{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

350lect35 - 35 Cavity radiation and thermal noise In a 1D...

This preview shows pages 1–2. Sign up to view the full content.

35 Cavity radiation and thermal noise Cavity radiance: Energy den- sity E ( f ) = N ( f ) V W ( f ) = 8 π f 2 c 3 hf e hf/KT - 1 J/m 3 Hz . in a 3D cavity in thermal equi- librium resides by equal amounts in the traveling wave components of the cavity modes arriving with speed c from the boundaries of the cavity subtending 4 π sterads. Multiplying E ( f ) by c/ 4 π we ob- tain L ( f ) = 2 f 2 c 2 hf e hf/KT - 1 W/m 2 /ster Hz , which is called radiance and rep- resents the power density per unit solid angle of the waves traveling within the cavity. Radiance L ( f ) also represents the spectrum of power radiated per unit solid angle by a unit area of a blackbody surface at temper- ature T (since non-reflective walls of a cavity will produce the same E ( f ) as partial-reflecting walls as mentioned earlier). In a 1D cavity of some length L — e.g. a TL with shorts at both ends as discussed in ECE 329 notes — the resonant frequencies are f m = c λ m = c 2 L/m = c 2 L m, where m = 1 , 2 , 3 , · · · which indicates that the mode density in f is N ( f ) = 2 L c modes Hz . Therefore the energy density in a 1D cavity in thermal equilibrium will be E ( f ) = N ( f ) L W ( f ) = 2 c hf e hf/KT - 1 J/m Hz in analogy with the energy density of 3D cavities. This energy density will reside by equal amounts in the traveling wave components of the 1D resonant modes arriving with speed c from the opposite ends of the 1D resonator. Power spectral content P ( f ) of each of these traveling wave components can thus be calculated as c/ 2 times 1 E ( f ) , i.e., P ( f ) = hf e hf/KT - 1 W Hz . 1 Note that per TEM plane wave, c ( 1 4 o ˜ | E | 2 + 1 4 μ o ˜ | H | 2 ) = ˜ | E | 2 4 η o + η o ˜ | H | 2 4 = ˜ | E | 2 2 η o , which confirms that the time-averaged stored energy density times c is indeed the time-averaged power transported per unit area.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

Page1 / 6

350lect35 - 35 Cavity radiation and thermal noise In a 1D...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online