How to calculate light transmission in dBm if it goes through SMF to MMF or MMF to SMF?

The question might be very basic, but I could not conclude it after thinking and searching about a couple of hours.

If a light in fiber optics is launched in an SMF, therefore, it is connected to an MMF with a connector, then how to calculate the transmitted light in dBm or mW or in any other unit provided that the radius of both fibers are given?

If I transmit the light just from opposite direction, then what would be the calculation of light transmission?

Note that, I have experimented both SMF to MMF and MMF to SMF light transmission and found that SMF to MMF gives better light transmission than MMF to SMF. But I want to find that why it is? what is the actual calculation behind it?

• The answer lies in the naming of the devices: a single mode fiber can only transmit a single mode (not quite, but that's the idea), a multi mode fiber can transmit many. If you launch the light from the SMF into the MMF, it will mostly make it, if you do the opposite, you are losing a of the modes that are simply not matched to the SMF. In other words... if you connect a small hose to a large one, it's all good, if you do the opposite, you only get a trickle. May 30 '16 at 18:32
• @CuriousOne Yes, I completely agreed with you and I have already visualized it in lab. But I want a mathematical model or formula to represent it or to calculate the final transmission of light. Thanks for your co-operation. May 30 '16 at 18:37
• For that you will need to understand how the light is distributed in your MMF and which modes get suppressed. I am sure someone has done the theory, but personally I wouldn't bother. That's what power measurements on the real thing are for. May 30 '16 at 18:39
• @CuriousOne You are absolutely right. I am in need of that theory. May 30 '16 at 18:44
• You have to calculate the modes of the receiving fiber, and have at hand the intensity distribution of the light at the end of the origin fiber. Calculate the overlap integral of the intensity distribution of each mode, then sum the results taking into account Fresnel losses. All that assumes that the fibers are butted up to one another. If there is some other coupling mechanism, that has to be taken into account. Sorry, it's not easy, and too involved to present here as an answer. Expensive computer programs are marketed to solve this (and related) questions. May 30 '16 at 19:08

After some more searching, I have found that the following formula would work fine to calculate the light transmission.

$Loss_{CD} = \begin{cases} - log_{10}(\frac{a_2}{a_1})^2 (dB),& \text{when } a_2\lt a_1\\ 0 (dB), & \text{otherwise} \end{cases}$

where $a_1$ and $a_2$ are the core radii of the transmitting and receiving fibers respectively.

So, when light goes from SMF to MMF, no loss in light occurs, but for connector, we could calculate it having 1dB loss.

On the other hand, when light goes in opposite direction, the loss could be calculated easily by the formula $- log_{10}(\frac{a_2}{a_1})^2 (dB)$ with 1dB connector loss for each connector.

Say, if -10 dBm light goes from SMF to MMF, then having 1dB loss, the transmitted light = -11 dBm. So, in watt, it could easily be calculated by the following formula:

$P_{(mW)} = 1mW \cdot 10^{\frac{P_{(dBm)}}{10}}$

For the light in opposite direction, the same procedure should be used.

Formula source: Optical Fiber Communications Principles and Practice (Thrid Edition) by John M. Senior assisted by M. Yousif Jamro.