193,000 1553.33 31 193,100 1552.52 32 193,200 1551.72 33 193,300 1550.92 34 193,400 1550.12 35 193,500 1549.32 36 193,600 1548.51 37 193,700 1547.72 38 193,800 1546.92 39 193,900 1546.12 40 194,000 1545.32 41 194,100 1544.53 42 194,200 1543.73 43 194,300 1542.94 44 194,400 1542.14 45 194,500 1541.35 46 194,600 1540.56 47 194,700 1539.77 48 194,800 1538.98 49 194,900 1538.19 50 195,000 1537.4 51 195,100 1536.61 52 195,200 1535.82 53 195,300 1535.04 54 195,400 1534.25 55 195,500 1533.47 56 195,600 1532.68 57 195,700 1531.9 58 195,800 1531.12 59 195,900 1530.33 60 196,000 1529.55 61 196,100 1528.77 62 196,200 1527.99 63 196,300 1527.22 64 196,400 1526.44 65 196,500 1525.66 66 196,600 1524.89 67 196,700 1524.11 68 196,800 1523.34 69 196,900 1522.56 70 197,000 1521.79 71 197,100 1521.02 72 197,200 1520.25 73 197,300 1519.48
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