Proposed link budget conventions for free-space optical communications

For the sake of simplicity and consistency, we describe the free-space power transmission by introducing the notion of receiver directivity. 1. Objections to Friis transmission equation The Friis transmission equation is given by

GGTx Rx PPRx Tx (1) LFSP with PTx and PRx the transmitted and received optical powers GRx and GRx are the transmitter and receiver gains respectively

LFSP the so-called “free-space path loss”.

LFSP is given by

2 4 R L   (2) FSP  with  the wavelength and R the link range.

There are two objections to this transmission model: 1) Only the Tx-gain should depend on the wavelength  . The  dependence is artificially

inserted in GRx and is cancelled with the introduction of the LFSP parameter. 2) The parameter “free-space path loss” has no meaning and should be avoided. More meaningful is the free-space coupling loss which is the fraction of power lost in free space.

2. Equation based on directivities

2.1. Simplification of Friis equation

We simplify the Friis equation by merging the Rx-gain GRx and the “free-space path loss”

LFSP and call the result Rx-directivity, noted for exampleRx :

PPGGL . (3) Rx Tx Tx Rx FSP Tx Rx

For reasons of symmetry, we note Tx the transmitter directivity. We obtain

PP. (4) RxTxTxR x coupling loss The term “directivity” keeps its usual definition for the Tx-antenna (Tx-directivity is equivalent to Tx-gain) but needs a new interpretation for the Rx-antenna. 2.2. Definitions

Tx-directivity: (usual definition) power density the antenna radiates in the direction of its strongest emission, relative to the power density radiated by an ideal isotropic radiator antenna radiating the same amount of total power.

Rx-directivity: angular extent (solid angle) of the receiver as seen by the transmitter, relative to the angular extent of a ideal receiver accepting power emitted by the transmitter in all directions.

Applying the definition of Tx-directivity for a Gaussian beam we have I   0 (5) Tx 2 PRTx 4 where I0 is the on-axis intensity in [W/m²] at the receiver. Using Gaussian beam relations, we find 8 Tx 2 (6) div where div is the 1/e2 half divergence angle (see Fig. 1). Note that the directivity (or gain) of a Gaussian beam is a factor 2 higher than that of a flat-top beam with half divergence θdiv.

Fig. 1. Illustration of a Gaussian beam with the 1/e² half divergence anglediv .

The Rx-directivity concept is illustrated in Fig. 2. Applying the above definition, we find

2  DR2    Rx 4 . (7) 2 D   4R Where D the Rx-diameter.

Fig. 2. Illustration of the concept of Rx-directivity.

Remarks: - Like for the Friis equation, the simplified equation (i.e., Eq. (4)) is accurate only for a high coupling loss

Tx Rx 1. (8)

- Because Rx is relative to an isotropic receiver, it can be expressed in dBi. - The notion of Equivalent Isotropically Radiated Power (EIRP) is unchanged.

2.3. Example: LEO-GEO link The proposed link-budget convention is applied in Table 1. The difference compared to a Friis link budget is easily noticed from the high negative Rx-Directivity in dBi.

Tx-Power (Average) 1 W θdiv 6.0E-6 rad R 4.0E+7m D 2.0E-1m Tx-Power 30.0 dBm

Tx-directivity Tx 113.5 dBi

Rx-directivity Rx -178.1 dBi Rx-Power -40.8 dBm Required Rx-Power for 1 Gbit/s at BER = 1E-9 -43.7 dBm Link Margin 2.9 dB Table 1: Typical power budget of a 1-Gbit/s LEO-GEO link.