Transmission loss between single-mode Gaussian antennas

Nicolas Perlot; Michael Rohde

doi:10.1364/OE.24.019491

1. Introduction

For applications such as laser communication, remote sensing or beam processing, it is often crucial to transmit single-mode beams over free-space distances with the least possible loss of power [1,2]. The coupling of Gaussian beams to waveguides, including various types of misalignment, has been well characterized in the literature [3–5]. However, approaching the coupling problem from the viewpoint of wireless transmission and transceiver design, there is a need to relate losses to antenna size and directivity. Showing the loss dependence on antenna type and propagation region is the intended novelty of this paper. Because we require the beam to be received as a single mode, we consider a spatially coherent receiver, not a “photon bucket”. The reciprocity of single-mode antennas implies that interchanging sender’s and receiver’s parameters does not modify the transmission loss.

Our analysis relies on the following assumptions. (i) Receiver is aligned to sender. (ii) Antennas are insensitive to polarization. (iii) Beams are monochromatic circular Gaussian and solve the paraxial wave equation. Starting with a general coupling efficiency equation, we derive transmission expressions for any single-mode Gaussian antennas and examine more specific antenna types. Results are given at first with distance variables, then with angular variables. After having shown how the introduced variables relate to conventional link budget parameters [6], we conclude by discussing the benefits of the results.

2. General transmission formula

The transmission $T$ is defined as the ratio of the received power $P_{R}$ to the sent power $P_{S}$ .

T \equiv \frac{P_{R}}{P_{S}} .

T

can be expressed as a free-space coupling efficiency with an emitted beam filtered by a receiver. Let

r

be a vector in the plane transverse to the propagation direction. In the receiver plane, the beam’s complex amplitude

E (r)

is superposed to the receiver’s complex aperture function

A_{R} (r)

leading to the transmission [4]

T = \frac{{| \int E (r) A_{R}^{*} (r) d r |}^{2}}{\int {| E (r) |}^{2} d r \int {| A_{R} (r) |}^{2} d r}

where * indicates the complex conjugate. At a distance

L

from the sender, the Gaussian beam can be put in the form [7]:

E (r) = \frac{\sqrt{I_{S}}}{1 + i α_{S} L} \exp [- \frac{π}{λ} (\frac{α_{S}}{1 + i α_{S} L}) r^{2}]

with

i^{2} = - 1

,

I_{S}

the on-axis intensity at the sender,

λ

the wavelength and

α_{S}

a complex parameter describing the sending antenna. For the convenience of symmetry, we consider for the receiver also a Gaussian antenna with complex parameter

α_{R}

:

A_{R} (r) = \exp (- \frac{π}{λ} α_{R} r^{2})

The antennas’ parameters are defined by

α_{S} \equiv \frac{1}{z_{S}} + i \frac{1}{f_{S}} and α_{R} \equiv \frac{1}{z_{R}} + i \frac{1}{f_{R}}

where

f_{S}

and

f_{R}

are the curvature radii of the respective antennas, defined positive for converging beam, and

z_{S} \equiv \frac{π w_{S}^{2}}{λ} and z_{R} \equiv \frac{π w_{R}^{2}}{λ}

where

w_{S}

and

w_{R}

are the aperture radii (at

1 / e

of amplitude decline) of the respective antennas. The terms of Eq. (2) can be written as

\int {| E (r) |}^{2} d r = P_{S}

\begin{matrix} \int {| A_{R} (r) |}^{2} d r = \frac{π w_{R}^{2}}{2} \\ = \frac{λ}{2 Re {α_{R}}} \end{matrix}

\begin{matrix} {| \int E (r) A_{R}^{*} (r) d r |}^{2} = \frac{I_{S}}{{| 1 + i α_{S} L |}^{2}} {| \int \exp [- \frac{π}{λ} (\frac{α_{S}}{1 + i α_{S} L} + α_{R}^{*}) r^{2}] d r |}^{2} \\ = \frac{2 P_{S}}{π w_{S}^{2} {| 1 + i α_{S} L |}^{2}} \frac{λ^{2}}{{| \frac{α_{S}}{1 + i α_{S} L} + α_{R}^{*} |}^{2}} \\ = \frac{2 P_{S} λ Re {α_{S}}}{{| α_{S} + α_{R}^{*} + i α_{S} α_{R}^{*} L |}^{2}} \end{matrix}

where

Re {\cdot}

is the real-part function. Introducing Eqs. (7)-(9) in Eq. (2), we arrive at

T = \frac{4 Re {α_{S}} Re {α_{R}}}{{| α_{S} |}^{2} {| α_{R} |}^{2} {| \frac{1}{α_{S}} + \frac{1}{α_{R}^{*}} + i L |}^{2}} .

Using the relation

{| x |}^{2} = x x^{*}

, we find

{| \frac{1}{α_{S}} + \frac{1}{α_{R}^{*}} + i L |}^{2} = {(L + \frac{Im {α_{R}}}{{| α_{R} |}^{2}} - \frac{Im {α_{S}}}{{| α_{S} |}^{2}})}^{2} + {(\frac{Re {α_{S}}}{{| α_{S} |}^{2}} + \frac{Re {α_{R}}}{{| α_{R} |}^{2}})}^{2}

where

Im {\cdot}

is the imaginary-part function. Equation (10) becomes

T = \frac{4 Re {α_{S}} Re {α_{R}}}{{| α_{S} |}^{2} {| α_{R} |}^{2}} {[{(L + \frac{Im {α_{R}}}{{| α_{R} |}^{2}} - \frac{Im {α_{S}}}{{| α_{S} |}^{2}})}^{2} + {(\frac{Re {α_{S}}}{{| α_{S} |}^{2}} + \frac{Re {α_{R}}}{{| α_{R} |}^{2}})}^{2}]}^{- 1} .

3. Analysis with distances

3.1. General expression

The signed distance from the antenna to the beam waist is given by [7]

z_{S b w} \equiv f_{S} {(1 + \frac{f_{S}^{2}}{z_{S}^{2}})}^{- 1} and z_{R b w} \equiv f_{R} {(1 + \frac{f_{R}^{2}}{z_{R}^{2}})}^{- 1} .

Using Eqs. (5) and (13), we conclude for the sender parameters that

\frac{Re {α_{S}}}{{| α_{S} |}^{2}} = \frac{f_{S} z_{S b w}}{z_{S}},

\frac{Im {α_{S}}}{{| α_{S} |}^{2}} = z_{S b w},

and reciprocally for the receiver. Introducing Eqs. (14) and (15) into Eq. (12), we obtain

T = \frac{4 f_{S} f_{R} z_{S b w} z_{R b w}}{z_{S} z_{R}} {[{(L + z_{R b w} - z_{S b w})}^{2} + {(\frac{f_{S} z_{S b w}}{z_{S}} + \frac{f_{R} z_{R b w}}{z_{R}})}^{2}]}^{- 1}

which has been inserted in Table 1 as the general formula in terms of distances. Other expressions listed in Table 1 are simplifications of Eq. (16) for special conditions that are discussed in the next sections.

Table 1. Transmission Expressions for Single-mode Gaussian Antennas

View Table

3.2. Matched receiver

An ideal receiver with a perfect match of the antenna to the incoming beam would achieve 100% transmission with a complex parameter defined by

α_{R, i d e a l} \equiv \frac{α_{S}}{1 + i α_{S} L} .

3.3. Identical antennas

With identical antennas (in opposite directions), we have

α_{R} = α_{S}^{*} .

Using the definition of Eq. (5), Eq. (18) yields

z_{R} = z_{S} \equiv z_{S R} .

f_{R} = - f_{S} \equiv - f_{S R} .

z_{R b w} = - z_{S b w} \equiv - z_{b w} .

Combining Eqs. (15), (17), and (18) gives the distance

L = 2 z_{b w}

for 100% transmission where the beam waist is in the link middle (

f_{S}

must be positive). Such a configuration of antennas was experimentally investigated by Arimoto in Ref [2].

3.4. Collimating antennas

For collimating (i.e. diffraction-limited) antennas, $f_{S} = f_{R} = \infty$ leading to $α_{S} = 1 / z_{S}$ and $α_{R} = 1 / z_{R}$ , where $z_{S}$ and $z_{R}$ are, in that case, the Rayleigh ranges of the Gaussian antennas. For given aperture radii, collimating antennas provide the highest transmission in the far field.

3.5. Refracting antennas

Refracting antennas have negligible diffraction, so $| f_{S} | < < z_{S}$ and $| f_{R} | < < z_{R}$ , and the distance to the beam waist is essentially the antenna’s curvature radius: $z_{S b w} \approx f_{S}$ and $z_{R b w} \approx f_{R}$ .

3.6. Far-field conditions

The far-field conditions for both antennas are

{\begin{cases} | α_{S} | L > > 1 \\ | α_{R} | L > > 1 \end{cases}

i.e.

L

is much longer than

z_{S}

(resp.

z_{R}

) or

| f_{S} |

(resp.

| f_{R} |

) whichever of both is the shortest.

4. Analysis with angles

We define the following antenna angles

θ_{S} \equiv \frac{λ}{π w_{S}}

β_{S} \equiv \frac{w_{S}}{f_{S}} with w_{S} < < f_{S}

and reciprocally for the receiver parameters.

θ_{S}

is the diffraction angle assuming a diffraction-limited antenna (i.e.

f_{S} = \infty

).

β_{S}

is the signed refraction angle of the curvature applied by the antenna to the wavefront. We additionally define the Fresnel angle

γ

as

γ \equiv \sqrt{\frac{λ}{π L}}

which is the maximum angle over which a beam can be effectively transmitted (i.e. coupled) whenever the antenna’s beam waist is much closer than the counter-antenna (i.e.

| z_{S b w} | < < L

, resp.

| z_{R b w} | < < L

). Applying Eqs. (23)-(25), Eq. (16) becomes

T = \frac{4 γ^{4}}{(θ_{S}^{2} + β_{S}^{2}) (θ_{R}^{2} + β_{R}^{2})} {[{(1 + \frac{γ^{2} β_{R} / θ_{R}}{θ_{R}^{2} + β_{R}^{2}} - \frac{γ^{2} β_{S} / θ_{S}}{θ_{S}^{2} + β_{S}^{2}})}^{2} + {(\frac{γ^{2}}{θ_{S}^{2} + β_{S}^{2}} + \frac{γ^{2}}{θ_{R}^{2} + β_{R}^{2}})}^{2}]}^{- 1}

which has been inserted in Table 1 as the general formula in terms of angles. For collimating antennas, we have

θ_{S} > > | β_{S} |

(resp.

θ_{R} > > | β_{R} |

), whereas for refracting antennas

| β_{S} | > > θ_{S}

(resp.

| β_{R} | > > θ_{R}

).

An object is in the far field of an antenna if the Fresnel angle $γ$ associated to the object’s distance $L$ is much smaller than the antenna’s angular resolution. The sender has an angular resolution of $λ | α_{S} |$ and thus operates in the far field if

θ_{S} \sqrt{θ_{S}^{2} + β_{S}^{2}} > > γ^{2}

and the reciprocal holds for the receiver.

5. Relation with Friis transmission equation

Conventional transmission equations for link budget assume collimating antennas in the far field [1,6]. Using our expression for collimating antennas in the near and far fields, we can identify the parameters of the Friis equation as follows, in terms of either distances or angles:

\begin{matrix} T = \frac{4 z_{S} z_{R}}{L^{2} + {(z_{S} + z_{R})}^{2}} \\ = \underset{Tx gain}{\underset{︸}{2 {(\frac{2 π w_{S}}{λ})}^{2}}} \underset{\begin{matrix} Isotropic \\ space loss \end{matrix}}{\underset{︸}{{(\frac{λ}{4 π L})}^{2}}} \underset{Rx gain}{\underset{︸}{2 {(\frac{2 π w_{R}}{λ})}^{2}}} \underset{Near-field loss}{\underset{︸}{\frac{L^{2}}{L^{2} + {(z_{S} + z_{R})}^{2}}}} \\ = \underset{\begin{matrix} Tx \\ gain \end{matrix}}{\underset{︸}{\frac{8}{θ_{S}^{2}}}} \underset{\begin{matrix} Isotropic \\ space loss \end{matrix}}{\underset{︸}{{(\frac{γ}{2})}^{4}}} \underset{\begin{matrix} Rx \\ gain \end{matrix}}{\underset{︸}{\frac{8}{θ_{R}^{2}}}} \underset{Near-field loss}{\underset{︸}{{[1 + {(\frac{γ^{2}}{θ_{S}^{2}} + \frac{γ^{2}}{θ_{R}^{2}})}^{2}]}^{- 1}}} \end{matrix}

where a near-field term could be identified as a loss. It should however be noted that for non-collimating antennas, the near-field term may be a gain (i.e. > 1). E.g., this is the case for identical converging (thus refracting) antennas when

L > f_{S R} > 0

.

6. Conclusion

The coupling loss between single-mode Gaussian antennas has been derived under the assumption of perfect alignment. Different antenna types and the far-field approximation were examined. Formulas should be consistent with the various published analyses of waveguide coupling losses that consider specific Gaussian antenna types.

Typically, near-field conditions are encountered in lab environments and far-field conditions characterize outdoor links. However, other scenarios can require antenna design considerations for both types of propagation conditions: one example could be laser links between a low-Earth-orbit satellite and a large astronomical telescope. We have shown how the Friis transmission equation can be extended to the near-field conditions, making power budget calculations for free-space links more accurate.

The reported loss dependences on either distances or angles provide practical interpretations. The angular analysis is useful for the far-field case or collimating antennas because, for these cases, the transmission loss is determined only by the ratio of the Fresnel angle $γ$ to the antenna directivity angle $\sqrt{θ^{2} + β^{2}}$ . This result implies that an increase of the receiver’s field-of-view over its diffraction limit (i.e. having $| β_{R} | > θ_{R}$ ) is associated with a power loss. We would like to stress that this is characteristic of single-mode reception and that incoherent (photon bucket) receivers would behave differently.

References and links

1. H. Hemmati, Near-Earth Laser Communications (CRC Press, 2009).

2. Y. Arimoto, “Optimum beam setting for near-field free-space optical communication system with bidirectional beacon tracking,” Proc. SPIE 8246, Free-Space Laser Communication Technologies XXIV, 82460S (2012).

3. N. Yu, “Coupling of a semiconductor laser to a single-mode fiber,” Scholar Archive, Paper 232 (1987).

4. E. G. Neumann, Single-Mode Fibers: Fundamentals (Springer, 1988).

5. S. Yuan and N. A. Riza, “General formula for coupling-loss characterization of single-mode fiber collimators by use of gradient-index rod lenses,” Appl. Opt. 38(15), 3214–3222 (1999). [CrossRef] [PubMed]

6. J. A. Shaw, “Radiometry and the Friis transmission equation,” Am. J. Phys. 81(1), 33–37 (2013). [CrossRef]

7. L. Andrews and R. Phillips, Laser Beam Propagation through Random Media, 2nd ed. (SPIE, 2005).

			In terms of distances		In terms of angles
			Near and far fields	Far field	Near and far fields	Far field
Antennas	General	different	$\begin{array}{l} \frac{4 f_{S} f_{R} z_{S b w} z_{R b w}}{z_{S} z_{R}} \\ \times [{(L + z_{R b w} - z_{S b w})}^{2} \\ {+ {(\frac{f_{S} z_{S b w}}{z_{S}} + \frac{f_{R} z_{R b w}}{z_{R}})}^{2}]}^{- 1} \end{array}$	$\frac{4 f_{S} f_{R} z_{S b w} z_{R b w}}{L^{2} z_{S} z_{R}}$	$\begin{array}{l} \frac{4 γ^{4}}{(θ_{S}^{2} + β_{S}^{2}) (θ_{R}^{2} + β_{R}^{2})} \\ \times [{(1 + \frac{γ^{2} β_{R} / θ_{R}}{θ_{R}^{2} + β_{R}^{2}} - \frac{γ^{2} β_{S} / θ_{S}}{θ_{S}^{2} + β_{S}^{2}})}^{2} \\ {+ {(\frac{γ^{2}}{θ_{S}^{2} + β_{S}^{2}} + \frac{γ^{2}}{θ_{R}^{2} + β_{R}^{2}})}^{2}]}^{- 1} \end{array}$	$\frac{4 γ^{4}}{(θ_{S}^{2} + β_{S}^{2}) (θ_{R}^{2} + β_{R}^{2})}$
	General	Identical	${[1 + \frac{z_{S R}^{2}}{f_{S R}^{2}} {(\frac{L}{2 z_{b w}} - 1)}^{2}]}^{- 1}$	${(\frac{2 f_{S R} z_{b w}}{L z_{S R}})}^{2}$	${[1 + {(\frac{θ_{S R}^{2} + β_{S R}^{2}}{2 γ^{2}} - \frac{β_{S R}}{θ_{S R}})}^{2}]}^{- 1}$	${(\frac{2 γ^{2}}{θ_{S R}^{2} + β_{S R}^{2}})}^{2}$
	Collimating	different	$\frac{4 z_{S} z_{R}}{L^{2} + {(z_{S} + z_{R})}^{2}}$	$\frac{4 z_{S} z_{R}}{L^{2}}$	$\frac{4 γ^{4}}{θ_{S}^{2} θ_{R}^{2}} {[1 + {(\frac{γ^{2}}{θ_{S}^{2}} + \frac{γ^{2}}{θ_{R}^{2}})}^{2}]}^{- 1}$	$\frac{4 γ^{4}}{θ_{S}^{2} θ_{R}^{2}}$
	Collimating	Identical	${[1 + {(\frac{L}{2 z_{S R}})}^{2}]}^{- 1}$	${(\frac{2 z_{S R}}{L})}^{2}$	${[1 + {(\frac{θ_{S R}^{2}}{2 γ^{2}})}^{2}]}^{- 1}$	${(\frac{2 γ^{2}}{θ_{S R}^{2}})}^{2}$
	Refracting	different	$\begin{array}{l} \frac{4 f_{S}^{2} f_{R}^{2}}{z_{S} z_{R}} \\ \times [{(L + f_{R} - f_{S})}^{2} \\ {+ {(\frac{f_{S}^{2}}{z_{S}} + \frac{f_{R}^{2}}{z_{R}})}^{2}]}^{- 1} \end{array}$	$\frac{4 f_{S}^{2} f_{R}^{2}}{L^{2} z_{S} z_{R}}$	$\begin{array}{l} \frac{4 γ^{4}}{β_{S}^{2} β_{R}^{2}} \\ \times [{(1 + \frac{γ^{2}}{θ_{R} β_{R}} - \frac{γ^{2}}{θ_{S} β_{S}})}^{2} \\ {+ {(\frac{γ^{2}}{β_{S}^{2}} + \frac{γ^{2}}{β_{R}^{2}})}^{2}]}^{- 1} \end{array}$	$\frac{4 γ^{4}}{β_{S}^{2} β_{R}^{2}}$
	Refracting	Identical	${[1 + \frac{z_{S R}^{2}}{f_{S R}^{2}} {(\frac{L}{2 f_{S R}} - 1)}^{2}]}^{- 1}$	${(\frac{2 f_{S R}^{2}}{L z_{S R}})}^{2}$	${[1 + \frac{β_{S R}^{2}}{θ_{S R}^{2}} {(\frac{β_{S R} θ_{S R}}{2 γ^{2}} - 1)}^{2}]}^{- 1}$	${(\frac{2 γ^{2}}{β_{S R}^{2}})}^{2}$

Transmission loss between single-mode Gaussian antennas

Abstract

1. Introduction

2. General transmission formula

3. Analysis with distances

3.1. General expression

3.2. Matched receiver

3.3. Identical antennas

3.4. Collimating antennas

3.5. Refracting antennas

3.6. Far-field conditions

4. Analysis with angles

5. Relation with Friis transmission equation

6. Conclusion

References and links

Cited By

Tables (1)

Equations (28)

Optics Express