Difference between RHC and LHC in receiving and transmitting antenna?

How do you characterize an antenna in receiving mode? The book said a LHC antenna in transmitting mode is RHC in receiving mode. Also if the antenna is LHC in transmitting mode, it's transmission characteristic is defined as

. But the same antenna is RHC in receiving mode and it"s receiving characteristic is defined as

?

Is it because the same antenna when characterized as LHC when transmitting in +z direction, when placed on the z axis as the receiving antenna, has to be characterized in -z direction and therefore is RHC antenna in receiving mode?

For RHC circular antenna in tx mode at the origin radiating in +z direction, the unit vector is

.

1) What if the same RHC antenna radiating on the +z axis, but back TOWARDS the origin? The unit vector should be

.

2) Also, the EM wave radiated by the RHC antenna from +z back towards the origin is represented by

. also.

Am I correct?

In fact this is the example from Antenna Theory by Balanis 3rd edition p78 to 79. I don't agree with the book. It uses spherical coordinates and I disagree with the assertion that the antenna transmitting from a distance from the origin towards the origin is

.

Also for RHC antenna in tx mode using as receiving antenna at the origin, I agree with the given equation. But that would give total loss in reception! I am confused, please help.

Thanks

correct! this is because RH and LH are defined based on propagation direction.

- - - Updated - - -

good thoughts!

1- radiation backward +z toward origin means -z!
2- yes.

Thanks for the reply. How about post #3?

I read through the book over and over. I finally catch the part I missed. The book said

HAS to be defined as in transmitting mode. So if the RHC transmitting antenna is at origin outward, then the incident wave has the unit vector of

.

This together with the mistake the book make that the incident wave towards the origin, the unit vector of the wave should be

due to the direction of propagation towards the origin. Then

.

Please confirm I got this one right. Thanks

Alan

PS: I know this is very tedious, but I really think this is important to clarify the polarity and the definition of the EM wave using a single coordination reference, not just RHC wave match with RHC antenna and hope it work out!